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Structural satisfaction in simple games
Mathematical Social Sciences 3 (1982) 397-401 North-Holland
397
Publishing Company
NOTE STRUCTURAL Christopher
SATISFACTION
IN SIMPLE GAMES
H. NEVISON
Department of Mathemlltics, Colgate University, Hamilton, NY 13346, U. S.A. Communicated by S.J. Brams
Received 1 May 1981 Revised I April 1982 Key words: Index of satisfaction; Zipke index; Brams-Lake
index; Banzhaf
index.
1. Introduction The concept of satisfaction in a voting (simple) game is introduced by Brams and Lake (1977) and Nevison (1978). The concept of satisfaction turns out to be closely connected with the idea of power in a simple game as developed by Banzhaf (1965, 1968) and others. Nevison (1978) has discussed this relationship of satisfaction to power in some detail. Here we present the proofs of some theorems on the limiting values of satisfaction for large games which closely parallel similar results for the Banzhaf family of measures of power (Dubey and Shapley, 1979). In addition, we compare axiomatic characterizations of measures of satisfaction and power.
2. Satisfaction and power We shall discuss certain measures of power and of satisfaction for simple games. A simple game may be defined as follows: Let 1={1,2,..., n} and let Y/ be a family of subsets of I such that if I+‘E rl and WC V, then VE uf. The pair (I, ti ) then denotes a simple game with characteristic function, U, for any SC I. The set I is interpreted as the set of pfayers or voters and y! is the set of winning coalitions. A voter i is critical to WE Y/ if ie W and W \ {i} $ FL We use the following notation. 01654896/82/0000-OOOOooo/$O2.75 0 1982 North-Holland
398
C. H. Nevcson / Structural satisfaction in simple games
Ci= pVtidV~
Y/
and W\{i}tf
fl)l*
We will discuss limit theorems for simple games which can be represented as weighted voting games. These are denoted by [c; or, cy, . . . , w,J (or (c; ol, . . . , con)) defining (I, s ) as follows: WE yf if and only if CiEWOi2 c (or >c). Our notation follows Dubey (1975). Nevison (1978) defines the Zfpke index of satisfaction (see also Nevison, Schoepke and Zicht (1978)): Z(i) = wi/2”-‘. The satisfaction for voter i is the probability that he is in the winning coalition, assuming each voter is equally likely to vote either way. Banzhaf (1965, 1968) simply uses Ci as his measure of a voter’s power. Other authors have normalized this raw Banzhaf index in various ways. The normalization used here is the absolute Banzhaf index: B(i) = Ci/2”- I, the probability that voter i is critical to the winning coalition. Nevison, Schoepke and Zicht (1978), Nevison (19;8), Brams and Lake (1977) and Straffin (1978), all discuss the close relationship between the Banzhaf indices of power and the Zipke or Brams-Lake indices of satisfaction. The relationship for the indices used here is B(i) = 2(2(i) - w/2”).
(1)
S%ce w/2” = Z(i) for i a dummy - a voter who is never critical - this equation suggests a useful definition of power in terms of satisfaction: Power is the gain in satisfaction which accrues to a player over that which could be expected by a dummy, due to the player’s strategic position in the game.
3. Axiomatic characterizations We define operations on simple games defined on the same set of players, I: For a game (I+ YI) we define the new game (I, n rt ) for any permutation 71of I by W~nfl ifandonlyif n-%‘E Y!. For two games (I, ti ) and (I, y ), (I, YI U Y) and (I, YI n y ) are also simple games on f. Theorem 3. (Dubey and Shapley, 1979). The unique function @from the set of all
simple games on I to IF?which sati: fies the following axioms is Qii(YI ) Al. If iisadummyfor(I, w), then &(M)=Q. A2. C,,,@i( H)= CIEIcifor ci definedfor (I, 11). A3. For any permutation n on I, #n(i)(n( Y/ )) = ei ( vt ). A4. @(IYr’U Y)+@(Ym
= Ci
Y)=#(Yi)+(p(Y).
