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This paper shows that bargainers may reach delayed agreements even in environments where there is no uncertainty about payoffs or feasible actions. Under such conditions, delay may arise when bargainers face direct forms of strategic uncertainty—that is, uncertainty about the opponent's play. The paper restricts the nature of this uncertainty in two important ways. First, it assumes on-path strategic certainty: Bargainers face uncertainty only after surprise moves. Second, it assumes Battigalli and Siniscalchi's (2002) rationality and common strong belief of rationality (RCSBR)—a requirement that bargainers are \"strategically sophisticated.\" The main result characterizes the set of outcomes consistent with on-path strategic certainty and RCSBR. It shows that these assumptions allow for delayed agreement, despite the fact that the bargaining environment is one of complete information. The source of delay is second-order optimism: Bargainers do not put forward \"good\" offers early in the negotiation process because they fear that doing so will cause the other party to become more optimistic about her future prospects.

Epistemic justifications of solution concepts often refer to type structures that are sufficiently rich. One important notion of richness is that of a complete type structure, i.e., a type structure that induces all possible beliefs about types. For instance, it is often said that, in a complete type structure, the set of strategies consistent with rationality and common belief of rationality are the set of strategies that survive iterated dominance. This paper shows that this classic result is false, absent certain topological conditions on the type structure. In particular, it provides an example of a finite game and a complete type structure in which there is no state consistent with rationality and common belief of rationality. This arises because the complete type structure does not induce all hierarchies of beliefs—despite inducing all beliefs about types. This raises the question: Which beliefs does a complete type structure induce? We provide several positive results that speak to that question. However, we also show that, within ZFC, one cannot show that a complete structure induces all second-order beliefs.

Strategic choice data from a carefully chosen set of ring-network games are used to obtain individual-level estimates of higher-order rationality. The experimental design exploits a natural exclusion restriction that is considerably weaker than the assumptions underlying alternative designs in the literature. In our data set, 93 percent of subjects are rational, 71 percent are rational and believe others are rational, 44 percent are rational and hold second-order beliefs that others are rational, and 22 percent are rational and hold at least third-order beliefs that others are rational.

We extend the epistemic analysis of dynamic games of Battigalli and Siniscalchi (J Econ Theory 88:188-230, 1999, J Econ Theory 106:356-391, 2002, Res Econ 61:165-184, 2007) from finite dynamic games to all simple games, that is, finite and infinite-horizon multistage games with finite action sets at nonterminal stages and compact action sets at terminal stages. We prove a generalization of Lubin's (Proc Am Math Soc 43:118-122, 1974) extension result to deal with conditional probability systems and strong belief. With this, we can provide a short proof of the following result: in every simple dynamic game, strong rationalizability characterizes the behavioral implications of rationality and common strong belief in rationality.

Suppose that each player in a game is rational, each player thinks the other players are rational, and so on. Also, suppose that rationality is taken to incorporate an admissibility requirement--that is, the avoidance of weakly dominated strategies. Which strategies can be played? We provide an epistemic framework in which to address this question. Specifically, we formulate conditions of rationality and mth-order assumption of rationality (RmAR) and rationality and common assumption of rationality (RCAR). We show that (i) RCAR is characterized by a solution concept we call a \"self-admissible set\"; (ii) in a \"complete\" type structure, RmAR is characterized by the set of strategies that survive m + 1 rounds of elimination of inadmissible strategies; (iii) under certain conditions, RCAR is impossible in a complete structure.

The epistemic conditions of rationality and mth-order strong belief of rationality (RmSBR; Battigalli and Siniscalchi, 2002) formalize the idea that players engage in contextualized forward-induction reasoning. This paper characterizes the behavior consistent with RmSBR across all type structures. In particular, in a class of generic games, R(m - 1)SBR is characterized by a new solution concept we call an m-best response sequence (m-BRS). Such sequences are an iterative version of extensive-form best response sets (Battigalli and Friedenberg, 2012). The strategies that survive m rounds of extensive-form rationalizability are consistent with an m-BRS, but there are m-BRS's that are disjoint from the former set. As such, there is behavior that is consistent with R(m - 1)SBR but inconsistent with m rounds of extensive-form rationalizability. We use our characterization to draw implications for the interpretation of experimental data. Specifically, we show that the implications are nontrivial in the three-repeated Prisoner's Dilemma and Centipede games.

This paper refines the game theoretic analysis of conversations in Asher et al. (J Philos Logic 46:355-404, 2017) by adding epistemic concepts to make explicit the intuitive idea that conversationalists typically conceive of conversational strategies in a situation of imperfect information. This 'epistemic' turn has important ramifications for linguistic analysis, and we illustrate our approach with a detailed treatment of linguistic examples.

In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models and type spaces. As shown in many papers (see Aumann and Brandenburger in Econometrica 36: 1161-1180,1995; Baltag et al. in Synthese 169: 301-333, 2009; Battigalli and Bonanno in Res Econ 53(2): 149-225,1999; Battigalli and Siniscalchi in J Econ Theory 106: 356-391,2002; Klein and Pacuit in Stud Log 102: 297-319, 2014; Lorini in J Philos Log 42(6): 863-904, 2013), both these frameworks can be used to express epistemic conditions for solution concepts in game theory. The main result of this paper is a formal comparison between the two and a statement of semantic equivalence with respect to two different logical systems: a doxastic logic for belief and an epistemic-doxastic logic for belief and knowledge. Moreover, a sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided. Finally, a probabilistic extension of the result is also presented. A further result of the paper is a study of the relationship between the epistemic-doxastic logic for belief and knowledge and the logic STIT (the logic of \"seeing to it that\") by Belnap and colleagues (Facing the future: agents and choices in our indeterminist world, 2001).

In this paper we focus on stochastic games with finitely many states and actions. For this setting we study the epistemic concept of common belief in future rationality, which is based on the condition that players always believe that their opponents will choose rationally in the future. We distinguish two different versions of the concept—one for the discounted case with a fixed discount factor \\[\\delta ,\\] and one for the case of uniform optimality, where optimality is required for all discount factors close enough to 1” . We show that both versions of common belief in future rationality are always possible in every stochastic game, and always allow for stationary optimal strategies. That is, for both versions we can always find belief hierarchies that express common belief in future rationality, and that have stationary optimal strategies. We also provide an epistemic characterization of subgame perfect equilibrium for two-player stochastic games, showing that it is equivalent to mutual belief in future rationality together with some “correct beliefs assumption”.

We characterize three interrelated solution concepts in epistemic game theory: permissibility, proper rationalizability, and iterated admissibility. We define the lexicographic epistemic model in a framework with incomplete information. Based on it, we give two groups of conditions; one characterizes permissibility and proper rationalizability, the other characterizes permissibility in an alternative way and iterated admissibility. In each group, the conditions for the latter are stronger than those for the former, which corresponds to the fact that proper rationalizability and iterated admissibility are incomparable but compatible refinements of permissibility within the complete information framework. The essential difference between the two groups is whether a full belief of rationality is needed.