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. R1 T=R1 BR_1 \ ^T = R_1 \ ^ B probability to keep the other players indifferent between strategies.

Example \rightarrow R1 T=q0+(1q)2=22q R_1 \ ^T = q0+(1-q)_2 = 2 - 2q

R1 B=3q+(1q)0=3qR_1 \ ^B = 3q + (1-q)0 = 3q

2=5q2=5q

q=25q=\frac{2}{5}

2020%%\% pitch, 80%80\% pickoff

So \rightarrow 8%8 \%  of Top left, 48%48 \% bottom right, 32 32 \ % bottom left, 12%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=12p=\frac{1}{2}. qq keeps indifferent between top & bottom. What is qq?

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




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