A similar set of axioms characterizes the Zipke index of satisfaction.
for (I, yf ):
C. H. Nevison
i Structural
satisfaction
in srrnple games
399
Theorem 3. The unique function @from the set of al/simple games on I to R” which satisfies the following axioms is @i(H ) = Z(i) for (I, ~1): Bl (Dictatorship). If iE W implres WE ~1,then &( rf ) = 1. B2 (Total satisfaction). C,, I @i( Y/ ) = C wE H1WI/p - 1. B3 (Symmetry). For any permutation n on I, &(i) (n yf) = A( H ). 84 (Additivity). For any two simple games (I, U) and (I, I). @(YN
Y)+qqYrn
Y)=@(Yf)+@(Y).
The proof of Theorem 3.2 is a straightforward modification of the proof of Theorem 1 in Dubey and Shapley (1979) and is omitted. The axioms for the Zipke index have a natural interpretation: Bl expresses the fact that a dictator is always satisfied; B2 normalizes the total satisfaction to be the mean number in a winning coalition; B3 is the obvious symmetry axiom; and B4 shows that satisfaction is based on the number of winning coalitions a player is in.
4. Satisfaction for large games We examine the limiting behavior of the Zipke index for an important class of weighted voting games and apply these results to obtain the limiting behavior for the absolute Banzhaf index. The presentation follows the spirit and notation of Dubey’s (1975) study of the relative Banzhaf index. Dubey and Shapley (1979) independently derive a proof for the limiting properties of the Banzhaf index similar to the arguments presented here. We derive the limiting behavior for the Zipke index for a sequence of games (P), where the number of players increases to infinity. The total n=k+ I,k+2,..., voting weights for the games are w(P) = 1 and the quota c is fixed. The first k players are major players whose voting weight is the same for every P. The number. increase in the minor players, remaining players, rn = [c; 01, . . . . 0J&,ok”+1,. . . . cr,“]where the ui are the fixed weights of the major players and the 4 are the weights of the minor players. The weights of the minor players may differ, but lim, __ max (ai”>=O. Under these conditions we have the following theorem. Theorem 4.1. If P,n=k+ 1, k+2 ,..., is a sequence of weighted voting games as defined above, then for a major player ., the limiting value of‘ the Zipke index is Z,(i) = !ir~IZn(i) = (Zl (i) + Zz (i))/2
where Z1 bna Zz are the Zipke indices J”rrthe games G, = [c - CU’~;U~, . . . , ok] and ez = (c- d2; CO,;...,ct)k). For a minor player j, Z,(j) = (w(G1)+ w(G~))/~~ + ’
CM. Nevison / Structural satisfaction in simple games
where W(Gi)is the number of winning coalitions in Gi. Proof. Let cy= I - Cfz, Oi. For any major player i let X, =&!$c$’ where 4 = 1 or 0 according to whether j votes with or against i. Let w(S) be the number of winning coalitions in P produced by combinations of a subset of major players, S, with minor players. In game P, we have Wi/2”--1 = EiE s w(S)/2” - ’ ) so Z,(i)=
1
ieS
limw(S)/2”-’
( nea)
. >
Lim, _ w(S)/2”- l = (l/2&- ’ ) lim,.,, P [Xn I c - m(S)], regarding each minor player independently equally likely to vote either way, where w(S) = zipSui. By the Central Limit Theorem (Feller, 1968; p. 254) the limit of this probability will be 0, +, or 1 according to whether c - U(S) is > , = , or < (r/2. (This application of the Central Limit Theorem is obvious if all the minor players have equal weight or, indeed, if max ai”< A/n for some constant A. The details of the general case stated in the theorem are omitted.) Thus the contribution to Z,(i) from S is 0, jk, or jk- ’ accord’ Jg to whether w(S) is c ,= ,or >c- ar/2 - exactly the contribution to (2, (i) + Z,(i))/2 from S. The first part of the theorem follows. The statement for a minor player has as similar proof. A similar limit theorem for the Brams-Lake index can be obtained as an immediate corollary. An application of (1) and Theorem 4.1 yields the following. Corollary 4.2 (Dubey, 1975). For a sequence of games rn as defined above, the limiting valuesfor the absolute Banzhaf index of power for a major player i and a minor player j are given by B(i) = (B,(i) + &z(i))/2
and
B(j) = 0,
where Bl and I$ are the Banzhaf indices for GI and G.
