Coalition Formation

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Category: Coalition Formation

Nash Bargaining with Coalitional Threats

(with Rajiv Vohra). March 2024. Supplementary Notes.

Summary. We axiomatically characterize bargaining outcomes in the presence of coalitional threats. As in Nash’s solution, these involve the product of payoffs net of disagreement points, but coalitional threats appear as conventional constraints, and are not netted out from payoffs as disagreement points are. This basic property is implied by a new “expansion axiom”  that is automatically satisfied in the standard bargaining problem. We then endogenize coalitional threats using internal consistency. For games with convex feasible sets of payoffs, the internally consistent solution coincides with one in which the only threat from each coalition is the use of their “standard” Nash solution, unconstrained by subcoalitions. For transferable-utility games, this observation uncovers a connection between the coalitional bargaining solution and the egalitarian solution of Dutta and Ray (1989, 1991).

Signaling and Discrimination in Collaborative Projects

with Paula Onuchic, American Economic Review 113 (1), 210-252 (2023).

Summary. We propose a model of collaborative work in pairs. Each potential partner draws an idea from a distribution that depends on their unobserved ability. The partners then choose to combine their ideas, or work separately. These decisions are based on the intrinsic value of their projects, but also on signaling payoffs, which depend on the public’s assessment of individual contributions to joint work. We study this equilibrium interaction, and argue equilibria with symmetric collaborative strategies are often fragile, in a sense made precise in the paper. In such cases, asymmetric equilibria exist: upon observing a collaborative outcome, the public ascribes higher credit to one of the partners based on payoff-irrelevant “identities.”

Groups in Conflict: Private and Public Prizes

(with Laura Mayoral),  Journal of Development Economics 154, https://doi.org/10.1016/j.jdeveco.2021.102759 (2022).

 Supplementary Appendix.

Summary. This paper studies costly conflict over private and public goods. Oil is an example of the former, political and civil rights an example of the latter. Our theory predicts that groups in conflict are likely to be small when the prize is private, and large when the prize is public. We examine these implications empirically by constructing a global dataset at the ethnic group level and studying conflict along ethnic lines. Our theoretical predictions find significant confirmation in this setting, and the analysis sheds new light on group size and collective action in the context of violent conflict.

Maximality in the Farsighted Stable Set

(with Rajiv Vohra)   Econometrica 87(5), 1763–1779 Online Appendix.

Summary. The stable set of von Neumann and Morgenstern can be extended to cover farsighted coalitional deviations, as proposed by Harsanyi (1974), and more recently reformulated by Ray and Vohra (2015). However,  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. 

The Farsighted Stable Set

(with Rajiv Vohra), Econometrica  83, 977–1011, 2015. Online Appendix.

Summary. We propose a definition of farsighted stability in coalitional games, in the spirit of von Neumann-Morgenstern stability and its modification by Harsanyi. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single-payoff allocation. We then conduct a comprehensive analysis of the existence and structure of farsighted stable sets in simple games.

Coalition Formation

(with Rajiv Vohra),  in Handbook of Game Theory Vol 4 (H.P. Young and S. Zamir, eds), Elsevier North Holland, 2014.

Summary. 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 noncooperatively 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.

Costly Conflict Under Complete Information

unpublished manuscript, June 2009.

Summary. This paper studies costly conflict in a world of complete information, in which society can commit to divisible transfers among all potentially warring groups. The difficulty in preventing conflict arises from the possibility that there may be several conflictual divisions of society, each based on a different marker, such as class, geography, religion, or ethnicity. It is shown that this diversity of societal markers is particularly conducive to social instability when potential conflict is over private, divisible resources. In contrast, when conflict is over public goods, such diversity promotes social stability.

Informal Insurance in Social Networks

(with Francis Bloch and Garance Genicot), Journal of Economic Theory 143, 36-58, 2008.

Summary. This paper studies bilateral insurance schemes across networks of individuals.  We investigate the structure of self-enforcing insurance networks. Network links play two distinct and possibly conflictual roles. They act as conduits for both transfers and information; affecting the scope for insurance and the severity of punishments upon noncompliance. Their interaction leads to a characterization of stable networks as suitably “sparse” networks. Thickly and thinly connected networks tend to be stable, whereas intermediate degrees of connectedness jeopardize stability.

On the Salience of Ethnic Conflict

(with Joan Esteban), American Economic Review 98:5, 2185–2202, 2008. Online Appendix.

Summary. “In much of Asia and Africa, it is only modest hyperbole to assert that the Marxian prophecy has had an ethnic fulfillment.” — Donald Horowitz (1985).

