# Relational contracts with hidden information and renegotiation

With Alexandra Charbi

**Abstract**: In a repeated principal-agent problem in which the agent has private information about her i.i.d. cost of effort (à la Levin 2003), we analyze relational contracts that the parties can renegotiate in a way that respects their relative bargaining power. On the equilibrium path, the agent’s productive effort is decreasing in her effort cost; in many cases with pooling for low-cost types and separating for high-cost types. We show that if a disagreement arises in a state in which she was to be rewarded, then it is optimal for the agent to destroy surplus, exerting costly effort to hurt the principal. In such an event, her counter-productive effort is optimally constant regardless of her effort cost, the principal does not fire her, and both parties anticipate agreeing to reward the agent in the next period. The welfare of the parties is increasing in the agent’s bargaining power; the principal, if he could choose, would elect to give the agent some bargaining power, but less than would be jointly optimal.

Working paper coming in June 2016

# Is multilateral enforcement vulnerable to bilateral renegotiation?

With S. Nageeb Ali and David Yilin Yang

**Abstract**: In a multilateral enforcement regime, a player who cheats on one partner is punished by many partners. But if partners can renegotiate in private, they can subvert the power of the multilateral punishment. We introduce a new notion of "bilateral renegotiation proofness" that applies to multilateral games with private monitoring. For symmetric networked environments, we characterize an optimal bilateral renegotiation proof equilibrium. While players’ ability to renegotiate bilaterally is indeed socially costly, it is perhaps not as costly as one might expect. In densely connected communities, the proportional cost imposed by bilateral renegotiation declines as the number of participants grows, and vanishes in the limit.

# Ostracism and forgiveness

With S. Nageeb Ali

Accepted for publication in *The American Economic Review*

**Abstract**: Many communities rely upon ostracism to enforce cooperation: if an individual shirks in one relationship, her innocent neighbors share information about her guilt in order to shun her, while continuing to cooperate among themselves. However, a strategic victim may herself prefer to shirk, rather than report others' deviations truthfully. If guilty players are to be permanently ostracized, then such deviations are so tempting that cooperation in any relationship is bounded by what the partners could obtain through bilateral enforcement. Ostracism can improve upon bilateral enforcement if tempered by forgiveness, through which guilty players are eventually readmitted to cooperative society.

Working paper version 2/8/2016

# A contract-theoretic model of conservation agreements

With Heidi Gjertsen, Theodore Groves, Eduard Niesten, Dale Squires, and Joel Watson

**Abstract**: We examine the structure and performance of conservation agreements, which are used across the world to protect natural resources. Key elements of these agreements are: (1) they are ongoing arrangements between a local community and an outside party, typically a non-governmental organization (NGO); (2) they feature payments in exchange for conservation services; (3) the prospects for success depend on the NGO engaging in costly monitoring to detect whether the community is foregoing short-term gains to protect the resource; (4) lacking a strong external enforcement system, they rely on self-enforcement; and (5) the parties have the opportunity to renegotiate at any time. We provide a novel model that contains these ingredients and we apply the model to assess the workings of real conservation agreements, using three case studies as representative examples. We characterize equilibrium play (including how punishments and rewards are structured) and we show how the parties' relative bargaining powers affect their ability to sustain cooperation over time. The model captures important features of real conservation agreements and identifies some of the features required for successful agreements.

# Enforcing cooperation in networked societies

With S. Nageeb Ali

**Abstract**: Which social norms and networks maximize cooperation in bilateral relationships? We study a network of players in which each link is a repeated bilateral partnership with two-sided moral hazard. The obstacle to community enforcement is that each player observes the behavior of her partners in their partnerships with her, but not how they behave in other partnerships. We introduce a new metric for the rate at which information diffuses in a network, which we call viscosity, and show that it provides an operational measure for how conducive a network is to cooperation. We prove that clique networks have the lowest viscosity and that the optimal equilibrium of the clique generates more cooperation and higher average utility than any other equilibrium of any other network. This result offers a strategic foundation for the perspective that tightly knit groups foster the most cooperation.

