Blend-in fairness and equal split

Dimitrov, Dinko; Sun, Ching-jen Article — Published Version Blend-in fairness and equal split International Journal of Game Theory Provided in Cooperation with: Springer Nature Suggested Citation: Dimitrov, Dinko; Sun, Ching-jen (2025) : Blend-in fairness and equal split, International Journal of Game Theory, ISSN 1432-1270, Springer, Berlin, Heidelberg, Vol. 54, Iss. 2, https://doi.org/10.1007/s00182-025-00949-z This Version is available at: https://hdl.handle.net/10419/323195 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL PAPER International Journal of Game Theory (2025) 54:27 https://doi.org/10.1007/s00182-025-00949-z Abstract Blending in with others is a possible self-serving motivation when people participate in cooperative situations. We use this motivation to formulate a corresponding fairness principle, combine it with rather weak standard axioms from cooperative game theory, and show that it leads to equal split of coalitional gains. The same normative principles characterize this solution when only cohesive games (where it is optimal for the coalition of all players to form) are considered. Keywords Blend-in fairness · Cohesive games · Cooperative games · Equal division solution JEL Classification: A13 · C71 · D63 · D91 1 Introduction A key question in coalitional game theory is how the total payoff a coalition can achieve through cooperation should be divided among its members. One of the most influential answers to this question within the context of cooperative transferable utility games (TU-games) has been provided by the Shapley value (cf. Shapley 1953). According to this single-valued solution, each player should be assigned a share of Accepted: 10 June 2025 © The Author(s) 2025 Blend-in fairness and equal split DinkoDimitrov1· Ching-jenSun2 We are grateful to Sylvain Béal, Rene van den Brink, Youngsub Chun, Yves Sprumont, two anonymous referees, and various seminar audiences for valuable suggestions. Dinko Dimitrov [email protected] Ching-jen Sun [email protected] 1 Chair of Economic Theory, Saarland University, Saarbrücken, Germany 2 Deakin Business School, Deakin University, Melbourne, Australia 1 3 D. Dimitrov, C.-j. Sun the payoff that is proportional to his marginal contributions in the corresponding game. From an axiomatic perspective, the Shapley value is the unique solution which is efficient (the entire gain when all players cooperate (i.e., the worth of the ‘grand’ coalition) is distributed among the players), symmetric (players receive equal payoffs if they are exchangeable in generating coalitional gains), satisfies the null player property (no contribution to any coalition results in zero payoff), and is additive over games. The experimental validity of the above principles was recently tested in de Clippel and Rozen (2022). The data analysis of these authors provides strong evidence for the symmetry and additivity axioms with the efficiency axiom being trivially satisfied due to the experimental design. However, no such evidence was found for the null player property; that is, even null players were assigned to payoffs which are significantly different from zero. The lack of experimental evidence for the null player property naturally draws one’s attention to another famous single-valued solution for TU-games, the equal division solution. For each game, the latter distributes the worth of the grand coalition equally among all players and it violates the null player property while satisfying the other three mentioned principles. As originally shown in van den Brink (2007), the equal division solution can be characterized by replacing, in the above axiomatic system, the null player property by a nullifying player requirement (each player is assigned zero payoff if the worth of each coalition containing that player is zero). We refer the reader to Alonso-Meijide et al. (2019) for an excellent and detailed survey of the corresponding strand of the literature, to Hernandez-Lamoneda et al. (2008) for a characterization of the class of all additive, symmetric, efficient and continuous solutions, as well as to Chapter 4 in Branzei et al. (2008) for an overview of other egalitarianism-based solution concepts. In this paper we follow a normative perspective on cooperative games (cf. Moulin 2003) and provide an axiomatic support for the idea that the equal division solution springs out of a fairness principle which is rather “... strongly shaped by cultural values” (cf. Young 1994, p. xii). That is, our work fits into the strand of literature emphasizing the context-dependence of fairness and equity “... not because of the lack of general