Game Theory

Displaying 48 Items

Category: Game Theory

Nash Bargaining in Coalitional Games

(with Rajiv Vohra). April 2025. Supplementary Notes.

Summary. 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 (possibly 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.”

Contracts with Interdependent Preferences

(with Marek Weretka), February 2025.

Summary.  A principal contracts with a team of agents with interdependent preferences. We characterize cost effective contracts, and relate the direction of co-movement in rewards — “joint liability” (positive) or “tournaments” (negative) — to the assumed structure of preference interdependence.  We identify two asymmetries. First, the optimal contract leans towards joint liability rather than tournaments, especially in larger teams, in a sense made precise in the paper. Second, when the mechanism-design problem is augmented by robustness constraints designed to eliminate multiple equilibria, the principal may prefer teams linked via adversarial rather than altruistic preferences.

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.

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.

Conveying Value Via Categories

(with Paula Onuchic), October 2019, revised December 2022. Forthcoming, Theoretical Economics.

A sender is about to come into possession of an object of heterogeneous quality. Prior to knowing that quality, she commits to a categorization. That is, she partitions the set of qualities into  subsets — some possibly singletons — and verifiably commits to reveal the element in which the quality belongs. The categories  must be monotone. Our main results fully describe the profit-maximizing categorization  for any pair of priors over object quality held by sender and receiver. We apply these results to the design of educational grades.

Too Good To Be True? Retention Rules for Noisy Agents

(with Francisco Espinosa), revised January 2022, forthcoming American Economic Journal: Microeconomics. Supplementary Appendix.

Summary. An agent who privately knows his type (good or bad) seeks to be retained by a principal. A principal seeks to retain good agents. Agents signal their type with some ambient noise, but can alter this noise, perhaps at some cost. Our main finding, that we examine in several extensions, is that in equilibrium,  the principal treats extreme signals in either direction with suspicion, and retains the agent if and only if the signal falls in some intermediate bounded set. In short, she follows the maxim: “if it seems too good to be true, it probably is.”  

A Principal-Agent Relationship With No Advantage to Commitment

(with Francisco Espinosa and Rajiv Vohra), in Pure and Applied Functional Analysis 6, 1043-1064 (2021).

Summary. This paper explores conditions under which the ability to commit 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. Dedicated to Ali Khan on the occasion of his 70th birthday.

Games of Love and Hate

(with Rajiv Vohra), Journal of Political Economy 128, 1789-1825 (2020).

Dedicated to Tapan Mitra — advisor, colleague and dear friend, whose sense of aesthetics, minimalism and rigor has been an inspiration to us. Tapan Mitra died on February 3, 2019.

Summary. A strategic situation with payoff-based externalities is one in which a player’s payoff is a function of her own action and the payoffs of other players. Every action profile therefore induces an interdependent utility system. A strategic situation is continuous if each such utility system is continuous. If each utility system is bounded, with a unique payoff solution for every action profile, we call the strategic situation coherent, and if the same condition also applies to every subset of players, we call the situation sub-coherent. A coherent, sub-coherent and continuous situation generates 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, in sharp contrast to the general prevalence of inefficient equilibria in the presence of externalities. 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 further show that our coherence, sub-coherence and continuity requirements are tight.

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. 

Certified Random: A New Order for Co-Authorship

(with Arthur Robson), American Economic Review 108, 489–520 (2018).

Summary. Certified random order (a) distributes the gain from first authorship evenly over the alphabet, (b) allows either author to signal when contributions are extremely unequal, (c) will invade an environment where alphabetical order is dominant, (d) is robust to deviations, (e) may be ex-ante more efficient than alphabetical order, and (f) is no more complex than the existing alphabetical system modified by occasional reversal of name order.

Information and Enforcement in Informal Credit Markets

(with Parikshit Ghosh), Economica 83, 59–90, 2016.

Summary. We study loan enforcement in informal credit markets with multiple lenders but no sharing of credit histories, and derive the dynamics of loan size and interest rates for relational lending. In the presence of a sufficient fraction of ‘natural defaulters’, the rest of the market can be incentivized against default by micro-rationing—sharper credit limits and possibly higher interest rates that serve as gateways into new borrowing relationships. When there are too few natural defaulters in the market, this can be supplemented by macro-rationing—random exclusion of some borrowers. When information collection is endogenized, multiple equilibria may arise. (Published version of unpublished notes from 2001.)

Anatomy of a Contract Change

with Rajshri Jayaraman and Francis de Vericourt, American Economic Review 106, 316-358, 2016. Online Appendix.

Summary. We study a contract change for tea pluckers. Base wages increased while incentive piece rates were lowered or kept unchanged. Yet, in the following month, output increased by 20–80%. This response contradicts the standard model, is only partly explicable by greater supervision, and appears to be “behavioral.” But in subsequent months, the increase is comprehensively reversed. Our findings suggest that behavioral responses may be ephemeral, and should ideally be tracked over an extended period. 

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.

 

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.

A Remark on Color-Blind Affirmative Action

(with Rajiv Sethi), Journal of Public Economic Theory 12, 399-406, 2010.

Summary. Elite educational institutions have turned to criteria that meet diversity goals without being formally contingent on applicant identity. Under weak and generic conditions, such color-blind affirmative action policies must be nonmonotone in student test scores.

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).

Group Decision-Making in the Shadow of Disagreement

(with Kfir Eliaz and Ronny Razin), Journal of Economic Theory 132, 236–273, 2007.

Summary.  A model of group decision-making is studied, in which one of two alternatives must be chosen. Our model is distinguished by three features: private information regarding valuations, differing intensities in preferences, and the option to declare neutrality to avoid disagreement. There is always an equilibrium in which the majority is more aggressive in pushing its alternative, thus enforcing their will via both numbers and voice. However, under general conditions an aggressive minority equilibrium inevitably makes an appearance, provided that the group is large enough. Such equilibria invariably display a “tyranny of the minority”: the increased aggression of the minority always outweighs their smaller number, leading to the minority outcome being implemented with larger probability than the majority alternative.

 

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”.

Bargaining Power and Enforcement in Credit Markets

(with Garance Genicot), Journal of Development Economics 79, 398-412, 2006.

Summary. In a credit market with enforcement constraints, we study the effects of a change in the outside options of a potential defaulter on the terms of the credit contract, as well as on borrower payoffs. The results crucially depend on the allocation of “bargaining power” between the borrower and the lender. We prove that there is a crucial threshold of relative weights such that if the borrower has power that exceeds this threshold, her expected utility must go up whenever her outside options come down. But if the borrower has less power than this threshold, her expected payoff must come down with her outside options.  These disparate findings within a single model permit us to interpret existing literature on credit markets in a unified way.

 

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.

Robert Rosenthal

(with Roy Radner), Journal of Economic Theory y 112, 365–368, 2003.

Summary. Robert Rosenthal died on February 25, 2002, of a sudden heart attack. He was just 58, in the prime of his professional life. He is missed and loved by the many friends, colleagues and students who knew him. Publications of Bob Rosenthal.

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.

Contractual Structure and Wealth Accumulation

(with Dilip Mookherjee), American Economic Review 92, 818–849, 2002. Online Appendix.

Summary. Can historical wealth distributions affect long-run output and inequality despite “rational” saving, convex technology and no externalities? We consider a model of equilibrium short-period financial contracts, where poor agents face credit constraints owing to moral hazard and limited liability. If agents have no bargaining power, poor agents have no incentive to save: poverty traps emerge and agents are polarized into two classes, with no interclass mobility. If instead agents have all the bargaining power, strong saving incentives are generated: the wealth of poor and rich agents alike drift upward indefinitely and “history” does not matter eventually.

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.

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.

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.

Inequality, Control Rights, and Rent Seeking: Sugar Cooperatives in Maharashtra

(with Abhijit Banerjee, Dilip Mookherjee and Kaivan Munshi), Journal of Political Economy 109, 138-190, 2001.

Summary. This paper presents a theory of rent seeking within farmer cooperatives in which inequality of asset ownership affects relative control rights of different groups of members. . Predictions concerning the effect of the distribution of local landownership on sugarcane price, capacity levels, and participation rates of different classes of farmers are confirmed by data from nearly 100 sugar cooperatives in the Indian state of Maharashtra over the period 1971–93.

