Nash Equilibrium and Robust Stability in Dynamic Games: A Small-Gain Perspective

This paper develops a novel methodology to study robust stability properties of Nash equilibrium points in dynamic games. Small-gain techniques in modern mathematical control theory are used for the first time to derive conditions guaranteeing uniqueness and global asymptotic stability of Nash equilibrium point for economic models described by functional difference equations. Specification to a Cournot oligopoly game is studied in detail to demonstrate the power of the proposed methodology

The anatomy of a business game

We describe in detail a business game, which has been used extensively in education for a decade. Although the business game is smaller than other games, it is fairly realistic as it includes decisions on investments, production, prices, and advertising. Furthermore, the game has dynamic properties, in that decisions and financial states of the firms carry over from one period to the next. There are not many such detailed descriptions of business games, although this is in demand. Such a complete mathematical description lays ground not only for alterations of the game, but also for developments of new games. It can also provide a link to models used in micro-economic theory.Business games; learning; microeconomics; modeling; science of simulation/gaming

Combinatorial and computational aspects of multiple weighted voting games

Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting weight vote in favour of or against a decision. A coalition of agents is winning if and only if the sum of weights of the coalition exceeds or equals a specified quota. We provide a mathematical and computational characterization of multiple weighted voting games which are an extension of weighted voting games1. We analyse the structure of multiple weighted voting games and some of their combinatorial properties especially with respect to dictatorship, veto power, dummy players and Banzhaf indices. Among other results we extend the concept of amplitude to multiple weighted voting games. An illustrative Mathematica program to compute voting power properties of multiple weighted voting games is also provided

Nash Equilibrium and Robust Stability in Dynamic Games: A Small-Gain Perspective

This paper develops a novel methodology to study robust stability properties of Nash equilibrium points in dynamic games. Small-gain techniques in modern mathematical control theory are used for the first time to derive conditions guaranteeing uniqueness and global asymptotic stability of Nash equilibrium point for economic models described by functional difference equations. Specification to a Cournot oligopoly game is studied in detail to demonstrate the power of the proposed methodology.Dynamic game; Cournot oligopoly; Nash equilibrium; Robust stability; Small gain

Automata games for multiple-model checking

3-valued models have been advocated as a means of system abstraction such that verifications and refutations of temporal-logic properties transfer from abstract models to the systems they represent. Some application domains, however, require multiple models of a concrete or virtual system. We build the mathematical foundations for 3-valued property verification and refutation applied to sets of common concretizations of finitely many models. We show that validity checking for the modal mu-calculus has the same cost (EXPTIME-complete) on such sets as on all 2-valued models, provide an efficient algorithm for checking whether common concretizations exist for a fixed number of models, and propose using parity games on variants of tree automata to efficiently approximate validity checks of multiple models. We prove that the universal topological model in [M. Huth, R. Jagadeesan, and D. A. Schmidt. A domain equation for refinement of partial systems. Mathematical Structures in Computer Science, 14(4):469-505, 5 August 2004] is not bounded complete. This confirms that the approximations aforementioned are reasonably precise only for tree-automata-like models, unless all models are assumed to be deterministic. © 2006 Elsevier B.V. All rights reserved

Extended Nonlocal Games

The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.Comment: PhD thesis, Univ Waterloo, 2017. 151 pages, 11 figure

A Framework for Studying Economic Interactions (with applications to corruption and business cycles)

Most economic models implicitly or explicitly assume that interactions between economic agents are 'global' - in other words, each agent interacts in a uniform manner with every other agent. However, localized interactions between microeconomic agents are a pervasive feature of reality. What are the implications of more limited interaction? One set of mathematical tools which appears useful in exploring the economic implications of local interactions is the theory of interacting particle systems. Unfortunately, the extant theory mainly addresses the long-time behavior of infinite systems, and focuses on the issue of ergodicity; many economic applications involve a finite number of agents and are concerned with other issues, such as the extent of shock amplification. In this paper, I introduce a framework for studying local interactions that is applicable to a wide class of games. In this framework, agents receive shocks which are stochastically independent; payoffs depend both upon the shocks and the strategies of other agents. In finite games, ergodicity is straightforward to determine. In finite games which evolve in continuous time, the stationary distribution (if it exists) may be computed easily; furthermore, in this class of games, I prove that any stationary distribution may be attained by suitable choice of payoff functions using shocks which are distributed uniform on (0, 1). In systems in which all interactions are global, I prove that nonlinear behavior can arise even in the infinite limit (thus demonstrating that laws of large numbers can fail in systems characterized by interaction), despite the fact that the only driving forces are agent-level iid disturbances. Using numerical methods, I investigate the properties of the processes as one passes from discrete to continuous time, as one alters the pattern of interaction, and as one increases the number of interacting agents. In so doing, I provide further evidence that the existence of local interactions can change the aggregate behavior of an economic system in fundamental ways, and that the form of that interaction has important implications for its dynamic properties.

Mathematical models of games of chance: Epistemological taxonomy and potential in problem-gambling research

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction