These pages are essentially my "Game Theory: An Introductory Sketch," with a few corrections. The second offering in 1997 caused me to consider a major rewriting and extension of it, which would incorporate more of the theory background for the examples, and some new ideas and examples. The outline in the frame to the left will suggest the scope of the revision I had in mind. However, none of the new material has been written. Nevertheless, I am returning the existing essays in this slightly changed form, moved mostly by the number and passion of the requests users have sent me. Well, the number, anyway. It had also seemed that there were errors to correct, and there were some, the original having been done in a great hurry; and in some of these cases I have taken the time to verify that the original was correct. If I have missed any substantive errors, please let me know.
So these pages join the mass of Web documents that are perpetually "under construction." The topics on the left that have been written have hyperlinks in the contents screen to the left. I hope one of these days I'll actually get around to writing the others.
Fifty years old and newly prominent, thanks to the 1994 Nobel Memorial prize to Nash, Selten and Harsanyi, game theory and its application to economics are finding their way into undergraduate curricula. The accessibility of the texts available has improved accordingly. However, as I began to teach game theory to students who, while very good students, mostly are not economics majors, I felt that the available texts are still not really accessible enough. More advanced game theory has been extensively covered elsewhere. Morton Davis' Game Theory: A Non-Technical Introduction seems to remain the only discussion at quite the right level of accessibility, but its range of coverage is no longer quite what I need, and never was really geared to economics.
Accordingly, this document may be useful as a supplement to the existing texts, as a means of making the basic ideas a bit more accessible. Anyway, that is the objective. Accordingly, I have tried to limit these pages to fairly elementary topics and to avoid mathematics other than numerical tables and a very little algebra. The presentation is as intuitive as I have been able to make it, and keep it brief. I welcome suggestions as to how to make it better.
Game theory is a distinct and interdisciplinary approach to the study of human behavior. The disciplines most involved in game theory are mathematics, economics and the other social and behavioral sciences. Game theory (like computational theory and so many other contributions) was founded by the great mathematician John von Neumann. The first important book was The Theory of Games and Economic Behavior, which von Neumann wrote in collaboration with the great mathematical economist, Oskar Morgenstern. Certainly Morgenstern brought ideas from neoclassical economics into the partnership, but von Neumann, too, was well aware of them and had made other contributions to neoclassical economics.
Since the work of John von Neumann, "games" have been a scientific metaphor for a much wider range of human interactions in which the outcomes depend on the interactive strategies of two or more persons, who have opposed or at best mixed motives. Among the issues discussed in game theory are
1) What does it mean to choose strategies "rationally" when outcomes depend on the strategies chosen by others and when information is incomplete?
2) In "games" that allow mutual gain (or mutual loss) is it "rational" to cooperate to realize the mutual gain (or avoid the mutual loss) or is it "rational" to act aggressively in seeking individual gain regardless of mutual gain or loss?
3) If the answers to 2) are "sometimes," in what circumstances is aggression rational and in what circumstances is cooperation rational?
4) In particular, do ongoing relationships differ from one-off encounters in this connection?
5) Can moral rules of cooperation emerge spontaneously from the interactions of rational egoists?
6) How does real human behavior correspond to "rational" behavior in these cases?
7) If it differs, in what direction? Are people more cooperative than would be "rational?" More aggressive? Both?
Thus, among the "games" studied by game theory are
(This list is extracted from an index of games discussed in Roy Gardner, Games for Business and Economics)
The key link between neoclassical economics and game theory was and is rationality. Neoclassical economics is based on the assumption that human beings are absolutely rational in their economic choices. Specifically, the assumption is that each person maximizes her or his rewards -- profits, incomes, or subjective benefits -- in the circumstances that she or he faces. This hypothesis serves a double purpose in the study of the allocation of resources. First, it narrows the range of possibilities somewhat. Absolutely rational behavior is more predictable than irrational behavior. Second, it provides a criterion for evaluation of the efficiency of an economic system. If the system leads to a reduction in the rewards coming to some people, without producing more than compensating rewards to others (costs greater than benefits, broadly) then something is wrong. Pollution, the overexploitation of fisheries, and inadequate resources committed to research can all be examples of this.
In neoclassical economics, the rational individual faces a specific system of institutions, including property rights, money, and highly competitive markets. These are among the "circumstances" that the person takes into account in maximizing rewards. The implications of property rights, a money economy and ideally competitive markets is that the individual needs not consider her or his interactions with other individuals. She or he needs consider only his or her own situation and the "conditions of the market." But this leads to two problems. First, it limits the range of the theory. Where-ever competition is restricted (but there is no monopoly), or property rights are not fully defined, consensus neoclassical economic theory is inapplicable, and neoclassical economics has never produced a generally accepted extension of the theory to cover these cases. Decisions taken outside the money economy were also problematic for neoclassical economics.
Game theory was intended to confront just this problem: to provide a theory of economic and strategic behavior when people interact directly, rather than "through the market." In game theory, "games" have always been a metaphor for more serious interactions in human society. Game theory may be about poker and baseball, but it is not about chess, and it is about such serious interactions as market competition, arms races and environmental pollution. But game theory addresses the serious interactions using the metaphor of a game: in these serious interactions, as in games, the individual's choice is essentially a choice of a strategy, and the outcome of the interaction depends on the strategies chosen by each of the participants. On this interpretation, a study of games may indeed tell us something about serious interactions. But how much?
In neoclassical economic theory, to choose rationally is to maximize one's rewards. From one point of view, this is a problem in mathematics: choose the activity that maximizes rewards in given circumstances. Thus we may think of rational economic choices as the "solution" to a problem of mathematics. In game theory, the case is more complex, since the outcome depends not only on my own strategies and the "market conditions," but also directly on the strategies chosen by others, but we may still think of the rational choice of strategies as a mathematical problem -- maximize the rewards of a group of interacting decision makers -- and so we again speak of the rational outcome as the "solution" to the game.
Recent developments in game theory, especially the award of the Nobel Memorial Prize in 1994 to three game theorists and the death of A. W. Tucker, in January, 1995, at 89, have renewed the memory of its beginnings. Although the history of game theory can be traced back earlier, the key period for the emergence of game theory was the decade of the 1940's. The publication of The Theory of Games and Economic Behavior was a particularly important step, of course. But in some ways, Tucker's invention of the Prisoners' Dilemma example was even more important. This example, which can be set out in one page, could be the most influential one page in the social sciences in the latter half of the twentieth century.
This remarkable innovation did not come out in a research paper, but in a
classroom. As S. J. Hagenmayer wrote in the
Philadelphia Inquirer ("Albert W. Tucker, 89, Famed Mathematician,"
Tucker began with a little story, like this: two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.
The strategies in this case are: confess or don't confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a "payoff table" of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:
Table 3-1
|
|
|
Al |
|
|
|
|
confess |
don't |
|
Bob |
confess |
10,10 |
0,20 |
|
don't |
20,0 |
1,1 |
|
The table is read like this: Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.
So: how to solve this game? What strategies are "rational" if both men want to minimize the time they spend in jail? Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it's best if I confess. Therefore, I'll confess."
But Bob can and presumably will reason in the same way -- so that they both confess and go to prison for 10 years each. Yet, if they had acted "irrationally," and kept quiet, they each could have gotten off with one year each.
What has happened here is that the two prisoners have fallen into something called a "dominant strategy equilibrium."
DEFINITION Dominant Strategy: Let an individual player in a game evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game.
DEFINITION Dominant Strategy Equilibrium: If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, that is a dominant strategy equilibrium.
This remarkable result -- that individually rational action results in both persons being made worse off in terms of their own self-interested purposes -- is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which (we suppose) individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them. That is the source of its power.
Having said that, we must also admit candidly that the Prisoners' Dilemma is a very simplified and abstract -- if you will, "unrealistic" -- conception of many of these interactions. A number of critical issues can be raised with the Prisoners' Dilemma, and each of these issues has been the basis of a large scholarly literature:
We will consider some of these points in what follows.
Game theory provides a promising approach to understanding strategic problems of all sorts, and the simplicity and power of the Prisoners' Dilemma and similar examples make them a natural starting point. But there will often be complications we must consider in a more complex and realistic application. Let's see how we might move from a simpler to a more realistic game model in a real-world example of strategic thinking: choosing an information system.
For this example, the players will be a company considering the choice of a new internal e-mail or intranet system, and a supplier who is considering producing it. The two choices are to install a technically advanced or a more proven system with less functionality. We'll assume that the more advanced system really does supply a lot more functionality, so that the payoffs to the two players, net of the user's payment to the supplier, are as shown in Table A-1.
Table A-1
|
|
|
User |
|
|
|
|
Advanced |
Proven |
|
Supplier |
Advanced |
20,20 |
0,0 |
|
Proven |
0,0 |
5,5 |
|
We see that both players can be better off, on net, if an advanced system is installed. (We are not claiming that that's always the case! We're just assuming it is in this particular decision). But the worst that can happen is for one player to commit to an advance system while the other player stays with the proven one. In that case there is no deal, and no payoffs for anyone. The problem is that the supplier and the user must have a compatible standard, in order to work together, and since the choice of a standard is a strategic choice, their strategies have to mesh.
Although it looks a lot like the Prisoners' Dilemma at first glance, this is a more complicated game. We'll take several complications in turn:
By the time Tucker invented the Prisoners' Dilemma, Game Theory was already a going concern. But most of the earlier work had focused on a special class of games: zero-sum games.
In his earliest work, von Neumann made a striking discovery. He found that if poker players maximize their rewards, they do so by bluffing; and, more generally, that in many games it pays to be unpredictable. This was not qualitatively new, of course -- baseball pitchers were throwing change-up pitches before von Neumann wrote about mixed strategies. But von Neumann's discovery was a bit more than just that. He discovered a unique and unequivocal answer to the question "how can I maximize my rewards in this sort of game?" without any markets, prices, property rights, or other institutions in the picture. It was a very major extension of the concept of absolute rationality in neoclassical economics. But von Neumann had bought his discovery at a price. The price was a strong simplifying assumption: von Neumann's discovery applied only to zero-sum games.
For example, consider the children's game of "Matching Pennies." In this game, the two players agree that one will be "even" and the other will be "odd." Each one then shows a penny. The pennies are shown simultaneously, and each player may show either a head or a tail. If both show the same side, then "even" wins the penny from "odd;" or if they show different sides, "odd" wins the penny from "even". Here is the payoff table for the game.
|
|
|
Odd |
|
|
|
|
Head |
Tail |
|
Even |
Head |
1,-1 |
-1,1 |
|
Tail |
-1,1 |
1,-1 |
|
If we add up the payoffs in each cell, we find 1-1=0. This is a "zero-sum game."
DEFINITION: Zero-Sum game If we add up the wins and losses in a game, treating losses as negatives, and we find that the sum is zero for each set of strategies chosen, then the game is a "zero-sum game."
In less formal terms, a zero-sum game is a game in which one player's winnings equal the other player's losses. Do notice that the definition requires a zero sum for every set of strategies. If there is even one strategy set for which the sum differs from zero, then the game is not zero sum.
Here is another example of a zero-sum game. It is a very simplified model of price competition. Like Augustin Cournot (writing in the 1840's) we will think of two companies that sell mineral water. Each company has a fixed cost of $5000 per period, regardless whether they sell anything or not. We will call the companies Perrier and Apollinaris, just to take two names at random.
The two companies are competing for the same market and each firm must choose a high price ($2 per bottle) or a low price ($1 per bottle). Here are the rules of the game:
1) At a price of $2, 5000 bottles can be sold for a total revenue of $10000.
2) At a price of $1, 10000 bottles can be sold for a total revenue of $10000.
3) If both companies charge the same price, they split the sales evenly between them.
4) If one company charges a higher price, the company with the lower price sells the whole amount and the company with the higher price sells nothing.
5) Payoffs are profits -- revenue minus the $5000 fixed cost.
Here is the payoff table for these two companies
Table 4-2
|
|
|
Perrier |
|
|
|
|
Price=$1 |
Price=$2 |
|
Apollinaris
|
Price=$1 |
0,0 |
5000, -5000 |
|
Price=$2 |
-5000, 5000 |
0,0 |
|
(Verify for yourself that this is a zero-sum game.) For two-person zero-sum games, there is a clear concept of a solution. The solution to the game is the maximin criterion -- that is, each player chooses the strategy that maximizes her minimum payoff. In this game, Appolinaris' minimum payoff at a price of $1 is zero, and at a price of $2 it is -5000, so the $1 price maximizes the minimum payoff. The same reasoning applies to Perrier, so both will choose the $1 price. Here is the reasoning behind the maximin solution: Apollinaris knows that whatever she loses, Perrier gains; so whatever strategy she chooses, Perrier will choose the strategy that gives the minimum payoff for that row. Again, Perrier reasons conversely.
SOLUTION: Maximin criterion For a two-person, zero sum game it is rational for each player to choose the strategy that maximizes the minimum payoff, and the pair of strategies and payoffs such that each player maximizes her minimum payoff is the "solution to the game."
