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The process of vaccinating the world population against COVID-19 is expected to take well over a year to complete. As vaccination progresses and population immunity increases, a counteracting relaxation of social distancing measures is observed. The result will be a prolonged period of high disease prevalence combined with a fitness advantage for vaccine-resistant variants, implying a considerably increased probability that a resistant variant will spread in the population. In this paper we propose a spatial vaccination strategy that has the potential to dramatically reduce this risk. Instead of spreading the vaccination effort equally 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 with others in their own region, spatial vaccination will reduce the number of encounters between infected people (the source of mutations) and vaccinated people (who facilitate the spread of vaccine-resistant strains). Thus with proper logistic preparations, a spatial vaccination campaign could be highly effective in reducing the global risk of vaccine-resistant variants.

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.

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.

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). (JEL D72)

Consider a principal with a good to allocate among a number of agents, each of whom wants the good. Each agent i knows the value the principal receives if he gives the good to i, but the principal does not. (This value need not coincide with the value to i of getting the good.) The principal can verify the agents’ private information at a cost, but cannot use transfers. There are a number of economic environments of interest that roughly correspond to this scenario; we discuss a few below. How does the principal maximize the expected gain from allocating the good less the costs of verification? We characterize optimal mechanisms for such settings. We construct an optimal mechanism with a particularly simple structure which we call a favored-agent mechanism. There is a threshold value and a favored agent, say i. If each agent other than i reports a value for the good below the threshold, then the good goes to the favored agent and no verification is required. If some agent other than i reports a value above the threshold, then the agent who reports the highest value is checked. This agent receives the good if and only if his claims are verified and the good goes to any other agent otherwise. In addition, we show that every optimal mechanism is essentially a randomization over optimal favored-agent mechanisms. In this sense, we can characterize the

We study the random Strotz model, a version of the Strotz (1955) model with uncertainty about the nature of the temptation that will strike. We show that the random Strotz representation is unique and characterize a comparative notion of “more temptation averse.” Also, we demonstrate an unexpected connection between the random Strotz model and a generalization of the Gul–Pesendorfer (GP) (2001) model of temptation which allows for the temptation to be uncertain and which we call random GP. In particular, a preference over menus has a random GP representation if and only if it also has a representation via a random Strotz model with sufficiently smooth uncertainty about the intensity of temptation. We also show that choices of menus combined with choices from menus can distinguish the random GP and random Strotz models.

We explore how allowing votes to be traded separately of shares may affect the efficiency of corporate control contests. Our basic set-up and the nature of the questions continue the work of Grossman and Hart (1980) , Harris and Raviv (1988) , and Blair, Golbe and Gerard (1989). We consider three cases with respect to the allowable price offers (for shares and for votes when they can be traded separately): unrestricted price offers, quantity-restricted price offers, and price offers contingent on winning. Our main results are characterizations of the equilibria and of the circumstances under which vote buying is harmful. We show that allowing votes to be traded separately of shares results in inefficiencies in all the cases we study. Similarly allowing quantity-restricted offers is also harmful, but allowing conditional offers is not in itself detrimental to efficiency. The paper also makes a methodological contribution to the analysis of takeover games with atomless shareholders. It provides a way of dealing with asymmetric equilibria that must be dealt with for a complete analysis and it proves existence of an equilibrium. Copyright 2012, Oxford University Press.

"My own behaviour baffles me. For I find myself not doing what I really want to do but doing what I really loathe." Saint PaulWhat behaviour can be explained using the hypothesis that the agent faces temptation but is otherwise a "standard rational agent"? In earlier work, Gul and Pesendorfer (2001) use a set betweenness axiom to restrict the set of preferences considered by Dekel, Lipman and Rustichini (2001) to those explainable via temptation. We argue that set betweenness rules out plausible and interesting forms of temptation including some which may be important in applications. We propose a pair of alternative axioms called DFC, desire for commitment, and AIC, approximate improvements are chosen. DFC characterizes temptation as situations in which given any set of alternatives, the agent prefers committing herself to some particular item from the set rather than leaving herself the flexibility of choosing later. AIC is based on the idea that if adding an option to a menu improves the menu, it is because that option is chosen under some circumstances. From this interpretation, the axiom concludes that if an improvement is worse (as a commitment) than some commitment from the menu, then the best commitment from the improved menu is strictly preferred to facing that menu. We show that these axioms characterize a natural generalization of the Gul-Pesendorfer representation. Copyright , Wiley-Blackwell.

