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1 econstor Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics van den Brink, Rene Working Paper Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games Tinbergen Institute Discussion Paper, No /1 Provided in Cooperation with: Tinbergen Institute, Amsterdam and Rotterdam Suggested Citation: van den Brink, Rene (2009) : Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games, Tinbergen Institute Discussion Paper, No /1 This Version is available at: Nutzungsbedingungen: Die ZBW räumt Ihnen als Nutzerin/Nutzer das unentgeltliche, räumlich unbeschränkte und zeitlich auf die Dauer des Schutzrechts beschränkte einfache Recht ein, das ausgewählte Werk im Rahmen der unter nachzulesenden vollständigen Nutzungsbedingungen zu vervielfältigen, mit denen die Nutzerin/der Nutzer sich durch die erste Nutzung einverstanden erklärt. Terms of use: The ZBW grants you, the user, the non-exclusive right to use the selected work free of charge, territorially unrestricted and within the time limit of the term of the property rights according to the terms specified at By the first use of the selected work the user agrees and declares to comply with these terms of use. zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics

2 TI /1 Tinbergen Institute Discussion Paper Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games René van den Brink VU University Amsterdam, and Tinbergen Institute.

3 Tinbergen Institute The Tinbergen Institute is the institute for economic research of the Erasmus Universiteit Rotterdam, Universiteit van Amsterdam, and Vrije Universiteit Amsterdam. Tinbergen Institute Amsterdam Roetersstraat WB Amsterdam The Netherlands Tel.: +31(0) Fax: +31(0) Tinbergen Institute Rotterdam Burg. Oudlaan PA Rotterdam The Netherlands Tel.: +31(0) Fax: +31(0) Most TI discussion papers can be downloaded at

4 Efficiency and Collusion Neutrality ofsolutionsforcooperativetu-games RenévandenBrink Department of Econometrics and Tinbergen Institute VU University De Boelelaan HV Amsterdam The Netherlands July 21, 2009 ThisresearchispartoftheResearchProgram StrategicandCooperativeDecisionMaking. Ithank Arantza Estévez-Fernández for comments on a previous draft.

5 Abstract Three well-known solutions for cooperative TU-games are the Shapley value, the Banzhaf value and the equal division solution. In the literature various axiomatizations of these solutions can be found. Axiomatizations of the Shapley value often use efficiency which is not satisfied by the Banzhaf value. On the other hand, the Banzhaf value satisfies collusion neutrality which is not satisfied by the Shapley value. Both properties seem desirable. However, neither the Shapley value nor the Banzhaf value satisfy both. The equal division solution does satisfy both axioms and, moreover, together with symmetry these axioms characterize the equal division solution. Further, we show that there is no solution that satisfies efficiency, collusion neutrality and the null player property. Finally, we show that a solution satisfies efficiency, collusion neutrality and linearity if and only if thereexistexogenousweightsfortheplayerssuchthatinanygametheworthofthe grand coalition is distributed proportional to these weights. Keywords: Efficiency, Collusion neutrality, Shapley value, Banzhaf value, Equal division solution, Impossibility. JEL code: C71

