We provide an epistemic analysis of arbitrary strategic games based on possibility correspondences. We first establish a generic result that links true common beliefs (and, respectively, common knowledge) of players' rationality defined by means of `monotonic' properties, with the iterated elimination of strategies that do not satisfy these properties. It allows us to deduce the customary results concerned with true common beliefs of rationality and iterated elimination of strictly dominated strategies as simple corollaries. This approach relies on Tarski's Fixpoint Theorem. We also provide an axiomatic presentation of this generic result. This allows us to clarify the proof-theoretic principles assumed in players' reasoning. Finally, we provide an alternative characterization of the iterated elimination of strategies based on the concept of a public announcement. It applies to `global properties'. Both classes of properties include the notions of rationalizability and the iterated elimination of strictly dominated strategies.
Epistemic analysis of strategic games (in short, games) aims at predicting the choices of rational players in the presence of (partial or common) knowledge or belief about the behaviour of other players. Most often it focusses on the iterated elimination of never best responses (a notion termed as rationalizability), the iterated elimination of strictly dominated strategies (IESDS) and on justification of the strategies selected in Nash and correlated equilibria.
Starting with Aumann [1987], Brandenburger and Dekel [1987] and Tan and Werlang [1988] a large body of literature arose that investigates the epistemic foundations of rationalizability by modelling the reasoning employed by players in choosing their strategies. Such an analysis, based either on possibility correspondences and partition spaces, or Harsanyi type spaces, is limited either to finite or compact games with continuous payoffs, or to twoplayer games, see, e.g., Battigalli and Bonanno [1999] or Ely and Peski [2006].
In turn, in the case of IESDS the epistemic analysis has focussed on finite games (with an infinite hierarchy of beliefs) and strict dominance either by pure or by mixed strategies, see, e.g. Brandenburger, Friedenberg and Keisler [2008].
In this paper we provide an epistemic analysis of arbitrary strategic games based on possibility correspondences. We prove a generic result that is concerned with monotonic program properties 1 used by the players to select optimal strategies.
More specifically, given a belief model for the initial strategic game, denote by RAT(φ) the property that each player i uses a property φ i to select his strategy (’each player i is φ i -rational’). We establish in Section 4 the following main result: Assume that each property φ i is monotonic. The set of joints strategies that the players choose in the states in which RAT(φ) is a true common belief is included in the set of joint strategies 1 The concepts of monotonic, global and local properties are introduced in Section 3.
that remain after the iterated elimination of the strategies that for player i are not φ i -optimal.
In general, transfinite iterations of the strategy elimination are possible. For some belief models the inclusion can be reversed.
This generic result covers the usual notion of rationalizability in finite games and a global version of the iterated elimination of strictly dominated strategies. For the customary, local version of the iterated elimination of strictly dominated strategies we justify in Section 5 the statement true common belief (or common knowledge) of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies for arbitrary games and transfinite iterations of the elimination process. Rationality refers here to the concept studied in Bernheim [1984].
Strict dominance is a non-monotonic property, so the use of monotonic properties allowed us to provide epistemic foundations for non-monotonic properties. However, weak dominance, another non-monotonic property, remains beyond the reach of this approach. A mathematical reason is that its global version is also non-monotonic (see Apt [2007c]), in contrast to strict dominance, the global version of which is monotonic. To provide epistemic foundations of weak dominance the only currently known approach is that of Brandenburger, Friedenberg and Keisler [2008] based on the lexicographic probability systems.
We also provide, in Section 6, an axiomatic presentation of the above generic result. This clarifies the logical underpinnings of the epistemic analysis and shows that the use of transfinite iterations can be naturally captured by a single inference rule that involves greatest fixpoints. Also, it shows that the relevant monotonic properties can be defined using positive formulae.
Finally, inspired by van Benthem [2007], we provide in Section 7 an alternative characterization of the strategies that remain after iterated elimination of strategies that for player i are not φ i -optimal, based on the concept of a public announcement due to Plaza [1989]. Here monotonicity is not needed and we obtain a generalization of van Benthem’s results to arbitrary strategic games and to other properties than rationalizability, notably a global version of weak dominance.
Apart of the necessity of the use of transfinite iterations when studying arbitrary strategic games, our analysis shows the relevance of two concepts of the underlying properties φ i used by the players to select their strategies. The first one is monotonicity which allows us to use Tarski’s Fixpoint Theorem. The second is globality, which intuitively means that each subgame obtained by iterated elimination of strategies is analyzed in the context of the given initial game. While the proposed epistemic analysis of arbitrary games based on possibility correspondences crucially depends on the use of monotonic properties, the one based on publ
