Dynamic Games – DEBRAJ RAY

Social Planning for Hedonistic Altruists

(with Carlo Andreatta), October 2026.

Summary. We study the welfare consequences of social planning for citizens who display hedonistic altruism towards the next generation. A social planner must effectively be more patient than its citizens, even if every citizen has the same discount factor, but the planner also exhibits time inconsistency and has systematic future bias. We presume that successive generations of planners cannot bind one another to a course of action. Our main result states that every equilibrium policy with positive consumption that is nondecreasing in initial output must be either fully Pareto optimal, or “consumption inefficient” in the sense of Malinvaud (1953) and Phelps (1962, 1965). Moreover, we provide sufficient conditions for either outcome to occur.

Past and Future: Backward and Forward Discounting

(with Nikhil Vellodi and Ruqu Wang), Journal of the European Economic Association 22 (2), 837–875, 2024. Online Appendix. This paper has been around for many years… here is an older version with Ruqu.

Summary. We study a model of time preferences in which agents discount both past and future payoffs to obtain their lifetime felicity. Agents derive utility from their current lifetime felicity, as well the anticipated felicity of a distinguished future self. These postulates permit an agent to anticipate future regret in current decisions, and generate a set of novel testable implications in line with empirical evidence.

The Social Equilibrium of Relational Arrangements

(with Parikshit Ghosh), forthcoming, Journal of Institutional and Theoretical Economics, Special Issue on Relational Contracts.

Summary. Building on Ghosh and Ray (1996), we study norms within partnerships that exhibit gradually increasing cooperation, thus serving to deter deviations. But socially beneficial gradualism may be undermined by partners renegotiating to greater cooperation from the outset. We show that incomplete information regard- ing partner patience ameliorates this tension even as it adds to the anonymity of the environment.

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.

Poverty and Self-Control

(with Doug Bernheim and Sevin Yeltekin), Econometrica 83 (5), 1877-1911, 2015. Online Appendix. A link to the 1999 version, which only had numerical results.

Summary. Poverty can perpetuate itself by undermining the capacity for self-control. Our main result demonstrates that low initial assets can limit self-control, trapping people in poverty, while those with high initial assets can accumulate indefinitely.

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.

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.

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.

Contracts and Externalities: How Things Fall Apart

(with Garance Genicot), Journal of Economic Theory 131, 71-100, 2006.

Summary. A single principal interacts with several agents, offering them contracts. The outside-option payoffs of the agents depend positively on how many uncontracted or “free” agents there are. We study how such a principal, unwelcome though he may be, approaches the problem of contract provision to agents when coordination failure among the latter group is explicitly ruled out. Agents cannot resist an “invasion” by the principal and hold to their best payoff. It is in this sense that “things [eventually] fall apart”.

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.

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.

The Time Structure of Self-Enforcing Agreements

Econometrica 70, 547–582, 2002.

Summary. A principal and an agent enter into a sequence of agreements. The principal faces an interim participation constraint at each date, but can commit to the current agreement; in contrast, the agent has the opportunity to renege on the current agreement. We show that every constrained efficient sequence must, after a finite number of dates, exhibit a continuation that maximizes the agent’s payoff over all such sequences.

Reinforcement Learning in Repeated Interaction Games

(with Jon Bendor and Dilip Mookherjee), Advances in Theoretical Economics 1, Issue 1, Article 3. Additional notes on extending the model to the probabilistic choice framework of Luce.

Summary. We study long run implications of reinforcement learning when two players repeatedly interact with one another over multiple rounds to play a finite action game. Within each round, the players play the game many successive times with a fixed set of aspirations used to evaluate payoff experiences as successes or failures. The probability weight on successful actions is increased, while failures result in players trying alternative actions in subsequent rounds. The learning rule is supplemented by small amounts of inertia and random perturbations to the states of players. Aspirations are adjusted across successive rounds on the basis of the discrepancy between the average payoff and aspirations in the most recently concluded round. We define and characterize pure steady states of this model, and establish convergence to these under appropriate conditions.

