Essays in Pure and Applied Game Theory
Open Access
- Author:
- Hu, Tai-Wei
- Graduate Program:
- Economics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- April 28, 2009
- Committee Members:
- Kalyan Chatterjee, Dissertation Advisor/Co-Advisor
Kalyan Chatterjee, Committee Chair/Co-Chair
Neil Wallace, Committee Chair/Co-Chair
Edward James Green, Committee Member
Stephen George Simpson, Committee Member - Keywords:
- effective randomness
zero-sum game
mixed strategy
frequency theory of probability
objective probability
Kolmogorov complexity
signaling games
monetary system
cash-in-advance
counterfeiting
game theory
decision theory
expected utility - Abstract:
- This dissertation consists of three chapters. Chapter 1 presents an axiomatization of expected utility from the frequentist perspective. It starts with a preference relation on the set of infinite sequences with limit relative frequencies. We consider three axioms parallel to the ones for the vN-M expected utility theory. Limit relative frequencies correspond to probability values in lotteries in the vN-M theory. This correspondence is used to show that each of our axioms is equivalent to the corresponding vN-M axiom in the sense that the former is an exact translation of the latter. As a result, a representation theorem is established: The preference relation is represented by an average of utilities with weights given by the relative frequencies. Chapter 2 reconsiders solutions to the problem of coexistence of money and higher-return substitute of media-of-exchange using imperfect recognizability. Most of the literature on imperfect recognizability focuses on pooling equilibrium and some of it assumes a zero cost of counterfeiting. We replace the zero-cost assumption with a positive cost of counterfeiting, and place the analysis within a signalling game framework in which the intuitive criterion is invoked. With these assumptions, there is no equilibrium in which only money is traded. In particular, if the cost of counterfeiting the substitutes is small, then there is no monetary equilibrium. Therefore, the counterfeiting of substitutes for money can be a threat to monetary systems. This result provides a new rationale for legal restrictions that prohibit substitute media-of-exchange. Chapter 3 gives a new theory of mixed strategies in zero-sum games. Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated two-person zero-sum games in which the stage games have no pure strategy equilibrium. Players are endowed with sets of feasible functions to generate their strategies. Two dimensions of complexity of these feasible functions are considered: one considers the computability relation and the other considers Kolmogorov complexity. Equilibrium existence is shown under a sufficient condition called mutual complexity. A close characterization of unpredictable strategies is obtained using the criterion stochasticity. In particular, this characterization implies that the failure of some statistical properties of randomness does not justify rejection of equilibrium play.