Conflict and Distribution

(with Joan Esteban), Journal of Economic Theory 87, 379-415, 1999.

Summary. We develop a behavioral model that links the level and pattern of social conflict to the society-wide distribution of individual characteristics. The model can be applied to groups that differ in characteristics such as wealth, ethnicity, religion, and political ideology. We settle questions of existence and uniqueness of conflict equilibrium. Conflict is seen to be closely connected with the bimodality of the underlying distribution of characteristics. However, in general, the conflictdistribution relationship is nonlinear and surprisingly complex. Our results on conflict patterns also throw light on the phenomena of extremism and moderation.

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.

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.

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.

Egalitarianism and Incentives

(with Kaoru Ueda), Journal of Economic Theory 71, 324-348, 1996.

Summary. A group of agents is collectively engaged in a joint productive activity. Each agent supplies an observable input, and output is then collectively shared among the members according a social welfare function. However, individual actions are taken on a selfish basis, and the collective decision is only made after inputs are chosen. This leads to inefficiency. The aim of this paper is to show formally that, contrary to popular belief, the degree of inefficiency decreases in the extent of egalitarianism embodied in the social welfare function.

Labor Tying

(with Anindita Mukherjee), Journal of Development Economics 47, 207-239, 1995.

Summary. The co-existence of seasonal fluctuations in income and imperfect credit markets suggests that tied contracts should dominate rural labor markets. However,  empirical observation from India suggests that this is far from being the case, and indeed, that there is a declining trend in  labor tying. In our model,  casual labor markets are always active despite the presence of  seasonality, and a variety of implications are derived that  link economic growth, changing information flows, and the decline of labor tying over 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.

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).

Game Theory and Economic Applications

Proceedings, New Delhi (edited with Bhaskar Dutta, Dilip Mookherjee, T. Parthasarathy, S. Raghavan, and Stef Tijs), Springer Verlag, 1992.

Summary. Brings together the proceedings of a conference on Game Theory at the Indian Statistical Institute, New Delhi.

 

Wages and Involuntary Unemployment in the Slack Season of a Village Economy

(with Anindita Mukherjee), Journal of Development Economics 37, 227-264, 1992.

Summary. We model slack season wages in a village economy, in the presence of involuntary unemployment. Our model draws its inspiration from sociological notions of ‘everyday peasant resistance’.  A continuum of equilibrium wage configurations is obtained. These configurations, barring one, involve wages exceeding reservation wages, despite the presence of involuntary unemployment.

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.

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.

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.

A Concept of Egalitarianism Under Participation Constraints

(with Bhaskar Dutta), Econometrica 57, 615-635, 1989.

Summary. We introduce a new solution concept for transferable-utility games in characteristic function form, when individuals collectively believe in equality as a desirable social goal, although in their private actions they behave selfishly. This latter consideration implies that an “egalitarian solution” must satisfy core-like participation constraints, while the former implies that such a solution is also a Lorenz-maximal element of the constrained set. Despite the well-known fact that the Lorenz ordering is incomplete, we show that the egalitarian solution is unique whenever it exists.

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.

A Consistent Bargaining Set

(with Bhaskar Dutta, Kunal Sengupta and Rajiv Vohra), Journal of Economic Theory 49, 93-112, 1989.

Summary. Both the core and the bargaining set fail to satisfy a natural requirement of consistency. In excluding imputations to which there exist objections, the core does not assess the “credibility” of such objections. The bargaining set goes a step further. Only objections which have no counter-objections are considered justified. However, the credibility of counter-objections is not similarly assessed. We formulate a notion of a consistent bargaining set in which each objection in a “chain” of objections is tested in precisely the same way as its predecessor. Various properties of the consistent bargaining set are also analyzed.

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.

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.

Nonpaternalistic Intergenerational Altruism

Journal of Economic Theory 41, 112-132, 1987.

Summary. The paper develops a concept of equilibrium behaviour  in a model of nonpaternalistic intergenerational altruism. When each generation’s utility depends on that of at least two successors, equilibria may be inefficient.