The Rolaids Relief Award goes annually to the top relief pitcher in both leagues.
It is the only award handed out that isn't voted on, but rather awarded to the player with the highest score based on the following formula:

Save = 2 pts, Tough Save (entering the game with the tying run on base) = 4 pts, Win = 2 pts, Loss = -2 pts, Blown Save = -2 pts

The scoring system was different when the award was first given out in 1976, and was changed in 1988. The current system was instituted in 2000.
1976 Rawly Eastwick Cincinnati 69 pts NL
Bill Campbell Minnesota 69 AL
1977 Rollie Fingers San Diego 77 NL
Bill Campbell Boston 79 AL
1978 Rollie Fingers San Diego 73 NL
Rich Gossage New York 63 AL
1979 Bruce Sutter Chicago 80 NL
Jim Kern Texas 79 AL
1980 Rollie Fingers San Diego 59 NL
Dan Quisenberry Kansas City 83 AL
1981 Bruce Sutter St. Louis 51 NL
Rollie Fingers Milwaukee 65 AL
1982 Bruce Sutter St. Louis 82 NL
Dan Quisenberry Kansas City 81 AL
1983 Al Holland Philadelphia 62 NL
Dan Quisenberry Kansas City 97 AL
1984 Bruce Sutter St. Louis 92 NL
Dan Quisenberry Kansas City 97 AL
1985 Jeff Reardon Montreal 78 NL
Dan Quisenberry Kansas City 81 AL
1986 Todd Worrell St. Louis 80 NL
Dave Righetti New York 100 AL
1987 Steve Bedrosian Philadelphia 87 NL
Dave Righetti New York 72 AL
1988 John Franco New York 91 NL
Dennis Eckersley Oakland 123 AL
1989 Mark Davis San Diego 126 NL
Jeff Russell Texas 106 AL
1990 John Franco New York 91 NL
Bobby Thigpen Chicago 151 AL
1991 Lee Smith St. Louis 135 NL
Bryan Harvey California 122 AL
1992 Lee Smith St. Louis 103 NL
Dennis Eckersley Oakland 159 AL
1993 Randy Myers Chicago 143 NL
Jeff Montgomery Kansas City 127 AL
1994 Rod Beck San Francisco 80 NL
Lee Smith Baltimore 81 AL
1995 Tom Henke St. Louis 104 NL
Jose Mesa Cleveland 140 AL
1996 Jeff Brantley Cincinnati 120 NL
John Wetteland New York 119 AL
1997 Jeff Shaw Cincinnati 116 NL
Randy Myers Baltimore 131 AL
1998 Trevor Hoffman San Diego 161 NL
Tom Gordon Boston 142 AL
1999 Billy Wagner Houston 117 NL
Mariano Rivera New York 129 AL
2000 Antonio Alfonseca Florida 125 NL
Todd Jones Detroit 118 AL
2001 Armando Benitez New York 130 NL
Mariano Rivera New York 135 AL
2002 John Smoltz Atlanta 164 NL
Billy Koch Oakland 136 AL
2003 Eric Gagne Los Angeles 165 NL
Keith Foulke Oakland 139 AL
2004 Eric Gagne Los Angeles 142 NL
Mariano Rivera New York 157 AL
2005 Chad Cordero Washington 137 NL
Mariano Rivera New York 135 AL
2006 Trevor Hoffman San Diego 124 NL
Francisco Rodriguez Anaheim 134 AL

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©Mike Fish