提出量化游戏技能与运气的新框架，可判断各类游戏的胜负主因。
Quantifying Skill and Chance: A Unified Framework for the Geometry of Games
- 将游戏建模为随机决策树，用技能和运气的控制力分离分析
- 30款游戏测试显示：象棋纯靠技能（S=1），硬币投掷全凭运气（S=-1）
- 适用于游戏设计、AI评估，适合关注公平性与预测稳定性的研究者
我们提出一种量化游戏中的技能与运气的框架，将二者视为对随机决策树的互补控制源。通过分解结果为技能控制力K和运气控制力L，定义了范围在[-1, 1]的技能-运气指数S(G)。对30款游戏的分析显示，从纯运气（如抛硬币，S = -1）到混合型（如双陆棋，S = 0，Sigma = 1.20）再到纯技能（如国际象棋，S = +1，Sigma = 0）构成连续谱系。扑克表现中等技能主导（S = 0.33），K = 0.40 ± 0.03，Sigma = 0.80。进一步引入波动率Sigma以衡量多轮后结果的不确定性。该框架可推广至一般随机决策系统，支持对玩家影响、游戏平衡及预测稳定性进行严谨比较，适用于游戏设计、人工智能评估与风险分析。
原文摘要 · Abstract (English)
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.
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