Games of incomplete information, ergodic theory, and the measurability of equilibria

Abstract

We present an example of a one-stage three player game of incomplete information played on a sequence space {0, 1} Z such that the players' locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1} Z , each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g. all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is as much a philosophical issue as a mathematical one. We consider the double-shift Bernoulli probability space B Z 2 , where T 1 is the shift in the first coordinate, T 2 is the shift in the second coordinate, and x i,j is the i, j-coordinate of x ∈ B Z 2 . Let C be a compact and convex set with compact subsets)) in A x 0,0 for all x ∈ B Z 2 and that the inability of measurable functions to satisfy this property (in expectation) is bounded below by a positive constant dependent on the sets (A b | b ∈ B). We give an example of a one-stage zero-sum game played on B Z 2 that would not have a value (but would have equilibria!) if this conjecture were valid.

Uniform Continuity of the Value of Zero-Sum Games with Differential Information

“…belongs), or more generally, by a -…eld z i of measurable sets (events) in (in which case i knows, given any event in z i ; whether it has occurred). It was shown by Simon (2003) that Bayesian Nash equilibrium (BNE) may fail to exist in games with di¤erential information, as a result of discontinuity of the expected payo¤ function in Bayesian strategies of all players simultaneously. The situation changes, however, when attention is con…ned to two-person zero-sum games with di¤erential information.…”

Robust Equilibria under Non-Common Priors

“…13 Assuming countability allows us to avoid measurability issues, in particular regarding existence of Bayesian Nash equilibria (cf. Simon [21]). …”

“…Weinstein and Yildiz [24] show that for any complete information type in the universal type space 21 and any rationalizable action profile a * of this game, there exist a dominance-solvable incomplete information game and a sequence of types drawn from this game such that (1) this sequence converges to the complete information type (with respect to the product topology in the universal type space) and (2) each type of the sequence plays a * , while they are not concerned with priors but working directly with interim beliefs. Our construction in Lemma 3.4 shows that, by attaching heterogeneous priors appropriately, such a dominance solvable incomplete information game can be made an (ε, N)-perturbation (where ε can be arbitrarily small and N arbitrarily large).…”

“…One might then ask whether we could use the further perturbation with a common prior so obtained for the purpose of Lemma 3.4. It is, however, not true; for, when the given a posteriori equilibrium is not a robust equilibrium under common prior (as in the example in Section 3.1), it is not possible, for vanishingly small ε, to have an (ε, N)-perturbation such that (1) it has a common prior, (2) it is dominance solvable, and (3) the unique rationalizable strat- 21 Recall that a complete information type is a (degenerate) type in the universal type space where it is common knowledge that payoffs are given by the complete information game. 22 While Weinstein and Yildiz [24] use a different setting and language, their construction as well as ours are in essence analogous to the information structure of Rubinstein's [20] email game.…”

Type space on a purely measurable parameter space

“…On the other hand, recently, Simon [13] showed that, there may be problem with the existence of measurable equilibrium of the games with incomplete information. Hence, a model, in which , the beliefs of the players are modeled by probability measures, is not necessarily appropriate for some problems.…”