5. Conclusion We have obtained limiting results and an axiomatization for the Zipke index of satisfaction analogous to similar results for the Banzhaf index of power. The characteristics of ‘power’ seem to depend directly on similar characteristics for ‘satisfaction’, the latter a mathematically simpler concept.
J.F. Banzhaf 111, Weighted voting doesn’t work: 317-343.
A mathematical
analysis,
Rutgers Law Rev. 19 (1965)
C. H. Nevison 1’S~ructurul satisfaction in simple games
4Ql
J.F. Banzhaf 111, One man, 3312 votes: A mathematical analysis of the electoral college, Villanova Lau Rev. 13 (1968) 304-332. S.J. Brams and M. Lake, Power and satisfaction in a representative democracy, in: Brams, Schotter and Schwiidiauer, eds., Applied Game Theory: Proc. Conf. at the Institute for Advanced
Studies,
Vienna, 1978 (Physica Verlag, Wiirzburg, West Germany, 1979). P. Dubey, Some results on values of finite and infinite games, Tech. Rept., Center for Applied Mathematics, Cornell University, Ithaca, New York (1975). P. Dubey and L.S. Shapley, Mathematical properties of the Banzhaf power index, Math. Oper. Res. 4 (1979) 99-131. W. Feller (1%8), An Introduction to Probability Theory and Its Applications, Vol. 4 (Wiley, New York, 3rd ed., 1968). C.H. Nevison, S. Schoepke and B. Zicht, A naieve approach to the Banzhaf index of power, Behav. Sci. 23 (1978) 130-131. P.D. Straffin, Probability models for power indices, in: Ordeshook, ed., Game Theory and Political Science (New York Univ. Press, New York, 1978).
397
Publishing Company
NOTE STRUCTURAL Christopher
SATISFACTION
IN SIMPLE GAMES
H. NEVISON
Department of Mathemlltics, Colgate University, Hamilton, NY 13346, U. S.A. Communicated by S.J. Brams
Received 1 May 1981 Revised I April 1982 Key words: Index of satisfaction; Zipke index; Brams-Lake
index; Banzhaf
index.
1. Introduction The concept of satisfaction in a voting (simple) game is introduced by Brams and Lake (1977) and Nevison (1978). The concept of satisfaction turns out to be closely connected with the idea of power in a simple game as developed by Banzhaf (1965, 1968) and others. Nevison (1978) has discussed this relationship of satisfaction to power in some detail. Here we present the proofs of some theorems on the limiting values of satisfaction for large games which closely parallel similar results for the Banzhaf family of measures of power (Dubey and Shapley, 1979). In addition, we compare axiomatic characterizations of measures of satisfaction and power.
2. Satisfaction and power We shall discuss certain measures of power and of satisfaction for simple games. A simple game may be defined as follows: Let 1={1,2,..., n} and let Y/ be a family of subsets of I such that if I+‘E rl and WC V, then VE uf. The pair (I, ti ) then denotes a simple game with characteristic function, U, for any SC I. The set I is interpreted as the set of pfayers or voters and y! is the set of winning coalitions. A voter i is critical to WE Y/ if ie W and W \ {i} $ FL We use the following notation. 01654896/82/0000-OOOOooo/$O2.75 0 1982 North-Holland
398
C. H. Nevcson / Structural satisfaction in simple games
Ci= pVtidV~
Y/
and W\{i}tf
fl)l*
We will discuss limit theorems for simple games which can be represented as weighted voting games. These are denoted by [c; or, cy, . . . , w,J (or (c; ol, . . . , con)) defining (I, s ) as follows: WE yf if and only if CiEWOi2 c (or >c). Our notation follows Dubey (1975). Nevison (1978) defines the Zfpke index of satisfaction (see also Nevison, Schoepke and Zicht (1978)): Z(i) = wi/2”-‘. The satisfaction for voter i is the probability that he is in the winning coalition, assuming each voter is equally likely to vote either way. Banzhaf (1965, 1968) simply uses Ci as his measure of a voter’s power. Other authors have normalized this raw Banzhaf index in various ways. The normalization used here is the absolute Banzhaf index: B(i) = Ci/2”- I, the probability that voter i is critical to the winning coalition. Nevison, Schoepke and Zicht (1978), Nevison (19;8), Brams and Lake (1977) and Straffin (1978), all discuss the close relationship between the Banzhaf indices of power and the Zipke or Brams-Lake indices of satisfaction. The relationship for the indices used here is B(i) = 2(2(i) - w/2”).
