Elsevier

Economics Letters

Volume 125, Issue 3, December 2014, Pages 440-443
Economics Letters

Solidarity within a fixed community

https://doi.org/10.1016/j.econlet.2014.10.023Get rights and content

Highlights

  • We study the consequences of a player becoming a null player.

  • Nullified solidarity: in this case, either all players weakly gain or all weakly lose.

  • We provide a new characterization of the equal division value.

  • It uses efficiency, the null game property, weak fairness, and nullified solidarity.

Abstract

We study the consequences of a solidarity property that specifies how a value for cooperative games should respond if some player forfeits his productivity, i.e., becomes a null player. Nullified solidarity states that in this case either all players weakly gain together or all players weakly lose together. Combined with efficiency, the null game property, and a weak fairness property, we obtain a new characterization of the equal division value.

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 N, and let N denote the set of non-empty and finite subsets of N. A (TU)-game is a pair (N,v) consisting of a set of players NN and a coalition function vV(N){f:2NRf()=0}, where 2N denotes the power set of N. Sometimes, for notational parsimony, we will write v instead of (N,v). Also for notational convenience, we will write the singleton {i} as i. Subsets of N are called coalitions; v(S) is called the worth of coalition S.

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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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.

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