cooperative TU game, value, axiomatic characterization, Shapley value,

Anna B. Khmelnitskaya

Crossref

Shapley value for constant-sum games

Research Papers in Economics

It is proved that Young’s [4] axiomatization for the Shapley value by marginalism, efficiency, and symmetry is still valid for the Shapley value defined on the class of nonnegative constant-sum games with nonzero worth of grand coalition and on the entire class of constant-sum games as well

Khmelnitskaya, Anna B.

NARCIS 

University of Twente Research Information

Abstract. It is proved that Young’s [4] axiomatization for the Shapley valueby marginalism, efficiency, and symmetry is still valid for the Shapley valuedefined on the class of nonnegative constant-sum games with nonzero worthof grand coalition and on the entire class of constant-sum games as well.Key words: cooperative TU game, value, axiomatic characterization, Shapleyvalue.Among single-valued solutions usually called values the most famous and themost appealing is the Shapley value [2]. Different axiomatizations for theShapley value defined on the entire space of games with fixed set of players areknown. Twomain of them are the classical one given by Shapley [2] and that ofYoung [4]. The original Shapley’s axiomatization exploits the additivity axiomthat being a very beautiful mathematical statement does not express any fair-ness property. The axiomatization of Young that characterizes the Shapleyvalue by marginalism, efficiency, and symmetry appears to be more attractivesince all the axioms present different reasonable properties of fair division.However, not always we consider the entire space of games. Sometimes due todifferent reasons we restrict consideration to some subclass of games, e.g. tononnegative or positive games, to simple games, to convex games, to constant-sum games, etc. And a reasonable question arises— is Young’s axiomatizationfor the Shapley value on the subclass of games under scrutiny still valid? InInt J Game Theory (2003) 32: 223–227DOI 10.1007/s001820300154 The research was supported by NWO (The Netherlands Organization for Scientific Research)grant NL-RF 047-008-010. I am thankful to Theo Driessen, Natalia Naumova and Elena Yanovskaya for interestingdiscussions and comments. The useful remarks of two anonymous referees are also appreciated.Shapley value for constant-sum gamesAnna B. KhmelnitskayaSPb Institute for Economics and Mathematics Russian Academy of Sciences, 1 Tchaikovsky St.,191187 St. Petersburg, Russia (e-mail: anna@AK3141.spb.edu)Received: May 2002general, the answer is negative. For instance, Young’s axiomatization for theShapley value considered on the subclass of all simple games (the Shapley-Shubik power index) is not correct. Indeed, in a simple game with three playersthe normalized Banzhaf power index presents a counterexample. In a simplegame with three players the Shapley-Shubik and normalized Banzhaf indicesagreewith each other, except in the casewhere the Shapley-Shubik index assignsto one player the value 2/3 and to each of the other players 1/6; in this case theBanzhaf index assigns to these players the values 3/5 and 1/5 respectively. Yet inthe case of simple games with three players the normalized Banzhaf indexappears to satisfy the conditions of Young (marginalism, efficiency, and sym-metry).1 In this note we prove that Young’s axiomatization is valid for theShapley value defined on the class of nonnegative constant-sum games withnonzeroworth of grand coalition and on the entire class of constant-sum gamesas well. Onemight argue that constant-sum games are not that appealing exceptfor their nice mathematical properties. On the other hand, there are indeedrelevant classes of nonnegative TUgames satisfying the constant sum conditionfrom outset, for example classes of simple majority games. In the literature,other solutions have also been related to the particular class of constant-sumgames, for example in [3] it is shown that for this class of games the modifiednucleolus coincides with the prenucleolus. The class of nonnegative constant-sum games with nonzero worth of grand coalition and in particular the Shapleyvalue defined on this class appear in the study of semiproportional values in [1].Now we recall some definitions and notation. A cooperative game withtransferable utility (TU game) is a pair hN ; vi, where N ¼ f1; . . . ; ng is a finiteset of n  2 players and v : 2N ! R is a characteristic function, defined on thepower set of N , satisfying vð;Þ ¼ 0. A subset S  N (or S 2 2N ) of s players iscalled a coalition, and the associated real number vðSÞ presents the worth ofthe coalition S. The set of all games with a fixed player set N we denote GN .For simplicity of notation and if no ambiguity appears, we write v instead ofhN ; vi when refer to a game. For any set of games G  GN , a value on G is amapping n : G! Rn that associates with each game v 2 G a vector nðvÞ 2 Rn,where the real number niðvÞ represents the payoff to player i in the game v.A value n is marginalist if, for all v 2 G, for every i 2 N , niðvÞ depends onlyupon the ith marginal utility vector fvðS [ iÞ  vðSÞgSNni, i.e.,niðvÞ ¼ /iðfvðS [ iÞ  vðSÞgSNniÞ;where /i : R2n1 ! R1.A value n is efficient if, for all v 2 G,Xi2NniðvÞ ¼ vðNÞ:A value n is symmetric