The equal split-off set for NTU-games

This paper introduces and studies the equal split-oﬀ set for cooperative games with nontransferable utility. We illustrate the new solution for the famous Roth-Shafer examples and present two axiomatic characterizations based on diﬀerent consistency properties on the class of exact partition games, i


Introduction
Nontransferable utility games arise when players in a cooperative game face the problem of allocating joint profits while having nonlinear utility functions over money.Also situations where the underlying infinitely divisible endowment is not of a monetary nature are accommodated.The opportunities of coalitions are represented by a set of attainable utility payoff allocations and the issue is to select a payoff allocation for the grand coalition while taking these opportunities into account.
We focus on egalitarianism in the context of nontransferable utility games.Since egalitarianism is a doctrine that favors the idea of equality, this requires the assumption that utility is not only comparable in an intrapersonal way, but also in an interpersonal way.In other words, we assume that utility is normalized in such a way that equating utilities for different players has a meaningful interpretation.This was also implied by the approach of Kalai and Samet (1985), who introduced and characterized an egalitarian solution for nontransferable utility games which recursively assigns equal payoffs to members of coalitions in games.We show that the equal split-off set intersects the core if and only if the underlying game is an exact partition game, i.e. there exists a core element for which all richest players, all richest and second richest players, and so on, are able to obtain their payoffs jointly by themselves.This generalizes the class of transferable utility games introduced by Llerena and Mauri (2017) and the corresponding result of Dietzenbacher and Yanovskaya (2021).On the class of exact partition games, we present two axiomatic characterizations of the equal split-off set based on weak versions of consistency properties that were employed by Peleg (1985) and Tadenuma (1992) to characterize the core.It turns out that nontransferable utility games induced by bargaining problems and bankruptcy problems are exact partition games.On the class of bargaining problems, the equal split-off set coincides with the Kalai (1977) solution.On the class of bankruptcy problems with nontransferable utility, the equal split-off set coincides with the constrained Kalai solution (cf.Albizuri et al. 2020).
This paper is organized in the following way.Section 2 presents preliminary notions and notations for nontransferable utility games.Section 3 introduces the equal split-off set as a solution for all nontransferable utility games and presents some elementary results.Section 4 introduces the class of exact partition games, shows that this class consists of all games where the equal split-off set intersects the core, and presents two axiomatic characterizations based on different consistency properties.Section 5 illustrates the equal split-off set for the class of bargaining problems and the class of bankruptcy problems with nontransferable utility.

Preliminaries
Let N be a nonempty and finite set.Denote 2 + denote the vector of all ones, i.e. e i = 1 for all i ∈ N .For all x, y ∈ R N + , x ≤ y denotes x i ≤ y i for all i ∈ N , and x < y denotes x i < y i for all i ∈ N .For all x ∈ R N + and all

