Incorporating Ignorance within Game Theory: An Imprecise Probability Approach

Abstract

Ignorance within non-cooperative games, reflected as a player’s uncertain prefer- ences towards a game’s outcome, is examined from a Bayesian point of view. This topic has had scarce treatment in the literature, which emphasises exogenous uncertainties caused by other players or nature and not by players themselves. That is primarily because a player’s endogenous uncertainty over an outcome poses significant challenges and complex sequences of reciprocal expectations. Therefore, it is often ignored, and preferences are either assumed from a continuous domain or set using introspection, resulting in non-optimal models. We here explore a solution concept based on recent research in imprecise probabilities and de Finetti’s approach to defining subjective probabilities, which utilises bets to assess beliefs. The resulting model allows players to be ignorant about their initial preferences and learn about them in repeated games. Furthermore, it permits improving the value of information in these situations. This model is proposed as a possible solution to the problem of utility inference in game- theoretic settings that include uncertainty over outcomes. We demonstrate it through motivating repeated-game problems modified to have uncertainty and through a sim- ulation over a case of extreme ignorance.