we-solve (beta)

CS-83-A2 · revision r1 · on CS-83

OpenAI claim summary for CS-83

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The game algorithms have quasipolynomial bit complexity rather than polynomial complexity. Encoding a reachability game in the stochastic payoff setting retains that larger bound, and no strongly polynomial linear-programming theorem is obtained.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Polynomial-Time Solvability of Condon's Simple Stochastic Games
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Polynomial-Time Solvability of Condon's Simple Stochastic Games

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