Axiomatization of the Farsighted Stable Set and the (Strong) Rational Expectations Farsighted Stable Set

Harsanyi (1974) criticized the von Neumann-Morgenstern notion of a stable set on the grounds that it implicitly assumes coalitions to be shortsighted in evaluating their prospects. He proposed a modification of the dominance relation to incorporate farsightedness. In doing so, however, Harsanyi retained another feature of the stable set: that a coalition S can impose any imputation as long as its restriction to S is feasible for S. This implicitly gives an objecting coalition complete power to arrange the payoffs of players elsewhere, which is clearly unsatisfactory. While this assumption is absolutely innocuous for the classical stable set, it is of crucial significance for farsighted dominance. Our proposed modification of the Harsanyi set respects “coalitional sovereignty.” The resulting farsighted stable set is very different, both from that of Harsanyi or of von Neumann and Morgenstern. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single payoff allocation. This condition is weaker than assuming that the relative interior of the core is non-empty, but roughly establishes an equivalence between core allocations and the union of allocations over all single payoff farsighted stable sets. We state two conjectures: that farsighted stable sets exist in all transferable-utility games, and that when a single-payoff farsighted stable set exists, there are no farsighted stable sets containing multiple payoff allocations.

The stable set of von Neumann and Morgenstern imposes credibility on coalitional deviations. Their credibility notion can be extended to cover farsighted coalitional deviations, as proposed by Harsanyi (1974), and more recently reformulated by Ray and Vohra (2015). However, the resulting farsighted stable set suffers from a conceptual drawback: while coalitional deviations improve on existing outcomes, coalitions might do even better by moving elsewhere. Or other coalitions might intervene to impose their favored moves. We show that every farsighted stable set satisfying some reasonable, and easily verifiable, properties is unaffected by the imposition of this stringent maximality requirement. These properties are satisfied by many, but not all, known farsighted stable sets.

Harsanyi (1974) and Ray and Vohra (2015) extended the stable set of von Neumann and Morgenstern to impose farsighted credibility on coalitional deviations. But the resulting farsighted stable set suffers from a conceptual drawback: while coalitional moves improve on existing outcomes, coalitions might do even better by moving elsewhere. Or other coalitions might intervene to impose their favored moves. We show that every farsighted stable set satisfying some reasonable and easily verifiable properties is unaffected by the imposition of these stringent maximality constraints. The properties we describe are satisfied by many, but not all, farsighted stable sets.

In the study of farsighted coalitional behavior, a central role is played by the von Neumann-Morgenstern (1944) stable set and its modification that incorporates farsightedness. Such a modification was first proposed by Harsanyi (1974) and has recently been re-formulated by Ray and Vohra (2015). The farsighted stable set is based on a notion of indirect dominance in which an outcome can be dominated by a chain of coalitional ‘moves’ in which each coalition that is involved in the sequence eventually stands to gain. However, it does not require that each coalition make a maximal move, i.e., one that is not Pareto dominated (for the members of the coalition in question) by another. Nor does it restrict coalitions to hold common expectations regarding the continuation path from every state. Consequently, when there are multiple continuation paths the farsighted stable set can yield unreasonable predictions. We resolve this difficulty by requiring all coalitions to have common rational expectations about the transition from one outcome to another. This leads to two related concepts: the rational expectations farsighted stable set (REFS) and the strong rational expectations farsighted stable set (SREFS). We apply these concepts to simple games and to pillage games to illustrate the consequences of imposing rational expectations for farsighted stability

This paper analyzes farsighted stable sets when agents have heterogeneous expectations over the dominance paths. We consider expectation functions satisfying two properties of path-persistence and consistency. We show that farsighted stable sets with heterogeneous expectations always exist and that any singleton farsighted stable set with common expectations is a farsighted stable set with heterogeneous expectations. We characterize singleton farsighted stable sets with heterogeneous expectations in one-to-one matching models and voting models, and show that the relaxation of the hypothesis of common expectations greatly expands the set of states which can be supported as singleton farsighted stable sets.

This paper analyzes farsighted stable sets when agents have heterogeneous expectations over the dominance paths. We consider expectation functions satisfying two properties of path-persistence and consistency. We show that farsighted stable sets with heterogeneous expectations always exist and that any singleton farsighted stable set with common expectations is a farsighted stable set with heterogeneous expectations. We characterize singleton farsighted stable sets with heterogeneous expectations in one-to-one matching models and voting models, and show that the relaxation of the hypothesis of common expectations greatly expands the set of states that can be supported as singleton farsighted stable sets.

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict $$\alpha $$ -core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.

Harsanyi (1974) criticized the von Neumann–Morgenstern (vNM) stable set for its presumption that coalitions are myopic about their prospects. He proposed a new dominance relation incorporating farsightedness, but retained another feature of the stable set: that a coalition S can impose any imputation as long as its restriction to S is feasible for it. This implicitly gives an objecting coalition complete power to arrange the payoffs of players elsewhere, which is clearly unsatisfactory. While this assumption is largely innocuous for myopic dominance, it is of crucial significance for its farsighted counterpart. Our modification of the Harsanyi set respects “coalitional sovereignty.” The resulting farsighted stable set is very different from both the Harsanyi and the vNM sets. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single-payoff allocation. This condition roughly establishes an equivalence between core allocations and the union of allocations over all single-payoff farsighted stable sets. We then conduct a comprehensive analysis of the existence and structure of farsighted stable sets in simple games. This last exercise throws light on both single-payoff and multi-payoff stable sets, and suggests that they do not coexist.

Farsighted stable sets in Hotelling’s location games

This paper revisits analysis of stable sets in two-player strategic-form games. Our two main contributions are (i) to establish a connection between myopic stable sets and stable matchings of an auxiliary two-sided matching problem and (ii) to identify a structural property of 2-player games, called the block partition property, which helps characterize strategy profiles that are indirectly dominated by a fixed profile. Our analysis also generalizes and unifies existing results on myopic and farsighted stable sets in 2-player games.

Myopic and farsighted stable sets in 2-player strategic-form games

We consider marriage problems where myopic and farsighted players in- teract. To study such problems, we use the pairwise myopic-farsighted stable set. Blocking occurs by coalitions of size one or two. We require that all blocking players should strictly improve. We pay particular attention to the question whether core elements survive in this environment. This is the case when all players are myopic as well as when all players are farsighted. It also holds for matching problems satisfying the top-coalition property. For general matching problems where all women are farsighted, there is only one core element that can belong to the pairwise myopic-farsighted stable set, the woman-optimal stable matching, so all other stable outcomes are excluded for sure. If the woman-optimal stable matching is dominated from the woman point of view by an individually rational matching, then the pairwise myopic- farsighted stable set cannot contain a core element. We show that blocking by coalitions of arbitrary size leads to identical results.

An abstract game satisfies Dominance Invariance if the indirect and the direct dominance relations, or myopic and farsighted dominance, are equivalent. Mauleon et al. (Int J Game Theory 43(4):925–943, 2014) study Dominance Invariance as an attractive condition that eliminates the differences between a farsighted solution concept and its myopic counterpart. We show that Dominance Invariance can also be used to eliminate the differences between various farsighted solution concepts in any abstract game. Together with an additional condition called No Infinite Chains, Dominance Invariance implies the existence and uniqueness of the farsighted stable set, its equivalence to the largest consistent set and its equivalence to the (strong) rational expectations farsighted stable set when the latter exists. This also implies that both the farsighted stable set and the largest consistent set do not suffer from the problem of maximality under these conditions.

Farsighted stability in patent licensing: An abstract game approach

Farsighted stability in average return games