6 1 Introduction AsituationinwhichafinitesetN ={1,...,n}ofnplayerscangeneratecertainpayoffsby cooperation can be described by a cooperative game with transferable utility (or simply a TU-game),beingapair(N,v)wherev:2 N IRisacharacteristicfunctiononNsatisfying v( )=0. Foranycoalition S N,v(S)istheworth ofcoalitions,i.e. themembersof coalitions canobtainatotalpayoffofv(s)byagreeingtocooperate. Apayoffvector x IR N ofann-playertu-game(n,v)isann-dimensionalvector giving a payoff x i IR to any player i N. A (single-valued) solution for TU-games is a function that assigns a payoff vector to every TU-game (N,v). Three well-known solutions for TU-games are the Shapley value, the Banzhaf value and the equal division solution. In the literature various axiomatizations of these solutions can be found. Most axiomatic characterizations of the Shapley value use efficiency. For example, the original characterization of the Shapley value characterizes it by efficiency, linearity, symmetry and the null player property, see Shapley (1953). Various characterizations of the Banzhaf value use some collusion neutrality axiom, see for example, Lehrer (1988), Haller (1994) and Malawski(2002) who characterize the Banzhaf value by linearity, symmetry, the null player property, the inessential game property and some collusion neutrality property. Collusion neutrality properties state that the sum of payoffs of two players does not change if these two players in some way collude and act as one player. Both efficiency and collusion neutrality seem to be desirable properties. Clearly, by the above mentioned characterizations of the Shapley value and Banzhaf value, there does not exist a solution satisfying efficiency, collusion neutrality, linearity, symmetry and the null player property. A solution that does satisfy both efficiency and collusion neutrality is the equal division solution. In fact, we show that together with symmetry these axioms characterize the equal division solution if there are at least three players. Since the equal division solution does not satisfy the null player property, the next question is whether there is a solution that satisfies efficiency, collusion neutrality and the null player property. It turns out that such a solution does not exist for games with at least three players. Finally, we show that a solution satisfies efficiency, collusion neutrality and linearity if and only if there exist exogenous weights for the players such that in any game the worth of the grand coalition is distributed proportional to these weights. Note that this implies that together with symmetry these axioms characterize the equal division solution but, as argued above, these axioms are not logically independent because we do not need linearity. The paper is organized as follows. In Section 2 we discuss some preliminaries on TU-games. In Section 3 we show that there is a unique solution satisfying efficiency, collusion neutrality andsymmetry,whichistheequaldivisionsolution. InSection4weshowthatthereisno 1

7 solution satisfying efficiency, collusion neutrality and the null player property. In Section 5 we characterize a class of proportional solutions by efficiency, collusion neutrality and linearity. Finally, in Section 6 we make some concluding remarks. 2 Preliminaries InthispaperwetakethesetofplayersN ={1,...,n}tobefixed,andthereforedenotea TU-game(N,v)justbyitscharacteristicfunctionv. Weassumethatthegamehasatleast threeplayers 2. Thecollectionofallcharacteristicfunctions(whichwewillthusrefertoas games)onn isdenotedbyg N. Theincreaseinworthwhenplayeri N joinscoalition S N\{i}iscalledthemarginalcontribution ofplayeritocoalitions ingamev G N andisdenotedby m S i(v)=v(s {i}) v(s). Assumingthatthe grandcoalition N formsinawaysuchthattheplayersenterthecoalition one by one, the Shapley value assigns to every player its expected marginal contribution tothecoalitionofplayersthatenterbeforehimgiventhatallordersofentrancehaveequal probability. Thus,theShapleyvalue (Shapley(1953))isthesolutionSh:G N IR N given by Sh i (v)= S N\{i} (n S 1)!( S )! n! m S i(v)foralli N. On the other hand, the Banzhaf value (introduced by Banzhaf(1965) to measure voting power in voting games and generalized by Owen (1975) and Dubey and Shapley (1979) to general TU-games) is the solution Ba:G N IR N that assigns to every player its expected marginal contribution given that every combination of the other players has equal probability of being the coalition that is already present when that player enters. Thus, it assigns to every player in a game its average marginal contribution, i.e. Ba i (v)= 1 2 n 1 S N\{i} m S i (v)foralli N. Playersi,j N aresymmetricingamevifv(s {i})=v(s {j})foralls N\{i,j}. Player i N is a null player in game v if v(s {i}) = v(s) for all S N\{i}. For v,w G N,thegame(v+w) G N isdefinedby(v+w)(s)=v(s)+w(s)foralls N. Forv G N andα IR,thegameαv G N isdefinedby(αv)(s)=αv(s)foralls N. Haller (1994) introduces some collusion neutrality properties which state that the sum 2 Wemakesomeremarksontwo-playergamesinthefinalsection. 2

