Game and Variance
The Mathematical Analysis of Games, Focusing on Variance
Tom Verhoeff
Eindhoven University of Technology
2009
Abstract
Verhoeff, 2009
---> DOWNLOAD <---
This paper provides a mathematical analysis of games, focusing on the role of variance. Games are classified into three categories based on uncertainty source: combinatorial games (move combinations), games of chance (stochastic processes), and strategic games (hidden information). Many real games blend these categories. The paper examines two examples - a simple strategic coin game and the solitaire game Yahtzee which involves chance. For the coin game, the optimal mixed strategy Nash equilibrium involving randomized play is derived. For Yahtzee, the optimal strategy's expected score of 254 and standard deviation of nearly 60 are calculated using Markov processes. The high variance indicates over 2000 games are needed to reliably attain the expected result. The paper highlights considering variance in addition to expected values when analyzing games and decisions under uncertainty. High variance reduces predictability and hinders planning. Mathematical techniques from combinatorics, probability, and game theory are valuable for understanding and optimizing decisions involving constraints and uncertainty, common in real-world situations akin to game scenarios. The role of variance is often overlooked but critical.
