Fall Back Equilibrium

Fall back equilibrium is a refinement of the Nash equilibrium concept. In the underly- ing thought experiment each player faces the possibility that, after all players decided on their action, his chosen action turns out to be blocked. Therefore, each player has to decide beforehand on a back-up action, which he plays in case he is unable to play his primary action. In this paper we introduce the concept of fall back equilibrium and show that the set of fall back equilibria is a non-empty and closed subset of the set of Nash equilibria. We discuss the relations with other equilibrium concepts, and among other results it is shown that each robust equilibrium is fall back and for bimatrix games also each proper equilibrium is a fall back equilibrium. Furthermore, we show that for bimatrix games the set of fall back equilibria is the union of finitely many polytopes, and that the notions of fall back equilibrium and strictly fall back equilibrium coincide. Finally, we allow multiple actions to be blocked, resulting in the notion of complete fall back equilibrium. We show that the set of complete fall back equilibria is a non-empty and closed subset of the set of proper equilibria.strategic game;equilibrium refinement;blocked action;fall back equilibrium

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

Modelling interactive behaviour, and solution concepts

The final chapter of this thesis extensively studies fall back equilibrium. This equilibrium concept is a refinement of Nash equilibrium, which is the most fundamental solution concept in non-cooperative game theory.

Modelling interactive behaviour, and solution concepts.

The final chapter of this thesis extensively studies fall back equilibrium. This equilibrium concept is a refinement of Nash equilibrium, which is the most fundamental solution concept in non-cooperative game theory.

Two Species Evolutionary Game Model of User and Moderator Dynamics

We construct a two species evolutionary game model of an online society consisting of ordinary users and behavior enforcers (moderators). Among themselves, moderators play a coordination game choosing between being "positive" or "negative" (or harsh) while ordinary users play prisoner's dilemma. When interacting, moderators motivate good behavior (cooperation) among the users through punitive actions while the moderators themselves are encouraged or discouraged in their strategic choice by these interactions. We show the following results: (i) We show that the $\omega$-limit set of the proposed system is sensitive both to the degree of punishment and the proportion of moderators in closed form. (ii) We demonstrate that the basin of attraction for the Pareto optimal strategy $(\text{Cooperate},\text{Positive})$ can be computed exactly. (iii) We demonstrate that for certain initial conditions the system is self-regulating. These results partially explain the stability of many online users communities such as Reddit. We illustrate our results with examples from this online system.Comment: 8 pages, 4 figures, submitted to 2012 ASE Conference on Social Informatic

Geometry and equilibria in bimatrix games

This thesis studies the application of geometric concepts and methods in the analysis of strategic-form games, in particular bimatrix games. Our focus is on three geometric concepts: the index, geometric algorithms for the computation of Nash equilibria, and polytopes. The contribution of this thesis consists of three parts. First, we present an algorithm for the computation of the index in degenerate bimatrix games. For this, we define a new concept, the “lex-index” of an extreme equilibrium, which is an extension of the standard index. The index of an equilibrium component is easily computable as the sum of the lex-indices of all extreme equilibria of that component. Second, we give several new results on the linear tracing procedure, and its bimatrix game implementation, the van den Elzen-Talman (ET) algorithm. We compare the ET algorithm to two other algorithms: On the one hand, we show that the Lemke-Howson algorithm, the classic method for equilibrium computation in bimatrix games, and the ET algorithm differ substantially. On the other hand, we prove that the ET algorithm, or more generally, the linear tracing procedure, is a special case of the global Newton method, a geometric algorithm for the computation of equilibria in strategic-form games. As the main result of this part of the thesis, we show that there is a generic class of bimatrix games in which an equilibrium of positive index is not traceable by the ET algorithm. This result answers an open question regarding sustainability. The last part of this thesis studies the index in symmetric games. We use a construction of polytopes to prove a new result on the symmetric index: A symmetric equilibrium has symmetric index +1 if and only if it is “potentially unique”, in the sense that there is an extended symmetric game, with additional strategies for the players, where the given symmetric equilibrium is unique