We examine the consequences of lobbying and vote buying, assuming this practice were allowed and free of stigma. Two lobbyists compete for the votes of legislators by offering up-front payments to the legislators in exchange for their votes. We analyze how the lobbyists’ budget constraints and legislators’ preferences determinethewinnerandthepayments.Whenlobbyistsarebudgetconstrainedthenthe preferences of all legislators can matter, and a lobbyist’s relative strength increases more steeply with a budget increase than with an increase of equal magnitude to the legislators’ original preferences for this lobbyist’s positio. When lobbyists are not budget constrained then only the preferences ofnearmedian legislators matter andthepreferencesoftheselegislatorsandthebudgetenterequallyindetermining the winner.

We examine the consequences of vote buying, assuming this practice were allowed and free of stigma. Two parties compete in a binary election and may purchase votes in a sequential bidding game via up‐front binding payments and/or campaign promises (platforms) that are contingent on the outcome of the election. We analyze the role of the parties’ and voters’ preferences in determining the winner and the payments to voters.

We endogenize preferences using the “indirect evolutionary approach”. Individuals are randomly matched to play a two-person game. Individual (subjective) preferences determine their behaviour and may differ from the actual (objective) pay-offs that determine fitness. Matched individuals may observe the opponents’ preferences perfectly, not at all, or with some in-between probability. When preferences are observable, a stable outcome must be efficient. When they are not observable, a stable outcome must be a Nash equilibrium and all strict equilibria are stable. We show that, for pure-strategy outcomes, these conclusions are robust to allowing almost perfect, and almost no, observability, with the notable exception that inefficient strict equilibria may fail to be stable with any arbitrarily small degree of observability (despite being stable with no observability).

We endogenize preferences using the “indirect evolutionary approach”. Individuals are randomly matched to play a two-person game. Individual (subjective) preferences determine their behaviour and may differ from the actual (objective) pay-offs that determine fitness. Matched individuals may observe the opponents' preferences perfectly, not at all, or with some in-between probability. When preferences are observable, a stable outcome must be efficient. When they are not observable, a stable outcome must be a Nash equilibrium and all strict equilibria are stable. We show that, for pure-strategy outcomes, these conclusions are robust to allowing almost perfect, and almost no, observability, with the notable exception that inefficient strict equilibria may fail to be stable with any arbitrarily small degree of observability (despite being stable with no observability). Copyright 2007, Wiley-Blackwell.

We propose a method to test a prediction of the distribution of a stochastic process. In a non-Bayesian, non-parametric setting, a predicted distribution is tested using a realization of the stochastic process. A test associates a set of realizations for each predicted distribution, on which the prediction passes, so that if there are no type I errors, a prediction assigns probability 1 to its test set. Nevertheless, these test sets can be "small", in the sense that "most" distributions assign it probability 0, and hence there are "few" type II errors. It is also shown that there exists such a test that cannot be manipulated, in the sense that an uninformed predictor, who is pretending to know the true distribution, is guaranteed to fail on an uncountable number of realizations, no matter what randomized prediction he employs. The notion of a small set we use is category I, described in more detail in the paper. Copyright 2006, Wiley-Blackwell.

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.

Ž. We extend Kreps’ 1979 analysis of preference for flexibility, reinterpreted by Kreps Ž. 1992 as a model of unforeseen contingencies. We enrich the choice set, consequently obtaining uniqueness results that were not possible in Kreps’ model. We consider several representations and allow the agent to prefer commitment in some contingencies. In the representations, the agent acts as if she had coherent beliefs about a set of possible future Ž. ex post preferences, each of which is an expected-utility preference. We show that this set of ex post preferences, called the subjectie state space, is essentially unique given the restriction that all ex post preferences are expected-utility preferences and is minimal even without this restriction. Because the subjective state space is identified, the way ex post utilities are aggregated into an ex ante ranking is also essentially unique. Hence when a representation that is additive across states exists, the additivity is meaningful in the sense that all representations are intrinsically additive. Uniqueness enables us to show that the size of the subjective state space provides a measure of the agent’s uncertainty about future contingencies and that the way the states are aggregated indicates whether these contingencies lead to a desire for flexibility or commitment.

We explore sequential voting in symmetric two-option environments. We show that the (informative) symmetric equilibria of the simultaneous voting game are also equilibria in any sequential voting structure. In unanimity games, (essentially) the whole set of equilibria is the same in all sequential structures. We also explore the relationship between simultaneous and sequential voting in other contexts. We illustrate several instances in which sequential voting does no better at aggregating information than simultaneous voting. The inability of the sequential structure to use additional information in voting models is distinct from that in the herd-cascade literature.

