We survey theoretical work on the use of evidence, including work in game theory and in mechanism design.

We explore two interrelated models of "hard information." In the evidence-acquisition model, an agent with private information searches for evidence to show the principal about her type. In the signal-choice model, a privately informed agent chooses an action generating a random signal whose realization may be correlated with her type. The signal-choice model is a special case, and as we show, under certain conditions, a reduced form of the evidence-acquisition model. We develop tools for characterizing optimal mechanisms for these models by giving conditions under which some aspects of the principal's optimal choices can be identified only from the information structure, without regard to the utility functions or the principal's priors.

We develop a theory of monotone comparative statics for models with adjustment costs. We show that comparative-statics conclusions may be drawn under the usual ordinal complementarity assumptions on the objective function, assuming very little about costs: only a mild monotonicity condition is required. We use this insight to prove a general Le Chatelier principle: under the ordinal complementarity assumptions, if short-run adjustment is subject to a monotone cost, then the long-run response to a shock is greater than the short-run response. We extend these results to a fully dynamic model of adjustment over time: the Le Chatelier principle remains valid, and under slightly stronger assumptions, optimal adjustment follows a monotone path. We apply our results to models of saving, production, pricing, labor supply and investment.

The COVID-19 pandemic demonstrated that the process of global vaccination against a novel virus can be a prolonged one. Social distancing measures, that are initially adopted to control the pandemic, are gradually relaxed as vaccination progresses and population immunity increases. The result is a prolonged period of high disease prevalence combined with a fitness advantage for vaccine-resistant variants, which together lead to a considerably increased probability for vaccine escape. A spatial vaccination strategy is proposed that has the potential to dramatically reduce this risk. Rather than dispersing the vaccination effort evenly throughout a country, distinct geographic regions of the country are sequentially vaccinated, quickly bringing each to effective herd immunity. Regions with high vaccination rates will then have low infection rates and vice versa. Since people primarily interact within their own region, spatial vaccination reduces the number of encounters between infected individuals (the source of mutations) and vaccinated individuals (who facilitate the spread of vaccine-resistant strains). Thus, spatial vaccination may help mitigate the global risk of vaccine-resistant variants.

We study statistical discrimination in a marriage market where agents, characterized by attractiveness (e

Two players with common interests exchange information to make a decision. But they fear scrutiny. Their unencrypted communications will be observed by another agent with different interests who can object to their decision. We show how the players can implement their ideal decision rule using a back and forth conversation. Such a subversive conversation reveals enough information for the players to determine their best decision but not enough information for the observer to determine whether the decision was against his interest. Our results show how conversations can maintain deniability even in the face of leaks, hacks

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.

We explore two highly interrelated models of “hard information.” In the evidence– acquisition model, an agent with private information searches for evidence to show to the principal about her type. In the signal–choice model, a privately informed agent chooses an action which generates a random signal whose realization may be correlated with her type. We show that the signal–choice model is a special case and, under certain conditions, a reduced form of the evidence–acquisition model. We develop tools for characterizing optimal mechanisms for these models by giving conditions under which some aspects of the principal’s optimal choices can be identified only from the information structure, without regard to the utility functions or the principal’s priors. We also give a novel result on conditions under which there is no value to commitment for the principal.

This paper considers a class of two-player symmetric games of incomplete information with strategic substitutes. First, we provide sufficient conditions under which there is either a unique equilibrium which is stable (in the sense of best-reply dynamics) and symmetric or a unique (up to permutations) asymmetric equilibrium that is stable (together with an unstable symmetric equilibrium). Thus, (i) there is always a unique stable equilibrium, (ii) it is either symmetric or asymmetric, and hence, (iii) a very simple local condition—stability of the symmetric equilibrium (i.e., the slope of the best-response function at the symmetric equilibrium)—identifies which case applies. Using this, we provide a very simple sufficient condition on primitives for when the unique stable equilibrium is asymmetric (and similarly for when it is symmetric). Finally, we show that the conditions guaranteeing the uniqueness described above also yield novel comparative statics results for this class of games.

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.

