# Lecture 14: Game Theory and Mixed Strategies Equilibrium

Two players with two strategies. You & him

**3 Types of Equilibrium** - Downward Strategy (DS) always best

**Prisoners Dilemma** - Thera is built in thereat if there are 2 periods. Thereat won't work because there is no period 3. Only when you don't know when the game ends, pacts work. A repeated game work only if there are infinite number of times. ("Long term cheating")

#### Example 1: Tree Diagram

Start at bottom of tree and work up. Player 2 will always confess so better for player 1 to confess so too. So DS is confess, confess.

#### Opposite Way

**Sub Game Perfect -** Doesn't matter who is on top of tree

**Second type of e is Nash Equilibrium** - Look for DS first

A Cournot Oligopoly is a Nash equilibrium - Stable is an outcome stable?

If play order is different, e is different. "First mover advantage."

**Mixed Strategies Equilibrium** - Last one to look for!

Confusing your opponent. Soccer. $R_1 \ ^T = R_1 \ ^ B$ probability to keep the other players indifferent between strategies.

**Example **$\rightarrow$ $R_1 \ ^T = q0+(1-q)_2 = 2 - 2q$

$R_1 \ ^B = 3q + (1-q)0 = 3q$

$2=5q$

$q=\frac{2}{5}$

$20%$$\%$ pitch, $80\%$ pickoff

So $\rightarrow$ $8 \%$ of Top left, $48 \%$ bottom right, $32 \ %$ bottom left, $12 \%$ top right.

**Asymmetric Payoff** - Ex. #9 is not symmetric to #1. If both ask some girl they get 0. Game of discoordination. Bruce is indifferent but Adam has a preference dating Carly. Adam needs to beep Bruce indifferent between left & right but he already is! So $p=\frac{1}{2}$. $q$ keeps indifferent between top & bottom. What is $q$?

- All DS are Nash - If we got DS, it is stable
- If there is a DS, there can't be a mixed strategy

Give kudos to your peers!