1. Mixed Strategies and Repeated Games
Game Theory 2
Mixed Strategies
and
Repeated Games

2. Mixed Strategies
Mixed Strategies: Players make random choices among two or more options based on sets of chosen probabilities
Pure Strategies: Players make specific choices among two or more options
Some games may have no Nash Equilibria in Pure Strategies
Every game has at least one Nash Equilibrium when mixed strategies are allowed
Examples
Matching Pennies
The Battle of the Sexes

3. Matching Pennies
Player B
Heads
Tails
Player A
1, -1
-1, 1

4. Mixed Strategies: Matching Pennies
Heads ½, Tails ½
¼ probability of getting any outcome
Expected payoff is zero for each player
Any other probabilities give at least one player a worse payoff
Example: Heads ¾ , Tails ¼
Probabilities
Heads, Heads 9/16
Tails, Tails 1/16
Heads, Tails 6/16
Payoffs
Player A: + ¼
Player B: - ¼

5. Battle of the Sexes
Jim
Wrestling
Opera
Joan
2, 1
0, 0
1, 2

6. Mixed Strategies: Battle of the Sexes
Nash Equilibrium
Jim: Wrestling 1/3 , Opera 2/3
Joan: Wrestling 2/3 , Opera 1/3
Expected Payoffs: 2/3
Proof
Say Jim chooses W 1/3, O 2/3 as above
Joan chooses W with probability p, O with (1-p)
Both choose O: 2/3(1-p)
Both Choose W: 1/3 p
Joan’s expected payoff:
2(1/3 p) + 1(2/3)(1-p) = (2/3)p + 2/3 – (2/3) p = 2/3
Independent of p: Joan cannot improve her payoff

7. Battle of the Sexes: Examples
Suppose Jim chooses W ½ , O ½
Joan chooses W p, O (1-p)
Joan’s Payoff: 2(1/2)p + 1(1/2)(1-p) = ½ p +1/2
Joan would choose p = 1 to max her payoff
When Joan chooses p = 1, Jim is better off with W 1, O 0
Suppose Jim chooses W ¼ , O ¾
Joan’s payoff: 2(1/4)p + 1(3/4)(1-p) = ½ p + ¾ - ¾ p
= ¾ - ¼ p
Joan chooses p = 0 to max her payoff
Jim is better off then to choose W = 0, O = 1
Pure Strategies preferred to all other mixed strategies

8. Why Mixed Strategies?
Mixed Strategy expected payoff is zero in Matching Pennies
In Battle of the Sexes, Pure Strategy equilibria yield higher payoffs
Possibilities
Equilibria when pure strategies fail (poker, for instance)
Players are risk lovers
Often not realistic
Would firms really expect a rival to randomly
Set prices
Choose advertising strategies
Enter or exit markets

9. Repeated Games Low Price High Price 10,10 100, -50 -50, 100 50, 50
Consider
Repeating
A Prisoners’
Dilemma
Game
Firm
Low Price 2
High Price
Firm 1
10,10
100, -50
-50, 100
50, 50

10. Infinitely Repeated Games
Strategies depend on present values of payoffs
Suppose the discount rate is 10%. The payoff for any player is
Π = π1 + π2/(1.1) + π3/(1.1)2 + π4/(1.1)3…..
Note: 1 + d + d2 + d3 +…. = 1/(1- d); d + d2 + d3 +…. = d/(1- d)
=> 1 + 1/(1.1) + 1/(1.1)2 + 1/(1.1)3… = 1/[1- (1/1.1)] = 11
Payoffs for infinitely repeated pricing game
Collusion forever: 50(11) = 550
Nash forever: 10(11) = 110
Cheat today, Nash after: (10) = 200
Defect from Nash, collude after: (10) = 450

11. The Tit for Tat Strategy
A player plays the same strategy played by its rival in the previous period
Collusion (cooperation) is the Nash Equilibrium when one player plays tit-for-tat with certainty
Collusion (cooperation) can be the Nash Equilibrium even if there is uncertainty about a player playing tit-for-tat
Players may also cycle through Collude, Cheat, Defect, Collude outcomes when one player plays tit-for-tat
Applies even for some (sufficiently low) probability that the game will end in the next period

12. Finitely Repeated Games
When a game has a known last period
Cheating always pays in the last period
Both players know this, such that whoever cheats first gets a higher payoff
The game “unravels” from the end backwards to the beginning
Consequently, the Low Price equilibrium is the Nash Equilibrium
If the end of the game is probabilistic (not known with certainty), High Price (cooperation) can be a Nash Equilibrium.