Solidarity within a fixed community☆
Introduction
Cooperative games with transferable utility (TU-game) are a simple and versatile tool to model the generation of worth in a society or community and to study the “fair” or “reasonable” allocation of this worth. A TU-game specifies a player set and the worth that can be generated by any subset of this player set. Shapley (1953) introduced a solution for this setup, which assigns to every player a payoff that measures his productivity within such a TU-game. Modern societies and organizations, however, base the allocation of wealth among their members not only on individual productivities but also on egalitarian or solidarity principles.
One such solidarity principle, population solidarity, is studied by Chun and Park (2012). Population solidarity requires that the arrival of new players should change the payoff of the original players in the same direction, i.e., either all original players lose together or gain together.
In this paper, we consider a similar solidarity property, nullified solidarity, that works on a fixed player set. Assume some player loses his productivity, i.e., becomes a null player. Then, a solidarity principle should rule out that some players gain while others lose. That is, nullified solidarity requires that whenever a player becomes a null player, the payoffs of all players should change in the same direction, i.e., all players lose together or all player win together.
There are (at least) two rival solution concepts for TU-games that rely on solidarity considerations to some or more extent. The equal division value (ED-value) distributes evenly the overall worth among the players, while according to the equal surplus division value (ES-value) the players first obtain what they can achieve for themselves alone and, then, gains from cooperation within the whole society are divided equally among them.
While population solidarity is satisfied by both the ES-value and the ED-value, it turns out that its sibling on a fixed player set is characteristic for the ED-value. More precisely, we obtain the following main result: the ES-value is the unique value that satisfies nullified solidarity, efficiency, the null game property, and weak fairness. Efficiency states that the worth of the grand coalition is shared. The null game property requires zero payoffs in the null game. According to weak fairness, all players should gain or lose equally, whenever all marginal contributions to coalitions of all players are changed by the same amount.
Further comparisons between the ED-value and the ES-value are conducted by van den Brink (2007), van den Brink and Funaki (2009), van den Brink et al. (2012), and Casajus and Huettner (2014). Béal et al. (2012) characterize the ED-value and the ES-value for graph games.
This paper is organized as follows. Basic definitions and notations are given in Section 2. Section 3 provides the new characterization of the ED-value.
Section snippets
Basic definitions and notation
Let the universe of players be given by the natural numbers , and let denote the set of non-empty and finite subsets of . A (TU)-game is a pair consisting of a set of players and a coalition function , where denotes the power set of . Sometimes, for notational parsimony, we will write instead of . Also for notational convenience, we will write the singleton as . Subsets of are called coalitions; is called the worth of coalition .
Solidarity on changing and fixed player sets
The solidarity principle is described informally by Thomson (2012) as follows:
“When the circumstances in which some group of agents find themselves change–the group could be the entire population of agents present or some subset–and if none of them bears any particular responsibility for the change, or deserves any particular credit for it, their welfare should be affected in the same direction: all members of the group should end up at least as well off as they were initially, or they should
Concluding remarks
In this paper, we reconsider the principle of solidarity used by Chun and Park (2012). While their characterization involves changing player sets, we use a solidarity principle that refers to a fixed player set. Their notion of solidarity and ours have in common that all players gain or all players lose if the productivity of some player is lost. The notions differ in that we keep the player present. While their population solidarity results in the equal split of surplus and allows the players
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Cited by (11)
Sign properties and axiomatizations of the weighted division values
2024, Journal of Mathematical EconomicsPlayers’ dummification and the dummified egalitarian non-separable contribution value
2023, Economics LettersThe proportional Shapley value and applications
2018, Games and Economic BehaviorCitation Excerpt :In other words, the player becomes dummy, while the worth of any coalition not containing him/her remains unchanged. The dummification is in essence similar to the nullification of a player studied by Gómez-Rúa and Vidal-Puga (2010), Béal et al. (2014) and Béal et al. (2016). The new axiom of proportional balanced contributions under dummification requires, for any two players, allocation variations after the dummification of the other player that are proportional to their stand-alone worths.
Values for environments with externalities – The average approach
2018, Games and Economic BehaviorAxiomatic characterizations under players nullification
2016, Mathematical Social SciencesCitation Excerpt :However the existence of a value satisfying these axioms is not always guaranteed. The closest articles in the literature are Gómez-Rúa and Vidal-Puga (2010) and Béal et al. (2014a). Apart from the aforementioned result, Gómez-Rúa and Vidal-Puga (2010) also consider the axiom of null balanced intracoalitional contributions for the class of TU-games with a coalition structure.
Preserving or removing special players: What keeps your payoff unchanged in TU-games?
2015, Mathematical Social SciencesCitation Excerpt :See also Radzik (2012) for recent developments. The class of Weighted Division values is studied by van den Brink (2009) and Béal et al. (2013, 2014). Other articles studying the Equal Division value and the Equal Surplus Division value on variable player sets are due to Chun and Park (2012) and van den Brink et al. (2012), while van den Brink and Funaki (2009) investigate the same values by imposing fixed player sets.
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We are grateful to Sylvain Ferrières for helpful comments on our paper. Financial support for Frank Huettner from the Deutsche Forschungsgemeinschaft (DFG) grant HU 2205/1-1 is gratefully acknowledged. Moreover, we gratefully acknowledge financial support by the National Agency for Research (ANR), research program DynaMITE: Dynamic Matching and Interactions: Theory and Experiments, contract ANR-13-BSHS1-0010.