principles of justice, but due to its effect on the interpretation of those principles” (cf. Konow 2001, p. 139). For instance, Alesina and Angeletos (2005) study the extent to which the difference in social perceptions regarding the fairness of market outcomes and the underlying sources of income inequality (Americans vs Europeans) are consistent with equilibrium behavior. Almås et al. (2020) provide experimental support for the differences between Americans and Scandinavians with respect to what kind of inequalities they consider fair and to the importance assigned to fairness relative to efficiency. Cappelen et al. (2007) consider a dictator game in which the distribution phase is preceded by a production phase and show how one may simultaneously estimate the prevalence of different fairness ideals and the weight people attach to fairness considerations. Finally, Gelfand et al. (2002) stress the differences when it comes to self-serving motivations in individualistic or in collectivistic cultures. These authors argue that, in the former, the self is served by enhancing one’s positive attributes to “stand out” and be better than others, while in 1 3 27 Page 2 of 21 Blend-in fairness and equal split collectivistic cultures the focus is rather on how individuals “blend in” and maintain interdependence with others. Since the Shapley value averages players’ contributions to every coalition they join, it can be axiomatically rooted via a corresponding marginality principle (cf. Young 1985) in rather individualistic societies. In the present paper we follow a possible self-serving motivation for collectivistic cultures, introduce a corresponding fairness principle (blend-in fairness), and show that it crucially shapes the characterization of the equal division solution. We start in Sect. 2 by providing the formal definitions of the mentioned standard axioms and first weaken efficiency and symmetry to a symmetric efficiency principle. The latter imposes on a single-valued solution the equal split of the worth of the grand coalition, provided that all players are symmetric in the corresponding game. A weak null player property additionally postulates null players to be assigned zero payoffs in games, where the total gain when all players cooperate is zero. These two properties, together with additivity, are clearly satisfied by both the Shapley value and the equal division solution. In Sect. 3 we develop a notion of cooperation equivalence of games for players. To fix ideas, consider two superadditive games (that is, games where the formation of larger coalitions is worthy) and a particular player. Imagine now that what the player realizes when comparing these two games is that there is only a renaming of the other players but no change in the corresponding coalitional worths. That is, in terms of the offered ‘blending-in with others’ possibilities, the two games are cooperation equivalent for that particular player. The blend-in fairness principle stated in Sect. 4 then requires a player to be assigned the same payoff in two superadditive games which are cooperation equivalent for him. This principle turns out to shape the characterization of the equal division solution both on the entire set of games (Theorem 1) and on the subset of cohesive games, where it is optimal that the grand coalition forms (Theorem 2). We conclude in Sect. 5 with some final remarks. The proofs of our characterization results are relegated to the Appendix, where we first introduce the class of partition games and show that they form a basis of the entire set of games (Theorem 0). Propositions 1–5 are then crucial for the proofs of Theorem 1 and Theorem 2 as they explain how additivity, symmetric efficiency, the weak null player property, and blend-in fairness generate the equal division solution on partition games. The Appendix also contains examples showing the independence of the utilized axioms. 2 Normative principles and solutions Let N be a set of n individuals. A coalition is any subset of N. A cooperative transferable utility game (TU-game) is a pair (N,v) , where v is the characteristic function of the game assigning a worthv(S) to each coalition S⊆N such that v(∅)=0 . The amount v(S) represents how much the members of S can share should they cooperate. In what follows, we fix the player set N and therefore denote the game (N,v) by its characteristic function v. The notation GN stands for the set of all TU-games on the player set N. 1 3 Page 3 of 21 27 D. Dimitrov, C.