We characterize optimal selling protocols/equilibria of a game in which an Agent first puts hidden effort to acquire information and then transacts with a Firm that uses this information to take a decision. We determine the equilibrium payoffs that maximize incentives to acquire information. Our analysis is similar to finding ex ante optimal self-enforcing contracts since information sharing, outcomes and transfers cannot be contracted upon. We show when and how selling and transmitting information gradually helps. We also show how mixing/side bets increases the Agent’s incentives.
An Agent who owns information that is potentially valuable to a Firm bargains for its sale, without commitment and certification possibilities, short of disclosing it. We propose a model of gradual persuasion and show how gradualism helps mitigate the hold-up problem (that the Firm would not pay once it learns the information). An example illustrates how it is optimal to give away part of the information at the beginning of the bargaining, and sell the remainder in dribs and drabs. The Agent can only appropriate part of the value of information. Introducing a third-party allows her to extract the maximum surplus.
We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to one for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players’) equilibrium payoffs is independent of the state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period’s state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.
We characterize belief-free equilibria in infinitely repeated games with incomplete information with N > 2 players and arbitrary information structures. This characterization involves a new type of individual rational constraint linking the lowest equilibrium payoffs across players. The characterization is tight: we define a set of payoffs that contains all the belief-free equilibrium payoffs; conversely, any point in the interior of this set is a belief-free equilibrium payoff vector when players are sufficiently patient. Further, we provide necessary conditions and sufficient conditions on the information structure for this set to be non-empty, both for the case of known-own payoffs, and for arbitrary payoffs.
This paper examines social learning when only one of the two types of decisions is observable. Because agents arrive randomly over time, and only those who invest are observed, later agents face a more complicated inference problem than in the standard model, as the absence of investment might reflect either a choice not to invest, or a lack of arrivals. We show that, as in the standard model, learning is complete if and only if signals are unbounded. If signals are bounded, cascades may occur, and whether they are more or less likely than in the standard model depends on a property of the signal distribution. If the hazard ratio of the distributions increases in the signal, it is more likely that no one invests in the standard model than in this one, and welfare is higher. Conclusions are reversed if the hazard ratio is decreasing. The monotonicity of the hazard ratio is the condition that guarantees the presence or absence of informational cascades in the standard herding model.
We apply the average cost optimality equation to zero-sum Markov games, by considering a simple game with one-sided incomplete information that generalizes an example of Aumann and Maschler (1995). We determine the value and identify the optimal strategies for a range of parameters.
We examine a repeated interaction between an agent, who undertakes experiments, and a principal who provides the requisite funding for these experiments. The agent’s actions are hidden, and the principal, who makes the offers, cannot commit to future actions. We identify the unique Markovian equilibrium (whose structure depends on the parameters) and characterize the set of all equilibrium payoffs, uncovering a collection of non-Markovian equilibria that can Pareto dominate and reverse the qualitative properties of the Markovian equilibrium. The prospect of lucrative continuation payoffs makes it more expensive for the principal to incentivize the agent, giving rise to a dynamic agency cost. As a result, constrained efficient equilibrium outcomes call for nonstationary outcomes that front-load the agent’s effort and that either attenuate or terminate the relationship inefficiently early.
We examine a repeated interaction between an agent, who undertakes experiments, and a principal who provides the requisite funding for these experiments. The repeated interaction gives rise to a dynamic agency cost — the more lucrative is the agent’s stream of future rents following a failure, the more costly are current incentives for the agent, giving the principal an incentive to reduce the continuation value of the project. We characterize the set of recursive Markov equilibria. We also find that there are non-Markov equilibria that make the principal better off than the recursive Markov equilibrium, and that may make both agents better off. Efficient equilibria front-load the agent’s effort, inducing as much experimentation as possible over an initial period, until making a switch to the worst possible continuation equilibrium. The initial phase concentrates the agent’s effort near the beginning of the project, where it is most valuable, while the eventual switch to the worst continuation equilibrium attenuates the dynamic agency cost.
We examine a repeated interaction between an agent, who undertakes experiments, and a principal who provides the requisite funding for these experiments. The agent’s actions are hidden, and the principal, who makes the offers, cannot commit to future actions. We identify the unique Markovian equilibrium (whose structure depends on the parameters) and characterize the set of all equilibrium payoffs, uncovering a collection of non-Markovian equilibria that can Pareto dominate and reverse the qualitative properties of the Markovian equilibrium. The prospect of lucrative continuation payoffs makes it more expensive for the principal to incentivize the agent, giving rise to a dynamic agency cost. As a result, constrained efficient equilibrium outcomes call for nonstationary outcomes that front-load the agent’s effort and that either attenuate or terminate the relationship inefficiently early.
We examine a repeated interaction between an agent, who undertakes experiments, and a principal who provides the requisite funding for these experiments. The repeated interaction gives rise to a dynamic agency cost — the more lucrative is the agent’s stream of future rents following a failure, the more costly are current incentives for the agent, giving the principal an incentive to reduce the continuation value of the project. We characterize the set of recursive Markov equilibria. We show that there are non-Markov equilibria that make the principal better off than the recursive Markov equilibrium, and that may make both players better off. Efficient equilibria front-load the agent’s effort, inducing as much experimentation as possible over an initial period, until making a switch to the worst possible continuation equilibrium. The initial phase concentrates the agent’s effort near the beginning of the project, where it is most valuable, while the eventual switch to the worst continuation equilibrium attenuates the dynamic agency cost.
This paper examines moral hazard in teams over time. Agents are collectively engaged in an uncertain project, and their individual efforts are unobserved. Free-riding leads not only to a reduction in effort, but also to procrastination. The collaboration dwindles over time, but never ceases as long as the project has not succeeded. In fact, the delay until the project succeeds, if it ever does, increases with the number of agents. We show why deadlines, but not necessarily better monitoring, help to mitigate moral hazard.