(1)
S%ce w/2” = Z(i) for i a dummy - a voter who is never critical - this equation suggests a useful definition of power in terms of satisfaction: Power is the gain in satisfaction which accrues to a player over that which could be expected by a dummy, due to the player’s strategic position in the game.
3. Axiomatic characterizations We define operations on simple games defined on the same set of players, I: For a game (I+ YI) we define the new game (I, n rt ) for any permutation 71of I by W~nfl ifandonlyif n-%‘E Y!. For two games (I, ti ) and (I, y ), (I, YI U Y) and (I, YI n y ) are also simple games on f. Theorem 3. (Dubey and Shapley, 1979). The unique function @from the set of all
simple games on I to IF?which sati: fies the following axioms is Qii(YI ) Al. If iisadummyfor(I, w), then &(M)=Q. A2. C,,,@i( H)= CIEIcifor ci definedfor (I, 11). A3. For any permutation n on I, #n(i)(n( Y/ )) = ei ( vt ). A4. @(IYr’U Y)+@(Ym
= Ci
Y)=#(Yi)+(p(Y).
A similar set of axioms characterizes the Zipke index of satisfaction.
for (I, yf ):
C. H. Nevison
i Structural
satisfaction
in srrnple games
399
Theorem 3. The unique function @from the set of al/simple games on I to R” which satisfies the following axioms is @i(H ) = Z(i) for (I, ~1): Bl (Dictatorship). If iE W implres WE ~1,then &( rf ) = 1. B2 (Total satisfaction). C,, I @i( Y/ ) = C wE H1WI/p - 1. B3 (Symmetry). For any permutation n on I, &(i) (n yf) = A( H ). 84 (Additivity). For any two simple games (I, U) and (I, I). @(YN
Y)+qqYrn
Y)=@(Yf)+@(Y).
The proof of Theorem 3.2 is a straightforward modification of the proof of Theorem 1 in Dubey and Shapley (1979) and is omitted. The axioms for the Zipke index have a natural interpretation: Bl expresses the fact that a dictator is always satisfied; B2 normalizes the total satisfaction to be the mean number in a winning coalition; B3 is the obvious symmetry axiom; and B4 shows that satisfaction is based on the number of winning coalitions a player is in.