Now let's look back at the game of matching pennies. It appears that this game does not have a unique solution. The minimum payoff for each of the two strategies is the same: -1. But this is not the whole story. This game can have more than two strategies. In addition to the two obvious strategies, head and tail, a player can "randomize" her strategy by offering either a head or a tail, at random, with specific probabilities. Such a randomized strategy is called a "mixed strategy." The obvious two strategies, heads and tails, are called "pure strategies." There are infinitely many mixed strategies corresponding to the infinitely many ways probabilities can be assigned to the two pure strategies.
DEFINITION Mixed strategy If a player in a game chooses among two or more strategies at random according to specific probabilities, this choice is called a "mixed strategy."
The game of matching pennies has a solution in mixed strategies, and it is to offer heads or tails at random with probabilities 0.5 each way. Here is the reasoning: if odd offers heads with any probability greater than 0.5, then even can have better than even odds of winning by offering heads with probability 1. On the other hand, if odd offers heads with any probability less than 0.5, then even can have better than even odds of winning by offering tails with probability 1. The only way odd can get even odds of winning is to choose a randomized strategy with probability 0.5, and there is no way odd can get better than even odds. The 0.5 probability maximizes the minimum payoff over all pure or mixed strategies. And even can reason the same way (reversing heads and tails) and come to the same conclusion, so both players choose 0.5.
We can now say more exactly what von Neumann's discovery was. Von Neumann showed that every two-person zero sum game had a maximin solution, in mixed if not in pure strategies. This was an important insight, but it probably seemed more important at the time than it does now. In limiting his analysis to two-person zero sum games, von Neumann had made a strong simplifying assumption. Von Neumann was a mathematician, and he had used the mathematician's approach: take a simple example, solve it, and then try to extend the solution to the more complex cases. But the mathematician's approach did not work as well in game theory as it does in some other cases. Von Neumann's solution applies unequivocally only to "games" that share this zero-sum property. Because of this assumption, von Neumann's brilliant solution was and is only applicable to a small proportion of all "games," serious and nonserious. Arms races, for example, are not zero-sum games. Both participants can and often do lose. The Prisoners' Dilemma, as we have already noticed, is not a zero-sum game, and that is the source of a major part of its interest. Economic competition is not a zero-sum game. It is often possible for most players to win, and in principle, economics is a win-win game. Environmental pollution and the overexploitation of resources, again, tend to be lose-lose games: it is hard to find a winner in the destruction of most of the world's ocean fisheries in the past generation. Thus, von Neumann's solution does not -- without further work -- apply to these serious interactions.
The serious interactions are instances of "nonconstant sum games," since the winnings and losses may add up differently depending on the strategies the participants choose. It is possible, for example, for rival nations to choose mutual disarmament, save the cost of weapons, and both be better off as a result -- so the sum of the winnings is greater in that case. In economic competition, increasing division of labor, specialization, investment, and improved coordination can increase "the size of the pie," leading to "that universal opulence which extends itself to the lowest ranks of the people," in the words of Adam Smith. In cases of environmental pollution, the benefits to each individual from the polluting activity is so swamped by others' losses from polluting activity that all can lose -- as we have often observed.
Poker and baseball are zero-sum games. It begins to seem that the only zero-sum games are literal games that human beings have invented -- and made them zero-sum -- for our own amusement. "Games" that are in some sense natural are non-constant sum games. And even poker and baseball are somewhat unclear cases. A "friendly" poker game is zero-sum, but in a casino game, the house takes a proportion of the pot, so the sum of the winnings is less the more the players bet. And even in the friendly game, we are considering only the money payoffs -- not the thrill of gambling and the pleasure of the social event, without which presumably the players would not play. When we take those rewards into account, even gambling games are not really zero-sum.
Von Neumann and Morgenstern hoped to extend their analysis to non-constant sum games with many participants, and they proposed an analysis of these games. However, the problem was much more difficult, and while a number of solutions have been proposed, there is no one generally accepted mathematical solution of nonconstant sum games. To put it a little differently, there seems to be no clear answer to the question, "Just what is rational in a non-constant sum game?" The well-defined rational policy in neoclassical economics -- maximization of reward -- is extended to zero-sum games but not to the more realistic category of non-constant sum games.
The maximin strategy is a "rational" solution to all two-person zero come games. However, it is not a solution for nonconstant sum games. The difficulty is that there are a number of different solution concepts for nonconstant sum games, and no one is clearly the "right" answer in every case. The different solution concepts may overlap, though. We have already seen one possible solution concept for nonconstant sum games: the dominant strategy equilibrium. Let's take another look at the example of the two mineral water companies. Their payoff table was:
Table 4-2 (Repeated)
|
|
|
Perrier |
|
|
|
|
Price=$1 |
Price=$2 |
|
Apollinaris
|
Price=$1 |
0,0 |
5000, -5000 |
|
Price=$2 |
-5000, 5000 |
0,0 |
|
We saw that the maximin solution was for each company to cut price to $1. That is also a dominant strategy equilibrium. It's easy to check that: Apollinaris can reason that either Perrier cuts to $1 or not. If they do, Apollinaris is better off cutting to 1 to avoid a loss of $5000. But if Perrier doesn't cut, Apollinaris can earn a profit of 5000 by cutting. And Perrier can reason in the same way, so cutting is a dominant strategy for each competitor.
But this is, of course, a very simplified -- even unrealistic -- conception of price competition. Let's look at a more complicated, perhaps more realistic pricing example:
Following a long tradition in economics, we will think of two companies selling "widgets" at a price of one, two, or three dollars per widget. the payoffs are profits -- after allowing for costs of all kinds -- and are shown in Table 5-1. The general idea behind the example is that the company that charges a lower price will get more customers and thus, within limits, more profits than the high-price competitor. (This example follows one by Warren Nutter).
Table 5-1
|
|
|
Acme Widgets |
||
|
|
|
p=1 |
p=2 |
p=3 |
|
Widgeon Widgets |
p=1 |
0,0 |
50, -10 |
40,-20 |
|
p=2 |
-10,50 |
20,20 |
90,10 |
|
|
p=3 |
-20, 40 |
10,90 |
50,50 |
|
We can see that this is not a zero-sum game. Profits may add up to 100, 20, 40, or zero, depending on the strategies that the two competitors choose. Thus, the maximin solution does not apply. We can also see fairly easily that there is no dominant strategy equilibrium. Widgeon company can reason as follows: if Acme were to choose a price of 3, then Widgeon's best price is 2, but otherwise Widgeon's best price is 1 -- neither is dominant.
We will need another, broader concept of equilibrium if we are to do anything with this game. The concept we need is called the Nash Equilibrium, after Nobel Laureate (in economics) and mathematician John Nash. Nash, a student of Tucker's, contributed several key concepts to game theory around 1950. The Nash Equilibrium conception was one of these, and is probably the most widely used "solution concept" in game theory.
DEFINITION: Nash Equilibrium If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.
Let's apply that definition to the widget-selling game. First, for example, we can see that the strategy pair p=3 for each player (bottom right) is not a Nash-equilibrium. From that pair, each competitor can benefit by cutting price, if the other player keeps her strategy unchanged. Or consider the bottom middle -- Widgeon charges $3 but Acme charges $2. From that pair, Widgeon benefits by cutting to $1. In this way, we can eliminate any strategy pair except the upper left, at which both competitors charge $1.
We see that the Nash equilibrium in the widget-selling game is a low-price, zero-profit equilibrium. Many economists believe that result is descriptive of real, highly competitive markets -- although there is, of course, a great deal about this example that is still "unrealistic."
Let's go back and take a look at that dominant-strategy equilibrium in Table 4-2. We will see that it, too, is a Nash-Equilibrium. (Check it out). Also, look again at the dominant-strategy equilibrium in the Prisoners' Dilemma. It, too, is a Nash-Equilibrium. In fact, any dominant strategy equilibrium is also a Nash Equilibrium. The Nash equilibrium is an extension of the concepts of dominant strategy equilibrium and of the maximin solution for zero-sum games.
It would be nice to say that that answers all our questions. But, of course, it does not. Here is just the first of the questions it does not answer: could there be more than one Nash-Equilibrium in the same game? And what if there were more than one?
Here is another example to try the Nash Equilibrium approach on.
Two radio stations (WIRD and KOOL) have to choose formats for their broadcasts. There are three possible formats: Country-Western (CW), Industrial Music (IM) or all-news (AN). The audiences for the three formats are 50%, 30%, and 20%, respectively. If they choose the same formats they will split the audience for that format equally, while if they choose different formats, each will get the total audience for that format. Audience shares are proportionate to payoffs. The payoffs (audience shares) are in Table 6-1.
Table 6-1
|
|
|
KOOL |
||
|
|
|
CW |
IM |
AN |
|
WIRD |
CW |
25,25 |
50,30 |
50,20 |
|
IM |
30,50 |
15,15 |
30,20 |
|
|
AN |
20,50 |
20,30 |
10,10 |
|
You should be able to verify that this is a non-constant sum game, and that there are no dominant strategy equilibria. If we find the Nash Equilibria by elimination, we find that there are two of them -- the upper middle cell and the middle-left one, in both of which one station chooses CW and gets a 50 market share and the other chooses IM and gets 30. But it doesn't matter which station chooses which format.
It may seem that this makes little difference, since
There are multiple Nash Equilibria in which neither of these things is so, as we will see in some later examples. But even when they are both true, the multiplication of equilibria creates a danger. The danger is that both stations will choose the more profitable CW format -- and split the market, getting only 25 each! Actually, there is an even worse danger that each station might assume that the other station will choose CW, and each choose IM, splitting that market and leaving each with a market share of just 15.
More generally, the problem for the players is to figure out which equilibrium will in fact occur. In still other words, a game of this kind raises a "coordination problem:" how can the two stations coordinate their choices of strategies and avoid the danger of a mutually inferior
Two-person games don't get us very far. Many of the "games" that are most important in the real world involve considerably more than two players -- for example, economic competition, highway congestion, overexploitation of the environment, and monetary exchange. So we need to explore games with more than two players.
Von Neumann and Morgenstern spent a good deal of time on games with three players, and some more recent authors follow their example. This serves to illustrate how even one more player can complicate things, but it does not help us much with realism. We need an analysis of games with N>3 players, where N can be quite large. To get that, we will simply have to pay our way with some simplifying assumptions. One kind of simplifying assumption is the "representative agent model." In this sort of model, we assume that all players are identical, have the same strategy options and get symmetrical payoffs. We also assume that the payoff to each player depends only on the number of other players who choose each strategy, and not on which agent chooses which strategy.
This "representative agent" approach shouldn't be pushed too far. It is quite common in economic theory, and economists are sometimes criticized for overdoing it. But it is useful in many practical examples, and the next few sections will apply it.
This section presents a "game" which extends the Prisoners' Dilemma in some interesting ways. The Prisoners' Dilemma is often offered as a paradigm for situations in which individual self-interested rationality leads to bad results, so that the participants may be made better off if an authority limits their freedom to choose their strategies independently. Powerful as the example is, there is much missing from it. Just to take one point: the Prisoners' Dilemma game is a two-person game, and many of the applications are many-person interactions. The game considered in this example extends the Prisoners' Dilemma sort of interaction to a group of more than two people. I believe it gives somewhat richer implications about the role of authority, and as we will see in a later section, its N-person structure links it in an important way to cooperative game theory.
As usual, let us begin with a story. Perhaps the story will call to mind some of the reader's experience. We suppose that six people are waiting at an airline boarding gate, but that the clerks have not yet arrived at the gate to check them in. Perhaps these six unfortunates have arrived on a connecting flight with a long layover. Anyway, they are sitting and awaiting their chance to check in, and one of them stands up and steps to the counter to be the first in the queue. As a result the others feel that they, too, must stand in the queue, and a number of people end up standing when they could have been sitting.
Here is a numerical example to illustrate a payoff structure that might lead to this result. Let us suppose that there are six people, and that the gross payoff to each passenger depends on when she is served, with payoffs as follows in the second column of Table 7-1. Order of service is listed in the first column.
|
Order served |
Gross Payoff |
Net Payoff |
|
First |
20 |
18 |
|
Second |
17 |
15 |
|
Third |
14 |
12 |
|
Fourth |
11 |
9 |
|
Fifth |
8 |
6 |
|
Sixth |
5 |
3 |
These payoffs assume, however, that one does not stand in line. There is a two-point effort penalty for standing in line, so that for those who stand in line, the net payoff to being served is two less that what is shown in the second column. These net payoffs are given in the third column of the table.
Those who do not stand in line are chosen for service at random, after those who stand in line have been served. (Assume WLOG that these six passengers are risk neutral.) If no-one stands in line, then each person has an equal chance of being served first, second, ..., sixth, and an expected payoff of 12.5. In such a case the aggregate payoff is 75.
But this will not be the case, since an individual can improve her payoff by standing in line, provided she is first in line. The net payoff to the person first in line is 18>12.5, so someone will get up and stand in line.
