Information, Control and Games Shi-Chung Chang EE-II 245, Tel: 2363-5251 ext. 245 scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm Office Hours: Mon/Wed 1:00-2:00 pm or by appointment Yi-Nung Yang (03 ) 2655201 ext. 5205, yinung@cycu.edu.tw
Mixed Strategy
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
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})
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
Linear function of p
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.
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)= q1+(1-q) (-1) = 2q-1 Player 1’s expected payoff if she chooses T E1 (T,2)= q(-1)+(1-q) 1 = 1-2q
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
Best response function
Best response in M-P game Mixed strategy equil. (1, 2) = ({1/2,1/2},{1/2,1/2})
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
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
Best response function in BoS
Best response in BoS game Mixed strategy equil. (p,q) = ({0,0},{2/3,1/3},{1,1})
Math optimization to the BoS game Player 1 max p1(2q1)+p2(q2) s.t. p1+p2=1 p10, p2 0 Z= p1(2q1)+p2(q2) + (1- p1+p2)-1p1-2p2 Player 2 does the same reasoning.