Note that SP(A) ⊆ WP(A).
A nontransferable utility game is a pair (N, V ) where N is a nonempty and finite set of players and V assigns to each coalition S ∈ 2 N \ {∅} a set of attainable payoff allocations is nonempty, closed, and bounded; The nonnegativity condition on the attainable payoff allocations was also asssumed by e.g.Asscher (1976), Asscher (1977), Greenberg (1985), and Lejano (2011).This convenient assumption is without loss of generality, i.e. our analysis is invariant under symmetric positive transformations of utility.The nonemptiness, closedness, boundedness, and comprehensiveness conditions are standard.The monotonicity condition was also assumed by e.g.Otten et al. (1998) and Hendrickx et al. (2002).Note that we do not assume that the sets of attainable payoff allocations are convex in order to allow for utility functions that are not necessarily of the Von Neumann-Morgenstern type.In line with Kalai and Samet (1985), we assume that utility is normalized in such a way that it is interpersonally comparable.
Let Γ all denote the class of all NTU-games.A solution σ on a class of games Γ ⊆ Γ all assigns to each (N, V ) ∈ Γ a set of payoff allocations σ(N, V ) ⊆ V (N ).Throughout this paper, Γ is the generic notation for a class of games and σ is the generic notation for a solution on Γ.The core is the solution on Γ all that assigns to each (N, V ) ∈ Γ all the set of payoff allocations 3 The equal split-off set In this section, we introduce the equal split-off set as a solution for all nontransferable utility games and present some elementary results.The equal split-off set for transferable utility games introduced by Branzei et al. (2006) is based on the computational algorithm of the egalitarian Dutta and Ray (1989) solution.We generalize this solution to nontransferable utility games in the following way.Consider an arbitrary nontransferable utility game for which we face the problem of selecting payoff allocations for the grand coalition.One of the coalitions with maximal attainable equal payoff allocation is selected and the members leave with these payoffs.The remaining players determine the attainable payoff allocations for each subgroup in coalition with the departed players.One of the subgroups with maximal attainable equal payoff allocation is selected and the members leave with these payoffs.This process continues and results in a payoff allocation for the players.The equal split-off set consists of all payoff allocations generated by this procedure.
Definition 1 (Equal split-off set) ) consists of all payoff allocations x ∈ R N + for which there exists an ordered partition {T 1 , . . ., T m } of N such that for all k ∈ {1, . . ., m}, Note that the equal split-off set is well-defined and nonempty for all nontransferable utility games.We illustrate this new solution by means of the examples introduced by Roth (1980) and Shafer (1980).These examples initiated an interesting and extensive discussion on the interpretation of solutions for nontransferable utility games.For details, we refer to Harsanyi (1980), Aumann (1985b), Hart (1985b), Roth (1986), andAumann (1986).Along the lines of this discussion, we compare the equal split-off set with the egalitarian Kalai and Samet (1985) solution.
Example 1 (cf.Roth 1980) Let (N, V p ) with N = {1, 2, 3} and p ∈ [0, 1 2 ] be the game given by Here, comp denotes the comprehensive hull and conv denotes the convex hull, i.e. the smallest containing comprehensive set and the smallest containing convex set, respectively.
} corresponding to ordered partition {{1, 2}, {3}} and the Kalai and Samet (1985) solution is ( 1 2 − 1 3 p, 1 2 − 1 3 p, 2 3 p).Note that, in contrast to the Kalai and Samet (1985) solution, the equal split-off set assigns the unique core element to this game.Moreover, Roth (1980) claims that ( 1 2 , 1 2 , 0) is the unique outcome of this game consistent with the hypothesis that the players are rational utility maximizers, because this outcome is strictly preferred by both players 1 and 2 over all other feasible outcomes, and it can be achieved without the cooperation of player 3.
2 , the game is completely symmetric with respect to the players and it is no longer the case that cooperation with player 3 offers strictly less to players 1 or 2 than cooperation with one another.The equal split-off set is 2 )} corresponding to ordered partitions {{1, 2}, {3}}, {{1, 3}, {2}}, and {{2, 3}, {1}}.The Kalai and Samet (1985) solution is ( 1 3 , 1 3 , 1 3 ).Note that the equal split-off set coincides with the core in this case, whereas the Kalai and Samet (1985) solution is not in the core.
In contrast to transferable utility games, equal split-off set allocations make players not necessarily leave with their payoffs in nonincreasing order, i.e. it does not generally hold that x i ≥ x j for all i ∈ T k and j ∈ T with k, ∈ {1, . . ., m} and k ≤ .This is shown by the following example.
Observations change significantly if we slightly restrict the domain of nontransferable utility games.Let Γ all denote the class of all NTU-games (N, V ) where for all S ∈ 2 N \ {∅}, • V (S) is nonleveled, i.e.SP(V (S)) = WP(V (S)).
Note that all nontransferable utility games in Γ all can be approximated by games in Γ all .
Let (N, V ) ∈ Γ all .For all x ∈ R N + , we define R x 0 = ∅ and for all k ∈ N, 2 ∈ 2 N \ {∅} consists of the richest and second richest players in x, and so on.
Lemma 1 implies that for each x ∈ ESOS(N, V ) the following holds: In Example 3, the equal split-off set consists of multiple core allocations.Another consequence of assuming nonleveled attainable sets of payoff allocations is that the equal split-off set is single-valued when it intersects the core.
Lemma 2 Suppose for the sake of contradiction that there exist

Exact partition games
In this section, we introduce the class of exact partition games, we show that this class consists of all games where the equal split-off set intersects the core, and we present two axiomatic characterizations based on different consistency properties.Llerena and Mauri (2017) introduced exact partition games in the transferable utility context.We generalize this definition to nontransferable utility games.A game is an exact partition game if there exists a core allocation for which all richest players, all richest and second richest players, and so on, are able to obtain their payoffs jointly by themselves.