8 of payoffs of two players does not change if they collude. He used these properties to axiomatize the Banzhaf value. Later, Malawski(2002) showed that several other collusion neutrality properties can be used. In this paper we consider collusion between two players wheretheyagreeto actasone inthesensethattheycontributetoacoalitiononlywhen they both are present. So, when players i,j N, i j, collude in game v G N, then insteadofgamevweconsiderthegamev ij G N givenby { v(s\{i,j}) if{i,j} S v ij (S)= v(s) if{i,j} S. (2.1) Finally,agameviscalledinessential ifv(s)= i Sv({i})forallS N,i.e. forevery player its marginal contribution to any coalition is the same. Various axiomatizations of the Shapley value and the Banzhaf value have been given in the literature. Some axioms that are usedin these axiomatic characterizationsare the following. Asolutionf:G N IR N satisfies efficiency if i N f i(v)=v(n)forallv G N ; linearity iff(αv+βw)=αf(v)+βf(w)forallv,w G N andα,β IR; symmetry iff i (v)=f j (v)wheneveriandj aresymmetricplayersinv G N ; thenullplayerproperty iff i (v)=0wheneveriisanullplayerinv G N ; theinessentialgameproperty iff i (v)=v({i})foralli N andinessentialgamesv; collusion neutrality iff i (v ij )+f j (v ij )=f i (v)+f j (v)foralli,j N and v G N, withv ij givenby(2.1). Most axiomatic characterizations of the Shapley value use efficiency. For example, the original characterization in Shapley(1953) characterizes it by efficiency, linearity, symmetry andthenullplayerproperty 3. TheBanzhafvaluesatisfieslinearity,symmetryandthenull player property, but it is not efficient. Malawski(2002) characterized the Banzhaf value by linearity, symmetry, the null player property, the inessential game property and collusion neutrality 4. 3 ThisaxiomatizationismoreoftenpresentedinthiswayalthoughShapley(1953)combinesefficiency and the null player property into a carrier axiom. 4 Asmentioned,othercollusionneutralitypropertiesthatareusedtoaxiomatizetheBanzhafvalueare stated in Haller(1994). The results in this paper also can be stated in terms of those neutrality properties. An axiomatization of the Banzhaf value with collusion properties in terms of inequalities can be found in Lehrer(1988). 3

9 InvandenBrink(2007)itisshownthatinseveralaxiomatizationsoftheShapleyvalue, replacing an axiom about null players(such as the null player property) by a similar axiom about nullifying players(being players whose presence in a coalition implies that the worth of the coalition is zero) yields axiomatic characterizations of the equal division solution ED:G N IR N whichisgivenby ED i (v)= v(n) n foralli N. In this paper we find another axiomatization of the equal division solution by combining axioms that characterize the Shapley value and the Banzhaf value. The proof of this axiomatization uses the unanimity basis for TU-games. The unanimity game of coalition T N, T, is the game u T G N given by u T (S) = 1 if T S, and u T (S) = 0 otherwise. Itiswell-knownthatthesetofunanimitygamesformabasisofG N : everygame v G N can be written as a linear combination of unanimity gamesv = T N v (T)u T T with v (T)= S T ( 1) T S v(s)beingtheharsanyidividends,seeharsanyi(1959). 3 Dropping the null player property: an axiomatization of the equal division solution Both efficiency and collusion neutrality seem to be desirable properties. Clearly, by axiomatizations of the Shapley value and the Banzhaf value mentioned before, there does not exist a solution satisfying efficiency, collusion neutrality, linearity, symmetry and the null player property. It turns out that for games with at least three players, dropping the null player property yields a characterization of the equal division solution. We can even state a stronger characterization result without linearity. Theorem1 Asolutionf:G N IR N satisfiesefficiency,collusionneutralityandsymmetryifandonlyifitistheequaldivisionsolution. Proof It is easy to verify that the equal division solution satisfies the three properties. To show uniqueness, we proceed by induction on the smallest cardinality of the coalitions with nonzero dividend in a game. Before starting to show uniqueness, we introduce the following notation. For any game v G N, define H(v) = {T N v (T) 0} as the set of coalitions with non-zero dividend, d(v) = min T H(v) T as the smallest cardinality of coalitionswithnon-zerodividend,andh(v)= {T H(v) d(v)= T } asthenumberof coalitions of smallest cardinality with non-zero dividend. 4

