**Homework number 2**

**1. **__Symmetric zero sum game __

A symmetric
two player game has *u _{1}(i,j)= u_{2}(j,i).*

For a zero sum
game matrix *A* it implies that *a _{i,j}= -a_{j,i}*.

Show that a
symmetric zero sum game has value zero.

2. __e__** -Nash:** Show, for every e > 0, an example of a two player game where there
is an e-Nash which in which both
players have a much higher payoff then in any Nash equilibrium.

3. ** Symmetric game:** Show that every symmetric game has a symmetric Nash
equilibrium.

**4. **__Action elimination:__

a. Show that in a two-player zero sum game,
elimination of actions of the MAX player can only reduce the value of the game.

b. Show an example of a general two player
game in which eliminating a given action of player 1 can increase player 1
payoff in **every Nash equilibrium**.

__The
homework is due April 20__