In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. This paper develops a refinement of self-confirming equilibrium in which players use information about opponents' payoffs in forming beliefs about the way that opponents play off of the equilibrium path. We show that this concept is robust to payoff uncertainty. We also discuss its relationship to other concepts and show that it is closely related to assuming almost common certainty of payoffs in an epistemic model with independent beliefs. Journal of Economic Literature Classification Numbers: C72, D84.

A long-standing conjecture is that winner-take-all games such as patent races lead to the survival of risk-takers and the extinction of risk-averters. In many species a winner-take-all game determines the males' right to reproduce, and the same argument suggests that males will evolve to be risk-takers. Psychological and sociological evidence buttresses the argument that males are more risk-taking than females. Using an evolutionary model of preference-formation, we investigate to what extent evolution leads to risk-taking in winner-take-all environments. Journal of Economic Literature Classification Numbers: C7, D8.

This is the third of three volumes containing edited versions of papers and commentaries presented at invited symposium sessions of the Tenth World Congress of the Econometric Society, held in Shanghai in August 2010. The papers summarize and interpret key developments in economics and econometrics, and they discuss future directions for a wide variety of topics, covering both theory and application. Written by the leading specialists in their fields, these volumes provide a unique, accessible survey of progress on the discipline. The first volume primarily addresses economic theory, with specific focuses on nonstandard markets, contracts, decision theory, communication and organizations, epistemics and calibration, and patents.

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. Journal of Economic Literature Classification Number: C72.

Game-theoretic analysis often leads to consideration of an infinite hierarchy of beliefs for each player. Harsanyi suggested that such a hierarchy of beliefs could be summarized in a single entity, called the player′s type . This paper provides an elementary construction, complementary to the construction already given in [J-F. Mertens and S. Zamir, Formulation of Bayesian analysis for games with incomplete information, Int. J. Game Theory 14 (1985), 1–29] of Harsanyi′s notion of a type. It is shown that if a player′s type is coherent then it induces a belief over the types of the other players. Imposing common knowledge of coherency closes the model of beliefs. We go on to discuss the question that often arises as to the sense in which the structure of a game-theoretic model is, or can be assumed to be, common knowledge. Journal of Economic Literature Classification Number: 026.

The iterated deletion of weakly dominated strategies has been advanced as a necessary requirement for “rational” play. However, this requirement relies on the assumption that the players have no doubts about their opponents' payoffs. We show that once such doubts are introduced, all that can be justified by an appeal to rationality is one round of deletion of weakly dominated strategies, followed by iterated deletion of strategies that are strongly dominated. This extends the Fudenberg, Kreps, and Levine (J. Econ. Theory 12 (1988), 354–380) study of the robustness of Nash equilibrium refinements to the robustness of solution concepts based only on rationality. Our results also clarify the relationship between various notions of what it means for payoff uncertainty to be “small.”

We discuss the goals and means of positive decision theory and the implications for how to do decision theory. We argue that the goal of positive economic theory generally is to provide predictions and understanding and that representation theorems and other results of decision theory should be seen as ways to achieve these goals. We also argue that the interpretation of a model is relevant to whether and how we use the model, that psychological considerations are not necessary for useful decision theory but can be helpful, and that nonchoice data, interpreted properly, can be valuable in predicting choice and therefore should not be ignored.

We consider extensions of games where some players have the option of signaling future actions by incurring costs. The main result is that in a class of games, if one player can incur costs, then forwards induction selects her most preferred outcome. Surprisingly, the player does not have to incur any costs to achieve this—the option alone suffices. However, when all players can incur costs, one player's attempt to signal a future action is vulnerable to a counter-signal by the opponent. This vulnerability to counter-signaling distinguishes signaling future actions from signaling types.

Sufficient conditions for the existence of a Nash equilibrium are given when preferences may violate the reduction of compound lotteries assumption (RCLA). Without RCLA decision makers may not be indifferent between compound lotteries which have the same probabilities of final outcomes. Therefore the conditions depend on how players perceive the game—whether they view themselves as moving first or second. We also review conditions under which the equilibria will be dynamically consistent.

This paper develops a decision-theoretic approach to normal-form refinements of Nash equilibrium and provides characterizations of (normal-form) perfect equilibrium and proper equilibrium. The approach relies on a theory of single-person decision-making that is a non-Archimedean version of subjective expected utility theory. Copyright 1991 by The Econometric Society.