Exam 16

Exercise 16. Consider the following bimatrix game:

$$C_1 =\left\lbrack \begin{array}{ccc} 9 & 7 & 3\\ 8 & 4 & 1\\ 5 & 3 & 2 \end{array}\right\rbrack$$

$$C_2 =\left\lbrack \begin{array}{ccc} 8 & 3 & 1\\ 9 & 6 & 5\\ 1 & 4 & 2 \end{array}\right\rbrack$$

close all;
clear;
clc;

matlab.lang.OnOffSwitchState = 1;

16.1 Find strictly dominated strategies (if any) and reduce the cost matrices accordingly.

This game is two-person non-cooperative game where the sets of pure strategies are finite, hence the sets of mixed stratergies are $X=\left\lbrace x;\epsilon ;\Re^m :x\ge 0,\sum_{i=1}^m x_i =1\right\rbrace ;$ and $Y=\left\lbrace y;\epsilon ;\Re^n :y\ge 0,\sum_{j=1}^n y_j =1\right\rbrace$ . $f_{2;} \not= -f_1$ (non-zero sum game), the cost functions are $f_1 \left(x,y\right)=x^T C_1 y$ and $f_2 \left(x,y\right)=x^T C_2 y$ , where $C_1$ and $C_2$ are $m\times n$ matrices.

16.2 Do pure strategies Nash equilibria exist? Why?

16.3 Evaluate the gap function at the point (x, y), where x = (1/3, 2/3) and y = (1/4, 3/4), in order to check if it is a Nash equilibrium.

16.4 Plot the polyhedra P and Q related to the game and find all the mixed strategies Nash equilibria.

16.5 Find the KKT multipliers related to the mixed strategies Nash equilibrium.