if, for all v 2 G, for any permutation p : N ! N , andfor all i 2 N ,npðiÞðvpÞ ¼ niðvÞ;where vpðSÞ ¼ vðpðSÞÞ for all S  N , S 6¼ ;.1This counterexample is due to Moshe’ Machover.224 A. B. KhmelnitskayaThroughout the remainder of the paper we restrict our consideration tononnegative constant-sum games of the classGþcN ¼ fv 2 GN jvðNÞ 6¼ 0; vðSÞ  0; vðSÞ þ vðNnSÞ ¼ vðNÞ; for all S  Ng:We prove below that the Shapley value defined on GþcN can be characterizedby three Young’s axioms of marginalism, efficiency, and symmetry. Our proofstrategy by induction is similar to that in Young. The proof of Young is notapplicable directly to the considered case since an important moment in hisproof is an expansion of a game v 2 GN via unanimity basis fuT gTNT 6¼;defined asuT ðSÞ ¼1; T  S;0; T 6 S; for all S  N ,but unanimity games uT , for all nonempty and non-singleton coalitionsT  N , do not belong to GþcN .Theorem 1. The only efficient, symmetric, and marginalist value defined on GþcNis the Shapley value.Proof: Every game v 2 GþcN , being a constant-sum game, appears to be a self-dual game, i.e., v ¼ v, where for any v 2 GN , a dual game v is defined asvðSÞ ¼ vðNÞ  vðNnSÞ; for all S  N :For any game v 2 GN presented via unanimity basis fuTgTNT 6¼;,v ¼XTNT 6¼;kT uT ;the dual game v can be presented via dual unanimity basis fuT gTNT 6¼; with thesame set of coefficients kT , T  N , T 6¼ ;, i.e.,v ¼XTNT 6¼;kT uT ;since it is easy to check that, for any two games v; v0 2 GN and any real a,ðvþ v0Þ ¼ v þ v0;ðavÞ ¼ av:Therefore, every constant-sum game v 2 GþcN being self-dual can be presentedas a linear combinationv ¼XTNT 6¼;kT wT ð1Þof games wT , T  N , T 6¼ ;, where for all S  N ,wT ðSÞ ¼uT ðSÞ þ uT ðSÞ2¼1; T  S,1=2; T \ S 6¼ ;; T 6 S,0; T \ S ¼ ; .8<:For all T  N , T 6¼ ;, wT 2 GþcN . Similarly to unanimity game uT , in any gamewT , T  N , T 6¼ ;, every player i =2 T is a null-player, i.e., all of his marginalShapley value for constant-sum games 225contributions are equal to zero. The Shapley values of both games wT and uTcoincide, i.e., for every i 2 N ,ShiðwT Þ ¼ ShiðuT Þ ¼1=t; i 2 T ,0; i =2 T .However, we cannot apply the induction procedure directly to the expansion(1) since for a nonnegative constant-sum game v 2 GþcN , not all coefficients kTin (1) are necessarily nonnegative (we can state only that at least one of themis positive), and deletion of a term in (1) may lead out of the class GþcN .To overcome the problem we consider another than (1) expansion of agame v 2 GþcN via games wT , T  N , T 6¼ ;. For an expression (1) and eacht ¼ 1; . . . ; n, definekt ¼ maxfmaxT :jT j¼tkT ; 0g; and kT ¼ kt  kT  0:Consider a symmetric gameu ¼Xnt¼1ktXTNT 6¼;jT j¼twT :One can easily see that u 2 GþcN . Since (1),v ¼ uXTNT 6¼;kT wT : ð2ÞObserve that deletion of any term under the summation sign in (2) does notmove out of GþcN sincekT  0, T  N , T 6¼ ;. Let now the index I of a gamev 2 GþcN be the minimum number of terms under the summation in anexpression (2), i.e.,v ¼ uXIk¼1kTk wTk ;where all kTk 6¼ 0. We proceed the remaining part of the proof by induction onthis index I .Let n be an efficient, symmetric, and marginalist value on GþcN .If I ¼ 0, then v ¼ u, and for symmetric game u the result follows directlyfrom efficiency and symmetry assumptions about both values n and theShapley value.Assume now that nðvÞ is the Shapley value whenever the index of v 2 GþcNis at most I , and consider some v 2 GþcN with the index equal to I þ 1. LetT ¼ \Iþ1k¼1Tk. For all i; j 2 T , symmetry implies that niðvÞ ¼ njðvÞ; the similarstatement is true for the Shapley value too. Hence, combined with therequirement of efficiency (both payoff vectors nðvÞ and ShðvÞ sum up to vðNÞ)it is sufficient to prove that niðvÞ ¼ ShiðvÞ when i =2 T . Define a gamevðiÞ ¼ uXk:i2TkkTk wTk :Obviously, the index of vðiÞ is at most I and, therefore, by inductionhypothesis, nðvðiÞÞ ¼ ShðvðiÞÞ. To complete the proof notice that both ith226 A. B. Khmelnitskayamarginal utility vectors relevant to the games v and vðiÞ coincide and, so, bymarginalism of both values n and the Shapley value, niðvÞ ¼ niðvðiÞÞ andShiðvÞ ¼ ShiðvðiÞÞ. hTo conclude with it is reasonable to note that Young’s axiomatization isvalid as well for the Shapley value defined on the entire class of constant-sumgamesGcN ¼ fv 2 GN j vðSÞ þ vðNnSÞ ¼ vðNÞ; for all S  Ng:Indeed, the last statement can be proved by the same way as it was done forthe case of the subclass of nonnegative constant-sum games GþcN or one canexploit the same proof as in Young but with replacement of unanimity gamesuT , T  N , T 6¼ ;, via games wT , T  N , T 6¼ ;.References1. Khmelnitskaya AB, Driessen TSH (2003) Semiproportional values for TU games. Math MethOper Res 57:495–5112. Shapley LS (1953) A value for n-person games. In: Tucker AW, Kuhn HW (eds.)Contributions to the theory of games II. Princeton University Press, Princeton, NJ,pp. 307–3173. Sudhölter P (1997) The modified nucleolus: properties and axiomatizations. Int J GameTheory 26:147–1824. Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14:65–72Shapley value for constant-sum games 227

Shapley value for constant-sum games

Abstract

Similar works

Full text

Available Versions

Crossref

Research Papers in Economics

NARCIS

University of Twente Research Information