4. Satisfaction for large games We examine the limiting behavior of the Zipke index for an important class of weighted voting games and apply these results to obtain the limiting behavior for the absolute Banzhaf index. The presentation follows the spirit and notation of Dubey’s (1975) study of the relative Banzhaf index. Dubey and Shapley (1979) independently derive a proof for the limiting properties of the Banzhaf index similar to the arguments presented here. We derive the limiting behavior for the Zipke index for a sequence of games (P), where the number of players increases to infinity. The total n=k+ I,k+2,..., voting weights for the games are w(P) = 1 and the quota c is fixed. The first k players are major players whose voting weight is the same for every P. The number. increase in the minor players, remaining players, rn = [c; 01, . . . . 0J&,ok”+1,. . . . cr,“]where the ui are the fixed weights of the major players and the 4 are the weights of the minor players. The weights of the minor players may differ, but lim, __ max (ai”>=O. Under these conditions we have the following theorem. Theorem 4.1. If P,n=k+ 1, k+2 ,..., is a sequence of weighted voting games as defined above, then for a major player ., the limiting value of‘ the Zipke index is Z,(i) = !ir~IZn(i) = (Zl (i) + Zz (i))/2
where Z1 bna Zz are the Zipke indices J”rrthe games G, = [c - CU’~;U~, . . . , ok] and ez = (c- d2; CO,;...,ct)k). For a minor player j, Z,(j) = (w(G1)+ w(G~))/~~ + ’
CM. Nevison / Structural satisfaction in simple games
where W(Gi)is the number of winning coalitions in Gi. Proof. Let cy= I - Cfz, Oi. For any major player i let X, =&!$c$’ where 4 = 1 or 0 according to whether j votes with or against i. Let w(S) be the number of winning coalitions in P produced by combinations of a subset of major players, S, with minor players. In game P, we have Wi/2”--1 = EiE s w(S)/2” - ’ ) so Z,(i)=
1
ieS
limw(S)/2”-’
( nea)
. >
Lim, _ w(S)/2”- l = (l/2&- ’ ) lim,.,, P [Xn I c - m(S)], regarding each minor player independently equally likely to vote either way, where w(S) = zipSui. By the Central Limit Theorem (Feller, 1968; p. 254) the limit of this probability will be 0, +, or 1 according to whether c - U(S) is > , = , or < (r/2. (This application of the Central Limit Theorem is obvious if all the minor players have equal weight or, indeed, if max ai”< A/n for some constant A. The details of the general case stated in the theorem are omitted.) Thus the contribution to Z,(i) from S is 0, jk, or jk- ’ accord’ Jg to whether w(S) is c ,= ,or >c- ar/2 - exactly the contribution to (2, (i) + Z,(i))/2 from S. The first part of the theorem follows. The statement for a minor player has as similar proof. A similar limit theorem for the Brams-Lake index can be obtained as an immediate corollary. An application of (1) and Theorem 4.1 yields the following. Corollary 4.2 (Dubey, 1975). For a sequence of games rn as defined above, the limiting valuesfor the absolute Banzhaf index of power for a major player i and a minor player j are given by B(i) = (B,(i) + &z(i))/2
and
B(j) = 0,
where Bl and I$ are the Banzhaf indices for GI and G.
5. Conclusion We have obtained limiting results and an axiomatization for the Zipke index of satisfaction analogous to similar results for the Banzhaf index of power. The characteristics of ‘power’ seem to depend directly on similar characteristics for ‘satisfaction’, the latter a mathematically simpler concept.
J.F. Banzhaf 111, Weighted voting doesn’t work: 317-343.
A mathematical
analysis,
Rutgers Law Rev. 19 (1965)
C. H. Nevison 1’S~ructurul satisfaction in simple games
4Ql
J.F. Banzhaf 111, One man, 3312 votes: A mathematical analysis of the electoral college, Villanova Lau Rev. 13 (1968) 304-332. S.J. Brams and M. Lake, Power and satisfaction in a representative democracy, in: Brams, Schotter and Schwiidiauer, eds., Applied Game Theory: Proc. Conf. at the Institute for Advanced
Studies,
Vienna, 1978 (Physica Verlag, Wiirzburg, West Germany, 1979). P. Dubey, Some results on values of finite and infinite games, Tech. Rept., Center for Applied Mathematics, Cornell University, Ithaca, New York (1975). P. Dubey and L.S. Shapley, Mathematical properties of the Banzhaf power index, Math. Oper. Res. 4 (1979) 99-131. W. Feller (1%8), An Introduction to Probability Theory and Its Applications, Vol. 4 (Wiley, New York, 3rd ed., 1968). C.H. Nevison, S. Schoepke and B. Zicht, A naieve approach to the Banzhaf index of power, Behav. Sci. 23 (1978) 130-131. P.D. Straffin, Probability models for power indices, in: Ordeshook, ed., Game Theory and Political Science (New York Univ. Press, New York, 1978).