10 Nextwestarttheproofofuniqueness. Considergamev G N. First,ifd(v)= N then visascaledunanimitygameofthe grandcoalition,i.e. v= v (N)u N with v (N) 0. (Notethatinthiscase v (N)=v(N).) Symmetryimpliesthatallplayersearnthesame payoff. Efficiencythendeterminesthatf i (v)= v(n) = v(n) =ED n n i (v)foralli N. Proceeding by induction, assume that f(v ) is uniquely determined for all v G N with d(v )>d(v). In order to use collusion neutrality, we use the following result from Malawski (2002) concerning collusion between two players in unanimity games. For every coalition R N andtwoplayersi,j N,i j,itholdsthat { u R ifeither[i,j R]or[i,j N\R] (u R ) ij = u R {i,j} otherwise. (3.2) We now start induction on h(v). First, assume that h(v) = 1. Then there is a unique T H(v) with T = d(v), i.e. T is the (unique) smallest cardinality coalition with non-zerodividend. Takeaspecificj T andh N\T. Collusionneutralityimpliesthat and f i (v)+f j (v)=f i (v ij )+f j (v ij )foralli N\T, (3.3) f h (v)+f g (v)=f h (v hg )+f g (v hg )forallg T\{j}, (3.4) while efficiency requires that f i (v)=v(n). i N (3.5) By(3.2)itfollowsthat (i) (u T ) ij =u T {i} foralli N\T, (ii) (u R ) ij iseitherequaltou R oru R {i,j} foreveryr N andi N\T, (iii) (u T ) hg =u T {h} forallg T\{j},and (iv) (u R ) hg iseitherequaltou R oru R {h,g} foreveryr N andg T\{j}. Therefore,d(v ij )>d(v)foralli N\T,andd(v hg )>d(v)forallg T\{j},andthus by the induction hypothesis, the equations(3.3) and(3.4) become f i (v)+f j (v)=ed i (v ij )+ED j (v ij )= 2v ij(n) n = 2v(N) n foralli N\T, (3.6) 5

11 and f h (v)+f g (v)=ed h (v hg )+ED g (v hg )= 2v hg(n) n = 2v(N) n forallg T\{j}. (3.7) Since the 1+(n T )+( T 1) = n equations given by (3.5), (3.6) and (3.7) are linearly independent 5, the values f i (v), i N, are uniquely determined and given by f i (v)= v(n) n =ED i (v). Next, proceeding by induction on h(v), assume that the result holds for every v G N withd(v ) d(v)andh(v )<h(v). Similarasabove, takeat H(v)with T =d(v). (Note that now there are h(v) > 1 of such coalitions.) Collusion neutrality, the induction hypothesis and efficiency yield the same equations (3.5), (3.6) and (3.7), and it can be similarlyshownthatf(v)isuniquelydeterminedbyf(v)=ed(v)wheneverh(v). Finally,wehavetoconsidergamev G N withh(v)=. Then,visthenull game,i.e. v(s)=0for all S N. Symmetry impliesthat all players earn the same payoff. With efficiencyitthenfollowsthatf i (v)=0=ed i (v)foralli N. Logical independence of the axioms of Theorem 1 is shown by the following solutions. 1. The Shapley value satisfies efficiency and symmetry. It does not satisfy collusion neutrality. 2. The Banzhaf value satisfies collusion neutrality and symmetry. It does not satisfy efficiency. 3. Thesolutionf:G N IR N thatassignsallpayofftoplayer1,i.e. f 1 (v)=v(n)and f i (v)=0foralli N\{1},satisfiesefficiencyandcollusionneutrality. Itdoesnot satisfy symmetry. 4 Dropping symmetry: an impossibility In the previous section we saw that dropping the null player property from our set of desirable properties yields an axiomatization of the equal division solution. The next questioniswhathappensifwedropsymmetryinsteadofthenullplayerproperty. Itturns out that there is no solution satisfying efficiency, collusion neutrality and the null player property. Note that for this impossibility result we do not need linearity. Theorem2 ThereisnosolutiononG N satisfyingefficiency,collusionneutralityandthe null player property. 5 Thisfollowsfromsomeeasybuttediouscomputations. 6

