Certified Resolving at Every Seat: Constructive Safety Certificates for Multiplayer Subgame Resolving

Exact solving of an imperfect-information game does not scale, so the standard recipe computes a blueprint offline by regret minimisation and recomputes the strategy inside the subgame play actually reaches. In two-player zero-sum games a gadget makes resolving safe: it lets the opponent cash in, at every entry, the counterfactual value it was guaranteed under the blueprint, so any equilibrium of it refines the blueprint without raising its exploitability. That guarantee rests on the zero-sum identity, which is gone as soon as a third seat is added: an unguarded update can leave the profile more exploitable than the blueprint. What survives is a per-player charge against the blueprint, anchored at the opponents' subgame best-response values. The charge is a maximum of affine functions of the resolver's sequence-form strategy, so the bound on the profile's Nash gap, the largest best-response gain over the players, is convex and piecewise linear; minimising it or maximising the resolver's own payoff under a Nash-gap budget is a linear program. The minimum-certificate program is the value of a two-player zero-sum gadget whose meta-adversary picks the victim, solvable by counterfactual regret minimisation. This gives a subgame-local resolver for independent strategies in multiplayer games whose every update carries an explicit certificate on the Nash gap of the whole profile. In constant-sum games the certified gain is at most $(n-1)\tau$ plus the opponents' Nash-gap slack minus the shift of their best-response values, and three-player Kuhn instances attain it to six decimal places. A two-sided variant equals the exact safety constraint whenever no opponent acts before the subgame. Every seat can run the same program, and a ledger of certified gains keeps the joint bound monotone under round-robin certified resolving. Across 3-player Kuhn, 3-player Leduc, and random games with up to six players, $2,244$ certified updates never exceed their certificate beyond the solver tolerance of $10^{-9}$; re-anchoring cuts the Nash gap by $41$–$57\%$ on 3-player Kuhn. Unsafe resolving raises the Nash gap in $18$–$70\%$ of configurations, by up to $0.17$, while under the common clipped budget the certified gain matches or beats the unsafe gain in $176$ of the $214$ poker configurations. Safety becomes an objective a resolver can optimise against, priced by the cap identity and the gap decomposition.