Forecasters’ predictions are routinely evaluated using statistical tests. This creates an incentive for forecasters to strategically distort their predictions in an effort to appear knowledgeable. A growing literature has shown the manipulative effects of these incentives: most known statistical tests are ineffective at discriminating between forecasters who are knowledgeable and forecasters who are uninformed but strategic. This adverse selection problem can be solved by restricting the domain of permissible forecasts. The starting point of this paper is a characterization providing sufficient and necessary conditions under which a domain of permissible forecasts can be effectively tested. These conditions have a natural Bayesian interpretation. The result is then applied to show that it is without loss of generality to restrict the attention to simple likelihood-ratio tests, to study maximal domains, and to provide conditions under which testing is robust to imperfect information and misspecification. The paper illustrates a novel connection between the problem of testing strategic forecasters, Bayesian learning, and the classical Neyman-Pearson paradigm. ⇤E-mail: luciano@caltech.edu Division of the Humanities and Social Sciences, Caltech, Pasadena, CA, 91125. I am grateful to Nabil Al-Najjar, Eddie Dekel, Federico Echenique, Johannes Horner, Nicolas Lambert, Larry Samuelson, Alvaro Sandroni and Max Stinchcombe for their helpful comments, and to the audiences at Yale, the University of Texas at Austin, and the 5th World Congress of the Game Theory Society. I thank the Cowles Foundation for Research in Economics, where part of this research was completed, for its support and hospitality. 1

Foundations for iterated admissibility (i.e., the iterated removal of weakly dominated strategies) need to confront a fundamental challenge. On the one hand, admissibility requires that a player consider every strategy of their opponents possible. On the other hand, reasoning that the opponents are rational requires ruling out certain strategies. Brandenburger, Friedenberg, Keisler's (BFK, Econometrica, 2008) foundations for iterated admissibility address this challenge with two ingredients: lexicographic beliefs and the concept of "assumption." However, BFK restrict attention to lexicographic beliefs whose supports are essentially disjoint. This restriction does not have a compelling behavioral rationale, or a clear intuitive interpretation. At the same time, it plays a crucial role in BFK's foundations for iterated admissibility—specifically, in their analysis of assumption. We provide an alternate characterization of assumption, which applies to all lexicographic beliefs. We also characterize two variants of assumption, based on two extensions of 'weak dominance' to infinite state spaces. These notions of assumption coincide with BFK's notion when the state space is finite and lexicographic beliefs have disjoint support; but they are different in more general settings. Leveraging these characterization results, we show that disjoint supports do not play a role in the foundations for iterated admissibility.

Epistemic game theory formalizes assumptions about rationality and mutual beliefs in a formal language, then studies their behavioral implications in games. Specifically, it asks: what do different notions of rationality and different assumptions about what players believe about…what others believe about the rationality of players imply regarding play in a game? Being explicit about these assumptions can be important, because solution concepts are often motivated intuitively in terms of players’ beliefs and their rationality; however, the epistemic analysis may show limitations in these intuitions, reveal what additional assumptions are hidden in the informal arguments, clarify the concepts or show how the intuitions can be generalized. A further premise of this chapter is that the primitives of the model— namely, the hierarchies of beliefs—should be elicitable, at least in principle. Building upon explicit assumptions about elicitable primitives, we present classical and recent developments in epistemic game theory and provide characterizations of a nonexhaustive, but wide, range of solution concepts.

Preliminaries Fix a finite type structure T = µ i) i∈I be a type structure that admits µ as a common prior and such that T µ i ⊆ T i for every i. Fix a player i and a type profile t * ∈ T µ. Define

In this Appendix, we show that we can reduce the principal’s problem to the choice of (p,q) functions as in the text. We begin with an arbitrary mechanism which could have multiple stages of cheap talk statements by the agents and checking by the principal, where who can speak and which agents are checked depend on past statements and the results from past checks, finally culminating in the allocation of the good, perhaps to no one. Think of such a dynamic mechanism as a game in extensive form between the agents and the principal where the principal is committed in advance to his strategy. The principal’s actions specify decisions about which agent or agents to check at various points and, ultimately, which (if any) to allocate the good to. Fix such a dynamic mechanism, deterministic or otherwise, and any equilibrium, say σ, in pure or mixed strategies. We show that the principal’s payoff in this mechanism can be duplicated or improved by the appropriate choice of (p,q) functions.

Under sequential voting, voting late enables conditioning on which candidates are viable, while voting early can influence the field of candidates. But the latter effect can be harmful: shrinking the field increases not only the likelihood that future voters vote for one's favorite candidate, but also that they vote for an opponent. Specifically, if one's favorite candidate is significantly better than all others, then early voting is disadvantageous and all equilibria are equivalent to simultaneous voting. Conversely, when some other candidate is almost as good, then any Markov, symmetric, anonymous equilibrium involves sequential voting (and differs from simultaneous voting).

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 if and only if her report is confirmed. All optimal mechanisms are essentially randomizations over optimal favored-agent mechanisms. (JEL D82)