Von Neumann-Morgenstern Stable Set Research Articles

Stable cores in information graph games

In an information graph situation, a finite set of agents and a source are the set of nodes of an undirected graph with the property that two adjacent nodes can share information at no cost. The source has some information (or technology), and agents in the same component as the source can reach this information for free. In other components, some agent must pay a unitary cost to obtain the information. We prove that the core of the derived information graph game is a von Neumann-Morgenstern stable set if and only if the information graph is cycle-complete, or equivalently if the game is concave. Otherwise, whether there always exists a stable set is an open question. If the information graph consists of a ring that contains the source, a stable set always exists and it is the core of a related situation where one edge has been deleted.

Von Neumann–Morgenstern stable set rationalization of choice functions

Abstract Two scenarios illustrate uses of von Neumann–Morgenstern stable sets in the construction of choice functions. A comparison is made to the construction of choice functions by the selection of maximal elements. A characterization is given of choice functions that are von Neumann–Morgenstern stable set rationalizable by acyclic, asymmetric binary relations. Two examples illustrate the use of the characterization.

Coalition formation and history dependence

Farsighted formulations of coalitional formation, for instance, by Harsanyi and Ray and Vohra, have typically been based on the von Neumann–Morgenstern stable set. These farsighted stable sets use 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. Dutta and Vohra point out that these solution concepts do not require coalitions to make optimal moves. Hence, these solution concepts can yield unreasonable predictions. Dutta and Vohra restricted coalitions to hold common, history‐independent expectations that incorporate optimality regarding the continuation path. This paper extends the Dutta–Vohra analysis by allowing for history‐dependent expectations. The paper provides characterization results for two solution concepts that correspond to two versions of optimality. It demonstrates the power of history dependence by establishing nonemptyness results for all finite games as well as transferable utility partition function games. The paper also provides partial comparisons of the solution concepts to other solutions.

Stable Cores in Information Graph Games

In an information graph situation, some agents that are connected by an undirected graph can share with no cost some information or technology that can also be obtained from a source. If an agent is not connected to an informed player, this agent pays a unitary cost to obtain this technology. A coalitional cost game can be defined from this situation, and the core of this game is known to be non- empty. We prove that the core of an information graph game is a von Neumann-Morgenstern stable set if and only if the graph is cycle- complete, or equivalently if the information graph game is concave. When the graph is not cycle-complete, whether there always exists a stable set is an open question. In this regard, we show that if the information graph consists of a ring that contains the source, then a stable set always exists and it is the core of a related information graph situation where one edge has been deleted.

Minimal Enforceability and Indirect Domination Relations in the Shapley-Scarf Economy

We consider the von Neumann-Morgenstern (vNM) stable sets based on a farsighted version of weak domination relation for the barter model with indivisible goods of Shapley and Scarf (1974). Recent research on farsighted stable sets has focused on coalitional sovereignty, a term used by Ray and Vohra (2015) regarding what coalitions can do to other coalitions, and they note the importance of this issue and how they may affect the results in characteristic function form games. However, defining a farsighted version of indirect weak domination introduces an additional issue of what coalitions can or cannot do from the inside, an issue which needed not be analyzed for the stronger versions of indirect domination nor for myopic domination relations in the literature. We first impose a minimality condition on the coalitions that can deviate in just one step, as coalitions in this model represent trading cycles, where such a minimality condition is often imposed. Under the domain of preferences defined by Klaus, Klijn, and Walzl (2010), each Pareto efficient allocation in the core and its Pareto indifferent allocations constitute a vNM stable set defined by a farsighted weak domination that respects the minimal enforceability condition, and those are the only essentially singleton such vNM stable sets.

A note on the relationship between the core and stable sets in three-sided markets

We analyze the extent to which two known results of the relationship between the core and the stable sets for two-sided assignment games can be extended to three-sided assignment games. We find that the dominant diagonal property is necessary for the core to be a stable set and, likewise, sufficient when each sector of the three-sided market has two agents. Unlike the two-sided case, the union of the extended cores of all the μ-compatible subgames with respect to an optimal matching μ may not be a von Neumann–Morgenstern stable set.

Von Neumann–Morgenstern stable sets of a patent licensing game: The existence proof

This paper provides the existence proof for stable sets of a game which may have empty cores. Given the number of licensees of a patented technology which is determined by the patent holder without any production facilities, a game with a coalition structure is formulated with the outcome expected in the subsequent market competition where any cartels are prohibited. Although the core is non-empty if and only if the grand coalition is formed with a condition, we provide, for each permissible coalition structure, the sufficient condition(s) for the existence of von Neumann–Morgenstern stable sets of the game. Under symmetric imputations, there exist stable sets for any permissible coalition structures, and each of those is completely characterized.

Competition and networks of collaboration

I develop a model of collaboration between tournament participants in which agents collaborate in pairs, and an endogenous structure of collaboration is represented by a weighted network. The agents are forward-looking and capable of coordination; they value collaboration with others and higher tournament rankings. I use von Neumann–Morgenstern stable sets as a solution. I find stable networks in which agents collaborate only within exclusive groups. Both an absence of intergroup collaboration and excessive intragroup collaboration lead to inefficiency. I provide a necessary and sufficient condition for the stability of efficient outcomes in winner-takes-all tournaments. I show that the use of transfers does not repair efficiency.

Online Appendix for 'Competition and Networks of Collaboration'

I develop a model of collaboration between tournament participants in which agents collaborate in pairs, and an endogenous structure of collaboration is represented by a weighted network. The agents are forward-looking and capable of coordination; they value collaboration with others and higher tournament rankings. I use von Neumann-Morgenstern stable sets as a solution. I find stable networks in which agents collaborate only within exclusive groups. Both an absence of intergroup collaboration and excessive intragroup collaboration lead to inefficiency. I provide a necessary and sufficient condition for the stability of efficient outcomes in winner-takes-all tournaments. I show that the use of transfers does not repair efficiency.

