基础博弈论大学英文讲义

Week 11: Game Theory

Required Reading: Schotter pp. 229 - 260

Lecture Plan

1. The Static Game Theory

Normal Form Games

Solution Techniques for Solving Static Games Dominant Strategy

Nash Equilibrium

2. Prisoner’s Dilemma

3. Decision Analysis

Maximim Criteria

Minimax Criteria

4. Dynamic One-Off Games

Extensive Form Games

The Sub-Game Perfect Nash Equilibrium

1. The static Game Theory

Static games: the players make their move in isolation without knowing what other players have done

1.1 Normal Form Games

In game theory there are two ways in which a game can be represented.

1st) The normal form game or strategic form game

2nd) The extensive form game

A normal form game is any game where we can identity the following three things:

1. Players:

2. The strategies available to each player.

3. The Payoffs. A payoff is what a player will receive at the end

of the game contingent upon the actions of all players in the game.

Suppose that two people (A and B) are playing a simple game. A will write one of two words on a piece of paper, “Top” or “Bottom”. At the same time, B will independently write “left” or “right” on a piece of paper. After they do this, the papers will be examined and they will get the payoff depicted in Table 1.

Table 1

If A says top and B says left, then we examine the top-left corner of the matrix. In this matrix, the payoff to A(B) is the first(Second) entry in the box. For example, if A writes “top” and B writes “left” payoff of A = 1 payoff of B = 2.

What is/are the equilibrium outcome(s) of this game?

1.2

Nash Equilibrium Approach to Solving Static Games

Nash equilibrium is first defined by John Nash in 1951 based on the work of Cournot in 1893.

A pair of strategy is Nash equilibrium if A's choice is optimal given B's choice, and B's choice is optimal given A's choice. When this equilibrium outcome is reached, neither individual wants to change his behaviour.

Finding the Nash equilibrium for any game involves two stages.

1) identify each optimal strategy in response to what the other players might do.

Given B chooses left, the optimal strategy for A is

Given B chooses right, the optimal strategy for A is

Given A chooses top, the optimal strategy for B is

Given A chooses bottom, the optimal strategy for B is

We show this by underlying the payoff element for each case.

2) a Nash equilibrium is identified when all players are player their optimal strategies simultaneously

In the case of Table 2,

If A chooses top, then the best thing for B to do is to choose left since the payoff to B from choosing left is 1 and the payoff from choosing right is 0. If B chooses left, then the best thing for A to do is to choose top as A will get a payoff of 2 rather than 0.

Thus if A chooses top B chooses left. If B chooses left, A chooses top. Therefore we have a Nash equilibrium: each person is making optimal choice, given the other person's choice.

If the payoff matrix changes as:

Table 2