Uncertainty quantification in game theory

Abstract

Abstract This work examines the applicability of Uncertainty Quantification (UQ) in Game Theory. We consider the classical games “matching pennies” and “Hawk an Dove” in situations involving uncertainty. The first game examined is “matching pennies”: a first situation concerns the game where the probabilities of choice between Heads and Tails are unknown and must be determined from observations. A second situation concerns fluctuations in the implementation of the Nash equilibrium. Instability is evidentiated and a strategy based on statistical estimation is introduced. A third situation considers random payoffs having an unknown distribution: observations are used to generate an UQ representation of the real distribution of the payoffs, without any supplementary assumption on the nature of the distribution. Finally, we analyze the effects of uncertainties on the associated replicator dynamics: UQ is applied to generate mean trajectories and mean orbits - in this step, we need to manipulate statistics of curves, which are objects defined by functions, belonging to infinitely dimensional vector spaces. The second game is “Hawk and Dove”. We examine the situation where the reward and the cost of an injury are both uncertain and only a small sample of values is available. The methods of UQ are applied to determine the mean evolution of the system and confidence intervals for the evolution of the fractions of Hawks and Doves. The UQ methods involved are described and simple examples are given to facilitate understanding and application to other situations.