Human strategic interaction requires reasoning about other people's behavior and mental states, combined with an understanding of their incentives. However, the ontogenic development of strategic reasoning is not well understood: At what age do we show a capacity for sophisticated play in social interactions? Several lines of inquiry suggest an important role for recursive thinking (RT) and theory of mind (ToM), but these capacities leave out the strategic element. We posit a strategic theory of mind (SToM) integrating ToM and RT with reasoning about incentives of all players. We investigated SToM in 3-to 9-y-old children and adults in two games that represent prevalent aspects of social interaction. Children anticipate deceptive and competitive moves from the other player and play both games in a strategically sophisticated manner by 7 y of age. One game has a pure strategy Nash equilibrium: In this game, children achieve equilibrium play by the age of 7 y on the first move. In the other game, with a single mixedstrategy equilibrium, children's behavior moved toward the equilibrium with experience. These two results also correspond to two ways in which children's behavior resembles adult behavior in the same games. In both games, children's behavior becomes more strategically sophisticated with age on the first move. Beyond the age of 7 y, children begin to think about strategic interaction not myopically, but in a farsighted way, possibly with a view to cooperating and capitalizing on mutual gains in long-run relationships.practical reasoning | child development | game theory S trategic environments determine outcomes as a function of the decision of many players. Behavior in these environments is directed by a capacity we term strategic theory of mind (SToM), the capacity to infer other people's mental processes and predict their behavior on the basis of knowledge of their incentives and assumption of their rationality. SToM requires two more primitive capacities. The first is "ordinary" theory of mind (ToM), which is a person's ability to "impute mental states to himself and others" (1). The second is recursive thinking (RT), which is the ability to use the output of one step of a reasoning process as input to a following step. In addition, SToM requires that agents reason about the incentives of all involved.To take an example of the combination of ToM, RT, and reasoning about incentives that will be relevant in the context of this study, suppose that Ann and Bob play a game with rules known to both players, where the rules imply that Ann has an incentive to lie to Bob when Bob believes her. Ann can conclude that she should lie to Bob. Similarly, Bob can conclude that Ann has an incentive to lie to him, and hence he will not believe her. However, in the same manner, Ann concludes that Bob will perform exactly the same reasoning, so that Bob will be skeptical in view of Ann's incentive to lie. Ann will use this output of her reasoning about Bob's incentives as input for her next step of reasoning, and conclu...

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A speaker attempts to persuade a listener to accept a request by presenting evidence. A persuasion rule specifies what evidence is persuasive. This paper compares static and dynamic rules. We present a single linear program (i) whose solution corresponds to the listener's optimal dynamic rule and (ii) whose solution with additional integer constraints corresponds to the optimal static rule. We present a condition-foresight-under which the optimal persuasion problem reduces to the classical maximum flow problem. This has various qualitative consequences, including the coincidence of optimal dynamic and static persuasion rules, elimination of the need for randomization, and symmetry of optimal static rules. . I am grateful to Rakesh Vohra for suggesting the relation of the persuasion problem to the set cover problem. I thank Bart Lipman and two anonymous referees for their helpful comments. Shihui Ma, Nyalleng Moorosi, and Benjamin Segal provided valuable research assistance. I thank Kemal Badur for helping to construct the figures for this paper. A grant-in-aid and single semester leave from the University of Minnesota are gratefully acknowledged. All errors are mine.

We study a seller's optimal mechanism for maximizing revenue when a buyer may present evidence relevant to her value. We show that a condition very close to transparency of buyer segments is necessary and sufficient for the optimal mechanism to be deterministic-hence, akin to classic third degree price discrimination-independently of nonevidence characteristics. We also find another sufficient condition depending on both evidence and valuations, whose content is that evidence is hierarchical. When these conditions are violated, the optimal mechanism contains a mixture of second and third degree price discrimination, where the former is implemented via sale of lotteries. We interpret such randomization in terms of the probability of negotiation breakdown in a bargaining protocol whose sequential equilibrium implements the optimal mechanism.

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