We show that in a class of I‐agent mechanism design problems with evidence, commitment is unnecessary, randomization has no value, and robust incentive compatibility has no cost. In particular, for each agent i, we construct a simple disclosure game between the principal and agent i where the equilibrium strategies of the agents in these disclosure games give their equilibrium strategies in the game corresponding to the mechanism but where the principal is not committed to his response. In this equilibrium, the principal obtains the same payoff as in the optimal mechanism with commitment. As an application, we show that certain costly verification models can be characterized using equilibrium analysis of an associated model of evidence.
An agent chooses among projects with random outcomes. His payoff is increasing in the outcome and in an observer's expectation of the outcome. With some probability, the agent will be able to disclose some information about the true outcome to the observer. We show that choice is inefficient in general. We illustrate this point with a characterization of the inefficiencies that result when the agent can perfectly disclose the outcome with some probability and can disclose nothing otherwise as in Dye (1985a). In this case, the agent favours riskier projects even with lower expected returns. On the other hand, if information can also be disclosed by a challenger who prefers lower beliefs of the observer, the chosen project is excessively risky when the agent has better access to information, excessively risk-averse when the challenger has better access, and efficient otherwise. We also characterize the agent's worst-case equilibrium payoff. We give examples of alternative disclosure technologies illustrating other forms the inefficiencies can take. For example, in a two-dimensional setting, we demonstrate a “hitting for the fences” effect where the agent systematically focuses on the “harder” dimension at the expense of success on the easier.
A principal allocates an object to one of I agents. Each agent values receiving the object and has private information regarding the value to the principal of giving it to him. There are no monetary transfers, but the principal can check an agent’s information at a cost. A favored–agent mechanism specifies a value v ∗ and an agent i ∗ . If all agents other than i ∗ report values below v ∗ , then i ∗ receives the good and no one is checked. Otherwise, whoever reports the highest value is checked and receives the good iff her report is confirmed. All optimal mechanisms are essentially randomizations over optimal favored–agent mechanisms.
We extend implementation theory by allowing the social choice function to depend on more than just the profile of preferences of the agents and by allowing agents to support their statements with hard evidence. We show that a simple condition on the evidence structure which is necessary for the implementation of a social choice function f when the preferences of the agents are state independent is also sufficient for implementation for any preferences (including state dependent) if the social planner can perform small monetary transfers beyond those called for by f and there are at least three players. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under an additional boundedness assumption. In both cases, transfers only occur off the equilibrium path. In the special but important case of allocation problems, under weak conditions, f can be implemented in a perfect information game with at least two players and no transfers. In all cases, the use of evidence enables implementation which is robust in the sense that the social planner needs very little information about the preferences, beliefs, and evidence of the agents and the agents need little information about each others’ preferences. Furthermore, our results still hold if evidence can be forged at an arbitrarily small but strictly positive cost. Finally, we relate our results to the classical work of Maskin (1977) and Moore and Repullo (1988) on implementation without evidence.
Consider an exchange economy with asymmetric information. What is the set of outcomes that are consistent with common knowledge of rationality and market clearing? To address this question we define an epistemic model for the economy that provides a complete description not only of the beliefs of each agent on the relationship between states of nature and prices but also of the whole system of interactive beliefs. The main result, Theorem 1, provides a characterization of outcomes that are consistent with common knowledge of rationality and market clearing (henceforth, CKRMC outcomes) in terms of a solution notion – Ex-Post Rationalizability – that is defined directly in terms of the parameters that define the economy. CKRMC manifests several intuitive properties that stand in contrast to the full revelation property of Rational Expectations Equilibrium. In particular, for a robust class of economies: (1) there is a continuum of prices that are consistent with CKRMC in every state of nature, and hence these prices do not reveal the true state, (2) the range of CKRMC outcomes is monotonically decreasing as agents become more informed about the economic fundamentals, and (3) trade is consistent with common knowledge of rationality and market clearing even when there is common knowledge that there are no mutual gains from trade.
The paper studies Bayesian games which are extended by adding pre-play communication. Let G be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q, that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of G, provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in G such that for each type ti of each player i the expected payoff of ti in q is larger than the expected payoff of ti in s. Journal of Economic Literature classification number: C7. Key words: communication, Bayesian games, sequential equilibrium.