Coalition Formation with Binding Agreements

(with Kyle Hyndman), Review of Economic Studies 74, 1125–1147, 2007.

Summary. We study coalition formation in “real time”, a situation in which coalition formation is intertwined with the ongoing receipt of payoffs. Agreements are assumed to be permanently binding: They can only be altered with the full consent of existing signatories. For characteristic function games we prove that equilibrium processes—whether or not these are history dependent—must converge to efficient absorbing states. For three-player games with externalities each player has enough veto power that a general efficiency result can be established. However, there exist four-player games in which all Markov equilibria are inefficient from every initial condition, despite the ability to write permanently binding agreements. Online Appendix.

Farsighted Network Formation

(with Bhaskar Dutta and Sayantan Ghosal), Journal of Economic Theory 122, 143 – 164, 2005.

Summary. This paper studies a model of dynamic network formation when individuals are farsighted: players evaluate the desirability of a “current” move in terms of its consequences on the entire discounted stream of payoffs. We define a concept of equilibrium which takes into account farsighted behavior of agents and allows for limited cooperation amongst agents.

Group Formation in Risk-Sharing Arrangements

 (with Garance Genicot), Review of Economic Studies 70, 87-113, 2003.

Summary. We study informal insurance within communities, explicitly recognizing the possibility that subgroups of individuals may destabilize insurance arrangements among the larger group. We therefore consider self-enforcing risk-sharing agreements that are robust not only to single-person deviations but also to potential deviations by subgroups. Variant on an Example in the paper. A conjecture related to the paper.

Coalition Formation as a Dynamic Process

(with Hideo Konishi), Journal of Economic Theory 110, 1–41, 2003.

Summary. We study coalition formation as an ongoing, dynamic process, with payoffs generated as coalitions form, disintegrate, or regroup.

Coalitional Power and Public Goods

(with Rajiv Vohra), Journal of Political Economy 109, 1355-1384, 2001.

Summary. We study the provision of public goods when all agents have complete information and can write binding agreements. The focus is on coalition formation as a potential source of inefficiency.

A Theory of Endogenous Coalition Structures

(with Rajiv Vohra), Games and Economic Behavior 26, 286–336, 1999.

Summary. Consider an environment with widespread externalities, and suppose that binding agreements can be written. We study coalition formation in such a setting. Our analysis proceeds by defining on a partition function an extensive-form bargaining game. We establish the existence of a stationary subgame perfect equilibrium. Our main results are concerned with the characterization of equilibrium coalition structures. We develop an algorithm that generates such a  structure. Our characterization results are especially sharp for symmetric partition functions.

Equilibrium Binding Agreements

(with Rajiv Vohra), Journal of Economic Theory 73, 30-78, 1997.

Summary. We study equilibrium binding agreements, the coalition structures that form under such agreements, and the efficiency of the outcomes that result. We analyze such agreements in a context where the payoff to each player depends on the actions of all other players. Thus a game in strategic form is a natural starting point. Unlike the device of a characteristic function, explicit attention is paid to the behavior of the complementary set of players when a coalition blocks a proposed agreement. A solution concept and its applications are discussed.

A Non-Cooperative Theory of Coalitional Bargaining

(with Kalyan Chatterjee, Bhaskar Dutta and Kunal Sengupta),  Review of Economic Studies 60, 463-477, 1993.

Summary. We explore a sequential-offers model of n-person coalitional bargaining with transferable utility and with time discounting. Our focus is on stationary equilibria of the resulting non-cooperative game. Efficient stationary equilibria converge to a point in the core as the discount factor approaches 1. For strictly convex games, this is the egalitarian solution of Dutta and Ray (Econometrica 1989).

Constrained Egalitarian Allocations

(with Bhaskar Dutta), Games and Economic Behavior 3, 403-422, 1991.

Summary. This paper proposes a constrained egalitarian solution concept for TU games which combines commitment for egalitarianism and promotion of individual interests in a consistent manner. The paper shows that the set of constrained egalitarian allocations is nonempty for weakly superadditive games. The solution is “almost” unique if the desirability relation between players is complete.

Credible Coalitions and the Core

International Journal of Game Theory 18, 185-187, 1989.

Summary. A problem with the concept of the core is that it does not explicitly capture the credibility of blocking coalitions, This notion is defined, and the concept of a modified core introduced, consisting of allocations not blocked by any credible coalition. The core and modified core are then shown to be identical. The concept of credibility is thus implicit in the definition of the core.