Variance Decompositions for Extensive-Form Games

The extent to which an individual or chance can influence the outcome of a game is a central question in the analysis of games. Consequently, the ability to characterize sources of variation in game outcomes may have significant implications in areas such as game design, law, and multi-agent reinforcement learning. We derive a closed-form expression and estimators for the variance in the outcome of a general multi-agent game that is attributable to a player or chance. We analyze poker hands to show that randomness in the cards dealt has surprisingly little influence on the outcomes of each hand. A simple example is given that demonstrates how variance decompositions can be used to measure other interesting properties of games.