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CS-162 · computer science · mathematics · Algorithmic game theory

Quasipolynomial stochastic mean-payoff decision

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For explicitly encoded finite turn-based zero-sum stochastic mean-payoff games with binary rational rewards and transition probabilities, can the states of value at least zero be computed deterministically in quasipolynomial bit time in the total encoding length? The statement concerns threshold decision, without asserting polynomial time or exact value output.

Short English statement

For explicitly encoded finite turn-based zero-sum stochastic mean-payoff games with binary rational rewards and transition probabilities, can the states of value at least zero be computed deterministically in quasipolynomial bit time in the total encoding length? The statement concerns threshold decision, without asserting polynomial time or exact value output.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-162-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-162

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 stochastic mean-payoff decision

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 finite stochastic games in which players move in turn admit deterministic quasipolynomial-bit-time zero-threshold solution for signed binary rewards and binary rational probabilities.

Mathematical statement of the claimed result

Manuscript 1.
Consider a finite explicitly encoded directed multigraph where every vertex has positive outdegree, integer edge rewards and vertices controlled by Max, Min or chance. Chance probabilities are binary rational numbers, and parallel edges, loops and zero probabilities are allowed. For behavioral strategies using the full history, define \[ \operatorname{MP}=\liminf_{T\to\infty}\frac1T\sum_{t<T}w(e_t),\qquad \operatorname{val}(i)=\sup_{\sigma}\inf_{\tau}\mathbb E_i^{\sigma,\tau}[\operatorname{MP}], \qquad U=\{i:\operatorname{val}(i)\ge0\}. \] The expectation follows the pathwise lower limit. The claim supplies a single deterministic Turing machine and absolute $C$ outputting exactly $U$ in \[ 2^{C(\log_2(L+2))^2} \] bit operations, where $L$ is the full binary input length, including exact rational arithmetic and output. Value-zero vertices are included.

Outline of the argument

For manuscript 1 (family 104; 2026-10-05), rational gain separation reduces sign decisions to one discounted monotone fixed point. Simultaneous boundary labels move two nearby integer comparison vectors through shrinking boxes, reaching enough precision to include zero-valued vertices correctly.

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

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