-j. Sun Two players i, j ∈N are called symmetric in a game v∈G N , if v(S∪{i})=v(S∪{j}) holds for every S⊆N\{i, j} . A player i∈N is a null player in v∈G N if v(S∪{i})=v(S) is valid for every S⊆N\{i} . The sum of two games v,w ∈G N is defined by (v+w)(S)=v(S)+w(S) for each S⊆N . It is an implicit assumption in cooperative game theory models that the grand coalition forms (all players cooperate) and the basic question is then how should the corresponding proceeds be distributed among all individuals. Thus, a (singlevalued) solution f:GN→Rn assigns a payoff vector to each game v∈G N . The next four normative principles that could be imposed on a solution f are standard in the literature. Efficiency For all v∈G N : Σi∈Nfi(v)=v(N) . Symmetry For all v∈G N : If i, j ∈N are symmetric in v, then fi(v)=fj(v) . Null player property For all v∈G N : If i∈N is a null player in v, then fi(v)=0 . Additivity For all v,w ∈G N : f(v+w)=f(v)+f(w) . The Shapley value is the unique solution which is efficient, symmetric, additive, and satisfies the null player property. This solution concept was introduced in Shapley (1953) and it assigns to each player the average of his marginal contributions in the corresponding game. Given a game v∈G N , the marginal contribution of player i to a coalition S⊆N\{i} is just the difference ∆i(S)=v(S∪{i})−v(S) . The Shapley value Sh is then defined by the condition Sh i(v)= 1 | N | ! ∑ R∈R ∆i(Si(R)) for each i∈ N, where R is the set of all |N|! orderings of N and Si(R) is the set of players preceding i in the ordering R. From the above four axioms, it is only the null player property that is violated by another famous procedure for sharing coalitional gains, the equal division solution. For each game v∈G N it is defined by ED i(v)=v ( N ) n for each i ∈N. That is, this solution distributes the worth of the grand coalition equally among all players. In the next section we present our axiomatic characterization of the equal division solution where all utilized axioms but one are also satisfied by the Shapley value. Besides additivity, these include the weak null player property and symmetric efficiency. Weak null player property For all v∈G N : If i∈N is a null player in v and v(N)=0 , then fi(v)=0 . Symmetric efficiency For all v∈G N : If all players are symmetric in v, then fi( v )= v ( N ) n for each i∈N . The weak null player property is clearly implied by the null player property and it is stronger than the null player in a null environment property introduced in Casajus 1 3 27 Page 4 of 21 Blend-in fairness and equal split and Huettner (2013); the latter imposes that a null player gets a non-negative payoff in a game in which the grand coalition has zero worth. On the other hand, the combination of efficiency and symmetry (but none of these properties taken separately) implies the symmetric efficiency axiom. Symmetric efficiency is stronger than the triviality axiom introduced in Chun (1989) and the weak symmetry axiom used in van den Brink (2007). 3 Cooperation equivalence of games Imagine a player who evaluates his participation in two different cooperative situations and claims equal payoffs in the correspondingly generated games. As already argued above, one possible way of rationalizing such a claim is to look at the ways in which the particular player maintains interdependence with others and blends in when working with them in the respective situations. The notion of cooperation equivalence of games we introduce below formalizes the idea that, from the viewpoint of a given player, two games offer the same possibilities when it comes to fitting in with other players. We use then this notion to formally state in Sect. 4 a corresponding blend-in fairness principle. In order to equip the reader with a suitable intuition for the notion of cooperation equivalence, let us consider two games in which three players cooperate on their investment decisions.1 Players 1 and 2 are domestic investors lacking any experience and suitable investment technologies, while player 3 is an experienced foreign investor who possesses 100 units of capital and is able to double any fraction of it he invests. There is no government restriction on such investments when made in cooperation with at least one domestic investor. However, the foreign investor when acting on his own is allowed to invest only 40 units (in the game v) or 30 units (in the game w). {1}{2}{3}{1 , 2}{1 , 3}{2 , 3} N v10 20 2 ×40 + 60 30 220 240 260 w 20 10 2 ×30 + 70 30 240 220 260 In the game v, players 1 and 2 would like to invest 10 and 20 units of capital, respectively. However, due to their lack of investment technology, one has v({1}) = 10 , v({2}) = 20 , and v({1,2}) = 30 . In a coalition containing player 3, the foreign investor is allowed to use the entire 100 units of his capital and each domestic coalitional member has access to the corresponding investment technology doubling the sum of individual capitals. The interpretation of the coalitional worths in the game w is analogous. Let us now compare the two games v and w from the viewpoint of the foreign investor. In both games he intends to invest 100 units of capital and this is exactly the amount he puts on the table when joining any non-empty coalition of domestic 1 Each of these games can be seen as an appropriate modification of the landlord game in Shapley and Shubik (1967). 