Aspiration-Based Reinforcement Learning in Repeated Interaction Games: An Overview

(with Jon Bendor and Dilip Mookherjee), International Game Theory Review 3, 159–174, 2001.

Summary. In models of aspiration-based reinforcement learning, agents adapt by comparing payoffs achieved from actions chosen in the past with an aspiration level. Though such models are well-established in behavioural psychology, only recently have they begun to receive attention in game theory and its applications to economics and politics. This paper provides an informal overview of a range of such theories applied to repeated interaction games.

Evolving Aspirations and Cooperation

(with Rajeeva Karandikar, Dilip Mookherjee, and Fernando Vega-Redondo), Journal of Economic Theory 80, 292-331, 1998.

Summary. A 2×2 game is played repeatedly by two satisficing players. The game considered includes the Prisoner’s Dilemma, as well as games of coordination and common interest. Each player has an aspiration at each date, and takes an action. The action is switched at the subsequent period only if the achieved payoff falls below aspirations; the switching probability depends on the shortfall. Aspirations are periodically updated according to payoff experience, but are occasionally subject to trembles. For sufficiently slow updating of aspirations and small tremble probability, it is shown that both players must ultimately cooperate most of the time.

Internally-Negotiation-Proof Equilibrium Sets: Limit Behavior for Low Discounting

Games and Economic Behavior 6, 162-177, 1994.

Summary. Recent literature in the theory of dynamic games addresses renegotiatioin-proof equilibria, For repeated games, I analyze the limit of renegotiation-proof equilibrium sets as discounting vanishes. The main result states that such limit sets must either be singletons or belong to the Pareto frontier of the convex hull of the feasible set of the stage game payoffs.

Collusive Market Structure Under Learning-by-Doing and Increasing Returns

(with Dilip Mookherjee), Review of Economic Studies 58, 993-1009, 1991.

Summary. Learning-by-doing and increasing returns are often perceived to have similar implications for market structure and conduct. We analyze this assertion in the context of an infinite-horizon, price-setting game.

Collective Dynamic Consistency in Repeated Games

Games and Economic Behavior 1, 295-326, 1989

We formalize the notion of collective dynamic consistency for noncooperative repeated games. Intuitively, we require that an equilibrium not prescribe any course of action in any subgame that players would jointly wish to renegotiate, given the restriction that any alternative must itself be invulnerable to subsequent deviations and renegotiation. While the appropriate definition of collective dynamic consistency is clear for finitely repeated games, serious conceptual difficulties arise when games are infinitely repeated.

Repeated Principal-Agent Games with Eviction

(with Bhaskar Dutta and Kunal Sengupta), in P. Bardhan (ed.), The Economic Theory of Agrarian Institutions, Clarendon Press, Oxford (1989).

Summary. We study repeated principal-agent problems in which the agent can be evicted and replaced by another identical agent. Thus current output, which is perfectly observed, can be used for incentives as well as efficiency wages. We describe conditions under which eviction threats will be used in equilibrium, in addition to output-based incentives.

Markov-Perfect Equilibrium in Altruistic Growth Economies With Production Uncertainty

(with Doug Bernheim), Journal of Economic Theory 47, 195-202, 1989.

Summary. This paper concerns the existence of Markov perfect equilibria in altruistic growth economies. Previous work on deterministic models has established existence only under extremely restrictive conditions. We show that the introduction of production uncertainly yields an existence theorem for aggregative infinite horizon models with very general forms of altruism.

Economic Growth With Intergenerational Altruism

(with Doug Bernheim), Review of Economic Studies 54, 227-243, 1987.

Summary. We consider the properties of equilibrium behavior in an aggregative growth model with intergenerational altruism. Various positive properties such as the cyclicity of equilibrium programs, and the convergence of equilibrium stocks to a steady state, are analyzed. Among other normative properties, it is established that under certain natural conditions, Nash equilibrium programs are efficient and “modified Pareto optimal”, in a sense made clear in the paper, but never Pareto optimal in the traditional sense.

Category: Dynamic Games