This leaves the average payoff at 11 for those who remain. Since the second person in line gets a net payoff of 15, someone will be better off to get up and stand in the second place in line.
This leaves the average payoff at 9.5 for those who remain. Since the third person in line gets a net payoff of 12, someone will be better off to get up and stand in the third place in line.
This leaves the average payoff at 8 for those who remain. Since the fourth person in line gets a net payoff of 9, someone will be better off to get up and stand in the fourth place in line.
This leaves the average payoff at 6.5 for those who remain. Since the fifth person in line gets a net payoff of 6, no-one else will join the queue. With 4 persons in the queue, we have arrived at a Nash equilibrium of the game. The total payoff is 67, less than the 75 that would have been the total payoff if, somehow, the queue could have been prevented.
Two people are better off -- the first two in line -- with the first gaining an assured payoff of 5.5 above the uncertain average payoff she would have had in the absence of queuing and the second gaining 2.5. But the rest are worse off. The third person in line gets 12, losing 0.5; the fourth 9, losing 3.5, and the rest get average payoffs of 6.5, losing 6 each. Since the total gains from queuing are 8 and the losses 16, we can say that, in one fairly clear sense, queuing is inefficient.
We should note that it is in the power of the authority (the airline, in this case) to prevent this inefficiency by the simple expedient of not respecting the queue. If the clerks were to ignore the queue and, let us say, pass out lots for order of service at the time of their arrival, there would be no point for anybody to stand in line, and there would be no effort wasted by queuing (in an equilibrial information state).
In the Queueing Game, the first person in line gets the best service. In a patent system, the first person to invent a device gets the patent. Let's apply that parallel and try to come up with a game-theoretic analysis of patenting. For this example, we will have to try to think a bit like economists, and use two economic concepts: the concept of diminishing returns to investment and the focus on the additional revenue as a result of one more unit of investment -- the "marginal revenue," in economist jargon. If you have studied those concepts, this will be your reminder. If you haven't studied them, take it slow, but don't worry -- it will all be explained.
Patents exist because inventions are easy to imitate. An inventor could spend all his wealth working on an invention, and once it is proven, other businessmen could imitate it and get the benefit of all that investment without making any comparable investment of their own. Because of this, if there were no patents, there would be little incentive to develop new inventions, and thus very few inventions. At least, that's the idea behind the patent system. But some economists point out that patents are limited in time and in scope, so that there probably isn't as much incentive to invent as we would need to get an "efficient" number of inventions. This suggests that we don't get enough inventions -- but that may be a hasty conclusion.
For this game example, let us think of some new invention. At the beginning of the game, a number of development labs are considering whether to invest in the development of the invention. It is not known whether the invention is actually possible or not. Research and development are required even to discover that. But everyone estimates that, if the invention is produced and patented, it will yield a profit of $10,000,000. To keep things simple, we ignore any other potential benefits and proceed as if the profits were the only net benefits of developing the invention. Also for simplicity, we suppose that the only decision a laboratory can make is to spend $1,000,000 on a development effort or to spend nothing. A lab is not allowed to invest more nor any positive amount less than $1,000,000.
If there are investments in development, the invention may or may not be successfully developed. Since we don't know whether the invention is possible or not, the development effort may fail, and we can only say how probable it is that the invention will be made. The probability that the invention is successfully developed depends on the amount invested by the whole group, as shown in Table 2. But what does this probability mean, in money? To answer that question, we compute the "expected value" of total revenue -- that is, the revenue of $10,000,000 if the invention is made times the probability. Thus, if the probability of success is fifty percent, the "expected revenue" is 0.5*$10,000,000 = $5,000,000. The more labs invest, the greater the probability of success is, up to a point. But the labs' investment is subject to "diminishing returns:" at each step, the additional $1,000,000 of investment increases the expected revenue by a smaller amount. This is shown in the last column, as the "additional expected revenue" goes from $3,000,000 to $2,000,000, and so on.
|
investment in millions |
probability |
expected revenue |
additional expected revenue |
|
0 |
0 |
0 |
0 |
|
1 |
.3 |
3,000,000 |
3,000,000 |
|
2 |
.5 |
5,000,000 |
2,000,000 |
|
3 |
.5667 |
5,667,000 |
667,000 |
|
4 |
.61 |
6,100,000 |
433,000 |
|
5 |
.61 |
6,100,000 |
0 |
|
6 |
.61 |
6,100,000 |
0 |
|
7 |
.61 |
6,100,000 |
0 |
How much should be invested, for maximum net benefits? Economic theory tells us that it makes sense to keep increasing the investment as long as the additional expected revenue (marginal benefit, in economist's jargon) is greater than the investment necessary (marginal cost, in economist's jargon). That means it is efficient to increase investment in development as long as adding one more lab will increase the expected revenues by at the amount the lab will invest, $1,000,000, but no further. The second lab adds 2 million, while the third adds only 667,000 in expected revenues. The third lab is a loser, and the efficient number of labs working on developing this invention is two.
But how are these payoffs distributed among the development labs? We assume they are distributed on the "horse-race" principle: only the lab that "comes in first" is a winner. That is, no matter how many labs invest, 100% of the profits go to one lab, the lab that completes a working prototype first. Since all of the labs invest the same lump sum $1,000,000, we shall assume that all who invest have an equal chance of getting the payoff, if there is any payoff at all. Thus, when two labs invest, each has a 50% chance at a 50% chance of a $10,000,000 payoff -- that is, overall, a 25% (50% times 50%) chance at the $10,000,000, for an expected value payoff of $2,500,000 and a profit of $1,500,000. How many will invest in this "horse race" model of invention? If an enterprise is considering investing when two others are already committed to investing, it can anticipate gaining a 1/3 chance at a 56.67% chance (overall, an 18.89% chance) at $10,000,000, for an expected value of $1,889,000 and a profit of $889,000. The third lab will invest. What about the fourth, fifth, sixth? To make a long story a little shorter, the sixth lab would gain a 1/6 chance at a 61% chance at $10,000,000, for an expected value of $10,166,666.67 and a profit of $166,666.67. The sixth (fourth, and fifth) labs will invest. However, a potential seventh lab will anticipate a 1/7 chance at a 61% chance at the $10,000,000 payoff. This is an expected value of $871,428.57 and a loss of $128,571.43. The seventh firm will not invest. Any set of strategies in which six firms invest in research to produce this invention is a Nash-equilibrium.
Thus, in equilibrium, six firms will contest this particular "horse race," and that is three times the efficient number of labs to work on developing this invention. The allocation of four more labs to this job increases the probability of success by just 11%, a change worth just over a million; but four million are spent doing it!
Notice that this model could be applied when the reward goes, not to the first innovation in time, but to the first on some other scale. For example, suppose that the information products developed are not patentable, but are slightly different in design, as different computer applications might be. Suppose also that only the one perceived as "best" can be sold, and the rest fail and investments in developing them are lost. Finally, suppose that enterprises that invest equally have equal chances of producing the "best" dominant product, the "killer app." This "star system" will create a tendency toward overallocation of resources to the production of the information products.
The last time this colloquium was offered, I assigned myself as homework a model of high school students' decisions which universities to apply to and attend, but I didn't get my homework in. Fortunately, I didn't have to give myself a grade. Here is my late homework.
In this game there are 100,000 players, seniors in high school. There are 100 universities. The universities are not players in the game -- just mindless, predictable automata. The students are ranked from the most to the least "promising," with the most "promising" getting 100,000 points, the second most "promising" getting 99,999 points, and so on down to the least "promising" student, who gets one point. Each university will admit 1000 students, and they will be the 1000 highest-ranking students who apply. The payoff to each student is the average "promise" ranking of the students who enroll in the same university she or he does.
Thus, suppose the "best" 1000 students enroll in Old Ivy University. Their average "promise" ranking is 99,500, so that is the payoff to every student at Old Ivy. (Well, actually, 99500.5, but we will round off to integers). Suppose the next ranked 1000 enroll in Pixel University. Their average ranking is 98,500, so that is the payoff to each student at Pixel. And so it goes.
The student's strategies are to apply to one and -- for simplicity -- only one university. We will assume that each student knows where she or he is in the "promise" ranking. Thus the student knows the best university that will accept her or him. We may assume each student will apply to and attend the university that will give her or him the best payoff, that is, the university with the highest average "promise" ranking, provided that the student is confident of being admitted. (We are ignoring tuition and also parents' preferences for a college nearer home).
This game has 100! distinct Nash equilibria, but, happily, they are all very similar to one another. Suppose, for example, that (as we have said before)
and so on, with each group of 1000 students ranked together applying to the same university. Then each university will admit the 1000 students that apply, and the payoffs will be highest to students enrolled in Old Ivy, second highest to students enrolled in Pixel, third highest to those enrolled in Pinhead, and so on. Every student knows what university to apply to and is enrolled in the university she or he applies to.
This is a Nash equilibrium. To see why, suppose a single student in the top thousand were to switch his or her application from Old Ivy to Pixel. The student who switches will be accepted, but that student's payoff drops from 99,500 to 98,500. Conversely, suppose a student in the second 1000 switches her or his application from Pixel to Old Ivy. She or he will not be accepted, so cannot improve the payoff by switching to a more highly ranked university.
Thus, the ranking of universities with Old Ivy at the top, Pixel second, and so on is a Nash Equilibrium; but it is not the only one. As we have said, there are 100! equilibria in this game. For example, there are equilibria in which Old Ivy is ranked last, instead of first. If Old Ivy were ranked last in terms of the average promise of its students, then only the 1000 worst students would bother applying to Old Ivy. No student who could get admitted to Pixel would bother applying to Old Ivy, since that would just reduce their payoff to 500, the minimum.
In other words, this game is a coordination game. So long as each group of students in the same thousand all apply to the same university, we have an equilibrium -- and it doesn't matter which university that is. If the best 1000 students happened to apply to Podunk State, Podunk State would be the best university in the country, and Harvard and MIT would be so much chopped liver. (Notice that it also doesn't depend on the quality of the faculty, the facilities, or the food in the lunchroom. All that matters is agreement among the students).
But it gets worse. Once all of the students have sorted themselves out into groups of 1000, each group with next-door promise rankings and attending separate universities, the payoffs to the students will range from a low of 500 to a high of 99,500. The average payoff will be 50,000. What would happen if the students were deprived of their decision to apply to one school or another, and instead were assigned to universities at random? Each university would then have an average promise ranking of -- average, that is, about 50,000. So that the average payoff to students would be 50,000. So all this struggle among the students hasn't changed the average student payoff at all. It has just taken from those who have less (promise) and given to those who have more (promise), like Robin Hood in reverse. If that seems discouraging, look at it this way: it's your decision. Harvard may be the best or Harvard may be the worst. It's the students who decide. The faculty and the trustees don't have any say at all. Just those high school seniors.
The queuing game gives us one example of how the Prisoners' Dilemma can be generalized, and I hope that it provides some insights on some real human interactions. But there is another simple approach to multi-person two-strategy games that is closer to textbook economics, and is important in its own right.
As an example, let us consider the choice of transportation modes -- car or bus -- by a large number of identical individual commuters. The basic idea here is that car commuting increases congestion and slows down traffic. The more commuters drive their cars to work, the longer it takes to get to work, and the lower the payoffs are for both car commuters and bus commuters.
Figure 10-1 illustrates this. In the figure, the horizontal axis measures the proportion of commuters who drive their cars. Accordingly, the horizontal axis varies from a lower limit of zero to a maximum of 1 or 100%. The vertical axis shows the payoffs for this game.The upper (green) line shows the payoffs for car commuters. We see that it declines as the proportion of commuters in their cars increases. The lower, red line shows the payoffs to bus commuters. We see that, regardless of the proportion of commuters in cars, cars have a higher payoff than busses. In other words, commuting by car is a dominant strategy in this game. In a dominant strategy equilibrium, all drive their cars. The result is that they all have negative payoffs, whereas, if all rode busses, all would have positive payoffs. If all commuters choose their mode of transportation with self-interested rationality, all choose the strategy that makes them individually better off, but all are worse off as a result.
This is an extension of the Prisoners' Dilemma , in that there is a dominant strategy equilibrium, but the choice of dominant strategies makes everyone worse off. But it probably is not a very "realistic" model of choice of transportation modes. Some people do ride busses. So let's make it a little more realistic, as in Figure 10-2:
The axes and lines in Figure 10-2 are defined as they were for Figure 10-1. In Figure 10-2, congestion slows the busses down somewhat, so that the payoff to bus commuting declines as congestion increases; but the payoff to car commuting drops even faster. When the proportion of people in their cars reaches q, the payoff to car commuting overtakes the payoff to bus-riding, and for larger proportions of car commuters (to the right of q), the payoff to car commuting is worse than to bus commuting.
