Martin W. Cripps

Strategic Experimentation with Exponential Bandits

This paper studies a game of strategic experimentation with two-armed bandits whose risky arm might yield a payoff only after some exponentially distributed random time. Because of free-riding, there is an inefficiently low level of experimentation in any equilibrium where the players use stationary Markovian strategies with posterior beliefs as the state variable. After characterizing the unique symmetric Markovian equilibrium of the game, which is in mixed strategies, we construct a variety of pure-strategy equilibria. There is no equilibrium where all players use simple cut-off strategies. Equilibria where players switch finitely often between the roles of experimenter and free-rider all lead to the same pattern of information acquisition; the efficiency of these equilibria depends on the way players share the burden of experimentation among them. In equilibria where players switch roles infinitely often, they can acquire an approximately efficient amount of information, but the rate at which it is acquired still remains inefficient; moreover, the expected payoff of an experimenter exhibits the novel feature that it rises as players become more pessimistic. Finally, over the range of beliefs where players use both arms a positive fraction of the time, the symmetric equilibrium is dominated by any asymmetric one in terms of aggregate payoffs.

The Design of Auctions and Tenders with Quality Thresholds: The Symmetric Case

Three designs of auctions with quality thresholds are considered. The quality threshold is not known with certainty by the bidders. In one scheme, quality plans submitted by potential bidders are evaluated first and price bids for only those plans that have been approved are invited. The second scheme has price bids first to yield a priority for scrutinizing quality. A third has simultaneous decisions on price and quality. The model yields an equivalence result. The use of reserve prices to produce social optimality is then investigated. The theory is used to discuss the recent auction for U.K. television franchises. Copyright 1994 by Royal Economic Society.

Imperfect Monitoring and Impermanent Reputations

We study the long-run sustainability of reputations in games with imperfect public monitoring. It is impossible to maintain a permanent reputation for playing a strategy that does not play an equilibrium of the game without uncertainty about types. Thus, a player cannot indefinitely sustain a reputation for non-credible behavior in the presence of imperfect monitoring.

Reputation with equal discounting in repeated games with strictly conflicting interests

We analyze reputation effects in two-player repeated games of strictly conflicting interests. In such games player 1 has a commitment action such that a best reply to it gives player 1 the highest individually rational payoff and player 2 the minmax payoff. Players have equal discount factors. With positive probability player 1 is a type who chooses the commitment action after every history. We show that player 1’s payoff converges to the maximally feasible payoff when the discount factor converges to one. This contrasts with failures of reputation effects for equal discount factors that have been demonstrated in the literature. Journal of Economic Literature Classification Number: C71.

Correlated equilibria and evolutionary stability

There is a considerable body of work which shows that the set of outcomes from rational play in a game is identical to a set of correlated equilibria. We show that the set of outcomes from evolutionary stable behavior, where nature serves as a correlating device, is identical to the set of strict correlated equilibria. This extends the result that ESS corresponds to strict Nash equilibrium, provided by R. Selten (Math. Social Sci. 5, 1983, 269–363), by allowing for correlation.

Markov bargaining games

I consider an alternating offer bargaining game which is played by a risk neutral buyer and seller, where the value of the good to be traded follows a Markov process. For these games the existence of a perfect equilibrium is proved and the set of equilibrium payoffs and strategies are characterised. The main results are (a) if the buyer is less patient than the seller, then there will be delays in the players reaching an agreement, the buyer is forced into a suboptimal consumption policy and the equilibrium is ex-ante inefficient, and (b) if the buyer is more patient than the seller, then there is a unique and efficient equilibrium where agreement is immediate.

Disappearing Private Reputations in Long-Run Relationships, Second Version

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps, Mailath, and Samuelson (2004) showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed players reputation is private. We also show that the rate at which reputations disappear is uniform across equilibria and that reputations disappear in sufficiently long discounted finitely-repeated games.

Imperfect Monitoring and Impermanent Reputations, Second Version

We study the long-run sustainability of reputations in games with imperfect public monitoring. It is impossible to maintain a permanent reputation for playing a strategy that does not play an equilibrium of the game without uncertainty about types. Thus, a player cannot indefinitely sustain a reputation for non-credible behavior in the presence of imperfect monitoring.

Optimal Complementary Auctions

This paper considers the situation where two products are sold by the same seller, but to disjoint sets of potential buyers. Externalities may arise from each market outcome to the other. The paper examines the nature of the sellers optimal mechanism, and, for example in the case of positive externalities, it is shown that the allocation decision in either market depends on the highest types in both markets. The optimal mechanism can be implemented by an indirect mechanism that essentially charges winning bidders for the value of their externalities. The analysis is applied to the sale of public sector franchises including exploration and development rights for oil and gas tracts. Copyright 2001 by Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research

Capital Market Risk and the Dynamics of the Income Distribution

This paper introduces a dynamic model of the wealth distribution with aggregate risk in the capital market; the model combines credit rationing and portfolio selection decisions. In a closed economy the long-run behaviour of wealth is independent of the initial income distribution when there is aggregate uncertainty, although further restrictions are necessary when there is no aggregate uncertainty. There can be credit rationing at the long-run equilibrium. In poor economies aggregate risk in the capital market slows growth, whereas in richer economies a risky capital market is good for income growth.