Rational expectations and farsighted stability

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

An alternative proof of the characterization of core stability for the assignment game

Solymosi and Raghavan (2001), characterize the stability of the core of the assignment game by means of a property of the valuation matrix. They show that the core of an assignment game is a von Neumann–Morgenstern stable set if and only if its valuation matrix has a dominant diagonal. While their proof makes use of graph-theoretical tools, the alternative proof presented here relies on the notion of the buyer–seller exact representative, as introduced by Núñez and Rafels in 2002.

An Alternative Proof of the Characterization of Core Stability for the Assignment Game

Solymosi and Raghavan (2001) characterize the stability of the core of the assignment game by means of a property of the valuation matrix. They show that the core of an assignment game is a von Neumann-Morgenstern stable set if and only if its valuation matrix has a dominant diagonal. Their proof makes use of some graph-theoretical tools, while the present proof relies on the notion of buyer-seller exact representative in Nunez and Rafels (2002).

Stable sets of allocations and the provision of public goods

In a market model with public projects, we introduce and study von Neumann–Morgenstern stable sets. Stability concepts derive from a procedure in which the blocking coalitions are responsible for a share of the total cost of the public project. This share is determined by a contribution measure. Moreover, the analysis is extended to the case in which traders can join a coalition with a grade of membership. In this framework, under proper assumptions, we provide a correspondence between stable sets of an economy with finitely many types and stable sets of an associated finite economy and prove uniqueness of stable sets when the dominance is defined over a suitable family of coalitions.

Stable sets in one-seller assignment games

We consider von Neumann–Morgenstern stable sets in assignment games with one seller and many buyers. We prove that a set of imputations is a stable set if and only if it is the graph of a certain type of continuous and monotone function. This characterization enables us to interpret the standards of behavior encompassed by the various stable sets as possible outcomes of well-known auction procedures when groups of buyers may form bidder rings. We also show that the union of all stable sets can be described as the union of convex polytopes all of whose vertices are marginal contribution payoff vectors. Consequently, each stable set is contained in the Weber set. The Shapley value, however, typically falls outside the union of all stable sets.

Characterization of the von Neumann–Morgenstern stable set in a non-cooperative model of dynamic policy-making with a persistent agenda setter

We generalize the dynamic bargaining game of Diermeier and Fong (2011) to arbitrary quota rules to provide a non-cooperative characterization of the von Neumann–Morgenstern stable set. Assuming that players are sufficiently patient and have strict preferences, a pure-strategy stationary equilibrium exists and in any such equilibrium, a policy is a steady state if and only if it is in the stable set. The result provides a non-cooperative foundation of stable sets in settings with dynamic policy-making and a persistent agenda setter.

Von Neumann-Morgenstern Farsightedly Stable Sets in Two-Sided Matching

We adopt the notion of von Neumann-Morgenstern farsightedly stable sets to predict which matchings are possibly stable when agents are farsighted in one-to-one matching problems. We provide the characterization of von Neumann-Morgenstern farsightedly stable sets: a set of matchings is a von Neumann-Morgenstern farsightedly stable set if and only if it is a singleton set and its element is a corewise stable matching. Thus, contrary to the von Neumann-Morgenstern (myopically) stable sets, von Neumann-Morgenstern farsightedly stable sets cannot include matchings that are not corewise stable ones. Moreover, we show that our main result is robust to many- to-one matching problems with responsive preferences.

A new old solution for weak tournaments

This note uncovers new properties of the von Neumann-Morgenstern solution in weak tournaments and majoritarian games. We propose a new procedure for the construction of choice sets from weak tournaments, based on stability criteria. The idea is to analyze versions of the tournament game introduced by Laffond, Laslier and Le Breton (1993) [The bipartisan set of a tournament game. Games and Economic Behavior 5, 182-201]. The exploration of a specific class of Markov perfect equilibria in these dynamic tournament games yields a new solution concept for weak tournaments - the A-stable set. The alternatives in an A-stable set constitute persistent, long-run policy outcomes in the corresponding tournament games. We find that, in any weak tournament, the class of A-stable sets coincides with that of von Neumann-Morgenstern stable sets.

An existence result for farsighted stable sets of games in characteristic function form

In this paper we show that every finite-player game in characteristic function form (not necessarily with side payments) obeying an innocuous condition (that the set of individually rational pay-off vectors is bounded) possesses a farsighted von-Neumann–Morgenstern stable set.

Farsighted house allocation

In this note we study von Neumann–Morgenstern farsightedly stable sets for Shapley and Scarf (1974) housing markets. Kawasaki (2010) shows that the set of competitive allocations coincides with the unique von Neumann–Morgenstern stable set based on a farsighted version of antisymmetric weak dominance (cf., Wako, 1999). We demonstrate that the set of competitive allocations also coincides with the unique von Neumann–Morgenstern stable set based on a farsighted version of strong dominance (cf., Roth and Postlewaite, 1977) if no individual is indifferent between his endowment and the endowment of someone else.

Noncooperative foundations of stable sets in voting games

This short note studies the noncooperative foundations of von Neumann–Morgenstern stable sets in voting games. To do so, we study stationary Markov equilibria (SMEs) of a noncooperative legislative bargaining game, based on underlying simple games. The following result emerges from such an exercise: Every stable set of the underlying simple game is the limit set of pure strategy, stage-undominated SMEs of the bargaining game when voters are sufficiently farsighted.