EE 228 a - Lecture 18 - Spring 2006 Non - Cooperative Games Scribed by Libin Jiang

This lecture consists of two parts. The first part is on comparing and choosing from Nash Equilibria (NE), when there are multiple NE in a game. The concepts to be covered include Pareto, Minimax and Risk-dominant. In the second part, we will apply game theory to three networks problems: Service Differentiation, Multi-Provider Network, and WiFi Pricing. It is shown that with game theoretical approaches, networks can be improved for the interest of both providers and users. I. Comparing Nash Equilibria In some games, there may be more than one Nash Equilibria. For examples in Fig 1, each game has two Nash Equilibria. So, there is a problem of comparing and choosing from different Nash Equilibria. Fig. 1 Multiple Nash Equilibria may exist Pareto There are two Nash Equilibria in Fig 2. Comparing the two, we have the following observations. Both players are strictly better off with the NE (T, L) than with the NE (B, R). So they can safely assume that (T, L) will be played (if they are both rational). Here, (T, L) is the unique Pareto NE. An NE is a ”Pareto NE” if the vector of rewards is not dominated component-wise by a different NE. Minimax In Fig 3, both (T, L) and (B, R) are Pareto: There is no other equilibrium where both players improve their rewards. In this case, player 1 does not know which Pareto NE player 2 will pick, and vice versa. As a result, both players maximize their minimum rewards. So, player 1 will pick B (the minimum reward when choosing B is 2, larger than 1, the minimum reward if choosing T); similarly, player 2 will pick R. The result is called ”Minimax”. Here, (B, R) happens to be a NE. Fig. 2 Pareto Fig. 3 Minimax