12 Proof Bycontradiction,supposethatsolutionf:G N IR N satisfiesefficiency,collusionneutrality and the null player property. Considerunanimitygamesu T, T N. Thenullplayerpropertyimpliesthat f k (u T )=0forallk N\T, (4.8) and with efficiency it then follows that f k (u T )=1. k T (4.9) Thus,if T =1,i.e. T ={i}forsomei N,thenthenullplayerpropertyandefficiency determinethepayoffsf j (u T )=0forj N\{i}andf i (u T )=1. If T =2,i.e. T ={i,j}forsomei,j N withi j,thenthenullplayerpropertyand efficienyimplythatf h (u T )=0forh N\{i,j},andf i (u T )+f j (u T )=1. (Notethatthe sameisimpliedbycollusionneutralityandthecase T =1consideredabove.) Next consider the case T = 3, i.e. T = {i,j,h} with i,j and h being three different players. (Recall that we only consider classes of games with at least three players.) By (4.9) we have that f i (u T )+f j (u T )+f h (u T ) = 1. In order to apply collusion neutrality, notice that ( ) u {i,j} = ( ) u hi {i,j} = ( ) u hj {i,h} = ( ) u ji {i,h} = ( ) u jh {j,h} = ( u ij {j,h} )ih = u T. Then, collusion neutrality implies that and f i (u T )+f j (u T )=f i (u {i,h} )+f j (u {i,h} )=f i (u {j,h} )+f j (u {j,h} ) (4.10) f i (u T )+f h (u T )=f i (u {i,j} )+f h (u {i,j} )=f i (u {j,h} )+f h (u {j,h} ) (4.11) f j (u T )+f h (u T )=f j (u {i,j} )+f h (u {i,j} )=f j (u {i,h} )+f h (u {i,h} ). (4.12) By(4.8)wehavethatthissystemofequationscanbereducedto and f i (u T )+f j (u T )=f i (u {i,h} )=f j (u {j,h} ) (4.13) f i (u T )+f h (u T )=f i (u {i,j} )=f h (u {j,h} ) (4.14) f j (u T )+f h (u T )=f j (u {i,j} )=f h (u {i,h} ). (4.15) Itfollowsthatf i (u T )+f j (u T )=f i (u {i,h} )=1 f h (u {i,h} )=1 f j (u T ) f h (u T ),where the first equality follows from(4.13), the second equality follows from(4.9) and the third equalityfollowsfrom(4.15). By(4.9)wethenhavethatf j (u T )=0. 7

13 Similarwecanderivefromtheotherequationsthatf i (u T )=f h (u T )=0,whichcontradicts efficiency. So, there does not exist a solution satisfying efficiency, collusion neutrality and thenullplayerproperty 6. 5 Dropping the null player property and symmetry: a characterization of a class of proportional solutions From the original axiomatization of the Shapley value given by Shapley(1953) combined with collusion neutrality, we might still consider what solutions are left if we drop both the nullplayerpropertyandsymmetry. Itturnsoutthatinthatcaseweareleftwithaclassof proportional solutions for which there exist exogenous weights for the players such that in anygametheworthofthe grandcoalition isdistributedproportionaltotheseweights 7. DefineX N :={λ IR N i N λ i=1}. Forλ X N wedefine f λ i(v)=λ i v(n)foralli N. (5.16) Theorem3 Asolutionf:G N IR N satisfiesefficiency,collusionneutralityandlinearity ifandonlyifthereexistsavectorofweightsλ X N suchthatf =f λ. Proof It iseasyto verifythatf λ, λ X N, satisfiesthe three properties. To show uniqueness, assumethatsolutionf:g N IR N satisfiesthethreeproperties. Wefirstproveuniqueness for unanimity games u T, T N. We do this by induction on T. If T = n, i.e. T =N,thenefficiencyimpliesthatthereexistsavectorλ X N suchthatf(u N )=λ= f λ (u N ). Proceedingbyinduction,supposethatf(u T )=λforallt N with T > T. Takea specificj T andh N\T. Since(u T ) ij =u T {i} wheni N\T,collusionneutrality implies that f i (u T )+f j (u T )=f i (u T {i} )+f j (u T {i} )foralli N\T (5.17) andsince(u T ) hg =u T {h} wheng T\{j},collusionneutralityalsoimpliesthat f h (u T )+f g (u T )=f h (u T {h} )+f g (u T {h} )forallg T\{j}. (5.18) 6 Onealsoobtainstheimpossibilitybyshowingdirectlythatthesystemofsixequations(4.10),(4.11) and(4.12)togetherwiththefourefficiencyequalitiesf i (u T )+f j (u T )+f h (u T )=f i (u {i,j} )+f j (u {i,j} )= f i (u {i,h} )+f h (u {i,h} )=f j (u {j,h} )+f h (u {j,h} )=1hasnosolution. 7 Althoughweuselinearity,thistheoremalsocanbestatedbyusingtheweakeradditivityaxiom. 8