Communication in Repeated Games with Costly Monitoring
with Michael Kahneman
We study repeated games with discounting where perfect monitoring is possible, but costly. It is shown that if players can make public announcements, then every payo. vector which is an interior point in the set of feasible and individually rational payo.s can be implemented in a sequential equilibrium of the repeated game when the discount factor is high enough. Thus, e.ciency can be approximated even when the cost of monitoring is high, provided that the discount factor is high enough. Key words: Repeated games, costly monitoring, Communication.
Correlation Without Mediation: Expanding the Set of Equilibria Outcomes by ‘Cheap’ Pre-Play Procedures
Let P be a correlated equilibrium distribution on the set of outcomes of a game G. Can P be implemented by some "cheap" pre-play procedure that does not involve a mediator? It is shown that if there are two distinct Nash equilibrium payoffs for each player in G and if P is rational (that is, consists of probabilities which are rational numbers) and generates for each player i an expected payoff which is above her worst Nash equilibrium payoff, then P can be virtually implemented in a sequential equilibrium of an extended game which is generated by adding a "cheap" pre-play phase.
On the Measurement of Inequality under Uncertainty
with Itzhak Gilboa and David Schmeidler
To take into account both ex ante and ex post inequality considerations, one has to deal with inequality and uncertainty simultaneously. Under certainty, much of the literature has focused on "comonotonically linear" indices: functionals that are linear on cones of income profiles that agree on the social ranking of the individuals. This family generalizes both the Gini index and the egalitarian index(minimal income). However, it does not include functionals such as the average of expected-Gini and Gini-of-expectation. In contrast, the family of min-of-means functionals is rich enough for this purpose.
Rationality, Nash Equilibrium and Backward Induction in Perfect Information Games
We say that a player is certain of an event A if she assigns probability 1 to A. There is common certainty (CC) of A if the event A occurred, each player is certain of A, each player is certain that every other player is certain of A, and so forth. It is shown that in a generic perfect-information game the set of outcomes that are consistent with common certainty of rationality (CCR) at the beginning of the game coincides with the set of outcomes that survive one deletion of weakly dominated strategies and then iterative deletion of strongly dominated strategies. Thus, the backward induction outcome is not the only outcome that is consistent with CCR. In particular, cooperation in Rosenthal's (1981) centipede game, and fighting in Selten's (1978) chainstore game are consistent with CCR at the beginning of the game. Next, it is shown that, if in addition to CCR, there is CC that each player assigns a positive probability to the true strategies and beliefs of the other players, and if there is CC of the support of the beliefs of each player, then the outcome of the game is a Nash equilibrium outcome.
Communication in Repeated Games with Partial Monitoring
with M. Kahneman
The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at least two other players.
Linear Measures, the Gini Index and the Income-Equality Tradeoff
with I. Gilboa
This paper provides an axiomatization of linear inequality measures representing binary relations on the subspace of income profiles having identical total income. Interpreting the binary relation as a policymaker's preference, we extend the axioms to the whole space and find that they characterize linear social evaluation functions. The axiomatization seems to suggest that a policymaker who has a linear measure of inequality on a subspace should have a linear evaluation on the whole space. An extension of the preferences reflected in the Gini index to the whole space is represented by a linear combination of total income and the Gini index. Journal of Economic Literature Classification numbers: D30, D31, D60, D63, D81
On the Relationship Between Mutation Rates and Growth Rates in a Changing Environment
with with E. Dekel and A. Rustichinni
The paper examines the relationship between the mutation rate and the rate of growth of the population when the environment is changing. It is shown that while the global maximum of the growth rate is attained at a non-zero mutation rate, zero is always a local maximum. This suggests that a population with an initially low mutation rate will evolve to a zero mutation rate.
The paper examines the asymptotic behavior of the set of equilibrium payoffs in a repeated game when there are bounds on the complexity of the strategies players may select. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. The main result is that in a zero-sum game, when the size of the automata of both players go together to infinity, the sequence of values converges to the value of the one-shot game. This is true even if the size of the automata of one player is a polynomial of the size of the automata of the other player. The result for the zero-sum games gives an estimate for the general case.