CS-141 · computer science · mathematics · Algorithmic game theory

Quasipolynomial mean-payoff/parity winning sets

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

Can the winning states for a conjunction of mean-payoff and parity objectives in an explicitly encoded finite turn-based game be computed deterministically in quasipolynomial time in its binary input length?

Short English statement

Sources

S1 Original question source or mathematical context. https://doi.org/10.4171/owr/2016/7.
S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-141-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for CS-141

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 104; 2026-10-05).

Quasipolynomial mean-payoff/parity winning sets

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 104; 2026-10-05), the asserted result is that mean-payoff parity games with arbitrary signed binary rewards and priorities have deterministic quasipolynomial-time winning-set computation.

Mathematical statement of the claimed result

Manuscript 1.

A finite directed arena has an outgoing edge at every vertex, a Max/Min ownership partition, binary integer vertex rewards $r(v)$ and nonnegative integer priorities $p(v)$. For pure strategies Max's objective on $\pi=v_0v_1\cdots$ is \[ \min\{p(v):v\text{ recurs infinitely often}\}\text{ even},\qquad \liminf_{T\to\infty}T^{-1}\sum_{t<T}r(v_t)\ge0. \] The claimed deterministic, uniform algorithm decides for each vertex whether Max has a strategy forcing both conditions against every strategy of Min in at most $2^{C(\log_2(L+2))^2}$ bit operations, with absolute $C$ and full binary input length $L$. Priority count and reward magnitudes are unrestricted.

Outline of the argument

For manuscript 1 (family 104; 2026-10-05), the ordinary mean-payoff solver is coupled to attractor and bounded-dominion recursion. Uniform prefix-payoff bounds permit strategies to be combined across attractors, and a decreasing recursion budget bounds the full bit complexity.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 104; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

OpenAI, Mean-payoff parity games in quasipolynomial time, 2026-10-05; citation key OAI:Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
Family 104: Lean scope.
OpenAI catalogue entry.

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Record details

Subjects: Algorithmic game theory
Collection: Top Open Problems
Computer Science rank: 141
Mathematics rank: 2983