Interactive Epistemology and Solution Concepts for Games with Asymmetric Information

Abstract

We use an interactive epistemology framework to provide a systematic analysis of some solution concepts for games with asymmetric information. We characterize solution concepts using expressible epistemic assumptions, represented as events in the universal type space generated by primitive uncertainty about the payoff relevant state, payoff irrelevant information, and actions. In most of the paper we adopt an interim perspective, which is appropriate to analyze genuine incomplete information. We relate ∆-rationalizability (Battigalli and Siniscalchi, 2003) to interim correlated rationalizability (Dekel, Fudenberg, and Morris, 2007) and to rationalizability in the interim strategic form. We also consider the ex ante perspective, which is appropriate to analyze asymmetric information about an initial chance move. We prove the equivalence between interim correlated rationalizability and an ex ante notion of correlated rationalizability.

Robust virtual implementation: Toward a reinterpretation of the Wilson doctrine

“…Even if a zeromeasure set of first-order beliefs is ruled out, -incentive compatibility is as restrictive as ex-post incentive compatibility. For measurability conditions, in contrast, ruling out certain zero-measure 8 Battigalli and Siniscalchi [5] show that -rationalizability coincides with Bayesian equilibria in all coherent type spaces and Battigalli et al [6] show that (a suitably defined) -rationalizability is equivalent to the interim correlated rationalizability of Dekel, Fudenberg, and Morris [12]. 9 To prove our sufficiency results (Theorems 3 and 4) we employ the assumption that Q i 's are finite sets, but this is not essential.…”

“…A&M measurability stipulates that the SCF must be constant on each element of the suitably defined, finest possible partition of the payoff-type space, which corresponds to the maximum possible separation of payoff types according to their interim preferences. 6 In a classic Bayesian environment, A&M measurability is usually perceived as very permissive, and even close to trivial. Hence, in such classic settings, the limitations of virtual implementation essentially amount to the incentive compatibility constraint.…”

“…Robust monotonicity amounts to requiring Bayesian monotonicity -a necessary condition in the classic approach -in every type space, which is a very restrictive condition (B&M [8]). 6 To find such a partition, one constructs an iterative algorithm that gradually refines the payoff-type space; see A&M [2] and our Section 3.2 for details. 7 Serrano and Vohra [24] earlier observed that virtual Bayesian implementation may fail in a classic Bayesian environment for exactly that reason, but they argued later (Serrano and Vohra [25]) that such failures are arguably "rare.…”

Correlated equilibrium and higher order beliefs about play

“…Our paper is also related to a recent literature on redundant types and solution concepts: Ely and Peski (2006), Dekel et al (2007), Liu (2009), Sadzik (2009), and Battigalli et al (2011. The main difference is that authors in this literature study incomplete-information games, with type structure and redundant types on the payoff states, while we focus on completeinformation games, with type structure and redundant types on the actions played in the game.…”

Implementation without incentive compatibility: Two stories with partially informed planners

“…15 See alsoBattigalli (1999Battigalli ( , 2003,Battigalli and Siniscalchi (2007), andBattigalli and Prestipino (2013). For the relationship among three different notions of rationalizability (belief-free, , and interim correlated), seeBattigalli et al (2011).16 In our framework, no player has a dominant strategy (except players with θ in the smashed antiques problem), implying that we have to rely on multiple rounds of iterative elimination of strictly dominated strategies.17 Our result shows that, for every i ∈ N and θ i ∈ i , R i (θ i ) = {θ i } and hence R(θ ) = {θ}.…”