Thus, the game no longer has a dominant strategy equilibrium. However, it has a Nash-equilbrium. When a fraction q of commuters drives cars, that is a Nash-equilibrium. Here is the reasoning: starting from q, if one bus commuter shifts to the car, that moves into the region to the right of q, where car commuters are worse off, so (in particular) the person who switched is worse off. On the other hand, starting from q, if one car commuter switches to the bus, that moves into the region to the left of q, where bus commuters are worse off, so, again, the switcher is worse off. No-one can be better off by individually switching from q.
This illustrates an important point: in a Nash-equilibrium, identical people may choose different strategies to maximize their payoffs. This Nash-equilibrium resembles some "supply- and- demand" type equilibria in economics, having been suggested by models of that type, but also differs in some important ways. In particular, it is inefficient, in this sense: if everyone were to ride the bus, moving back to the origin point in Figure 10-2 (as in Figure 10-1), everyone would be better off. As in the Prisoners' dilemma, though, they will not do so when they act on the basis of individual self-interest without coordination.
This example is an instance of "the tragedy of the commons." The highways are a common resource available to all car and bus commuters. However, car commuters make more intensive use of the common resource, causing the resource to be degraded (in this instance, congested). Yet the car commuters gain a private advantage by choosing more intensive use of the common resource, at least while the resource is relatively undegraded. The tragedy is that this intensive use leads to the degradation of the resource to the point that all are worse off.
In general, "the tragedy of the commons" is that all common property resources tend to be overexploited and thus degraded, unless their intensive use is restrained by legal, traditional, or (perhaps) philanthropic institutions. The classical instance is common pastures, on which, according to the theory, each farmer will increase her herds until the pasture is overgrazed and all are impoverished. Most of the applications have been in environmental and resource issues. The recent collapse of fisheries in many parts of the world seems to be a clear instance of "the tragedy of the commons."
All in all, it appears that the Tragedy of the Commons is correctly understood as a multiperson extension of the Prisoners' Dilemma along the lines suggested in Figures 10-1 and 10-2, and, conversely, that the Prisoners' Dilemma is a valuable tool in understanding the many tragedies of the commons that we face in the modern world.
We now turn to a different application of the same technique for generalizing two-person, two-strategy games to many persons.
by Roger A. McCain
These notes are suggested by some recent discussions of the economic theory of persistent slumps, such as the Great Depression of the 1930's and the European unemployment of the 1980's and 1990's. Clearly, Keynesian ideas are a central (and still controversial) aspect of any such theory, but there are a wide variety of interpretation of Keynes, both favorable and critical, and recently one group in that broad tradition have begun to call themselves "Post-Walrasian." The Post-Walrasian view has several distinctive insights. The following list is not represented as either complete or final, but I hope it will do as a starting point:
These notes will attempt to give a game-theoretic framework for such a theory at about the level of simplicity of the introductory text. The discussion will be "simple" in that -- among other things -- it will not consider the multiple levels of analysis in the model itself, although aspects of the model will point in that direction.
Game theory seems promising for this purpose in part because the concept of Nash equilibrium is known to include a possibility of multiple equilibria. In my 1980 textbook I gave an illustration of games with multiple equilibria, some better than others, called the Heave-Ho Game. Here is the "little story" that goes with the game:
Two people are driving on a back-road short-cut and they come to a place where the road is obstructed by a fallen tree. Together, they are capable of moving the tree off the road, but only if each motorist heaves as hard as he can. If they do coordinate their decisions and heave, they can get the road clear, get back in their car, and continue to their destination. Let us say that their payoffs in that case are both +5. If neither motorist heaves, they have to turn back and arrive very late at their destination. In that case, we shall say that the payoffs are 0 for each motorist. However, if one motorist heaves and the other slacks off, making less than an all-out effort, the tree is shifted only a little -- it remains in the road -- the the motorist who heaved gets a painful muscle-strain as a result of the effort. Thus, the payoffs in that case are 0 (for the slacker) and -5 for the motorist who heaves. The payoff table is shown in Table 11-1.
Table 11-1
|
|
|
|
|
|
heave |
slack off |
|
heave |
5,5 |
-5,0 |
|
slack off |
0,-5 |
0,0 |
I hope it is clear that there are two Nash equilibria, at the upper left and the lower right. Starting from the upper right, either the column player or the row player will be worse off -- going from 5 to 0 -- if he changes strategies unilaterally. Starting from the lower right, again, either the column player or the row player will be worse off, going from 0 to -5, if he changes strategies unilaterally. However, the other two outcomes are not equilibria: from the upper right, for example, the row player will be better off to switch from a "heave" strategy, with a -5 payoff, to a "slack off" strategy, with a 0 payoff, if the other player does not change. Symmetrically, the lower left is not an equilibrium either.[1]
This reasoning -- that we have equilibrium if neither player can be better off by changing strategy unilaterally -- is consistent with the definition of Nash equilibrium. However, even though the lower right outcome is a Nash equilibrium, it is clearly inferior to the upper left outcome. In fact, it is Pareto-inferior, which means that a coordinated change of strategies from (slack off, slack off) to (heave, heave) would make some participants better off (in this case, both are better off) and nobody worse off. This illustrates that the game has multiple equilibria, with some equilibria superior to others. It also illustrates the importance of coordination: can they make the coordinated change of strategies they need to make them both better off? Because of this issue, the Heave-Ho game illustrates a pure coordination game.
With only two decision-makers to coordinate, it ought to be relatively easy. But even in this simple case, coordination might fail because of mistrust. If each of the motorists suspects that the other is a slacker, each will consider the choice as being between -5 and 0, rather than a choice between 0 and +5, and choose to slack off. Pessimism and loss aversion can have the same effect. A very pessimistic way of choosing strategies (in this situation) is to maximize the minimum payoff. "If I heave, my minimum payoff is -5. If I slack off, my minimum payoff is 0, and that is better than -5. I'll slack off."
We can also see how institutions might help to solve the coordination problem. An institution that might help in this case is a social convention. Suppose it were widely believe that, in cases of this kind, "gentlemen don't slack off. Just isn't done, don't y'know." If this convention were widespread enough that both motorists believed that the other motorist would subscribe to it -- and therefore would not slack off -- that would be enough to assure each motorist that the other one would slack off, and then each would behave like a gentleman -- and heave!
Because the Heave-Ho game is symmetrical, it doesn't quite tell the whole story. Let us make the example just a little more complex. Suppose that one of the motorists -- the "bridegroom" -- has more to lose by being late than the other motorist, the "hitchhiker." The payoff table for this modified game is Table 11-2.
Table 11-2
|
|
|
|
|
|
heave |
slack off |
|
heave |
5,5 |
-5,0 |
|
slack off |
0,-5 |
0,-4 |
In this case, the column player is the bridegroom and the row player is the hitchhiker. It is still true that they both gain in a coordinated switch of strategies from (slack off, slack off) to (heave, heave). But now the bridegroom gains 9, while the hitchhiker, who is in less of a hurry, gains only 5. Seeing that difference, the hitchhiker might demand some compensation for his cooperation -- turning the decision into a bargaining session. Bargaining outcomes can be unpredictable, and contribute to distrust, so this temptation could be another reason why coordination might fail.
Despite all this, it should be relatively easy for just two people to coordinate their strategy. In economics, we are often concerned with cases in which very large numbers of people must coordinate their strategies. We are also really concerned with macroeconomics, rather than back-road driving, and so it is time to bring the economics to the fore. Accordingly, we consider an investment game with N potential investors -- N very large -- and two strategies: each investor can choose a high rate of investment or a low one. We suppose that the payoff to a low rate of investment is always zero, but the payoff to a high rate of investment depends on how many other investors choose a high rate of investment.
The payoffs in the Investment Game are shown in Figure 9-1. The figure is an xy diagram with the profitability of investment on the vertical axis. The horizontal axis measures the proportion of all investors who choose the high-investment strategy, which varies from a minimum of n=zero to a maximum of n=N. The number of investors who choose a low-investment strategy is N-n. An increase in the number of investors choosing a high-investment strategy raises overall investment, and that stimulates demand, which in turn increases the profitability of investment. For the high-investment strategy, the payoff increases more rapidly. Thus, in Figure 1, the thick gray line ab gives the profitability of the high-investment strategy, and the cross-hatched line fg gives the profitability of the low-investment strategy.
An economist naturally assumes that something important happens when two lines cross. In this case y, the proportion at which the two lines cross, is a watershed rather than an equilibrium. Suppose that, at a given time, the number of investors choosing the high-investment strategy is greater than (to the right of) y, but less than N. This is not an equilibrium, since the profit-maximizing strategy for all investors is then the high-investment strategy. On the other hand, suppose that the number of investors choosing the high-investment strategy is less than (to the left of) y, but positive. This is not an equilibrium either, since the profit-maximizing strategy for all investors is then the low-investment strategy. There are 3 equilibria. If all investors choose the high-investment strategy, then we are at the right-hand extreme of the diagram, and the high-investment strategy is the profitable one -- so this is an equilibrium. Similarly, if all investors choose the low-investment strategy, we are at the left extreme of the diagram, and the low-investment strategy is the profitable choice, and this is an equilibrium. Finally, when exactly y investors choose the high-investment strategy, both strategies are equally profitable, so there is no reason for anyone to deviate from it. This, too, is an equilibrium, but it is an unstable one -- any random deviation from it will lead on to one of the extremes.[2]
This example illustrates some key points. Once again we have two equilibria -- and one is better than the other. In the high-investment equilibrium, profits are higher for all investors. The logic behind the example is that the investment community sometimes settles into the low-investment equilibrium, though. When that happens we have a depression or long-term stagnation.
However, there is still a good deal going on behind the scenes. We have said that higher overall investment leads to higher profits, because it stimulates demand. That is a Keynesian idea. But the Keynesian tradition tells us that there are several other sources of spending that stimulate demand. To take them into account we would need to look at things from a more traditionally Keynesian perspective. We will reserve that to an appendix, though. The appendix links these ideas to those in the macroeconomic principles textbook, so it may be of interest to students who have studied macroeconomics; but it is not, in itself, game theory.
All of the examples so far have focused on non-cooperative solutions to "games." We recall that there is, in general, no unique answer to the question "what is the rational choice of strategies?" Instead there are at least two possible answers, two possible kinds of "rational" strategies, in non-constant sum games. Often there are more than two "rational solutions," based on different definitions of a "rational solution" to the game. But there are at least two: a "non-cooperative" solution in which each person maximizes his or her own rewards regardless of the results for others, and a "cooperative" solution in which the strategies of the participants are coordinated so as to attain the best result for the whole group. Of course, "best for the whole group" is a tricky concept -- that's one reason why there can be more than two solutions, corresponding to more than concept of "best for the whole group."
Without going into technical details, here is the problem: if people can arrive at a cooperative solution, any non-constant sum game can in principle be converted to a win-win game. How, then, can a non-cooperative outcome of a non-constant sum game be rational? The obvious answer seems to be that it cannot be rational: as Anatole Rapoport argued years ago, the cooperative solution is the only truly rational outcome in a non-constant sum game. Yet we do seem to observe non-cooperative interactions every day, and the "noncooperative solutions" to non-constant sum games often seem to be descriptive of real outcomes. Arms races, street congestion, environmental pollution, the overexploitation of fisheries, inflation, and many other social problems seem to be accurately described by the "noncooperative solutions" of rather simple nonconstant sum games. How can all this irrationality exist in a world of absolutely rational decision makers?
There is a neoclassical answer to that question. The answer has been made explicit mostly in the context of inflation. According to the neoclassical theory, inflation happens when the central bank increases the quantity of money in circulation too fast. Then, the solution to inflation is to slow down or stop increasing in the quantity of money. If the central bank were committed to stopping inflation, and businessmen in general knew that the central bank were committed, then (according to neoclassical economics) inflation could be stopped quickly and without disruption. But, in a political world, it is difficult for a central bank to make this commitment, and businessmen know this. Thus the businessmen have to be convinced that the central bank really is committed -- and that may require a long period of unemployment, sky-high interest rates, recession and business failures. Therefore, the cost of eliminating inflation can be very high -- which makes it all the more difficult for the central bank to make the commitment. The difficulty is that the central bank cannot make a credible commitment to a low-inflation strategy.
Evidently (as seen by neoclassical economics) the interaction between the central bank and businessmen is a non-constant sum game, and recessions are a result of a "noncooperative solution to the game." This can be extended to non-constant sum games in general: noncooperative solutions occur when participants in the game cannot make credible commitments to cooperative strategies. Evidently this is a very common difficulty in many human interactions.
Games in which the participants cannot make commitments to coordinate their strategies are "noncooperative games." The solution to a "noncooperative game" is a "noncooperative solution." In a noncooperative game, the rational person's problem is to answer the question "What is the rational choice of a strategy when other players will try to choose their best responses to my strategy?"
