Rationality, uncertainty aversion and equilibrium concepts in normal and extensive form games

This thesis contributes to a re-examination and extension of the equilibrium concept in normal and extensive form games. The equilibrium concept is a solution concept for games that is consistent with individual rationality and various assumptions about players' knowledge about the nature of their strategic interaction. The thesis argues that further consistency conditions can be imposed on a rational solution concept. By its very nature, a rational solution concept implicitly defines which strategies are non-rational. A rational player's beliefs about play by non-rational opponents should be consistent with this implicit definition of non-rational play. The thesis shows that equilibrium concepts that satisfy additional consistency requirements can be formulated in Choquet-expected utility theory, i.e. non-expected utility theory with non-additive or set-valued beliefs, together with an empirical assumption about players' attitude toward uncertainty. Chapter 1 introduces the background of this thesis. We present the conceptual problems in the foundations of game theory that motivate our approach. We then survey the decision-theoretic foundations of Choquet-expected utility theory and game-theoretic applications of Choquet-expected utility theory that are related to the present approach. Chapter 2 formulates this equilibrium concept for normal form games. This concept, called Choquet-Nash Equilibrium, is shown to be a generalization of Nash Equilibrium in normal form games. We establish an existence result for finite games, derive various properties of equilibria and establish robustness results for Nash equilibria. Chapter 3 extends the analysis to extensive games. We present the equivalent of subgame-perfect equilibrium, called perfect Choquet Equilibrium, for extensive games. Our main finding here is that perfect Choquet equilibrium does not generalize, but is qualitatively different from subgame-perfect equilibrium. Finally, in chapter 4 we examine the centipede game. It is shown that the plausible assumption of bounded uncertainty aversion leads to an 'interior' equilibrium of the centipede game

Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence

Recent advances in quantum computing and in particular, the introduction of quantum GANs, have led to increased interest in quantum zero-sum game theory, extending the scope of learning algorithms for classical games into the quantum realm. In this paper, we focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update (a generalization of the multiplicative weights update method) and its continuous analogue, Quantum Replicator Dynamics. When each player selects their state according to quantum replicator dynamics, we show that the system exhibits conservation laws in a quantum-information theoretic sense. Moreover, we show that the system exhibits Poincare recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often. Our analysis generalizes previous results in the case of classical games.Comment: NeurIPS 202

Complexity Theory, Game Theory, and Economics: The Barbados Lectures

This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some recent citations to v1 Revised v3 corrects a few typos in v

Ambiguity, Pessimism, and Religious Choice

Using a relatively mild restriction on the beliefs of the MMEU-apreference functional, in which the decision maker’s degree of ambiguity and degree of pessimism are each parameterized, we present a rather general theory of religious choice in the decision theory tradition, one that can resolve dilemmas, address the many Gods objection, and address the inherent ambiguity. Using comparative static analysis, we are able to show how changes in either the degree of ambiguity or the degree of pessimism can lead a decision maker to “convert” from one religion to another. We illustrate the theory of religious choice using an example where the decision maker perceives three possible religious alternatives.Religion; Decision Theory; Ambiguity; Optimism; Pessimism