A Bayesian Game without epsilon equilibria

We present a three player Bayesian game for which there is no epsilon equilibria in Borel measurable strategies for small enough epsilon, however there are non-measurable equilibria.

💡 Research Summary

The paper constructs a three‑player Bayesian game in which no ε‑equilibrium exists among Borel‑measurable strategies for sufficiently small ε (specifically ε ≤ 1/1000), yet equilibria do exist when non‑measurable strategies are allowed. The authors achieve this by exploiting a non‑amenable semigroup action on a Cantor‑type probability space, thereby breaking the usual Harsanyi‑ε‑equilibrium existence results that rely on measurability.

The underlying state space is built from the free semigroup (G^{+}) generated by two symbols (T_{1}) and (T_{2}). Each element of the space (X = {0,1}^{G^{+}}) is an infinite binary labeling of the semigroup’s elements. The shift maps (T_{i}) act on (X) by moving the label along the semigroup, and the product Bernoulli(½) measure (m) makes every shift measure‑preserving. The game’s full state space is (\Omega = D \times X) with (D={r,g}); the colour component determines which of two “colour” states (red or green) the game is in at a given point.

Three players are defined: a green player (G_{0}) and two red players (R_{1},R_{2}). Their information partitions are deliberately coarse. (G_{0}) cannot distinguish between ((g,x)) and ((r,x)); each red player cannot distinguish between a red point ((r,x)) and the green pre‑image ({g}\times T_{i}^{-1}(x)). Consequently each player’s sigma‑algebra is generated by these information sets, and a strategy must be Borel‑measurable with respect to that sigma‑algebra.

Each player has two pure actions. The red players’ payoffs at red states are simple 2×2 matrices with entries 300 or 100, depending on the value of the coordinate (x_{e}) (the label of the identity element). The green player’s payoff at green states is a 2×2×2 tensor; when (x_{e}=0) choosing (b_{0}) yields 1000 or 2000 depending on the red actions, while choosing (b_{1}) yields the opposite high numbers, and the whole structure is reversed when (x_{e}=1). Thus the green player’s action strongly influences the red players’ incentives and vice‑versa.

A crucial combinatorial device is the “parity rule”. Define sets (A_{0}) and (A_{1}) as the points where (G_{0}) plays (b_{0}) or (b_{1}) with probability at least 0.95; let (A_{M}) be the remainder. The parity rule states that if (T_{1}(x)\in A_{i}) and (T_{2}(x)\in A_{j}) and the identity label (x_{e}=k), then (x) must belong to (A_{i+j+k\ (\text{mod }2)}). This rule captures the logical consistency that would be required in any ε‑equilibrium: the incentives at a point propagate through the semigroup shifts, forcing a coherent parity assignment across the whole space.

Lemma 1 shows that at any point at least one red player receives an incentive of at least 80 to choose a specific pure action; consequently mixing (randomizing) is heavily discouraged. Lemma 2 quantifies this: in any Borel‑measurable ε‑equilibrium with ε ≤ 1/1000, the green player’s mixing probability is below (1.6\times10^{-4}) and the parity rule can be violated on at most 0.4 % of the space.

The authors then exploit the non‑amenability of the free semigroup. Because (G^{+}) lacks Følner sequences, there is no way to construct Borel sets that are almost invariant under both shifts while also satisfying the parity rule on a set of large measure. A careful measure‑theoretic argument shows that any Borel candidate for (A_{0},A_{1}) would either have to be too small (contradicting the high probability requirement) or would violate the parity rule on a set of measure exceeding the bound imposed by Lemma 2. Hence no Borel‑measurable ε‑equilibrium can exist for ε ≤ 1/1000; in particular, there is no Harsanyi‑ε‑equilibrium for such ε.

Nevertheless, if measurability is dropped, the authors invoke the axiom of choice to select a non‑measurable “parity‑consistent” function (f:X\to{0,1}) that satisfies the parity rule everywhere. Defining the green player’s pure strategy as (b_{f(x)}) and the red players’ pure strategies accordingly yields a (non‑measurable) Bayesian equilibrium. This equilibrium is pure almost everywhere, i.e., each player uses a deterministic action at almost every state, but the strategy cannot be expressed as a Borel‑measurable function.

The paper situates its contribution within a line of research initiated by Hellman (2014) and extended by Hellman‑Levy (2016), who exhibited two‑player games with similar phenomena under amenable group actions. The present work shows that the phenomenon persists—and can be made stronger—when the underlying action is non‑amenable and when three players are involved, thereby providing a clean separation between the existence of Harsanyi‑ε‑equilibria and the existence of Bayesian equilibria.

In summary, the authors demonstrate a concrete Bayesian game where the usual measurability assumptions preclude any ε‑equilibrium for small ε, while abandoning those assumptions restores equilibrium existence via non‑measurable strategies. This result highlights the delicate interplay between information structures, group‑theoretic properties of the underlying action, and the measurability requirements that are often taken for granted in equilibrium analysis. It opens new questions about the robustness of equilibrium concepts under weaker measurability conditions and about the role of amenability in guaranteeing equilibrium existence.