Result 104, Theoretical computer science

Quasipolynomial algorithms for mean-payoff, stochastic and parity games

Gives deterministic algorithms using 2O((log⁡(L+2))2)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.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

In finite turn-based mean-payoff games, players move through a graph while rewards accumulate, aiming to control their long-run average. The reported algorithms address optimal play and two separate complications: chance and recurring-priority requirements.

What changes?

The unreviewed manuscripts report deterministic algorithms taking 2 to the power O((log(L+2)) squared) bit operations, where L is the complete binary input length. They handle unrestricted binary-encoded signed integer rewards, rational chance probabilities in the stochastic extension, and binary priorities in the separate parity extension. Ordinary games yield exact values and optimal positional strategies. Stochastic games yield vertices with nonnegative expected pathwise liminf average; parity games yield vertices enforcing both nonnegative liminf average and parity.

What does that help mathematicians do?

In ordinary games, exact values and strategies depending only on the current vertex give both players a benchmark and a way to achieve it against an opponent. The stochastic result includes value exactly zero, distinguishing break-even from negative values without numerical tolerance. Its payoff uses the lower limiting average on each path before taking expectation. The parity extension additionally requires a condition on priorities recurring forever, identifying where both objectives can be enforced together.

Are there practical applications?

The immediate value is foundational: the claimed quasipolynomial bound measures computation against the full encoded input, not just the number of graph vertices. Large rewards, probability denominators and priorities therefore remain accounted for. This is a theoretical resource guarantee, not evidence that these algorithms are fast on practical instances.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time

October 5, 2026 20 pages

We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games with signed integer rewards and rational chance-transition probabilities encoded in binary. It computes exactly the vertices of nonnegative value, including value zero, for the expectation of the pathwise liminf mean payoff. The algorithm uses exact rational arithmetic and 2O((log⁡(L+2))2)2^{O((\log(L+2))^2)} bit operations, where L is the complete binary input length.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026,
  author = {{OpenAI}},
  title = {{Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf}{OAI:Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026}},
  year = {2026}
}

Mean-payoff parity games in quasipolynomial time

October 5, 2026 14 pages

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 2O((log⁡(L+2))2)2^{O((\log(L+2))^2)} bit operations, where L is the complete input length.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Deterministic quasipolynomial-time mean-payoff games

September 25, 2026 27 pages

We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses 2O((log⁡(L+2))2)2^{O((\log(L+2))^2)} bit operations. A reduction also computes the exact rational value at every vertex and globally optimal positional strategies for both players within the same quasipolynomial bound.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026,
  author = {{OpenAI}},
  title = {{Deterministic quasipolynomial-time mean-payoff games}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf}{OAI:Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026}},
  year = {2026}
}

Randomized quasipolynomial-time mean-payoff games

September 25, 2026 41 pages Main result formalized in Lean

We give a randomized algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses 2O((log⁡(L+2))2)2^{O((\log(L+2))^2)} bit operations on every random tape and is correct with probability at least 7/8. A polynomial-time check certifies the winning regions and positional strategies for both players or reports failure. Independent repetition therefore gives an always-correct algorithm with the same expected quasipolynomial bit bound.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026,
  author = {{OpenAI}},
  title = {{Randomized quasipolynomial-time mean-payoff games}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf}{OAI:Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/104.md.

Quasipolynomial algorithms for mean-payoff, stochastic and parity games

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization checks that a specified two-step execution of Truffet's elimination procedure terminates with a feasible but nonoptimal output. It is a finite counterexample to that proposed optimization step.

The formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights. It succeeds with probability at least 7/87/8 and runs in quasipolynomial bit time on every random tape. Self-loops, parallel edges, and history-dependent strategies are allowed, and winning means a nonnegative liminf average payoff.

Comparator links

Result Comparator statement
Finite counterexample to Truffet's optimization procedure TruffetCounterexample.lean
Randomized quasipolynomial mean-payoff algorithm RandomizedMeanPayoff.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.