1 3 Page 5 of 21 27 D. Dimitrov, C.-j. Sun players. Moreover, when looking at the domestic investors in the two games, he realizes that there is only a renaming of these players (player 1 becomes player 2 and vice versa) but no change in the worth of any coalition containing some of these players. Notice additionally that each of these games is superadditive as, exemplified with respect to the game v, the inequality v(S∪T)≥v(S)+v(T) holds for all S, T ⊆N with S∩T=∅ . In other words, the domestic investors and the foreign investor are “incentivized” in both games to form larger coalitions; hence, it seems natural for a player to focus rather on larger coalitions when evaluating his participation in such situations. Correspondingly, we declare the two superadditive games as being cooperation equivalent for the foreign investor. Let us denote by GN sa , GN sa ⊂G N , the set of all superadditive games on the player set N. Consider two games v,w ∈G N sa , and fix a player i∈N whose viewpoint we would like to describe. Suppose further that there exists a bijection σ:N→N with σ(i)=i such that v(S)=w(σ(S)) holds for all S⊆N with j∈S for some j∈N\{i} . In other words, when evaluating the two superadditive games, player i focusses on the coalitions each of the other players might belong to and realizes only a permutation of the players’ names but no changes at all in the corresponding coalitional worths. We call the games v and w cooperation equivalent for playeri. 4 Blend-in fairness leads to equal split The blend-in fairness principle we introduce below relies on the notion of cooperation equivalence of games for players. More precisely, it requires from a solution to assign the same payoff to a player in superadditive games which are cooperation equivalent for him. Blend-in fairness For all v,w ∈G N sa and i∈N : If v and w are cooperation equivalent for player i, then fi(v)=fi(w) . We would like to mention the fact that (1) it is a specific bijection σ ( satisfying σ(i)=i for the corresponding player i∈N ) we use in the formulation of cooperation equivalence of games and (2) the implication refers only to the payoff of player i in the two games. The reader might then wonder whether there is a logical relation (on the class of superadditive games) between the above axiom and the anonymity property of single-valued solutions requiring that a solution should not discriminate between the players solely on the basis of their “names”. Notice that, in the definition of blend-in fairness, it might happen that v({i})=w({σ(i)}) for player i whose viewpoint the bijection σ describes. Clearly then, the Shapley value violates the newly introduced requirement, while satisfying anonymity. On the other hand, one can define a dictatorship solution dk:GN sa →Rn with respect to a pre-specified player k∈N , where for v∈G N sa one has d k j(v)= { v ( N )if j = k , 0otherwise. As it can be easily seen, dk satisfies blend-in fairness and violates anonymity. Hence, these two requirements are independent. 1 3 27 Page 6 of 21 Blend-in fairness and equal split Our main characterization result is stated in Theorem 1 below and it basically says that, along with additivity and the weak versions of standard requirements, it is the blend-in fairness that leads to equal split of coalitional gains. Theorem 1 A solution f:GN→Rn satisfies additivity, the weak null player property, symmetric efficiency, and blend-in fairness if and only if it is the equal division solution. It is worth mentioning that the equal division solution also satisfies a stronger version of the blend-in fairness axiom requiring that for all pairs of (not necessarily superadditive) cooperation equivalent games for some particular player, this player receives the same payoff in both games. As we show in the proof of Theorem 1, there is no need to impose this stronger version on a solution since its weaker version stated only with respect to superadditive games suffices for providing the characterization of the equal division solution on the entire set of games. Observe additionally that two of the other axioms we utilize, symmetric efficiency