Definition 2 (Exact partition games)
A game (N, V ) ∈ Γ all is an exact partition game if there exists Example 4 Let (N, V ) with N = {1, 2, 3} be the game given by The equal split-off set is ESOS(N, V ) = {(6, 3, 1)} corresponding to ordered partition . This means that (N, V ) is an exact partition game.
Let Γ exp denote the class of all exact partition games.In Example 4, we observe that the equal split-off set of an exact partition game intersects the core.We show that the equal split-off set intersects the core of a game if and only if it is an exact partition game.Then Lemma 2 implies that the equal split-off set of an exact partition game is single-valued.
Suppose for the sake of contradiction that there exists S ∈ 2 Hence, x ∈ ESOS(N, V ).
The following punctual properties are satisfied by the equal split-off set on the class of exact partition games.

Strong feasible richness
x R x k ∈ V (R x k ) for all (N, V ) ∈ Γ, all x ∈ σ(N, V ), and all k ∈ N.
Core selection Nonemptiness requires that a solution assigns to all games at least one payoff allocation.Feasible richness requires that the richest players are able to obtain their payoffs by themselves.Strong feasible richness requires that the richest players, the richest and second richest players, and so on, are able to obtain their payoffs jointly by themselves.Equal payoff stability requires that no coalition is better off by an attainable equal payoff allocation, i.e. for all coalitions there exists a member whose allocated payoff is at least the maximal attainable equal payoff.Core selection requires that only core elements are assigned.Note that strong feasible richness implies feasible richness, and core selection implies equal payoff stability.Clearly, the equal split-off set satisfies nonemptiness and strong feasible richness.
By Lemma 2 and Lemma 3, the equal split-off set satisfies core selection on the class of exact partition games.In fact, an axiomatic characterization of the equal split-off set in terms of nonemptiness, strong feasible richness, and core selection on the class of exact partition games is directly obtained.

Theorem 1
The equal split-off set is the unique solution for exact partition games satisfying nonemptiness, strong feasible richness, and core selection.
Proof.Clearly, the equal split-off set satisfies nonemptiness and strong feasible richness.By Lemma 2 and Lemma 3, the equal split-off set satisfies core selection on Γ exp .
By core selection and strong feasible richness, Theorem 1 generalizes the corresponding result of Calleja et al. (2021) for transferable utility games.The empty solution, which assigns to all exact partition games the empty set, satisfies strong feasible richness and core selection, but does not satisfy nonemptiness.
The core satisfies nonemptiness and core selection, but does not satisfy strong feasible richness.The equal payoff solution, which assigns to all exact partition games the maximal attainable equal payoff allocation of the grand coalition, satisfies nonemptiness and strong feasible richness, but does not satisfy core selection.Hence, the properties in Theorem 1 are independent.
The equal split-off set is not the unique solution for exact partition games satisfying nonemptiness, feasible richness, and equal payoff stability.The solution which assigns (6, 2, 2) to the game in Example 4, and the equal split-off set to all other exact partition games, also satisfies these properties.However, this solution does not apply feasible richness in a coherent way to nontransferable utility games with variable population.In other words, it does not satisfy the relational property of consistency.Suppose that we apply a certain solution to select payoff allocations for the grand coalition and consider one such assigned payoff allocation.Some players leave with their payoffs and the remaining players reevaluate their payoffs on the basis of a reduced game.The solution is consistent if it assigns the same payoffs to the remaining players in the reduced game as in the original game.Peleg (1985) axiomatically characterized the core for nontransferable utility games using the consistency property where the attainable payoff allocations for the remaining players in the reduced game are the attainable payoff allocations in coalition with any subgroup of departed players in the original game when these departed players are assigned their original payoffs.This generalizes the consistency property for transferable utility games of Davis and Maschler (1965) and we refer to it as max-consistency.In order to axiomatically characterize the equal split-off set for exact partition games, we use the weaker version which only requires consistent payoff allocations when all richest players leave, to which we refer as rich-restricted max-consistency.
Rich-restricted max-consistency ) for all (N, V ) ∈ Γ and all x ∈ σ(N, V ) with R x 1 = N , where Theorem 2 The equal split-off set is the unique solution for exact partition games satisfying nonemptiness, feasible richness, equal payoff stability, and rich-restricted max-consistency.