# 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