1. Information, Control and Games
Shi-Chung Chang
EE-II 245, Tel: ext. 245
Office Hours: Mon/Wed 1:00-2:00 pm or by appointment
Yi-Nung Yang
(03 ) ext. 5205,

2. Mixed Strategy

3. Mixed Strategy
Definition: A mixed strategy of a player in a strategic game is a probability distribution over the player’s actions.
i(ai) : the probability assigned by player i’s mixed strategy i to her actions ai.
Pure strategy i(ai) =1

4. Matching Pennies p=1(Head), (1-p)= 1(Tail)
q=2(Head), (1-q)= 2(Tail)
Mixed Strategy pair (1, 2) = ({p,1-p},{q,1-q})

5. Expected Payoffs For player 1 to the strategy pair (1, 2)
if she play T, the expected payoff is E1(T,2)=[q  u1(T,L)+ (1-q) u1(T,R)]
if she play B, the expected payoff is E1(B,2)=[q  u1(B,L)+ (1-q) u1(B,R)]
Player 1’s expected payoff pE1 (T,2)+(1-p) E1(B,2)
linear function of p

6. Linear function of p

7. Best Response
If E1 (T,2)>E1(B,2), player 1 chooses T (pure strategy)
If E1 (T,2)<E1(B,2), player 1 chooses B (pure strategy)
If E1 (T,2)=E1(B,2), all mixed strategies (p,1-p) yields the same expected payoff.

8. Matching Pennies No Nash equil. in pure strategy game
Given player 2’s mixed strategy
Player 1’s expected payoff if she chooses H E1 (H,2)= q1+(1-q) (-1) = 2q-1
Player 1’s expected payoff if she chooses T E1 (T,2)= q(-1)+(1-q) 1 = 1-2q

9. Best Response in Matching Pennies
If 2q-1<0 (q<1/2), E1 (H,2)< E1 (T,2) Player 1 should choose T, i.e., p=0
If 2q-1>0 (q>1/2), E1 (H,2)> E1 (T,2) Player 1 should choose H, i.e., p=1
If 2q-1=1-2q (q=1/2), E1 (H,2)= E1 (T,2) Player 1 either one, i.e., p=1/2

10. Best response function

11. Best response in M-P game
Mixed strategy equil. (1, 2) = ({1/2,1/2},{1/2,1/2})

12. Mixed Strategy Nash Equil.
* is a mixed strategy Nash equilibrium if for each player i and every mixed strategy i of player i , the expected payoff to player i of i is at least as large as the expected payoff to player i of (i -i) Ei (*) Ei (i,-i) for every mixed strategy i of player i

13. Mixed Strategy in BoS game
Player 1’s expected payoff
to B: 2  q+0(1-q)=2q
to S: 0  q+1(1-q)=1-q
Best response
if 2q>1-q (q>1/3), then she should choose B
if 2q<1-q (q<1/3), then she should choose S

14. Best response function in BoS

15. Best response in BoS game
Mixed strategy equil. (p,q) = ({0,0},{2/3,1/3},{1,1})

16. Math optimization to the BoS game
Player 1 max p1(2q1)+p2(q2) s.t. p1+p2=1 p10, p2  0 Z= p1(2q1)+p2(q2) + (1- p1+p2)-1p1-2p2
Player 2 does the same reasoning.