and the weak null player property, rather indicate the formation of the grand coalition, the reader might ask whether a corresponding characterization could be reached when only cohesive games are considered. In a cohesive game v∈G N it is optimal for the grand coalition to form since by definition v(N)≥ΣK k=1v(Sk) holds for every partition {S1,...,S K} of N (cf. Osborne and Rubinstein (1994), p. 258). Theorem 2 shows that there are the same four normative principles2 which characterize the equal division solution on the mentioned subclass (denoted by GN coh ) of games. Theorem 2 A solution f:GN coh →Rn satisfies additivity, the weak null player property, symmetric efficiency, and blend-in fairness if and only if it is the equal division solution. As we show in the Appendix, the proof of Theorem 2 relies on the fact that all games used in the proof of Theorem 1 are cohesive (with the partition games serving as a basis of the entire set of games being even superadditive (cf. Remark 0 in Sect. 6.1). 5 Concluding remarks The notion of blend-in fairness introduced in this paper relies on the very basic idea that individuals in cooperative situations are usually construed as being fundamentally connected to others. Such a fundamental connection in superadditive games is mirrored by the formation of larger coalitions; in other words, players’ “incentives” to look at individual worths when comparing two such games are rather moderate. The latter interpretation naturally leads to the developed notion of cooperation equiv2 Additivity, the weak null player property, and symmetric efficiency should then be stated with respect to cohesive games. There is no need to change the formulation of blend-in fairness since it is anyway stated with respect to superadditive games and superadditive games are also cohesive. 1 3 Page 7 of 21 27 D. Dimitrov, C.-j. Sun alence of games with blend-in fairness requiring a player to be assigned the same payoff in two superadditive games that are cooperation equivalent for that player. As already elaborated in Sect. 4, the standard anonymity axiom (satisfied by the Shapley value) and blend-in fairness (violated by the Shapley value) are independent requirements. Nevertheless, blend-in fairness can alternatively be seen as a local version of anonymity in the sense that (1) anonymity in the corresponding superadditive games holds with the possible exception for the player under consideration and (2) the payoff equivalence is required only for that particular player (whereas it holds for all players in the classical axiom of anonymity). In other words, an axiom in the spirit of anonymity (blend-in fairness) combined with (small modifications of) classical axioms satisfied by the Shapley value turns out to have rather dramatic consequences. It seems therefore reasonable to further study and axiomatically characterize solutions that satisfy the blend-in fairness axiom and are as close as possible to the formal definition of the Shapley value. Notice that the Shapley value violates blend-in fairness due to the fact that it particularly accounts for players’ marginal contributions to the empty set and to singleton coalitions. Hence, any solution respecting only the marginal contributions to coalitions of size at least two would satisfy our fairness axiom. One particular example of a symmetrically efficient and additive solution satisfying blend-in fairness is provided in Sect. 6.3. Appendix In Sect. 6.1 we introduce the collection of partition games and show that they form a basis of the space of all games. This fact is then used in Sect. 6.2 to show how blend-in fairness shapes the characterization of the equal division solution (Theorem 1 and Theorem 2). Section 6.3 contains examples for the independence of the utilized axioms. Partition games Let T⊆N be a nonempty coalition. The partition game over T is the game uT defined as follows. (1) If T=N , then for each coalition S⊆N , u T(S) := {0if (1) S∈ {∅,T,N} or (2) |S|>|T| with ( S, T )partitioning N , − 1otherwise. (2) If T=N , then for each coalition S⊆N , u N(S) := {0if S = N , 1otherwise. 