Incorporating Ignorance within Game Theory: An Imprecise Probability Approach - Fares, Bernard (Trinity College Dublin. School of Computer Science & Statistics. Discipline of Statistics, 2023)
Ignorance within non-cooperative games, reflected as a player's uncertain preferences towards a game's outcome, is examined from a probabilistic 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. Decisions under ignorance could be optimised by permitting a player to compute rational strategies with respect to elicited lower and upper expectations of an uncertain outcome, allowing them to update these strategies when new observations are available, and helping them assess the impact and value of acquired information. Therefore, this dissertation aims to develop a complete framework for decision optimisation within strategic settings that include uncertainty. We 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. An in-depth literature review of game theory and imprecise probabilities is provided, focusing on existing normative theories and their plausible generalisations. The motivation behind a solution permitting ignorance is presented, and foundational issues related to existing approaches are argued. Afterwards, we introduce a framework that allows a risk-neutral player with constant marginal utilities for money to incorporate and dynamically learn about uncertain outcomes. This framework is then generalised to cover risk-averse players whose marginal utilities across outcomes are state-dependent. The resulting framework is proposed as a possible solution to the problem of utility induction and decision-making in game-theoretic settings that include uncertainty. It is analysed and demonstrated through motivating examples modified to include uncertainty. Each example's correlated equilibria's convex polytope is computed and compared to its uncertainty-free equivalent. Exceptional cases such as extreme ignorance are also examined and assessed through a Monte Carlo simulation where we demonstrate that, in repeated games, vacuous lower and upper previsions converge to one linear value that reflects the true expected preference over the uncertain outcome. Moreover, inadequate value of information under uncertainty is considered, and a model to assess the impact of information patterns on strategic interactions is proposed. This model enables a player to compute their expected and actual values of a piece of information with respect to a Pareto-efficient strategy. We showcase it within a game that includes uncertainty by applying utility diagnostics to two types of players, pessimistic and optimistic. Finally, since the foundations of the normative game theory introduced by Von Neumann and Morgenstern assume that all outcomes are known, the consistency of its axiomatic rules under ignorance is reviewed. We show that uncertainty can alter relevant games' zero-sum and symmetry properties and propose an approach to force these properties.
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Keywords: Game theory; Imprecise probability; Uncertain utility; Nonparametrics; Value of information