Extend sequential minimax duality to countably infinite horizons

Establish a general minimax theorem for sequential gamble spaces (Y, \hat Z, T) with countably infinite time horizon T = ∞ that yields minimax duality—equivalently, equality between the game-theoretic upper expectation and the measure-theoretic upper expectation—in composite settings, including the bounded law of large numbers with per-round outcomes Y = [-1,1]. Specifically, prove that for bounded measurable variables X on Y^∞, the equality sup_{P ∈ Δ_0^∞(\hat Z)} E_P X = inf_{Z ∈ mdcl(Z_∞)} sup_{ω ∈ Y^∞} (X(ω) − Z(ω)) (or, equivalently, sup_{P ∈ Δ(Y^∞)} inf_{Z ∈ mdcl(Z_∞)} E_P[X − Z] = inf_{Z ∈ mdcl(Z_∞)} sup_{P ∈ Δ(Y^∞)} E_P[X − Z]) holds under broad, verifiable conditions that cover the LLN example.

Background

The paper proves a general minimax theorem for finite time horizons (Theorem 4.8) and uses it to convert measure-theoretic results into game-theoretic ones. However, many central game-theoretic results (e.g., the bounded LLN) are infinite-horizon statements, and the author explicitly notes that the infinite-horizon case is not yet covered.

The author further highlights in the future-work discussion that extending the finite-time minimax framework to countably infinite time appears plausible and desirable, particularly for indicators of tail events such as the LLN. A general infinite-horizon minimax theorem would establish price equality and minimax duality in these composite sequential settings, enabling systematic lifting of measure-theoretic statements to game-theoretic statements for T = ∞.