1 3 27 Page 8 of 21 Blend-in fairness and equal split Lemma 3 Fix T⊂N , k∈T with |T|≥2 , and let f:GN→Rn satisfy additivity, the weak null player property, and symmetric efficiency. Then, (1) [ |T|=|N|/2 and fk(uT)=0 ] ⇒ f k ( u T\{k})=0 ; (2) [ |T|=|N|/2 and f k ( uT )= fk ( u N\T)=0 ] ⇒ f k ( u T\{k})=0 . Proof Fix T⊂N with |T|≥2 and k∈T . Clearly then, the coalitions T and T\{k} are nonempty. Recall further that u T ( N )= u N\T( N )=0 holds. (1) Notice that |T|=|N|/2 implies either ( |T|>|N|/2 and |T\{k}| ≥ |N|/2 ) or ( |T|<|N|/2 and |T\{k}| <|N|/2 ). We consider these two possibilities separately. (1.1) ( |T|>|N|/2 and |T\{k}| ≥ |N|/2 ): For the two partition games uT and uT\{k} we have: uT(S)=0 for S∈ {∅,T,N} and uT(S)=−1 , otherwise; uT\{k}( S )=0 for S∈ {∅,T\{k},N} and uT\{k}( S )=−1 , otherwise. Define the games cT and cT\{k} by c T(S)= { − 1if S = T , 0otherwise, and c T\{k}(S)= { − 1if S = T\{k} , 0otherwise, and proceed as follows. For the game uT+cT we have (uT+cT)(S)=0 for S∈ {∅,N} and (uT+cT)(S)=−1 , otherwise. That is, all players are symmetric in uT+cT and thus, by symmetric efficiency, fk(uT+cT)=0 in particular holds. By assumption, fk(uT)=0 holds as well and thus, by additivity, fk( c T)=0 (8) follows. Considering now the game c T +cT\{k} with ( cT + c T\{k})( S )=−1 for S∈{T,T\{k}} and ( c T+ c T\{k})( S )=0 , otherwise, we have that player k is a null player in the game. By ( cT + c T\{k})( N )=0 and the weak null player property, fk( c T+ c T\{k})=0 (9) follows. We have then from (8), (9), and additivity that f k ( c T\{k})=0 (10) should hold. Notice finally that for the game uT\{k}+cT\{k} we have ( u T\{k}+ c T\{k})( S )=0 for S∈ {∅,N} and ( u T\{k}+ c T\{k})( S )=−1 , otherwise. In other words, all players are symmetric in uT\{k}+cT\{k} and thus, by 1 3 Page 15 of 21 27 D. Dimitrov, C.-j. Sun symmetric efficiency, fk( u T\{k}+ c T\{k})=0 in particular holds. By (10) and additivity, f k ( uT \{ k })=0 follows. (1.2) ( |T|<|N|/2 and |T\{k}| <|N|/2 ): Recall that the coalitions T and T\{k} are nonempty and observe further that the coalitions N\T and N\(T\{k}) are nonempty as well. For the two partition games uT and uT\{k} we have: uT(S)=0 for S∈ {∅,T,N \T,N} and uT(S)=−1 , otherwise; uT\{k}( S )=0 for S∈ {∅,T\{k},N\(T\{k}),N} and uT\{k}( S )=−1 , otherwise. Define the games cT and cT\{k} by c T(S)= { − 1if S∈{T,N \T} , 0otherwise, and c T\{k}(S)= { − 1if S∈{T\{k},N \ ( T\{k} ) } , 0otherwise, and proceed as follows. For the game uT+cT we have (uT+cT)(S)=0 for S∈ {∅,N} and (uT+cT)(S)=−1 , otherwise. That is, all players are symmetric in uT+cT and thus, by symmetric efficiency, fk(uT+cT)=0 in particular holds. By assumption, fk(uT)=0 holds as well and thus, by additivity, fk( c T)=0 (11) follows. Consider now the game c T +cT\{k} with ( cT + c T\{k})( S )=−1 for S∈{T\{k},T,N\T,N\(T\{k})} and ( c T+ c T\{k})( S )=0 , otherwise. Clearly, player k is a null player in this game. By ( cT + c T\{k})( N )=0 and the weak null player property, fk( c T+ c T\{k})=0 (12) follows. We have then from (11), (12), and additivity that f k ( c T\{k})=0 (13) should hold. Notice finally that for the game uT\{k}+cT\{k} we have ( u T\{k}+ c T\{k})( S )=0 for S∈ {∅,N} and ( u T\{k}+ c T\{k})( S )=−1 , otherwise. In other words, all players are symmetric in uT\{k}+cT\{k} and thus, by symmetric efficiency, f k ( u T\{k}+ c T\{k})=0 in particular holds. By (13) and additivity, fk( u T\{k})=0 follows. (2) Notice that |T|=|N|/2 implies |T|=|N\T|=|N|/2 . Consider then the partition games uT , uN\T , and uT\{k} recalling that they are defined as follows: uT(S)=0 for S∈ {∅,T,N} and uT(S)=−1 , otherwise; uN\T( S )=0 for S∈ {∅,N\T,N} and uT(S)=−1 , otherwise; uT\{k}( S )=0 for 1 3 27 Page 16 of 21 Blend-in fairness and equal split S∈ {∅,T\{k},N\(T\{k}),N} and uT\{k}( S )=−1 , otherwise. Define the games cT , cN\T , and cT\{k} by c T(S)= { − 1if S = T , 0otherwise, c N\T(S)= { − 1if S = N\T , 0otherwise, and c T\{k}(S)= { − 1if S∈{T\{k},N \ ( T\{k} ) } , 0otherwise, and proceed as follows. For the game uT+cT we have (uT+cT)(S)=0 for S∈ {∅,N} and (uT+cT)(S)=−1 , otherwise. That is, all players are symmetric in uT+cT and thus, by symmetric efficiency, fk(uT+cT)=0 in particular holds. By assumption, fk(uT)=0 holds as well and thus, by additivity, fk( c T)=0 (14) follows. Applying the same argument with respect to the game uN\T+cN\T and recalling f k ( u N\T)=0 holds by assumption, we get fk( c N\T)=0. (15) Consider now the game c T +cN\T+cT\{k} with ( cT + c N\T+ c T\{k})( S )=−1 for S∈{T\{k},T,N\T,N\(T\{k})} and ( c T+ c N\T+ c T\{k})( S )=0 , otherwise. Clearly, player k is a null player in this game. By ( cT + c N\T+ c T\{k})( N )=0 and the weak null player property, fk( c T+ c N\T+ c T\{k})=0 (16) follows. We have then from (14), (15), (16), and additivity that f k ( c T\{k})=0 (17) should hold. Notice finally that for the game uT\{k}+cT\{k} we have ( u T\{k}+ c T\{k})( S )=0 for S∈ {∅,N} and ( u T\{k}+ c T\{k})( S )=−1 , otherwise. In other words, all players are symmetric in uT\{k}+cT\{k} and thus, by symmetric efficiency, fk( u T\{k}+ c T\{k})=0 in particular holds. By (17) and additivity, f k ( u T\{k})=0 follows. 1 3 Page 17 of 21 27 D. Dimitrov, C.