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Mean-payoff parity games in quasipolynomial time

OpenAI · October 5, 2026

Lean

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Abstract

We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games. It computes all vertices from which a player can enforce both nonnegative liminf mean payoff and the parity condition, with arbitrary signed binary rewards and unrestricted binary priorities. The running time is $2^{O((\log(L+2))^2)}$ bit operations, where L is the complete input length.

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Notes

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope notes for family 104 (lean/docs/104.md, release of 6 Oct 2026). The notes describe the randomized zero-threshold theorem and a finite counterexample to Truffet's elimination procedure, and list only the two September 25 mean-payoff papers; this paper's mean-payoff parity reduction is not formalized. [family-level]

Digests and expositions

  • Latent Seminar, Quasipolynomial algorithms for mean-payoff games (7 October 2026). Explains the four manuscripts' potential comparisons, recursive methods, stochastic discounting, and parity strategy composition, explicitly without independently verifying the complete proofs. [family-level]
  • AI Proved What?, Who wins in the long run? (Episode 104, October 2026; status updated 19 October 2026). Provides an animated exposition and transcript covering all four claims, the binary-weight complexity distinction, and the reported limits of Lean coverage. [AI-generated per source] [family-level]

News and blogs

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026,
  author = {{OpenAI}},
  title = {{Mean-payoff parity games in quasipolynomial time}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/mean-payoff-parity.pdf}{OAI:Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/104.md. The Lean statements have not been checked by people against the paper's claims.

Family 104: Quasipolynomial algorithms for mean-payoff, stochastic and parity games

Gives deterministic algorithms using $2^{O((\log(L+2))^2)}$ bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.

Other papers in this family: