Recent Papers

Nash Bargaining in Coalitional Games, with Debraj Ray, Revised April 2025

Supplementary Notes

Abstract.  We revisit Nash’s axiomatic approach to bargaining when both individuals and coalitions of individuals have outside options. As in Nash, our solution maximizes a (pos- sibly weighted) product of payoffs net of individual disagreements, but coalitional threats appear as conventional constraints that are not netted out. We embed this solution into a setting with cross-coalitional externalities, and develop a “Nash-in-Nash” theory of viable coalitional structures. Every coalition follows its coalitional Nash solution but interacts noncooperatively with other coalitions, leading to a recursive determination of both threats and solutions. We discuss applications to public goods provision, R&D coalitions, and cartels in oligopolistic competition. Finally, for transferable utility characteristic functions, we connect the coalitional Nash solution to a notion of “pragmatic egalitarianism.”

Signaling, Screening, and Core Stability, with Yusuke Kamishiro and Roberto Serrano, Journal of Economic Theory, 213, 1-31 (2023), https://doi.org/10.1016/j.jet.2023.105715.
Abstract. his paper provides a noncooperative approach to core stability in an economy with incomplete information. We study the perfect Bayesian equilibria of an extensive form mechanism that extends the one used by Serrano and Vohra (1997) to implement the core of a complete-information economy. This leads to a version of the core that we refer to as the sequential core, which allows for information to be transmitted among the agents during the process of coalition formation. Such information flows include proposals that can be viewed as signaling devices and/or screening contracts. Equilibrium refinements are then used to provide justifications for the coarse core and the fine core. As a robustness check, we show that the sequential core corresponds to stationary PBE outcomes of an infinite horizon bargaining game.

A Principal-Agent Relationship with No Advantage to Commitment, with Francisco Espinosa and Debraj Ray, Pure and Applied Functional Analysis 6, 1043-1064 (2021)
Abstract.  This paper explores conditions under which the ability to commit to a menu of contracts in a principal-agent relationship creates no additional benefit for the principal, over and above simultaneous interaction without commitment. A central assumption is that the principal's payoff depends only on the payoff to the agent and her type. 

Games of Love and Hate, with Debraj Ray, Journal of Political Economy, 128 (5), 1789-1825 (2020), https://doi.org/10.1086/705552
Abstract. A strategic situation with payoff-based externalities is one in which a player's payoff depends on her own action and others' payoffs. We place restrictions on the resulting interdependent utility system that generate a standard normal form, referred to as a game of love and hate. Our central theorem states that every equilibrium of a game of love and hate is Pareto optimal. While externalities are restricted to flow only through payoffs there are no other constraints: they could be positive or negative, or of varying sign. We examine the philosophical implications of the restrictions that underlie this theorem.

Maximality in the Farsighted Stable Set, with Debraj Ray, Econometrica, 87, 1763-1779 (2019)
Supplementary Material
Previous Version (with additional results, Sept. 2018)
Abstract. 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 b y 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.

Rational Expectations and Farsighted Stability, with Bhaskar Dutta, Theoretical Economics, 12, 1191-1227 (2017)
Abstract. 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. Consequently, when there are multiple continuation paths the farsighted stable set can yield unreasonable predictions. We restrict coalitions to hold common, history independent expectations that incorporate maximality regarding the continuation path. This leads to two related solution 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.

The Farsighted Stable Set, with Debraj Ray, Econometrica, 83, 977-1011 (2015)
Supplementary Material
Abstract. 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 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 co-exist.

Coalition Formation, with Debraj Ray, in Handbook of Game Theory, Vol. 4, Eds., P. Young and S. Zamir, North Holland: Elsevier (2014)
Abstract. This chapter surveys a sizable and growing literature on coalition formation. We refer to theories in which one or more groups of agents (``coalitions") deliberately get together to jointly determine within-group actions, while interacting non-cooperatively across groups. The chapter describes a variety of solution concepts, using an umbrella model that adopts an explicit real-time approach. Players band together, perhaps disband later and re-form in shifting alliances, all the while receiving payoffs at each date according to the coalition structure prevailing at the time. We use this model to nest two broad approaches to coalition formation, one based on cooperative game theory, the other based on noncooperative bargaining. Three themes that receive explicit emphasis are agent farsightedness, the description of equilibrium coalition structures, and the efficiency implications of the various theories.