14 With the induction hypothesis the above two equations yield f i (u T )+f j (u T )=λ i +λ j foralli N\T (5.19) and f h (u T )+f g (u T )=λ h +λ g forallg T\{j}. (5.20) Then f i (u T ) = f g (u T )+f j (u T )+ f i (u T )+f h (u T ) i N g T\{j} i N\(T {h}) = (λ h +λ g f h (u T ))+f j (u T )+ (λ i +λ j f j (u T ))+f h (u T ) g T\{j} i N\(T {h}) = λ g +( T 1)(λ h f h (u T ))+f j (u T ) g T\{j} + λ i +(n T 1)(λ j f j (u T ))+f h (u T ) i N\(T {h}) = i Nλ i +( T 2)(λ h f h (u T ))+(n T 2)(λ j f j (u T )), where the second equality follows from(5.19) and (5.20). With efficiency it follows that i N f i(u T )=1= i N λ i,andthus ( T 2)(λ h f h (u T ))+(n T 2)(λ j f j (u T ))=0. (5.21) Forj T andh N\T,(5.19)yieldsthat f j (u T )+f h (u T )=λ j +λ h. (5.22) Solving(5.21)and(5.22)yieldsf j (u T )=λ j, f h (u T )=λ h,andwith(5.19)and(5.20)this yieldsthatf i (u T )=λ i foralli N. Since every game v G N can be written as a linear combination of unanimity games v = T N v (T)u T, with v (T) = S T ( 1) T S v(s) being the Harsanyi dividends, T uniqueness for arbitrary v G N then follows since linearity of f implies that f i (v) = f i ( T N v (T)u T ) = T N v (T)f i (u T ) = T Nλ i v (T) = λ i T N v (T) = λ i v(n) T T T T foralli N. NotethatasacorollaryitfollowsthataddingsymmetrytotheaxiomsofTheorem3yields that all weights λ i should be equal, and thus characterizes the equal division solution. 9

15 However, these axioms are not logically independent. In Section 3 we showed that for this characterization linearity is not necessary since it is sufficient to require efficiency, collusion neutrality and symmetry. Also note the difference between the solutions characterized in this section for TUgames and the(unique) proportional solution for bankruptcy problems. In that solution the total estate is distributed proportionally to the claims of the agents, while in the proportional solutions defined by(5.16) for TU-games, the weights that are used are exogenous andneednotberelatedtothegame. Logical independence of the axioms of Theorem 3 is shown by the following solutions. 1. The Shapley value satisfies efficiency and linearity. It does not satisfy collusion neutrality. 2. The Banzhaf value satisfies collusion neutrality and linearity. It does not satisfy efficiency. 3. Thesolution f:g N IR N givenby f(v)=f(v)ifv(n) 10and f(v)=ed(v)if v(n)>10,withf asgivenattheendofsection3,satisfiesefficiencyandcollusion neutrality. It does not satisfy linearity. A final question we consider is what solutions satisfy efficiency and collusion neutrality. The answer is that these are kind of proportional solutions, but the weights λ that determine whatshareintheworthofthegrandcoalitiontheplayersgetdependsontheworthofthe grand coalition. Theorem 4 A solution f satisfies efficiency and collusion neutrality if and only if there isafunctionl:ir X N suchthatf =f L(v(N)). So,intwogamesv,w G N withv(n)=w(n)thepayoffdistributionsarethesame. The proofofthistheoremgoesalongsimilarlinesastheproofoftheorem1andistherefore omitted. 6 Concluding remarks In this paper we have studied the possibilities of having solutions for TU-games that satisfy efficiency and collusion neutrality. We have seen that for games with at least three players, additionally requiring the symmetry property characterizes the equal division solution. So, the equal division solution is the unique symmetric solution that satisfies these two properties (that distinguish the Shapley value and Banzhaf value from each 10

