Balanced contributions for TU games with awards and applications

Abstract

Calleja et al. [Calleja, P., Borm, P., Hendrickx, R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730–747] introduced multi-issue allocation situations with awards. In this paper, we extend the classical model of cooperative games with transferable utility to the cooperative games with transferable utility and awards. We define a run-to-the-bank rule for cooperative games with transferable utility and awards and characterise it in terms of a property of balanced contributions. We apply our main result to bankruptcy problems and multi-issue allocation situations with awards.

Introduction

Game theory is the study of decision making in multi-person situations, where the outcome depends on everyone’s choice. In the cooperative approach, it is assumed that binding agreements are possible, and it abstracts away from the detailed bargaining procedures. The selection of the appropriate cooperative outcome is usually based on a set of desired postulates or axioms, which, when applied to a class of problems, characterise one or another solution concept. One important axiom in the literature is balanced contributions. This is based on a principle of reciprocity, as introduced by Myerson (1980), and is often used in the literature on the Shapley value. Myerson’s property of balanced contributions asserts that for any two players the gain or loss to each player when the other “leaves” the game should be equal.

One significant application of cooperative game theory is the bankruptcy theory. Bankruptcy problems are situations where we have to divide an estate among a set of agents, but the estate is not enough to satisfy all the quantities the agents demand. Due to this insufficiency, bankruptcy rules appear. These rules, whose definition depends on the context, divide the estate adequately. There are many situations which can be described by means of bankruptcy models. One of the more classical examples is the case of a firm which goes bankrupt and has to divide the active (estate) among the claimants (agents) taking into account that the active is not sufficient to satisfy the claims.

In spite of being already studied in the Talmud, the first formal analysis of bankruptcy problems appears in O’Neill (1982). In this paper, he not only associates a cooperative game to each bankruptcy problem but also defines the run-to-the-bank rule, which coincides with the Shapley (Shapley, 1953) value of this game and is characterised with a property he calls consistency.

Multi-issue allocation situations were introduced by Calleja et al. (2005) as an extension to bankruptcy situations. In a multi-issue allocation situation, the players do not have a single claim on the estate, but multiple claims. This multidimensionality of claims is not the result of some exogenously given difference in status or priority (e.g., Kaminski, 2000). Rather, the various claims result from different issues, which all have the same status.

Calleja et al. (2005) generalise O’Neill’s run-to-the-bank (recursive completion) rule to the class of multi-issue allocation situations and show that this coincides with the Shapley value (cf Shapley, 1953) of the corresponding multi-issue allocation game. In fact, they define two such rules and games, based on the so-called proportional approach and the queue approach. As it is done in González-Alcón et al. (2003), who introduce and characterise an alternative run-to-the-bank rule for multi-issue allocation situations, in this paper, we focus on the more pessimistic queue approach. Similar results can be obtained for the proportional approach.

Following O’Neill’s characterisation of the run-to-the-bank rule for bankruptcy situations by the property of consistency, Calleja et al. (2005) characterise their rule for multi-issue allocation situations in a similar fashion. Because the underlying idea of a player “leaving” the game is not easy to implement in multi-issue allocation situations, they have to extend their domain to the class of the so-called multi-issue allocation situations with awards. In such situations, the awarded agents are still part of the game, but any solution must give them their predetermined award.

The consideration of a set of awarded agents has perfect sense in a variety of situations, not only in multi-issue allocation situations. Suppose, for example, the problem of the cost allocation of one project in which some agents are invited to participate with the compromise of being allocated with some fixed quantity. In this way, with the objective of providing a general framework, this paper extends the classical model of cooperative games with transferable utility (TU games) to the more general model of TU games with awards in which any solution must allocate to the awarded players their fixed award. For this class of games, we define a run-to-the-bank rule and characterise in terms of a property of balanced contributions. As application of our main result, we obtain characterizations for the run-to-the-bank rule for bankruptcy problems and for the run-to-the-bank rules by Calleja et al. (2005) in terms of properties of balanced contributions with a similar flavour.