شناخت شناسی تعاملی در بازی همراه با عدم قطعیت بازده

We adopt an interactive epistemology perspective to analyse dynamic games with partially unknown payoff functions. We consider solution procedures that iteratively delete strategies conditional on private information about the state of nature. In particular we focus on a weak and a strong version of the ΔΔ-rationalizability solution concept, where ΔΔ represents given restrictions on players’ beliefs about state of nature and strategies [Battigalli, P., 2003. Rationalizability in infinite, dynamic games of incomplete information. Research in Economics 57, 1–38; Battigalli, P., Siniscalchi, M., 2003. Rationalization and incomplete information. Advances in Theoretical Economics 3 (Article 3). http://www.bepress.com/bejte/advances/vol3/iss1/art3]. We first show that weak ΔΔ-rationalizability is characterized by initial common certainty of rationality and of the restrictions ΔΔ, whereas strong ΔΔ-rationalizability is characterized by common strong belief in rationality and the restrictions ΔΔ (cf. [Battigalli, P., Siniscalchi, M., 2002. Strong belief and forward induction reasoning. Journal of Economic Theory 106, 356–391]). The latter result allows us to obtain an epistemic characterization of the iterated intuitive criterion. Then we use the framework to analyse the robustness of complete-information rationalizability solution concepts to the introduction of “slight” uncertainty about payoffs. If the set of conceivable payoff functions is sufficiently large, the set of strongly rationalizable strategies with slight payoff uncertainty coincides with the set of complete-information, weakly rationalizable strategies.

Games with payoff uncertainty are situations of strategic interaction where players’ payoff functions are not common knowledge. Following Harsanyi’s approach (Harsanyi, 1967–68), such situations are usually studied as Bayesian games. A Bayesian game adds to the primitives of the model (players set, rules of interactions, possible payoff functions, players’ private information about such functions) a list of “types” for each player, whereby a type determines (implicitly) a whole hierarchy of beliefs about the unknown payoff parameters. In economic applications the analysis is often simplified by assuming that the correspondence from types to hierarchies of beliefs is trivial. For example, it is often assumed that each player’s first-order beliefs, i.e. his beliefs about the payoff parameter vector, say θθ, are solely determined by his private information about θθ according to some mapping, and that all these mappings are common knowledge.The signalling game depicted in Fig. 1 provides an example. Players 1 and 2 move sequentially: Player 1 chooses either u (up), thus terminating the game, or d (down); after d, Player 2 chooses between a, b, or c. There are three possible pairs of payoff functions (mappings from complete sequences of actions to payoffs pairs) indexed by a state of nature  2 {, , }. Player 1 knows the true  and Player 2 does not. This is represented by drawing three possible game trees with distinguished roots ,  and respectively and joining the three possible decision nodes of Player 2 with a dashed line signifying his ignorance of the true game tree. The model described so far is what we call a game with payoff uncertainty. A very simple way to obtain a Bayesian game is to specify an initial belief of the uninformed Player 2 (e.g. Pr2() = 1 2 , Pr2() = 1 3 , Pr2( ) = 16 as in Fig. 1) and to assume that it is common knowledge that (Player 1 knows the true  and) Player 2 holds this initial belief. But one could consider much more complicated Bayesian games. For example, if Player 1 were unsure about the initial belief about  of Player 2, one should introduce several possible types of Player 2, corresponding to possible beliefs about , and a belief of Player 1 about such types. Were this belief not known to Player 2, one should multiply the possible types of Player 1, and so on. Equilibria of Bayesian games are very sensitive to the precise specification of higher-order beliefs (see Weinstein and Yildiz, 2007). This is especially problematic because economic modellers find it hard to provide non- arbitrary specifications of the fine details of hierarchical beliefs. Battigalli (1999, 2003), Battigalli and Siniscalchi (2003) propose a different approach to the analysis of dynamic games with payoff uncertainty: instead of specifying a (more or less complex) type space `a la Harsanyi, they suggest to take as given some restrictions