Conversely, games in which the participants can make commitments to coordinate their strategies are "cooperative games," and the solution to a "cooperative game" is a "cooperative solution." In a cooperative game, the rational person's problem is to answer the question, "What strategy choice will lead to the best outcome for all of us in this game?" If that seems excessively idealistic, we should keep in mind that cooperative games typically allow for "side payments," that is, bribes and quid pro quo arrangements so that every one is (might be?) better off. Thus the rational person's problem in the cooperative game is actually a little more complicated than that. The rational person must ask not only "What strategy choice will lead to the best outcome for all of us in this game?" but also "How large a bribe may I reasonably expect for choosing it?"
Cooperative games are particularly important in economics. Here is an example that may illustrate the reason why. We suppose that Joey has a bicycle. Joey would rather have a game machine than a bicycle, and he could buy a game machine for $80, but Joey doesn't have any money. We express this by saying that Joey values his bicycle at $80. Mikey has $100 and no bicycle, and would rather have a bicycle than anything else he can buy for $100. We express this by saying that Mikey values a bicycle at $100.
The strategies available to Joey and Mikey are to give or to keep. That is, Joey can give his bicycle to Mikey or keep it, and Mikey can give some of this money to Joey or keep it all. it is suggested that Mikey give Joey $90 and that Joey give Mikey the bicycle. This is what we call "exchange." Here are the payoffs:
Table 12-1
|
|
|
Joey |
|
|
|
|
give |
keep |
|
Mikey |
give |
110, 90 |
10, 170 |
|
keep |
200, 0 |
100, 80 |
|
EXPLANATION: At the upper left, Mikey has a bicycle he values at $100, plus $10 extra, while Joey has a game machine he values at $80, plus an extra $10. At the lower left, Mikey has the bicycle he values at $100, plus $100 extra. At the upper left, Joey has a game machine and a bike, each of which he values at $80, plus $10 extra, and Mikey is left with only $10. At the lower right, they simply have what they begin with -- Mikey $100 and Joey a bike.
If we think of this as a noncooperative game, it is much like a Prisoners' Dilemma. To keep is a dominant strategy and keep, keep is a dominant strategy equilibrium. However, give, give makes both better off. Being children, they may distrust one another and fail to make the exchange that will make them better off. But market societies have a range of institutions that allow adults to commit themselves to mutually beneficial transactions. Thus, we would expect a cooperative solution, and we suspect that it would be the one in the upper left. But what cooperative "solution concept" may we use?
We have observed that both participants in the bike-selling game are better off if they make the transaction. This is the basis for one solution concept in cooperative games.
First, we define a criterion to rank outcomes from the point of view of the group of players as a whole. We can say that one outcome is better than another (upper left better than lower right, e.g) if at least one person is better off and no-one is worse off. This is called the Pareto criterion, after the Italian economist and mechanical engineer, Vilfredo Pareto. If an outcome (such as the upper left) cannot be improved upon, in that sense -- in other words, if no-one can be made better off without making somebody else worse off -- then we say that the outcome is Pareto Optimal, that is, Optimal (cannot be improved upon) in terms of the Pareto Criterion.
If there were a unique Pareto optimal outcome for a cooperative game, that would seem to be a good solution concept. The problem is that there isn't -- in general, there are infinitely many Pareto Optima for any fairly complicated economic "game." In the bike-selling example, every cell in the table except the lower right is Pareto-optimal, and in fact any price between $80 and $100 would give yet another of the (infinitely many) Pareto-Optimal outcomes to this game. All the same, this was the solution criterion that von Neumann and Morgenstern used, and the set of all Pareto-Optimal outcomes is called the "solution set."
If we are to improve on this concept, we need to solve two problems. One is to narrow down the range of possible solutions to a particular price or, more generally, distribution of the benefits. This is called "the bargaining problem." Second, we still need to generalize cooperative games to more than two participants. There are a number of concepts, including several with interesting results; but here attention will be limited to one. It is the Core, and it builds on the Pareto Optimal solution set, allowing these two problems to solve one another via "competition."
When we looked at "Choosing an Information Technology," one of the two introductory examples, we came to the conclusion that it is more complex than the Prisoners' Dilemma in several ways. Unlike the Prisoners' Dilemma, it is a cooperative game, not a noncooperative game. Now let's look at it from that point of view.
When the information system user and supplier get together and work out a deal for an information system, they are forming a coalition in game theory terms. (Here we have been influenced more by political science than economics, it seems!) The first decision will be whether to join the coalition or not. In this example, that's a pretty easy decision. Going it alone, neither the user nor the supplier can be sure of a payoff more than 0. By forming a coalition, both choosing the advanced system, they can get a total payoff of 40 between them. The next question is: how will they divide that 40 between them? How much will the user pay for the system? We need a little more detail about this game before we can go on. The payoff table above was net of the payment. It was derived from the following gross payoffs:
Table A-2
|
|
|
User |
|
|
|
|
Advanced |
Proven |
|
Supplier |
Advanced |
-50,90 |
0,0 |
|
Proven |
0,0 |
-30,40 |
|
The gross payoffs to the supplier are negative, because the production of the information system is a cost item to the supplier, and the benefits to the supplier are the payment they get from the user, minus that cost. For Table A-1, I assumed a payment of 70 for an advanced or 35 for a proven system. But those are not the only possibilities in either case.
How much will be paid? Here are a couple of key points to move us toward an answer:
Using that information, we get Figure A-1:
The diagram shows the net payoff to the supplier on the horizontal axis and the net payoff to the user on the horizontal axis. Since the supplier will not agree to a payment that leaves her with a loss, only the solid green diagonal line -- corresponding to total payoffs of 40 to the two participants -- will be possible payoffs. But any point on that solid line will satisfy the two points above. In that sense, all the points on the line are possible "solutions" to the cooperative game, and von Neumann and Morgenstern called it the "solution set."
But this "solution set" covers a multitude of sins. How are we to narrow down the range of possible answers? There are several possibilities. The range of possible payments might be influenced, and narrowed, by:
There are game-theoretic approaches based on all these approaches, and on combinations of them. Unfortunately, this leads to several different concepts of "solutions" of cooperative games, and they may conflict. One of them -- the core, based on competitive pressures -- will be explored in these pages. We will have to leave the others for another time.
There is one more complication to consider, when we look at the longer run. What if the supplier does not continue to support the information system chosen? What if the supplier invests to support the system in the long run, and the user doesn't continue to use it? In other words, what if the commitments the participants make are limited by opportunism?
We will need a bit of language to talk about cooperative games with more than two persons. A group of players who commit themselves to coordinate their strategies is called a "coalition." What the members of the coalition get, after all the bribes, side payments, and quids pro quo have cleared, is called an "allocation" or "imputation."
(The problem of coalitions also arises in zero-sum games, if there are more than two players. With three or more players, some of the players may profit by "ganging up" on the rest. For example, in poker, two or more players may cooperate to cheat a third, dividing the pelf between themselves. This is cheating, in poker, because the rules of poker forbid cooperation among the players. For the members of a coalition of this kind, the game becomes a non-zero sum game -- both of the cheaters can win, if they cheat efficiently).
"Allocation" and "imputation" are economic terms, and economists are often concerned with the efficiency of allocations. The standard definition of efficient allocation in economics is "Pareto optimality." Let us pause to recall that concept. In defining an efficient allocation, it is best to proceed by a double-negative. An allocation is inefficient if there is at least one person who can do better, while no other person is worse off. (That makes sense -- if somebody can do better without anyone else being made worse off, then there is an unrealized potential for benefits in the game). Conversely, the allocation is efficient in the Paretian sense if no-one can be made better off without making someone else worse off.
The members of a coalition, C, are a subset of the set of players in the game. (Remember, a "subset" can include all of the players in the game. If the subset is less than the whole set of players in the game, it is called a "proper" subset). If all of the players in the game are members of the coalition, it is called the "grand" coalition. A coalition can also have only a single member. A coalition with just a single member is called a "singleton coalition."
Let us say that the members of coalition C get payoff A. (A is a vector or list of the payoffs to all the members of C, including side payments, if any). Now suppose that some of the members of coalition C could join another coalition, C'; with an allocation of payoffs A'. The members of C who switch to C' may be called "defectors." If the payoffs to defectors in A' are greater than those in A, then we say that A' "dominates" A through coalition C. In other words: an allocation is dominated if some of the members of the coalition can do better for themselves by deserting that coalition for some other coalition.
We can now define one important concept of the solution of a cooperative game. The core of a cooperative game consists of all undominated allocations in the game. In other words, the core consists of all allocations with the property that no subgroup within the coalition can do better by deserting the coalition.
Notice that an allocation in the core of a game will always be an efficient allocation. Here, again, the best way to show this is to reason in double-negatives -- that is, to show that an inefficient allocation cannot be in the core. To say that the allocation A is inefficient is to say that a grand coalition can be formed in which at least one person is better off, and no-one worse off, than they are in A. Thus, any inefficient coalition is dominated through the grand coalition.
Now, two very important limitations should be mentioned. The core of a cooperative game may be of any size -- it may have only one allocation, or there may be many allocations in the core (corresponding either to one or many coalitions), and it is also possible that there may not be any allocations in the core at all. What does it mean to say that there are no allocations in the core? It means that there are no stable coalitions -- whatever coalition may be formed, there is some subgroup that can benefit by deserting it. A game with no allocations in the core is called an "empty-core game."
I said that the rational player in a cooperative game must ask "not only 'What strategy choice will lead to the best outcome for all of us in this game?' but also 'How large a bribe may I reasonably expect for choosing it?'" The core concept answers this question as follows" "Don't settle for a smaller bribe than you can get from an other coalition, and don't make any commitments until you are sure."
We will now consider two applications of the concept of the core. The first is a "market game," a game of exchange. We then return to a game we have looked at from the noncooperative viewpoint: the queuing game.
Economists often claim that "increasing competition" (an increasing number of participants on both sides of the market, demanders and suppliers) limits monopoly power. Our market game is designed to bring out that idea.
The concept of the core, and the effect of "increasing competition" on the core, can be illustrated by a fairly simple numerical example, provided we make some simplifying assumptions. We will assume that there are just two goods: "widgets" and "money." We will also use what I call the benefits hypothesis -- that is, that utility is proportional to money. In other words, we assume that the subjective benefits a person obtains from her or his possessions can be expressed in money terms, as is done in cost-benefit analysis. In a model of this kind, "money" is a stand-in for "all other goods and services." Thus, people derive utility from holding "money," that is, from spending on "all other goods and services," and what we are assuming is that the marginal utility of "all other goods and services" is (near enough) constant, so that we can use equivalent amounts of "money" or "all other goods and services" as a measure of the utility of widgets. Since money is transferable, that is very much like the "transferable utility" conception originally used by Shubik in his discussions of the core.
We will begin with an example in which there are just two persons, Jeff and Adam. At the beginning of the game, Jeff has 5 widgets but no money, and Adam has $22 but no widgets. The benefits functions are
|
Jeff |
|
Adam |
||||
|
widgets |
benefits |
|
widgets |
benefits |
||
|
|
total |
marginal |
|
|
total |
marginal |
|
1 |
10 |
10 |
|
1 |
9 |
9 |
|
2 |
15 |
5 |
|
2 |
13 |
4 |
|
3 |
18 |
3 |
|
3 |
15 |
2 |
|
4 |
21 |
3 |
|
4 |
16 |
1 |
|
5 |
22 |
1 |
|
5 |
16 |
0 |
Adam's demand curve for widgets will be his marginal benefit curve, while Jeff's supply curve will be the reverse of his marginal benefit curve. These are shown in Figure 13-1.
Figure 13-1
Market equilibrium comes where p=3, Q=2, i.e. Jeff sells Adam 2 widgets for a total payment of $6. The two transactors then have total benefits of
|
|
Jeff |
Adam |
|
widgets |
18 |
13 |
|
money |
6 |
16 |
|
total |
24 |
29 |
The total benefit divided between the two persons is $24+$29=$53.
Now we want to look at this from the point of view of the core. The "strategies' that Jeff and Adam can choose are unilateral transfers -- Jeff can give up 0, 1, 2, 3, 4, or 5 widgets, and Adam can give up from 0-22 dollars. Presumably both would choose "zero" in a noncooperative game. The possible coalitions are a) a grand coalition of both persons, or b) two singleton coalitions in which each person goes it alone. In this case, a cooperative solution might involve a grand coalition of the two players. In fact, a cooperative solution to this game is a coordinated pair of strategies in which Jeff gives up some widgets to Adam and Adam gives up some money to Jeff. (In more ordinary terms, that is, of course, a market transaction). The core will consist of all such coordinated strategies such that a) neither person (singleton coalition) can do better by going it alone, and b) the coalition of the two cannot do better by a different coordination of their strategies. In this game, the core will be a set of transactions each of which fulfills both of those conditions.
Let us illustrate both conditions: First, suppose Jeff offers to sell Adam one widget for $10. But Adam's marginal benefit is only nine -- Adam can do better by going it alone and not buying anything. Thus, "one widget for $10" is not in the core. Second, suppose Jeff proposes to sell Adam one widget for $5. Adam's total benefit would then be 22-5+9=26, Jeff's 5+21=26. Both are better off, with a total benefit of 52. However, they can do better, if Jeff now sells Adam a second widget for $3.50. Adam now has benefits of 13+22-8.50=26.50, and Jeff has benefits of 18+8.50=26.50, for a total benefit of 53. Thus, a sale of just one widget is not in the core. In fact, the core will include only transactions in which exactly two widgets are sold.
