Characterizing Solution Concepts in Terms of Common Knowledge of Rationality

Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known [Aumann 1987; Brandenburger and Dekel 1987]. We show how to get analogous characterizations of sequential equilibrium and (trembling hand) perfect equilibrium, as a consequence of recent results of Halpern [2009]. Arguably the major goal of epistemic game theory is to characterize solution concepts epistemically. Characterizations of Nash equilibrium, correlated equilibrium, and ∗Supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR under grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1-0055. †The Israel Pollak academic chair at the Technion; work supported in part by Israel Science Foundation under grant 1520/11. rationalizability in terms of common knowledge of rationality are well known [Aumann 1987; Brandenburger and Dekel 1987]. We show how to get analogous characterizations of sequential equilibrium and (trembling hand) perfect equilibrium as a consequence of recent results of Halpern [2009]. 1 A review of earlier results To explain our results, we briefly review the relevant earlier results. We assume that the reader is familiar with standard solution concepts such as Nash equilibrium, correlated equilibrium, and rationalizability; see [Osborne and Rubinstein 1994] for a discussion. Let Γ = (N,S, (ui)i∈N) be a finite strategic-form game, where N = {1, . . . , n} is the set of players, S = ×i∈NSi is a finite set of strategy profiles, and ui : S → IR is player i’s utility function. For ease of exposition we assume that Si ∩ Sj = ∅ for i 6= j. Let a model of Γ be a tuple M = (Ω, s, (Pri)i∈N), where Ω is a set of states of Γ, s associates with each state ω ∈ Ω a pure strategy profile s(ω) ∈ S, and Pri is a probability distribution on Ω, describing i’s initial beliefs. Let si(ω) denote player i’s strategy in the profile s(ω), and let s−i(ω) denote the strategy profile consisting of the strategies of all players other than i. For S ∈ Si, let [S] = {ω ∈ Ω : si(ω) = S} be the set of states at which player i chooses strategy S. Similarly, let [~ S−i] = {ω ∈ Ω : s−i(ω) = ~ S−i} and [~ S] = {ω ∈ Ω : s(ω) = ~ S}. For simplicity, we assume that [~ S] is measurable for all strategy profiles ~ S, and that Pri([Si]) > 0 for all strategies Si ∈ Si and all players i ∈ N . As usual, we say that a player is rational at state ω (in a model M of Γ) if his strategy at ω is a best response in Γ given his beliefs at ω. We view Pri as i’s prior belief, intuitively, before i has been assigned or has chosen a strategy. We assume that i knows his strategy at ω, so his beliefs at ω are the result of conditioning Pri on [si(ω)]. Given our assumption that Pri([si(ω)]) > 0, the conditional probability Pri | [si(ω)] is well defined. Note that we can view Pri as inducing a probability Pr S i on strategy profiles ~ S ∈ S by simply taking Pri (~ S) = Pri([~ S]); we similarly define Pr S i (Si) = Pri([Si]) and Pr S i (~ S−i) = Pri([S−i]). Let Pr S i,ω = Pr S i | si(ω). Intuitively, at state ω, player i knows his strategy si(ω), so his distribution Pr S i,ω on strategies at ω is the result of conditioning his prior distribution on strategies Pri on this information. Formally, i is rational at ω if, for all strategies S ∈ Si, we have that ∑ ~ S′ −i∈S−i Pri,ω(~ S ′ −i)ui(si(ω), ~ S ′ −i) ≥ ∑ ~ S′ −i∈S−i Pri,ω(S ′ −i)ui(S, ~ S ′ −i). We say that player i is rational in model M if i is rational at every state ω in M . Finally, For simpliciity, we assume in this paper that Ω is finite, and all subsets of Ω are measurable.