on players’ initial and updated beliefs about  and their opponents’ strategies, and then iteratively delete private information-strategy pairs that are inconsistent with progressively higher levels of mutual certainty of rationality and of the restrictions
. For example, in the signalling game above, the modeller may find it reasonable to assume that Pr2() is larger than Pr2( ), and there is common certainty of this fact;
would then be the set of beliefs profiles such that Pr2() > Pr2( ). The resulting solution concept, called
-rationalizability, is therefore parametrized by the assumed restrictions
.2 Battigalli and Siniscalchi (2003) specifically focus on a strong version of
-rationalizability, akin to extensive form rationalizability (Pearce, 1984), that also captures a forward induction principle. Indeed, they show that, when
reflects agreement of beliefs with a given probability distribution  on the terminal nodes of a signalling game, then non-emptiness of the strong
-rationalizability solution is equivalent to  passing the Iterated Intuitive Criterion of Cho and Kreps (1987). Battigalli (1999, 2003) also considers a weak version of the solution concept that only relies on initial common certainty of rationality and the restrictions
. These papers present several examples and economic applications of the approach and report about other applications in the literature. To illustrate, independently of Player 2’s initial beliefs, the signalling game depicted in Fig. 1 has a pooling (sequential) equilibrium where each type of Player 1 chooses u and Player 2 would choose a after d. To see this, note that the best response of Player 2 is the action whose label is the Latin equivalent of the ex-post most likely  (i.e. a if  is most likely, etc.); since beliefs off the equilibrium path are not determined via Bayes’ rule, we may have Pr2(|d)  max{Pr2(|d), Pr2(|d)}, making action a a (sequential) best response. This outcome is a fortiori weakly
-rationalizable for every
that allows Pr2(|d)  max{Pr2(|d), Pr2(|d)}. However, the pooling-equilibrium outcome is not strongly
-rationalizable; this follows from a forward-induction argument. Action d is dominated for type , and undominated for types  and . If Player 2 believes in the rationality of Player 1 whenever possible, then Pr2(|d) = 0 and a is deleted. Therefore the best response of types  and is d, which yields two for sure. (The
-restrictions may, or may not, pin down the choice between b and c. If
implies Pr2(|d) > Pr2( |d), then Player 2 best responds with b.) The above-mentioned papers by Battigalli and Siniscalchi report that
-rationalizability is characterized by explicit and transparent assumptions on players’ rationality and interactive beliefs. The present paper provides the interactiveepistemology analysis supporting these claims. Following Battigalli and Siniscalchi (1999, 2002), assumptions about rationality and interactive beliefs – including assumptions on how players revise their beliefs when they observe unexpected moves – are represented as events in a universal space of states of the world, whereby each state of the world ! specifies the state of nature  and how players would behave and think conditional on every partial history of the game (including those that are counterfactual at !). The weak and strong versions of
-rationalizability are shown to be the behavioural consequences of two different sets of assumptions on rationality and interactive beliefs. Given the above-mentioned equivalence result of Battigalli and Siniscalchi (2003), the epistemic characterization of strong
-rationalizability also yields an epistemic characterization of the Iterated Intuitive Criterion. The interactive-epistemology framework is then used to explore the robustness of the complete information versions of these solution concepts to the introduction of “slight” payoff uncertainty. In particular, it is shown that if the set of conceivable payoff functions is sufficiently large, the set of strongly rationalizable strategies with slight payoff uncertainty coincides with the set of complete-information, weakly rationalizable strategies. This paper is arranged in five sections, including the present one, plus an Appendix A. In Section 2 we introduce the elements of the analysis: dynamic games with payoff uncertainty, systems of conditional beliefs, epistemic models for dynamic games, (sequential) rationality and belief operators. Section 3 is devoted to the epistemic characterization of weak and strong
-rationalizability. Section 4 contains the analysis of robustness of the complete information versions of weak and strong rationalizability to the introduction of “slight payoff uncertainty”. Section 5 summarizes the results and discusses the related literature. The Appendix A collects all the proofs.