We can check for this in the following way. If the "benefits hypothesis" is correct, the only transactions in the core will be transactions that maximize the total benefits for the two persons. When the two person shift from a transaction that does not maximize benefits to one that does, they can divide the increase in benefits among themselves in the form of money, and both be better off -- so a transaction that does not maximize benefits cannot satisfy condition b) above. From Table 13-1,
|
Quantity Sold |
benefit of widgets |
money |
total |
|
|
|
to Jeff |
to Adam |
|
|
|
0 |
22 |
0 |
22 |
44 |
|
1 |
21 |
9 |
22 |
52 |
|
2 |
18 |
13 |
22 |
53 |
|
3 |
15 |
15 |
22 |
52 |
|
4 |
10 |
16 |
22 |
48 |
|
5 |
0 |
16 |
22 |
38 |
and we see that a trade of 2 maximizes total benefits.
But we have not figured out the price at which the two units will be sold. This is not necessarily the competitive "supply-and-demand" price, since the two traders are both monopolists and one may successfully hold out for a better-than-competitive price.
Here are some examples:
|
Quantity Sold |
Total Payment |
Total Benefits |
|
|
|
|
Jeff's |
Adam's |
|
2 |
12 |
18+12=30 |
22-12+13=22 |
|
2 |
5 |
18+5=23 |
22-5+13=30 |
|
2 |
8 |
18+8=28 |
22-8+13=27 |
What all of these transactions have in common is that the total benefits are maximized -- at 53 -- but the benefits are distributed in very different ways between the two traders. All the same, each trader does no worse than the 22 of benefits he can have without trading at all. Thus each of these transactions is in the core.
It will be clear, then, that there are a wide range of transactions in the core of this two-person game. We may visualize the core in a diagram with the benefits to Jeff on the horizontal axis and benefits to Adam on the vertical axis. The core then is the line segment ab. Algebraically, it is the line BA=53-BJ, where BA means "Adam's benefits" and BJ means "Jeff's benefits," and the line is bounded by BA>=22 and BJ>=22. The competitive equilibrium is at C.
Figure 13-2
The large size of the core is something of a problem. The cooperative solution must be one of the transactions in the core, but which one? In the two-person game, there is just no answer. The "supply-and-demand" approach does give a definite answer, shown as point C in the diagram. According to the supply-and-demand story, this equilibrium comes about because there are many buyers and many sellers. In our example, instead, we have just one of each, a bilateral monopoly. That would seem to be the problem: the core is large because the number of buyers and sellers is small.
So what happens if we allow the number of buyers and sellers to increase until it is very large? To keep things simple, we will continue to suppose that there are just two kinds of people -- jeffs and adams -- but we will consider a sequence of games with 2, 3, ..., 10, ..., 100,... adams and an equal number of jeffs and see what happens to the core of these games as the number of traders gets large.
First, suppose that there are just two jeffs and two adams. Each jeff and each adam has just the same endowment and benefit function as before.
What coalitions are possible in this larger economy? There could be two one-to-one coalitions of a jeff and an adam. Two jeffs or two adams could, in principle, form a coalition; but since they would have nothing to exchange, there would be little point in it. There could also be coalitions of two jeffs and an adam, two adams and a jeff, or a grand coalition of both jeffs and both adams.
We want to show that this bigger game has a smaller core. There are some transactions in the core of the first game that are not in this one.
Here is an example: In the 2-person game, an exchange of 12 dollars for 2 widgets is in the core. But it is not in the core of this game. At an exchange of 12 for 2, each adam gets total benefits of 23, each jeff or 30. Suppose then that a jeff forms a coalition with 2 adams, so that the jeff sells each adam one widget for $7. The jeff gets total benefits of 18+7+7=32, and so is better off. Each adam gets benefits of 15+9=24, and so is better off. This three-person coalition -- which could not have been formed in the two-person game -- "dominates" the 12-for-2 allocation and so the 12-for-2 allocation is not in the core of the 4-person game. (Of course, the other jeff is out in the cold, but that's his look-out -- the three-person coalition are better off. But, in fact, we are not saying that the three-person coalition is in the core either. It probably isn't, since the odd jeff out is likely to make an offer that would dominate this one).
This is illustrated by the diagram in Figure 13-3. Line segment de shows the trade-off between benefits to the jeffs and the adams in a 3-person coalition. It means that, from any point on line segment fb, a shift to a 3-person coalition makes it possible to move to the northwest -- making all members of the coalition better off -- to a point on fe. Thus all of the allocations on fb are dominated, and not in the core of the 4-person game.
Figure 13-3
Here is another example: in the two-person game, an exchange of two widgets for five dollars is in the core. Again, it will not be in the core of a four-person game. Each jeff gets benefits of 23 and each adam of 30. Now, suppose an adam proposes a coalition with both jeffs. The adam will pay each jeff $2.40 for one widget. The adam then has 30.20 of benefits and so is better off. Each jeff gets 23.40 of benefits and is also better off. Thus the one-adam-and-two-jeffs coalition dominates the 2-for-5 coalition, which is no longer in the core. Figure 13-4 illustrates the situation we now have. The benefit trade-off for a 2-jeff-one-adam coalition is shown by line gj. Every allocation on ab to the left of h is dominated. Putting everything together, we see that allocations on ab to the left of h and to the right of f are dominated by 3-person coalitions, but the 3-person coalitions are dominated by the 2-person coalitions between h and f. (Four-person coalitions function like pairs of two-person coalitions, adding nothing to the game).
Figure 13-4
We can now see the core of the four-person game in Figure 13-4. It is shown by the line segment hf. It is limited by BA>=27, BJ>=24. The core of the four-person game is part of the core of the two-person game, but it is a smaller part, because the four-person game admits of coalitions which cannot be formed in the two-person game. Some of these coalitions dominate some of the coalitions in the core of the smaller game. This illustrates an important point about the core. The bigger the game, the greater the variety of coalitions that can be formed. The more coalitions, often, the smaller the core.
Let us pursue this line of reasoning one more step, considering a six-person game with three jeffs and three adams. We notice that a trade of two widgets for $8 is in the core of the four-person game and we will see that it is not in the core of the 6-person game. Beginning from the 2-for-8 allocation, a coalition of 2 jeffs and 3 adams is proposed, such that each jeff gives up three widgets and each adam buys two, at a price of 3.50 each. The results are shown in Table 13-3
|
Type |
old allocation |
new allocation |
||||
|
|
widgets |
money |
total benefit |
widgets |
money |
total benefit |
|
Jeff |
4 |
8 |
26 |
3 |
11.4 |
26.4 |
|
Adam |
2 |
14 |
27 |
2 |
14.4 |
27.4 |
We see that both the adams and the jeffs within the coalition are better off, so the two-and-three coalition dominates the two-for-eight bilateral trade. Thus the two-for-eight trade is not in the core of the six-person game.
What is in it? This is shown by Figure 13-5. As before, the line segment ab is the core of the two-person game and line segment gj is the benefits trade-off for the coalition of two jeffs and one adam. Segment kl is the benefits trade-off for the coalition of of two jeffs and thee adams. We see that every point on a, b except point h is dominated, either by a 2-jeff 1-adam coalition or by a two-jeff three-adam coalition. The core of a six-player game is exactly one allocation: the one at point h. And this is the competitive equilibrium! No coalition can do better than it.
Figure 13-5
If we were to look at 8, 10, 100, 1000, or 1,000,000 player games, we would find the same core. This series of examples illustrates a key point about the core of an exchange game: as the number of participants (of each type) increases without limit, the core of the game shrinks down to the competitive equilibrium. This result can be generalized in various ways. First, we should observe that in some games, any game with a finite number of players has more than one allocation in the core. This game has been simplified by only allowing players to trade in whole numbers of widgets. That is one reason why the core shrinks to the competitive equilibrium so soon in our example. We may also eliminate the benefits hypothesis, assuming instead that utility is nontransferable and not proportional to money. We can also allow for more than two kinds of players, and get rid of the "types" assumption completely, at the cost of much greater mathematical complexity. But the general idea is simple enough. With more participants, more kinds of coalitions can form, and some of those coalitions dominate coalitions that could form in smaller games. Thus a bigger game will have a smaller core; in that sense "more competition limits monopoly power." But (in a market game) the supply-and-demand is the one allocation that is always in the core. And this provides us with a new understanding of the unique role of the market equilibrium.
We have seen that the market game has a non-empty core, but some very important games have empty cores. From the mathematical point of view, this seems to be a difficulty -- the problem has no solution. But from the economic point of view it may be an important diagnostic point. The University of Chicago economist Lester Telser has argued that empty-core games provide a rationale for government regulation of markets. The core is empty because efficient allocations are dominated -- people can defect to coalitions that can promise them more than they can get from an efficient allocation. What government regulation does in such a case is to prohibit some of the coalitions. Ruling out some coalitions by means of regulation may allow an efficient coalition to form and to remain stable -- the coalitions through which it might be dominated are prohibited by regulation.
In another segment, we have looked at a game that has an inefficient noncooperative equilibrium: the queuing game. We shall see that the Queuing Game also is an empty-core game. Recalling that every allocation in the core is Pareto Optimal, and that Pareto Optimality in this game presupposes a grand coalition of all players to refrain from starting a queue, it will suffice to show that the grand coalition is unstable against a defection of a single agent to form a singleton coalition and form a one-person queue.
It is easy to see that the defector will be better off if the rump coalition (the five remaining in a coalition not to queue) continues its strategy of not contesting for any place in line. Then the defector gets a net payoff of 18 with certainty, better than the average payoff of 12.5 she would get with the grand coalition -- and this observation is just a repetition of the argument that the grand coalition is not a Nash equilibrium. But the rump coalition needs not simply continue with its policy of noncontesting. For example, it can contest the first position in line, while continuing the agreement to allocate the balance at random. This would leave the aggressor with a one-sixth chance of the first place, but she can do no worse than second, so her expected payoff would then be (1/6)(18)+(5/6)(15)=15.5. She will not be deterred from defecting by this possible strategy response from the rump coalition.
That is not the only strategy response open to the rump coalition. Table 13-4 presents a range of strategy alternatives available to the rump coalition:
|
contest the first |
payoff to |
average |
|
no places |
18 |
11 |
|
one place |
15.5 |
11.167 |
|
two places |
13.5 |
11.223 |
|
three places |
12 |
11.2 |
|
four places |
11.167 |
11.167 |
|
five places |
11.167 |
11.167 |
These are not the only strategy options available to the rump coalition. For example, the rump coalition might choose to contest just the first and third positions in line, leaving the second uncontested. But this would serve only to assure the defector of a better outcome than she could otherwise be sure of, making the members of the rump coalition worse off. Thus, the rump coalition will never choose a strategy like that, and it cannot be relevant to the defector's strategy. Conversely, we see that the rump coalition's best response to the defection is to contest the first two positions in line, but no more -- leaving the defector better off as a result of defecting, with an expected payoff of 13.5 rather than 12.5. If follows that the grand coalition is unstable under recontracting.
To illustrate the reasoning that underlies the table, let us compute the payoffs for the case in which the rump coalition contests the first two positions in line, the optimal response. 1) The aggressor has a one/sixth chance of first place in line for a payoff of 18, one/sixth of second place for 15, and four-fifths chance of being third, for 12. (The aggressor must still stand in line to be sure of third place, rather than worse, although that position is uncontested). Thus the expected payoff is 18/6+15/6+4*12/6, for 13.5. 2a) With one chance in six, the aggressor is first, leaving the rump coalition to allocate among themselves rewards of 15 (second place in queue), 14, 11, 8, and 5 (third through last places without standing in the queue). Each of these outcomes has a conditional probability of one-fifth for each of the five individuals in the rump coalition. This accounts for expectations of one in thirty (one-sixth times one-fifth) of each of those rewards. 2b) With one chance in six, the aggressor is second, and the rump coalition allocate among themselves, at random, payoffs of 18 (first place in queue), 14, 11, 8 and 5 (as before) accounting for yet further expectations of one in thirty of each of these rewards. 2c) With four chances in six, the aggressor is third -- without contest -- and the members of the rump coalition allocate among themselves, at random, rewards of 18, 15, (first two places in the queue), 11, 8, and 5 (last three places without queuing). 2d) Thus the expected payoff of a member of the rump coalition is (15+14+11+8+5)/30+(18+14+11+8+5)/30+4*(18+15+11+8+5)/30, or 11.233.