16 other). Additionally requiring the null player property instead of symmetry yields an impossibility. In both results linearity of the solution is not necessary. Other relations between the Shapley value and equal division solution are given in van den Brink(2007). Finally, we showed that a solution satisfies efficiency, collusion neutrality and linearity if andonlyifthereexistexogenousweightsfortheplayerssuchthatinanygametheworth of the grand coalition is distributed proportional to these weights. We summarize these results in Table 1. Properties/Solutions Sh Ba ED f λ,λ X N Impossibility Efficiency x x x x Collusion neutrality x x x x Symmetry x x x Null player property x x Linearity x x x Table 1: Characterizing properties of solutions x We remark that collusion neutrality can be replaced by other(but similar) axioms that reflect collusion between two players going to act as one, see also Footnote 4. Note that the multiplicative normalization of the Banzhaf value(i.e. dividing the worth of the grand coalition proportional to the Banzhaf values of the players) as axiomatizedinvandenbrinkandvanderlaan(1998)doesnotsatisfycollusionneutralitynor linearity, while the additive normalization of the Banzhaf value(i.e. adding or subtractingfromthebanzhafvalueofeveryplayerthesameamounttoobtainanefficientpayoff vector) as considered in Ruiz, Valenciano and Zarzuelo (1998) does not satisfy collusion neutrality nor the null player property. In the proofs of Theorems 1 and 2 we needed the player set to contain at least three players. For two-player games there are more solutions satisfying efficiency, collusion neutrality and symmetry(and linearity). An example is the Shapley value which on the classofone-andtwo-playergamesisequaltothebanzhafvalue. Moreover,fortwoplayer games there exist solutions satisfying efficiency, collusion neutrality and the null player property(and linearity), which is again illustrated by the Shapley value. References Banzhaf, J.F. (1965), Weighted Voting Doesn t Work: A Mathematical Analysis, RutgersLawReview,19,

17 Brink, R. van den (2007), Null players or nullifying players: the difference between the Shapley value and equal division solutions, Journal of Economic Theory, 136, Brink, R. van den,andg. van derlaan (1998) Axiomatizations of the Normalized Banzhaf Value and the Shapley Value, Social Choice and Welfare, 15, Dubey, P., and LS. Shapley (1979), Mathematical Properties of the Banzhaf Power Index, Mathematics of Operations Research, 4, Haller, H. (1994), Collusion Properties of Values, International Journal of Game Theory, 23, Harsanyi, J.C. (1959), A Bargaining Model for Cooperative n-person Games, in ContributionstotheTheoryofGamesIV(A.W.TuckerandR.D.Luce,Eds.),pp Princeton: Princeton UP. Lehrer, E. (1988), An Axiomatization of the Banzhaf Value, International Journal of Game Theory, 17, Malawaski, M. (2002), Equal Treatment, Symmetry and Banzhaf Value Axiomatizations, International Journal of Game Theory, 31, Owen G.,(1975), Multilinear Extensions and the Banzhaf Value, Naval Research Logistics Quarterly, 22, Ruiz, L., F. Valenciano, and J.M. Zarzuelo (1998) The family of least-square values for TU games, Games and Economic Behavior 24, Shapley, L.S. (1953), A Value for n-person Games, Annals of Mathematics Studies 28 (Contributions to the Theory of Games Vol.2)(eds. H.W. Kuhn and A.W. Tucker), Princeton University Press,