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Quantifying Skill and Chance: A Unified Framework for the Geometry of Games

David H. Silver

TL;DR

The paper introduces a unified framework for quantifying skill and chance in sequential decision processes by defining the Skill--Luck Index $S(\mathcal{G}) = \frac{K - L}{K + L}$ with leverages for skill $K$ and luck $L$, and a volatility measure $\Sigma$ to capture outcome uncertainty over turns. It provides exact and approximate computational methods, demonstrates the framework on 30 games including chess, backgammon, poker, and Baccarat, and reveals a spectrum from pure luck ($S = -1$) to pure skill ($S = +1$) with balanced cases like backgammon ($S = 0$). The authors extend the framework to a two-dimensional Skill--Luck--Volatility map, validate with case studies, and discuss design implications for game balance, AI evaluation, and legal classification. Beyond games, the approach applies to any sequential decision system with chance and choice, offering guidance for MARL, risk assessment, and automated game design with targeted $S(\mathcal{G})$ and $\Sigma$ profiles.

Abstract

We introduce a quantitative framework for separating skill and chance in games by modeling them as complementary sources of control over stochastic decision trees. We define the Skill-Luck Index S(G) in [-1, 1] by decomposing game outcomes into skill leverage K and luck leverage L. Applying this to 30 games reveals a continuum from pure chance (coin toss, S = -1) through mixed domains such as backgammon (S = 0, Sigma = 1.20) to pure skill (chess, S = +1, Sigma = 0). Poker exhibits moderate skill dominance (S = 0.33) with K = 0.40 +/- 0.03 and Sigma = 0.80. We further introduce volatility Sigma to quantify outcome uncertainty over successive turns. The framework extends to general stochastic decision systems, enabling principled comparisons of player influence, game balance, and predictive stability, with applications to game design, AI evaluation, and risk assessment.

Quantifying Skill and Chance: A Unified Framework for the Geometry of Games

TL;DR

The paper introduces a unified framework for quantifying skill and chance in sequential decision processes by defining the Skill--Luck Index with leverages for skill and luck , and a volatility measure to capture outcome uncertainty over turns. It provides exact and approximate computational methods, demonstrates the framework on 30 games including chess, backgammon, poker, and Baccarat, and reveals a spectrum from pure luck () to pure skill () with balanced cases like backgammon (). The authors extend the framework to a two-dimensional Skill--Luck--Volatility map, validate with case studies, and discuss design implications for game balance, AI evaluation, and legal classification. Beyond games, the approach applies to any sequential decision system with chance and choice, offering guidance for MARL, risk assessment, and automated game design with targeted and profiles.

Abstract

We introduce a quantitative framework for separating skill and chance in games by modeling them as complementary sources of control over stochastic decision trees. We define the Skill-Luck Index S(G) in [-1, 1] by decomposing game outcomes into skill leverage K and luck leverage L. Applying this to 30 games reveals a continuum from pure chance (coin toss, S = -1) through mixed domains such as backgammon (S = 0, Sigma = 1.20) to pure skill (chess, S = +1, Sigma = 0). Poker exhibits moderate skill dominance (S = 0.33) with K = 0.40 +/- 0.03 and Sigma = 0.80. We further introduce volatility Sigma to quantify outcome uncertainty over successive turns. The framework extends to general stochastic decision systems, enabling principled comparisons of player influence, game balance, and predictive stability, with applications to game design, AI evaluation, and risk assessment.

Paper Structure

This paper contains 53 sections, 5 theorems, 8 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Setting.
  4. Policies.
  5. Leverage.
  6. The Skill--Luck Index
  7. Properties.
  8. Computation.
  9. Theoretical Properties
  10. Computational Methods
  11. Algorithm 1: Computing Minimax Value.
  12. Algorithm 2: Computing Skill Leverage.
  13. Algorithm 3: Computing Luck Leverage.
  14. Approximations for large games.
  15. Toy Example
  16. ...and 38 more sections

Key Result

Theorem 1

For any game $\mathcal{G}$, both $K \geq 0$ and $L \geq 0$.

Figures (1)

  • Figure 1: The Skill--Luck Spectrum: PCA Biplot. Each point represents a game, positioned via principal component analysis of game features including $S(\mathcal{G})$ and volatility $\Sigma$. Node color encodes $S(\mathcal{G})$: blue (luck-heavy) to red (skill-heavy). Red arrows show original feature loadings. Games span from pure luck (coin toss, war: $S = -1.0$) to pure skill (chess, go: $S = +1.0$), with balanced cases (backgammon: $S = 0.0$). Inset shows zoomed view of crowded region (dashed rectangle).

Theorems & Definitions (13)

  • Definition 1: Luck Leverage
  • Definition 2: Skill Leverage
  • Definition 3: Skill--Luck Index
  • Theorem 1: Non-negativity
  • proof
  • Theorem 2: Boundedness
  • proof
  • Theorem 3: Extreme Cases
  • proof
  • Proposition 1: Symmetry for Symmetric Games
  • ...and 3 more