All of the game examples so far are relatively simple in that time plays no part in them, however complex they may be in other ways. The passage of time can make at least three kinds of differences. First, people may learn -- new information may become available, that affects the payoffs their strategies can give. Second, even when people do not or cannot commit themselves at first, they may commit themselves later -- and they may have to decide when and if to commit. Of course, this blurs the distinction we have so carefully set up between cooperative and noncooperative games, but life is like that. Third, there is the possibility of retaliation against other players who fail to cooperate with us. That, too, blurs the cooperative-noncooperative distinction. That means, in particular, that repeated games -- and particularly repeated prisoners' dilemmas -- may have quite different outcomes than they do when they are played one-off. But we shall leave the repeated games out as an advanced topic and move on to study sequential games and the problems that arise when people can make commitments only in stages, at different points in the game. I personally find these examples interesting and to the point and they are somewhat original.
There are some surprising results. One surprising result is that, in some games, people are better off if they can give up some of their freedom of choice, binding themselves to do things at a later stage in the came that may not look right when they get to this stage. An example of this (I suggest) is to be found in Marriage Vows. This provides a good example of what some folks call "economic imperialism" -- the use of economics (and game theory) to explain human behavior we do not usually think of as economic, rational, or calculating -- although you do not really need to know any economics to follow the example in Marriage Vows. Another example along the same line (although the main application is economics in a more conventional sense) is The Paradox of Benevolent Authority, which tries to capture, in game-theoretic terms, a reason why liberal societies often try to constrain their authorities rather than relying on their benevolence.
Also, the following example will have to do with relations between an employer and an employee: A Theory of Burnout. For an example in which flexibility is important, so that giving up freedom of choice is a bad idea, and another non-imperialistic economic application of game theory, see The Essence of Bankruptcy. Of course, that note is meant to discuss bankruptcy, not to exemplify it!
This example is an attempt to use game theory to "explain" marriage vows. But first (given the nature of the topic) it might be a good idea to say something about "explanation" using game theory.
One possible objection is that marriage is a very emotional and even spiritual topic, and game theory doesn't say anything about emotions and spirit. Instead game theory is about payoffs and strategies and rationality. That's true, but it may be that the specific phenomenon -- the taking of vows that (in some societies, at least) restrict freedom of choice -- may have more to do with payoffs and strategies than with anything else, and may be rational. In that case, a game-theoretic model may capture the aspects that are most relevant to the institution of marriage vows. Second, game-theoretic explanations are never conclusive. The most we can say is that we have a game-theoretic model, with payoffs and strategies like this, that would lead rational players to choose the strategies that, in the actual world, they seem to choose. It remains possible that their real reasons are different and deeper, or irrational and emotional. That's no less true of investment than of marriage. Indeed, from some points of view, their "real reasons" have to be deeper and more complex -- no picture of the world is ever "complete." The best we can hope for is a picture that fits fairly well and contains some insight. I think game theory can "explain" marriage vows in this sense.
In some sequential games, freedom of choice can be a problem. We have seen this in the previous examples. These are games that give one or more players possibilities for "opportunism." That is, some players are able to make their decisions in late stages of the game in ways that exploit the decisions made by others in early stages. But those who make the decisions in the early stages will then avoid decisions that make them vulnerable to opportunism, with results that can be inferior all around. In these circumstances, the potential opportunist might welcome some sort of restraint that would make it impossible for him to act opportunistically at the later stage. Jon Elster made the legend of "Ulysses and the Sirens" a symbol for this. Recall, in the legend, Ulysses wanted to hear the sirens sing; but he knew that a person who would hear them would destroy himself trying to go to the sirens. Thus, Ulysses decided at the first stage of the game to have himself bound to the mast, so that, at the second stage, he would not have the freedom to choose self-destruction. Sequential games are a bit different from that, in that they involve interactions of two or more people, but the games of sequential commitment can give players reason to act as Ulysses did -- that is, to rationally choose at the first step in a way that would limit their freedom of choice at the second step. That is our strategy in attempting to "explain" marriage vows.
Here is the "game." At the first stage, two people get together. They can either stay together for one period or two. If they take a vow, they are committed to stay together for both periods. During the first period, each person can choose whether or not to "invest in the relationship." "Investing in the relationship" means making a special effort in the first period that will only yield the investor benefits in the second period, and will yield benefits in the second period only if the couple stay together. At the end of the first period, if there has been no vow, each partner decides whether to remain together for the second period or separate. If either prefers to separate, then separation occurs; but if both choose to remain together, they remain together for the second period. Payoffs in the second period depend on whether the couple separate, and, if they stay together, on who invested in the first period.
The payoffs are determined as follows: First, in the first stage, the payoff to one partner is 40, minus 30 if that partner "invests in the relationship," plus 20 if the other partner "invests in the relationship." Thus, investment in the relationship is a loss in the first period -- that's what makes it "investment." In the second period, if they separate, both partners get additional payoffs of 30. Thus, each partner can assure himself or herself of 70 by not investing and then separating. However, if they stay together, each partner gets an additional payoff of 20 plus (if only the other partner invested) 25 or (if both partners invested) 60.
Notice that the total return to investment to the couple over both periods is disproportionately greater if both persons invest -- that is, it is 2*20-2*30 in the first period plus 20+2*60 = 80 if both invest, but is 20+25=45 for one partner and 20 for the other partner if only one invests. The difference 80-65=15 reflects the assumption that the investments are complementary -- that each partner's investment reinforces and increases the "productivity" of the other person's investment.
These ground rules lead to the payoffs in Figure 19, in which "his" decisions are indicated by nodes labeled H for he, "her" decisions are denoted S for she, "her" payoffs are to the left in each pair and "his" are to the right. This complicated game of incomplete information has just four proper subgames, and all of them are basic. We will take the basic subgames in order from the top.
Exercise: Circle the four basic subgames of the game in Figure 19.
Since the decision to invest (or not) precedes the decision to separate (or not) we have to work backward to solve this game. Suppose that there are no vows and both partners invest. Then we have the first basic subgame as shown in Figure 20. This is a game of incomplete information and may be better analyzed by looking at it in the normal form, which is shown in Table 2. Clearly, in this subgame, to remain together is a dominant strategy for both partners, so we can identify 110, 110 as the payoffs that will in fact occur in case both partners invest.
Figure 19. Relationship
Figure 20. Basic Proper Subgame 1
Table 2. The Above Subgame in Normal Form
|
|
He |
||
|
stay |
go |
||
|
She |
stay |
110, 110 |
60, 60 |
|
go |
60, 60 |
60, 60 |
|
Now take the other symmetrical case and suppose that neither partner invests. We then have the subgame shown in Figure 21 and Table 3. Here, again, we have a clear dominant strategy, and it is to separate. The payoffs of symmetrical non-investment are thus 70,70.
Figure 21. Basic Proper Subgame 4
Table 3. Basic Proper Subgame 4 in Normal Form
|
|
He |
||
|
stay |
go |
||
|
She |
stay |
60, 60 |
70, 70 |
|
go |
70, 70 |
70, 70 |
|
Now suppose that only one partner invests, and (purely for illustrative purposes!) we consider the case in which "he" invests and "she" does not. We then have the subgame shown in Figure 22 and Table 4. Here again, separation is a dominant strategy, so the payoffs for the subgame where "she" invests and "he" does not are 115,40. A symmetrical analysis will give us payoffs of 40, 115 when "she" invests and "he" does not.
Putting these subgame outcomes together, we have reduced the more complex game to the one in Figure 23 and Table 5. This game resembles the Prisoners' Dilemma, in that non-investment is a dominant strategy, but when both players play their dominant strategies, both are worse off than they would be if both played the non-dominant strategy. Anyway, we identify 70, 70 as the subgame perfect equilibrium payoffs in the absence of marriage vows.
Figure 22. Basic Proper Subgame 2
Table 4. Basic Proper Subgame 2 in Normal Form
|
|
He |
||
|
stay |
go |
||
|
She |
stay |
105,30 |
115,40 |
|
go |
115,40 |
115,40 |
|
Figure 23. The Reduced Relationship Game
Table 5. The Reduced Relationship Game
|
|
He |
||
|
invest |
Don't |
||
|
She |
invest |
110,110 |
40,115 |
|
Don't |
115,40 |
70,70 |
|
But now suppose that, back at the beginning of things, the pair have the option to take, or not to take, a vow to stay together regardless. If they take the vow, only the "stay together" payoffs would remain as possibilities. If they do not take the vow, we know that there will be a separation and no investment, so we need consider only that possibility. In effect, there is now an earlier stage in the game. Using what we have already figured out and "reasoning backward," a partly reduced game will look like Figure 24.
Figure 24. A Partly Reduced Game with a Vow
We have already solved the lower proper subgame of this game. The upper proper subgame is simpler, since there are no go-or-stay games to be solved — having taken the vow, neither has the choice of going — but the payoffs here are taken from the stay-stay strategies in Figure 19. Once again, the upper proper subgame is a game of incomplete information, and it is shown in normal form in Table 6. This is a game with two Nash equilibria — where both sweethearts choose the same strategies, invest or do not invest — but since the "invest, invest" strategy is better for both, it is a Schelling point. If we identify the Schelling point at "invest, invest" as the likeliest outcome of this game, we have the fully reduced game shown in Figure 25.
Table 6. The Subgame with the Vow
|
|
He |
||
|
invest |
Don't |
||
|
She |
invest |
110,110 |
30,105 |
|
Don't |
105,30 |
60,60 |
|
Figure 24. A Fully Reduced Marriage Game
The Schelling point Nash equilibrium is for each player to take the vow and invest, and thus the payoff that will occur if a vow can be taken is 110, 110, the "efficient" outcome. In effect, willingness to take the vow is a "signal" that the partner intends to invest in the relationship -- if (s)he didn't, it would make more sense for him (her) to avoid the vow. Both partners are better off if the vow is taken, and if they had no opportunity to bind themselves with a vow, they could not attain the blissful outcome at the upper left.
Thus, when each partner decides whether or not to take the vow, each rationally expects a payoff of 110 if the vow is taken and 70 if not, and so, the rational thing to do is to take the vow. Of course, this depends strictly on the credibility of the commitment. In a world in which marriage vows become of questionable credibility, this reasoning breaks down, and we are back at Table 5, the Prisoners' Dilemma of "investment in the relationship." Some sort of first-stage commitment is necessary. Perhaps emotional commitment will be enough to make the partnership permanent -- emotional commitment is one of the things that is missing from this "rational" example. But emotional commitment is hard to judge. One of the things a credible vow does is to signal emotional commitment. If there are no vows that bind, how can emotional commitment be signaled? That seems to be one of the hard problems of living in modern society!
There is a lot of common sense here that your mother might have told you -- anyway my mother would have! What the game-theoretic analysis gives us is an insight on why Mom was right, after all, and how superficial reasoning can mislead us. If we compare the payoffs in Figure 24, we can observe that, given the investment choices made, no-one is ever better off in the upper proper subgame (vow) than in the lower (no vow). And except for the invest, invest strategies, both parties are worse off with the vow than without it. Thus I might reason -- wrongly! -- that since, ceteris paribus, I am better off with freedom of choice than without it, I had best not take the vow. But this illustrates a pitfall of "ceteris paribus" reasoning. In this comparison, ceteris are not paribus. Rather, the outcomes of the various subgames -- "ceteris" -- depend on the payoff possibilities as a whole. The vow changes the whole set of payoff possibilities in such a way that "ceteris" are changed -- non paribus -- and the outcome improved. The set of possible outcomes is worse but the selection of outcomes among the available set is so much improved that both parties are almost twice as well off as they would be had they not agreed to restrain their freedom of choice.
In other words: Cent' Anni!
The "Prisoners' Dilemma" is without doubt the most influential single analysis in Game Theory, and many social scientists, philosophers and mathematicians have used it as a justification for interventions by governments and other authorities to limit individual choice. After all, in the Prisoners' Dilemma, rational self-interested individual choice makes both parties worse off. A difficulty with this sort of reasoning is that it treats the authority as a deus ex machina -- a sort of predictable, benevolent robot who steps in and makes everything right. But a few game theorists and some economists (influenced by Game Theory but not strictly working in the Game Theoretic framework) have pointed out that the authority is a player in the game, and that makes a difference. This essay will follow that line of thought in an explicitly Game-Theoretic (but very simple) frame, beginning with the Prisoners' Dilemma. Since we begin with a Prisoners' Dilemma, we have two participants, whom we will call "commoners," who interact in a Prisoners' Dilemma with payoffs as follows:
|
|
|
Commoner 1 |
|
|
|
|
cooperate |
defect |
|
Commoner 2 |
cooperate |
10,10 |
0,15 |
|
defect |
15,0 |
5,5 |
|
The third player in this game is the "authority," and she (or he) is a very strange sort of player. She can change the payoffs to the commoners. The authority has two strategies, "penalize" or "don't penalize." If she chooses "penalize," the payoffs to the two commoners are reduced by 7. If she chooses "don't penalize," there is no change in the payoffs to the two commoners.
