When Are We Near Nash? Certifying Solutions in General-Sum Games

Boning Li, Longbo Huang

Heuristic solvers for general-sum extensive-form games produce a stream of iterates, and practice reports the NashGap of the returned profile. We ask what a solver can know about its own iterates from the regret accumulators and the terminal-distribution accumulators its updates already maintain. In two-player games the NashGap of a weighted average strategy equals, at the maximizing player, the weighted external regret plus a signed value correction. Bounding the regret by the running counterfactual-regret accumulator gives an online certificate whose cost per checkpoint is linear in the number of terminal histories. The certificate stays sound under alternating updates and in games with three or more players. The mismatch between what alternation evaluates and what the averages record enters as one signed comparator term, maximized in a single backward pass per player whose comparator weight is nonzero. The bound is read off the regret state, with no best response against the average. Run a portfolio of training-free regret-matching solvers, output the iterate with the smallest certificate, and verify that iterate once. On the benchmark's biased Leduc family a pool of sixteen…