-j. Sun Remark 3 Notice that the games cT , cT\{k} , and cN\T constructed in the proof of Lemma 3 are not superadditive but cohesive. Proposition 4 If a solution f:GN→Rn satisfies additivity, the weak null player property, symmetric efficiency, and blend-in fairness, then fi(uT)=0 for each i∈N and T⊂N . Proof We proceed by induction on the cardinality of T⊂N . Initialization: For |T|=|N|−1 , fi(uT)=0 for each i∈N follows from Proposition 3. Induction Hypothesis: Suppose that fi(uT)=0 holds for each i∈N and each T⊂N with |T|>t . Take T⊂N with |T|=t and i∈N\T . Notice that if t=1 , then the assertion directly follows from Proposition 2. Suppose now that t≥2 holds. Since |T∪{i}| =t+1>t , we get f i ( u T∪{i})=0 by the induction hypothesis. By the same reasoning and for the case when t+1=|N|/2 , we additionally get f i ( u N\(T∪{i}))=0 due to |N\(T∪{i})|=t+1 . Applying Lemma 3 results then in fi(uT)=0 . Since player i∈N\T was arbitrary chosen, we conclude that fi(uT)=0 holds for each i∈N\T . Hence, it remains to be shown that fi(uT)=0 holds for each i∈T as well. For this, fix i∈T , denote by T|T| the set of all coalitions of size |T| and by T|T|( i ) the set of all coalitions in T|T| containing player i. Notice that we have i/∈T′ for each T′∈T |T|\T|T|( i ) . Hence, as already shown above, fi(uT′)=0 holds for each T′∈T |T|\T|T|( i ) . Consider now the game u=Σ T ′∈T|T|u T ′ and notice that, when |T|≥|N|/2 holds, we have u (S)=    0if S∈ {∅,N} , −  T|T|  +1 if |S|=|T|, −T| T | otherwise, while, when |T|<|N|/2 is the case, we have u (S)=    0if S∈ {∅,N} , −  T|T|  +1 if |S| ∈ {|T|,|N\T|} , −T| T | otherwise. Observe that, in either of these cases, all players are symmetric in u. By symmetric efficiency, fi(u)=0 . By fi(uT′)=0 for each T′∈T |T|\T|T|(i) and additivity, T|T|( i ) x i=0 should hold. We have then xi=0 and thus, fi(uT)=0 follows. Remark 4 Notice that the game u constructed in the proof of Proposition 4 is superadditive and thus, cohesive. 1 3 27 Page 18 of 21 Blend-in fairness and equal split Proposition 5 If a solution f:GN→Rn satisfies symmetric efficiency, then f i (u N )=1 n for each i∈N . Proof Recall that uN(N)=1 and uN(S)=0 for each S=N holds. Clearly then, all players are symmetric in uN(N) and thus, by symmetric efficiency, f i ( uN )= 1 n for each i∈N follows. □ Propositions 1–5 show that, thanks to additivity, the weak null player property, symmetric efficiency, and blend-in fairness, one has fi(uT)=EDi(uT) for each partition game uT ( T⊆N , T=∅ ) and each i∈N . Proof of Theorem 1 It can be easily verified that the equal division solution satisfies the four axioms. As for the reverse implication, let T be a non-empty coalition, b a real number, and define the game ub T as follows: (1) If T=N , then for each coalition S⊆N , u b T(S) := {0if (1) S∈ {∅,T,N} or (2) |S|>|T| with ( S, T )partitioning N , − 1otherwise. (2) If T=N , then for each coalition S⊆N , u b N(S) := {0if S = N , botherwise. In view of Propositions 1–5, we conclude for each i∈N that f i(ub T)= {0if T⊂N , b n if T=N. Theorem 0 further implies the existence of real numbers (a T ){T⊆N,T =∅} such that, for v∈G N , one has v=Σ T⊆N,T =∅ aTu b T . By the additivity of f, fi( v )=Σ T⊆N,T =∅ f i( a T u b T) (18) follows. We further make use of the following result. Claim ΣT⊂N,T =∅ f i( a T u b T)=0 . □ Proof of the Claim Notice first that Proposition 1–5 and the additivity of f give us fi(aub T)=0 for each i∈N , each non-empty T⊂N , and any a≥0 ; in particular, we have fi(aTub T)=0 for each i∈N and each non-empty T⊂N whenever aT≥0 holds. Suppose now that we have aT<0 for some non-empty T⊂N . Notice then that, with z being the zero game, aTub T+(−aT)ub T=z holds and thus, due to the additivity of f and f(z)=0 following from symmetric efficiency, we get fi(aTub T)=0 for each i∈N . Hence, the assertion follows. 1 3 Page 19 of 21 27 D. Dimitrov, C.-j. Sun The combination of (18) with the above Claim gives us f i ( v )= fi ( aNub N)= fi ( ubaN N)= fi ( uv ( N ) N)= v ( N ) n for each i∈N . □ Proof of Theorem 2 Recall that partition games are superadditive (Remark 0) and thus, cohesive, and that all games constructed in the corresponding proofs of Propositions 1–5 are cohesive as well (Remarks 1–4). Since the zero game is also superadditive, the proof of Theorem 2 is fully analogous to the proof of Theorem 1. □ Independence of the axioms In what follows, we assume that the player set contains at least three players and construct four examples where the correspondingly defined solution satisfies all axioms but the mentioned one. Observe that all solutions utilized to show axioms’ logical independence on GN can also be used to show the corresponding