Existence, convergence and efficiency analysis of nash equilibrium and its applications

Abstract

Game theory deals with strategic interactions among multiple players, where each player tries to maximize/minimize its utility/cost. It has been applied in a broad array of areas such as economics, transportation, engineering, psychology, etc. Nash equilibrium, a fundamental concept in the realm of noncooperative game theory, is defined as the action profile of all players where none of the players can improve its utility/cost by a unilateral move. However, it is widely known that a Nash equilibrium often exhibits a suboptimal behavior compared with the socially optimal assignment. Moreover, in repeated games where each player makes its decision based on the available information at each stage, it is possible that the action profiles of all players fail to converge to a Nash equilibrium. This thesis presents research results on existence, convergence and efficiency analysis of Nash equilibrium in variety classes of games and their applications. First, we discuss a repeated noncooperative multiple choices congestion game in which players have limited information about each other and make their decisions simultaneously. Congestion games are a class of games in game theory, in which the utility of each player depends on the resources each player chooses and the number of players choosing the same resource. At each stage, players can calculate their best choice if they know the number of players choosing each resource. However, in most cases, each player does not know other players' strategies before it makes its decision. Therefore, a player may need to estimate the number of players choosing each resource. We introduce a consensus protocol to estimate the percentage of players selecting each resource. At each stage, each player may exchange information with its neighbors randomly and independently. We show that the congestion game under investigation has at least one pure strategy Nash equilibrium and the almost sure convergence to a pure Nash equilibrium can be ensured after some sort of inertia is adopted. Then, we apply our results to the trip timing and routing problem in traffic congestion control. By introducing different dynamic pricing schemes, the social optimum is achieved and players' choices are spread out, respectively. Simulation results based on the real traffic data of Singapore validate the effectiveness of the designed pricing schemes. Next, we analyze the efficiency loss of Nash equilibrium in a nonatomic congestion game, where there is a continuum of players, each of which is infinitesimally small. A network with one origin-destination pair is formulated, where each edge is assigned a latency function. To characterize the worst-case efficiency loss of all possible Nash flow, price of anarchy (POA), defined as the worst possible ratio between the total latency of Nash flow and that of the socially optimal flow, is adopted. In order to improve the POA, a scaled marginal-cost is designed to affect players' choices. All players in the noncooperative congestion game are divided into groups based…