The authority also has two other peculiar characteristics:
Now suppose that the authority chooses the strategy "penalize" if, and only if, one or both of the commoners chooses the strategy "defect." The payoffs to the commoners would then be
|
|
|
Commoner 1 |
|
|
|
|
cooperate |
defect |
|
Commoner 2 |
cooperate |
10,10 |
-7,8 |
|
defect |
8,-7 |
-2,-2 |
|
But the difficulty is that this does not allow for the authority's flexibility and benevolence. Is that indeed the strategy the authority will choose? The strategy choices are shown as a tree in Figure 1 below. In the diagram, we assume that commoner 1 chooses first and commoner 2 second. In a Prisoners' Dilemma, it doesn't matter which participant chooses first, or they both choose at the same time. What is important is that the authority chooses last.
What we see in the figure is that the authority has a dominant strategy: not to penalize. No matter what the two commoners choose, imposing a penalty will make them worse off, and since the authority is benevolent -- she "feels their pain," her payoffs being the sum total of theirs -- she will always have an incentive to let them off, not to penalize. But the result is that she cannot change the Prisoners Dilemma. Both commoners will choose "defect," the payoffs will be (5,5) for the commoners, and 10 for the authority.
Perhaps the authority will announce that she intends to punish the commoners if they choose "defect." But they will not be fooled, because they know that, whatever they do, punishment will reduce the payoff to the authority herself, and that she will not choose a strategy that reduces her payoffs. Her announcements that she intends to punish will not be credible.
EXERCISE In this example, a punishment must fall on both commoners, even if only one defects. Does this make a difference for the result? Assume instead that the authority can impose a penalty on one and not the other, so that the authority has 4 strategies: no penalty, penalize commoner 1, penalize commoner 2, penalize both. What are the payoffs to the authority in the sixteen possible outcomes that we now have? Under what circumstances will a benevolent authority penalize? What are the equilibrium outcomes in this more complicated game?
There are two ways to solve this problem. First, the authority might not be benevolent. Second, the authority might not be flexible.
Non-benevolent authority:
We might change the payoffs to the authority so that the authority no longer "feels the pain" of the commoners. For example, make the payoff to the authority 1 if both commoners cooperate and zero otherwise. We might call an authority with a payoff system like this a "Prussian" authority, since she values "order" regardless of the consequences for the people, an attitude sometimes associated with the Prussian state. She then has nothing to lose by penalizing the commoners whenever there is defection, and announcements that she will penalize defection become credible. EXERCISE Suppose the authority is sadistic; that is, the authority's payoff is 1 if a penalty is imposed and 0 otherwise. What will be the game equilibrium in this case?
Non-flexible authority:
If the authority can somehow commit herself to imposing the penalty in some cases and not in others, perhaps by posting a bond greater than the 15 point cost of a penalty, then the announcement of an intention to penalize would become credible. The announcement and commitment would then be a strategy choice that the authority would make first, rather than last. Let's say that at the first step, the authority has two strategies: commit to a penalty whenever any commoner chooses "defect," or don't commit. We then have a tree diagram like Figure 2. What we see in Figure 2 is that if the authority commits, the outcome will be cooperation and a payoff of 20 for her, at the top; but if she does not commit, the outcome will be at the bottom -- both commoners defect and the payoff will be -4 for the authority. So the authority will choose the strategy of commitment, if she can, and in that case the rational, self-interested action of the commoners will lead to cooperation and good results. But, if the commoners irrationally defect, or if they don't believe the commitment and defect for that reason, then the authority is boxed in. She has to impose a penalty even though it makes everyone worse off. In short, she cannot be flexible.
What we have seen here are two principles that play an important part in modern macroeconomics. Many modern economists apply these principles to the central banks that control the money supply in modern economies. They are
The principle of "rules rather than discretion."
That is, the authority should act according to rules chosen in advance, rather than responding flexibly to events as they occur. In the case of the central banks, they should control the money supply or the interest rate on public debt (there is controversy about which) according to some simple rule, such as increasing the money supply at a steady rate or raising the interest rate when production is close to capacity, to prevent inflation. If some groups in the economy push their prices up, the monetary authority might be tempted to print money, which would cause inflation and help other groups to catch up with their prices, and perhaps reduce unemployment. But this must be avoided, since the groups will come to anticipate it and just push their prices up all the faster.
The principle of credibility.
It is not enough for the authority to be committed to the simple rule. The commitment must be credible if the rule is to have its best effect.
The difficulty is that it may be difficult for the authority to commit itself and to make the commitment credible. This can be illustrated by another application: dealing with terrorism. Some governments have taken the position that they will not negotiate with terrorists who take hostages, but when the terrorists actually have hostages, the pressure to make some sort of a deal can be very strong. What is to prevent a sensitive government from caving in -- just this once, of course! And potential terrorists know those pressures exist, so that the commitments of governments may not be credible to them, even when the governments have a "track record" of being tough.
This may have an effect on the way we want our institutions to function, at the most basic, more or less constitutional level. For example, in countries with strong currencies, like Germany and the United States, the central bank or monetary authority is strongly insulated from democratic politics. This means that the pressures for a more "flexible" policy expressed by voters are not transmitted to the monetary authority -- or, anyway, they are not as strong as they might otherwise be -- so the monetary authority is more likely to commit itself to a simple rule and the commitment will be more credible.
Are these "conservative" or "liberal" ideas? Some would say that they are conservative rather than liberal, on the grounds that liberals believe in flexibility -- considering each case on its own merits, and making the best decision in the circumstances, regardless of unthinking rules. But it may be a little more complex than that. This and the previous essay have considered particular cases in which commitment and rules work better than flexibility. There may be many other cases in which flexibility is needed. I should think that the "liberal" approach would be to consider the case for commitment and for rules rather than discretion on its merits in each instance, rather than relying on an unthinking rule against rules! Anyway, conservative or liberal or radical (as it could be!), the theory of games in extended form is now a key tool for understanding the role of commitment and rules in any society.
As an illustration of the concepts of sequential games and subgame perfect equilibrium, we shall consider a case in the employment relationship. This game will be a little richer in possibilities than the economics textbook discussion of the supply and demand for labor, in that we will allow for two dimensions of work the principles course does not consider: variable effort and the emotional satisfactions of "meaningful work." We also allow for a sequence of more or less reliable commitments in the choice of strategies.
We consider a three-stage game. At the first stage, one player in the game, the "worker," must choose between two kinds of strategies, that is, two "jobs." In either job, the worker will later have to choose between two rates of effort, "high" and "low." In either job, the output is 20 in the case of high effort and 10 if effort is low. We suppose that the first job is a "meaningful job," in the sense that it meets needs with which the worker sympathizes. As a consequence of this, the worker "feels the pain" of unmet needs when her or his output falls below the potential output of 20. This reduces her or his utility payoff when she or he shirks at the lower effort level. Of course, her or his utility also depends on the wage and (negatively) on effort. Accordingly, in Job 1 the worker's payoff is
wage - 0.3(20-output) - 2(effort)
where effort is zero or one. The other job is "meaningless," so that the worker's utility does not depend on output, and in this job it is
wage - 2(effort)
At the second stage of the game the other player, the "employer," makes a commitment to pay a wage of either 10 or 15. Finally, the worker chooses an effort level, either 0 or 1.
The payoffs are shown in Table 17-1.
|
|
|
|
Job |
|||
|
|
|
|
1 |
2 |
||
|
|
|
effort |
0 |
1 |
0 |
1 |
|
wage |
high |
|
-5, 12 |
5, 13 |
-5,15 |
5,13 |
|
low |
|
0,7 |
10,8 |
0,10 |
10,8 |
|
In each cell of the matrix, the worker's payoff is to the right of the comma and the employer's to the left. Let us first see what is "efficient" here. The payoffs are shown in Figure 1. Payoffs to the employer are on the vertical axis and those to the worker on the horizontal axis. Possible payoff pairs are indicated by stars-of-David. In economics, a payoff pair is said to be "efficient," or equivalently, "Pareto-optimal," if it is not possible to make one player better off without making the other player worse off. The pairs labeled A, B, and C have that property. They are (10,8), (5,13) and (-5,15). The others are inefficient. The red line linking A, B, and C is called the utility possibility frontier. Any pairs to the left of and below it are inefficient.
Now let us explore the subgame perfect equilibrium of this model. First, we may see that the low wage is a "dominant strategy" for the employer. That is, regardless which strategy the worker chooses -- job 1 and low effort, job 2 and high effort, and so on -- the employer is better off with low wages than with high. Thus the worker can anticipate that the wages will be low. Let us work backward. Suppose that the worker chooses job 2 at the first stage. This limits the game to the right-hand side of the table, which has a structure very much like the Prisoners' Dilemma. In this subgame, both players have dominant strategies. The worker's dominant strategy is low effort, and the Prisoners' Dilemma-like outcome is at (0,10). This is the outcome the worker must anticipate if he chooses Job 2.
What if he chooses Job 1? Then the game is limited to the left-hand side. In this game, too, the worker, like the employer, has a dominant strategy, but in this case it is high effort. This subgame is not Prisoners' Dilemma-like, since the equilibrium -- (10,8) -- is an efficient one. This is the outcome the worker must expect if she or he chooses Job 1, "meaningful work."
But the worker is better off in the subgame defined by "nonmeaningful work," Job 2. Accordingly, she will choose Job 2, and thus the equilibrium of the game as a whole (the subgame perfect equilibrium) is at (0,10). It is indicated by point E in the figure, and is inefficient.
Why is meaningful work not chosen in this model? It is not chosen because there is no effective reward for effort. With meaningful work, the worker can make no higher wage, despite her greater effort. Yet she does not reduce her effort because doing so brings the greater utility loss of seeing the output of meaningful work decline on account of her decision. The dilemma of having to choose between a financially unrewarded extra effort and witnessing human suffering on account of one's failure to make the effort seems to be a very stylized account of what we know as "burnout" in the human service professions.
Put differently, workers do not choose meaningful work at low wages because they have a preferable alternative: shirking at low effort levels in nonmeaningful jobs. Unless the meaningful jobs pay enough to make those jobs, with their high effort levels, preferable to the shirking alternative, no-one will choose them.
Inefficiency in Nash equilibria is a consequence of their noncooperative nature, that is, of the inability of the players to commit themselves to efficiently coordinated strategies. Suppose they could do so -- what then? Suppose, in particular, that the employer could commit herself or himself, at the outset, to pay a high wage, in return for the worker's commitment to choose Job 1. There is no need for an agreement about effort -- of the remaining outcomes, in the upper left corner of the table, the worker will choose high effort and (5,13), because of the "meaningful" nature of the work. This is an efficient outcome.
And that, after all, is the way markets work, isn't it? Workers and employers make reciprocal commitments that balance the advantages to one against the advantages to the other? It is, of course, but there is an ambiguity here about time. There is, of course, no measurement of time in the game example. But commitments to careers are lifetime commitments, and correspondingly, the wage incomes we are talking about must be lifetime incomes. The question then becomes, can employers make credible commitments to pay high lifetime income to workers who choose "meaningful" work with its implicit high effort levels? In the 1960's, it may have seemed so; but in 1995 it seems difficult to believe that the competitive pressures of a profit-oriented economic system will permit employers to make any such credible commitments.
This may be one reason why "meaningful work" has generally been organized through nonprofit agencies. But under present political and economic conditions, even those agencies may be unable to make credible commitments of incomes that can make the worker as well off in a high-effort meaningful job as in a low-effort nonmeaningful one. If this is so, there may be little long-term hope for meaningful work in an economy dominated by the profit system.
Lest I be misunderstood, I do not mean to argue that a state-organized system would do any better. There is an alternative: a system in which worker incomes are among the objectives of enterprises, that is, a cooperative system. It appears to be possible that such a system could generate meaningful work. There is empirical evidence that cooperative enterprises do in fact support higher effort levels than either profit-oriented or state organizations.
Of course, some nonmeaningful work has to be done, and it remains true that when nonmeaningful work is done it is done inefficiently and at a low effort level, that is, at E in the figure. In other words, the fundamental source of inefficiency in this model is the inability of the workers to make a credible commitment to high effort levels. If high effort could somehow be assured, then (depending on bargaining power) a high-effort efficient outcome would become a possibility in the nonmeaningful work subgame, and this in turn would eliminate the worker's incentive to choose nonmeaningful work in order to shirk. (If worker bargaining power should enforce the outcome at C, which is Pareto-optimal, the shirking nonmeaningful strategy would still dominate meaningful work). However, it does seem that it is very difficult to make commitments to high effort levels credible, or enforceable, in the context of profit-oriented enterprises.
It may be, then, the the problem of finding meaningful work and of burn-out in fields of meaningful work is a relatively minor aspect of the far broader question of effort commitment in modern economic systems. Perhaps it will do nevertheless as an example of the application of subgame perfect equilibrium concepts to an issue of considerable interest to many modern university students.
The bank's expected repayment thus is
.9(1100) + .05(500) + .05(0) = 1015 > 1010
.9(1900) + .05(1500) + .05(1500) = 1860 > 1500
And so they, too, accept the contract. Thus the loan is made,
