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Fast Algorithm for Calculating Probability of Chess Winning Streaks

Guoqing Diao

TL;DR

The paper addresses exact probability calculations for streaks in sequences of $n$ games given per-game outcomes, deriving recurrence-based procedures for three streak types. It develops a recurrence framework for the pure streak via $p_{m,k}$, extends to non-losing streaks by aggregating outcomes, and handles the in-between streak with a three-outcome model using $D_{m,k+1/2}$ and $g_{m,k+1/2}$ alongside a tractable approximation for $h_{a,b,k}$. The contributions include closed-form-like recurrences for exact at-least-one-streak probabilities, a fast C implementation, and numerical studies validating accuracy and scalability to large $n$ and $k$. The approach provides conservative bounds under independence and broad applicability to other domains with known per-game probabilities, such as online gaming or sports analytics.

Abstract

Motivated by the controversy in the chess community, where Hikaru Nakamura, a renowned grandmaster, has posted multiple impressive winning streaks over the years on the online platform chess.com, we derive the probabilities of various types of streaks in online chess and/or other sports. Specifically, given the winning/drawing/losing probabilities of individual games, we derive the probabilities of "pure" winning streaks, non-losing streaks, and "in-between" streaks involving at most one draw over the course of games played in a period. The performance of the developed algorithms is examined through numerical studies.

Fast Algorithm for Calculating Probability of Chess Winning Streaks

TL;DR

The paper addresses exact probability calculations for streaks in sequences of games given per-game outcomes, deriving recurrence-based procedures for three streak types. It develops a recurrence framework for the pure streak via , extends to non-losing streaks by aggregating outcomes, and handles the in-between streak with a three-outcome model using and alongside a tractable approximation for . The contributions include closed-form-like recurrences for exact at-least-one-streak probabilities, a fast C implementation, and numerical studies validating accuracy and scalability to large and . The approach provides conservative bounds under independence and broad applicability to other domains with known per-game probabilities, such as online gaming or sports analytics.

Abstract

Motivated by the controversy in the chess community, where Hikaru Nakamura, a renowned grandmaster, has posted multiple impressive winning streaks over the years on the online platform chess.com, we derive the probabilities of various types of streaks in online chess and/or other sports. Specifically, given the winning/drawing/losing probabilities of individual games, we derive the probabilities of "pure" winning streaks, non-losing streaks, and "in-between" streaks involving at most one draw over the course of games played in a period. The performance of the developed algorithms is examined through numerical studies.

Paper Structure

This paper contains 5 sections, 15 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Algorithms
  3. Numerical Studies
  4. Limitations
  5. Applications to other fields

Figures (2)

  • Figure 1: Probabilities of having "pure" winning streaks of $k$ or more games out of a total of 30,000 games with winning probabilities of individual games from: (1) $U(0.85, 1)$; (2) $U(0.8, 1)$; (3) $U(0.75, 1)$; and (4) $U(0.7, 1)$.
  • Figure 2: Probabilities of having "in-between" streaks of $k+1/2$ (i.e., allowing for at most one draw) or more games out of a total of 10,000 games with drawing probabilities of individual games from $U(0, 0.2)$ and losing probabilities following: (1) $U(0, 0.15)$; (2) $U(0, 0.2)$; (3) $U(0, 0.25)$; and (4) $U(0, 0.3)$.