Abstract
The main purpose of this paper is to present an analytical framework that can be used to study rationalizable strategic behavior in general situations—i.e., arbitrary strategic games with various modes of behavior. We show that, under mild conditions, the notion of rationalizability defined in general situations has nice properties similar to those in finite games. The major features of this paper are (1) our approach does not require any kind of technical assumptions on the structure of the game, and (2) the analytical framework provides a unified treatment of players’ general preferences, including expected utility as a special case. In this paper, we also investigate the relationship between rationalizability and Nash equilibrium in general games.
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Notes
Bernheim (1984, Proposition 3.2) and Tan and Werlang (1988) studied the properties of rationalizable strategies in compact (metric) and continuous games. There are also a few exceptional examples on infinite games such as Arieli’s (2010) analysis of rationalizability in continuous games where every player’s strategy set is a Polish space and the payoff function of each player is bounded and continuous and Zimper’s (2006) discussions on a variant of “strong point-rationalizability” in games with metrizable strategy sets. See also Jara-Moroni (2012) and Yu (2010, 2014) for discussions on rationalizability in games with a continuum of players.
See, e.g., Bergemann and Morris (2005a, b), Bergemann et al. (2011), and Kunimoto and Serrano (2011). In particular, Bergemann et al. (2011) and Kunimoto and Serrano (2011) considered infinite mechanisms (game forms) for which transfinite rounds of deletion of never-best replies or dominated strategies are necessary.
Eichberger and Kelsey (2011) showed that some experimental results which contradict Nash equilibrium can be explained by the hypothesis that subjects view their opponents’ behavior as ambiguous.
We here adopt the conventional game-theory framework which includes the component of payoff functions for players; our analysis of this paper is applicable to a more general framework with players’ preference orderings over consequences of the game. (In particular, we keep payoff functions/preference orderings in the framework only for the purpose of discussing the notion of Nash equilibrium in the usual way.) Throughout this paper, we consider only the sets which satisfy the ZFC axioms; see, e.g., Jech (2003, p. 3).
In the game-theory literature, players are typically assumed to be Bayesian rational, that is, each player forms a prior over the space of states of the world and maximizes the expected value of some fixed vNM index on outcomes. The model of situation also allows representing player’s beliefs as other forms of subjective expected utility preferences such as Borgers’s (1993) ordinal expected utility and the state-dependent utility preferences discussed in Morris and Takahashi (2011).
Definition 1 can be viewed as a generalization of the best response property; indeed, we may define \(\beta \left( t_{i}\right) \) as a choice set for type \(t_{i}\), so that it can be used to model and analyze different behavioral patterns and decision rules for \(t_{i}\). Throughout this paper, for simplicity we focus on pure strategies; we can apply our analytical framework to the mixed extensions of finite games by allowing for using mixed strategies in finite games.
We thank a referee for drawing our attention to Basu and Weibull’s concept of the curb set. It is easy to verify that the largest rationalizable set \(R^{*}\) (in Proposition 1) is a curb set.
Lipman (1994) demonstrated that, in infinite games, we may need the transfinite induction to analyze the strategic implication of “common knowledge of rationality.” See also Chen et al.’s (2007) Example 1 for the reason why we may need a transfinite process for iterated deletion of strictly dominated strategies in general games.
In fact, we can alternatively construct a concrete IENBR procedure similar to one constructed in Chen et al. (2007). We thank a referee for providing us with useful comments and suggestions that lead to this proof for the existence of an IENBR procedure.
Note that each quasi-procedure in \({\mathcal {Q}}\) can be viewed as an element, which satisfies the property (i)–(ii), in the power set of \(2^{S}\). By the Axiom Schema of Separation [see, e.g., Jech (2003, p. 3)], \({\mathcal {Q}}\) is a set in ZFC.
We thank a referee for pointing out this to us.
In this paper, we impose no essential condition for the relationship between the preference relation \(\succsim _{t_{i}}\) of a type \(t_{i}\) and the payoff function \(u_{i}\)—i.e., the only link between payoff functions and the preferences of types is given by the very weak Diracability condition (C2) in this section and the strong monotonicity condition (C3) in Sect. 5.
The game of this example is in the class of Reny’s (1999) better-reply secure games.
Nevertheless, as demonstrated in Example 6, the expected utility preference model with a finitely additive probability charge may violate C3.
Chen and Luo (2012) showed that rationalizability under general preferences can be indistinguishable from the outcome of the IESDS procedure for a class of (in)finite games where each player’s strategy space is compact Hausdorff and each player’s payoff function is continuous and “concave-like.” The indistinguishability result in Proposition 4 does not rely on the structure of strategic game.
We take a point of view that an epistemic model is a pragmatic and convenient framework to be used for doing epistemic analysis; see Aumann and Brandenburger (1995, Sect. 7a) for related discussions. There are many examples of well-defined type spaces: Mertens and Zamir (1985) constructed a compact Hausdorff type space, Brandenburger and Dekel (1993) constructed a Polish type space, Heifetz and Samet (1998) provided an alternative “topology-free” construction of type space, and Epstein and Wang (1996) constructed a compact Hausdorff (nonprobabilistic) type space in a setting of “regular” preferences. In this paper, we are mainly concerned with the analysis of the game-theoretic solution concept of rationalizability in general situations. In particular, we do not assume that preferences have utility function representations.
This formalism can be easily applied to Aumann’s definition of knowledge by the possibility correspondence in a semantic framework and the notion of “belief with probability one” in a probabilistic model, for instance. Some readers may prefer the term “believes \(E\)” rather than “knows \(E\).” In this paper, we do not particularly distinguish between “knowledge” and “belief.”
Throughout this paper, C1 is perhaps the only essential behavioral assumption under which the rationalizability defined in general situations possesses nice properties as in the case of finite games. C2 can be removed if one does not care about its relationship with the Nash equilibrium. Morris and Takahashi (2011) did not impose this condition in their analysis. C3 is rather mild and innocuous, and the condition is satisfied by almost all preference models discussed in the literature.
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We are grateful to the editor and two anonymous referees for very useful and helpful comments and suggestions. We thank Stephen Morris for pointing out this research topic to us. We also thank Murali Agastya, Geir Asheim, Rohan Dutta, Yossi Greenberg, Duozhe Li, Jian Li, Ming Li, Bart Lipman, Takashi Kunimoto, Bin Miao, Dag Einar Sommervoll, Xiang Sun, Yeneng Sun, Satoru Takahashi, Ben Wang, Licun Xue, Chih-Chun Yang, Chun-Lei Yang, Haomiao Yu, Yongchao Zhang, and participants at NUS theory workshops and seminars at Chinese University of Hong Kong, BI Norwegian Business School, Fudan University, McGill University, Ryerson University, and Shanghai University of Finance and Economics for helpful comments and discussions. This paper was presented at the 2012 Game Theory Congress in Istanbul, Turkey, and the 12th SAET Conference in Brisbane, Australia. Financial supports from National University of Singapore and BI Norwegian Business School are gratefully acknowledged. The usual disclaimer applies.
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Chen, YC., Luo, X. & Qu, C. Rationalizability in general situations. Econ Theory 61, 147–167 (2016). https://doi.org/10.1007/s00199-015-0882-y
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DOI: https://doi.org/10.1007/s00199-015-0882-y