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Closed form for multicandidate first-to-ahead-by-k voting (generalized gambler’s ruin)

Derive a closed-form expression for the winning probability in the first-to-ahead-by-k voting process when multiple candidates race simultaneously with dependent counts (the multinomial generalization of gambler’s ruin), rather than restricting analysis to a two-candidate approximation.

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Background

MAKER uses first-to-ahead-by-k voting to select actions from multiple LLM samples at each step. The authors model the selection probability via a simplified two-candidate race because the full multicandidate setting—simultaneous dependent races—is analytically challenging.

They explicitly state that no closed form is known for the general multicandidate case, motivating a precise derivation that would strengthen theoretical guarantees and scaling laws for error correction under MDAPs.

References

This process is a generalization of the classic gambler's ruin problem , but with simultaneous dependent races between all pairs of candidates . Since there is no known closed form for this general case, the analysis is simplified by assuming the worst case, i.e., that a correct candidate with probability $p$ races against a single alternative with probability $1 - p$.

— Solving a Million-Step LLM Task with Zero Errors (2511.09030 - Meyerson et al., 12 Nov 2025) in Section 3.2 (First-to-ahead-by-k Voting and Scaling Laws)