A survey of consistency properties in cooperative game theory

The main purpose of this survey paper is to review the axiomatic characterizations of the Shapley value, the prekernel, the prenucleolus, and the core by means of a consistency property in terms of the reduced games. Whenever possible, new results and new proofs are addded

The consistency principle for games in strategic form

Game Theory

A comparison of the average prekernel and the prekernel

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral setting

Consistency and potentials in cooperative TU-games: Sobolev's reduced game revived

It was a quarter of a century ago that Sobolev proved the reduced game (otherwise called consistency) property for the much-discussed Shapley value of cooperative TU-games. The purpose of this paper is to extend Sobolev's result in two ways. On the one hand the unified approach applies to the enlarged class consisting of game-theoretic solutions that possess a so-called potential representation; on the other Sobolev's reduced game is strongly adapted in order to establish the consistency property for solutions that admit a potential. Actually, Sobolev's explicit description of the reduced game is now replaced by a similar, but implicit definition of the modified reduced game; the characteristic function of which is implicitly determined by a bijective mapping on the universal game space (induced by the solution in question). The resulting consistency property solves an outstanding open problem for a wide class of game-theoretic solutions. As usual, the consistency together with some kind of standardness for two-person games fully characterize the solution. A detailed exposition of the developed theory is given in the event of dealing with so-called semivalues of cooperative TU-games and the Shapley and Banzhaf values in particular

Two axiomatizations of the kernel of TU games: bilateral and converse reduced game properties

We provide two axiomatic characterizations of the kernel of TU games by means of both bilateral consistency and converse consistency with respect to two types of two-person reduced games. According to the first type, the worth of any single player in the two-person reduced game is derived from the difference of player's positive (instead of maximum) surpluses. According to the second type, the worth of any single player in the two-person reduced game either coincides with the two-person max reduced game or refers to the constrained equal loss rule applied to an appropriate two-person bankruptcy problem, the claims of which are given by the player's positve surpluses

A Noncooperative View of Consistent Bankruptcy Rules

We introduce a game form that captures a noncooperative dimention of the consistency property of bankruptcy rules. Any consistent and monotonic rule is fully characterized by a bilateral principle and consistency. Like the consistency axiom, our game form, together with the bilateral principle, yields the corresponding consistent bankruptcy rule as a result of a unique outcome of Nash equilibria. The result holds for a large class of consistent monotone rules, including the Constrained Equal Award, the Proportional Rule, and many other well known rules. Moreover, all the subgame perfect equilibria are coalition-proof in the associated game in strategic form.

A COMPARISON OF THE AVERAGE PREKERNEL AND THE PREKERNEL

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral settings

A comparison of the average prekernel and the prekernel.

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral settings

Consistency, converse consistency, and aspirations in TU-games

In problems of choosing ‘aspirations’ for TU-games, we study two axioms, ‘MW-consistency’ and ‘converse MW-consistency.’ In particular, we study which subsolutions of the aspiration correspondence satisfy MW-consistency and/or converse MW-consistency. We also provide axiomatic characterizations of the aspiration kernel and the aspiration nucleolus