No Target-Free Equilibrium Learning: Adaptive Phase Transitions in Zero-Sum Games

Learning a game has no target-independent statistical complexity. Under the same noisy payoff observations, values and low-exploitability decisions may be accurately learned while point equilibrium recovery remains statistically unresolved. We first prove a dimension-uniform accuracy–sensitivity obstruction for every globally Lipschitz single-valued strategy rule, together with an exact two-by-two separation of value, decision, branch, and point risks. We then introduce TARGET, an implementable procedure that discovers an unknown simple equilibrium interface, estimates its information normal and side targets, and adaptively allocates samples. Uniformly over a fixed quantitative class and compact phase ranges, TARGET recovers the active chart, converges to the oracle Gaussian score, and attains the known-geometry minimax interface-side risk and the least-favorable-pair point and confidence benchmarks. Matching lower bounds show that geometric oracle information has zero first-order value, while an explicit competing-interface family identifies the sharp adaptation boundary. Finally, we derive the exact price of sampling before the downstream equilibrium target is known: one experiment is simultaneously oracle optimal for several targets if and only if their normalized information profiles coincide. A certified generator and a fixed-seed study of 320 heterogeneous games verify implementation and exhibit dimension-free collapse onto the predicted information-normal curves.