# A theory of disagreement in repeated games with bargaining

With Joel Watson

Published in *Econometrica*, 81(6):2303–2350, November 2013.

**Abstract**: This paper proposes a new approach to equilibrium selection in repeated games with transfers, supposing that in each period the players bargain over how to play. Although the bargaining phase is cheap talk (following a generalized alternating-offer protocol), sharp predictions arise from three axioms. Two axioms allow the players to meaningfully discuss whether to deviate from their plan; the third embodies a “theory of disagreement”—that play under disagreement should not vary with the manner in which bargaining broke down. Equilibria that satisfy these axioms exist for all discount factors and are simple to construct; all equilibria generate the same welfare. Optimal play under agreement generally requires suboptimal play under disagreement. Whether patient players attain efficiency depends on both the stage game and the bargaining protocol. The theory extends naturally to games with imperfect public monitoring and heterogeneous discount factors, and yields new insights into classic relational contracting questions.

# Robust collusion with private information

Published in *The Review of Economic Studies*, 79(2):778–811, April 2012.

**Abstract**: The game-theoretic literature on collusion has been hard pressed to explain why a cartel should engage in price wars, without resorting to either impatience, symmetry restrictions, inability to communicate, or failure to optimize. This paper introduces a new explanation that relies on none of these assumptions: if the cartel's member firms have private information about their costs, price wars can be optimal in the face of complexity. Specifically, equilibria that are robust to payoff-irrelevant disruptions of the information environment generically cannot attain or approximate efficiency. An optimal robust equilibrium must allocate market shares inefficiently, and may call for price wars under certain conditions. For a two-firm cartel, cost interdependence is a sufficient condition for price wars to arise in an optimal robust equilibrium. That optimal equilibria are inefficient generically applies not only to collusion games, but also to the entire

*separable payoff environment*(Chung & Ely 2006)—a class that includes most typical economic models.

Published article (free access)

# Attainable payoffs in repeated games with interdependent private information

# Efficiency in repeated trade with hidden valuations

With Susan Athey

Published in *Theoretical Economics*, 2(3):299-354, September 2007

**Abstract**: We analyze the extent to which efficient trade is possible in an ongoing relationship between impatient agents with hidden valuations (i.i.d. over time), restricting attention to equilibria that satisfy ex post incentive constraints in each period. With ex ante budget balance, efficient trade can be supported in each period if the discount factor is at least one half. In contrast, when the budget must balance ex post, efficiency is not attainable, and furthermore for a wide range of probability distributions over their valuations, the traders can do no better than employing a posted price mechanism in each period. Between these extremes, we consider a "bank" that allows the traders to accumulate budget imbalances over time, but only within a bounded range. We construct non-stationary equilibria that allow traders to receive payoffs that approach efficiency as their discount factor approaches one, while the bank earns exactly zero expected profits. For some probability distributions there exist equilibria that yield exactly efficient payoffs for the players and zero profits for the bank, but such equilibria require high discount factors.

Published article (free access)

# “Token” equilibria in sensor networks with multiple sponsors

With Sameer Tilak and Tony Fountain

Published in the Proceedings of the Workshop on Stochasticity in Distributed Systems (StoDiS'05), San Jose, CA, December 19, 2005

**Abstract**: When two sponsoring organizations, working towards separate goals, employ wireless sensor networks for a finite period of time, it can be efficiency-enhancing for the sponsors to program their sensors to cooperate. But if each sensor privately knows whether it can provide a favor in any particular period, and the sponsors cannot contract on ex post payments, then no favors are performed in any Nash equilibrium. Allowing the sponsors to contract on ex post payments, we construct equilibria based on the exchange of "tokens" that yield significant cooperation and increase expected sponsor payoffs. Increasing the sponsors' liability is beneficial because it enables them to use more tokens.

Working paper 5/22/2006 (newer and better than the StoDiS version)