independence on GN coh as well. This is due to the fact that each non-superadditive game constructed in these examples is cohesive. Additivity The solution fA , given by fA(v)=ED(v) if v∈G N sa and fA(v)=Sh(v) if v∈G N\GN sa , satisfies all axioms but additivity. As to see the latter fact, let N={1,2,3} and consider the games v and w defined as follows: v(S)=1 if S∈ {{1,2},N} , and v(S)=0 , otherwise; w(S)=1 if S∈ {{3},N} , and w(S)=0 , otherwise. The game v is superadditive but w is not and thus, we have fA( v )=(1 3 , 1 3 , 1 3) and fA( w )=(1 6 , 1 6 , 4 6) . For the game (v+w) we have (v+w)(N)=2 , (v+w)(S)=1 if S∈ {{3},{1,2}} , and (v+w)(S)=0 , otherwise. Since (v+w) is not a superadditive game, we get fA( v + w )=(4 6 , 4 6 , 4 6)= fA( v )+ fA( w ) . Weak null player property Denote by R3 i the set of all permutations of the player set N at which player i∈N is at least the third player in the corresponding order. Notice that R3 i=R3 j holds for all i, j ∈N and set r : =R3 i . For i∈N , let g i(v)= 1 r ∑ R ∈R 3 i ∆i(Si(R)) . Define then the solution fWNP by f WNP i(v)=gi(v)+v ( N )−Σ i∈Ngi ( v ) n for each i ∈N. In other words, the payoff fWNP i(v) is just the sum of the average of player i’s marginal contributions to coalitions of size at least two ( gi(v) ) and the equal division of the excess v(N)−Σi∈Ngi(v) . This solution clearly satisfies additivity and symmetric efficiency. In order to see that it satisfies blend-in fairness as well, recall that σ(i)=i holds for the permutation σ guiding the cooperation equivalence of two superadditive games v and w for player i∈N . Given that v(S)=w(σ(S)) holds for all S⊆N with j∈S for some j∈N\{i} , fWNP i(v)=fWNP i(w) follows. Notice finally that fWNP violates the weak null player property. As to see it, take N={1,2,3} and consider the superadditive game v defined by v(S)=0 if S∈ {{1},{2,3},N} , and v(S)=−1 , otherwise. Notice that player 1 is a null player in v and v(N)=0 . We get g(v) = (0,1,1) and 1 3 27 Page 20 of 21 Blend-in fairness and equal split fWNP( v )=(−2 3 , 1 3 , 1 3) in violation of the weak null player property requiring player 1 to get zero payoff in the game v. Symmetric efficiency The solution fSE , defined by fSE(v)=0 for each v∈G N , satisfies all axioms but symmetric efficiency. Blend-in fairness The Shapley value violates blend-in fairness while satisfying all other axioms. 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References Alesina A, Angeletos G-M (2005) Fairness and redistribution. Am Econ Rev 95(4):960–980 Almås I, Cappelen A, Tungoddden B (2020) Cutthroat capitalism versus cuddly socialism: are Americans more meritocratic and efficiency-seeking than scandinavians? J Polit Econ 128(5):1753–1788 Alonso-Meijide JM, Costa J, García-Jurado I (2019) Values, nullifiers and dummifiers. In: Algaba E, Fragnelli V, Sánchez-Soriano J (eds) Handbook of the Shapley value. Chapman and Hall/CRC Press, Boca Raton, pp 75–92 Branzei R, Dimitrov D, Tijs S (2008) Models in cooperative game theory, 2nd edn. Springer, Germany Cappelen A, Hole A, Sørensen E, Tungodden B (2007) The pluralism of fairness ideals: an experimental approach. Am Econ Rev 97(3):818–827 Casajus A, Huettner F (2013) Null players, solidarity, and the egalitarian Shapley values. J Math Econ 49:58–61 Chun Y (1989) A new axiomatization of the Shapley value. Games Econ Behav 1(2):119–130 de Clippel G, Rozen K (2022) Fairness through the lens of cooperative game theory: an experimental approach. Am Econ J Microecon 14(3):810–836 Gelfand M, Higgins M, Nishii L, Raver J, Dominiguez A, Murakami F, Yamaguchi S, Toyama M (2002) Culture and egocentric perceptions of fairness in conflict and negotiation. J Appl Psychol 87(5):833–845 Hernandez-Lamoneda L, Juarez R, Sanchez-Sanchez F (2008) Solutions without dummy axiom for TU cooperative games. Econ Bull 3(1):1–9 Konow J (2001) Fair and square: the four sides of distributive justice. J Econ Behav Organ 46:137–164 Moulin H (2003) Fair division and collective welfare. MIT Press, Cambridge Osborne M, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge Shapley LS (1953) A value for [CDATA[n]] n -person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II, pp 307–317 Shapley LS, Shubik M (1967) Ownership and the production function. Q J Econ 81(1):88–111 van den Brink R (2007) Null or nullifying players: the difference between the Shapley value and equal division solutions. J Econ Theory 136:767–775 Young P (1985) Monotonic solutions of cooperative games. Int J Game Theory 14(2):65–72 Young P (1994) Equity: in theory and practice. Princeton University Press, Princeton Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1 3 Page 21 of 21 27