Download presentation

Presentation is loading. Please wait.

Published byKendra Hamby Modified over 3 years ago

1
Simultaneous- Move Games with Mixed Strategies Zero-sum Games

2
Mixed Strategy Random choice of the pure strategies Pure strategy of probability distribution Suppose Player one has 2 strategies, A and B. Let p in [0,1] be the probability for Player one to play A, then 1-p is the probability to play B. When p=1, one plays A purely and when p=0, one plays B purely.

3
Expected payoff Suppose E plays DL with p=0.75, when N plays DL, E ’ s expected payoff is 0.75x50+0.25x90=60. Suppose N also plays mixed strategies with prob. q=0.2 for playing DL. Then E ’ s expected payoff is 0.2x50+0.8x80=74 for playing DL 0.2x90+0.8x20=34 for CC. 0.75x74+0.25x34=63.66 for mixed with p=0.75 Navratilova DLCC Evert DL50, 5080, 20 CC90, 1020, 80

4
For any p, E ’ s expected payoff is 50p+90(1-p)=90-40p when N plays DL 80p+20(1-p)=20+60p when N plays CC Note 90-40p>20+60p when p<0.7

5
Minimax Method p E ’ s payoff N ’ s DL N ’ s CC 0.7 90 20

6
If N mixes her strategies, E ’ s payoff is between two lines. For p<=0.7, N playing CC purely will minimize E ’ s payoff (Note E can guarantee a payoff equal to 20+60p) For p>=0.7, N playing DL purely will minimize E ’ s payoff (Note E can guarantee a payoff equal to 90-40p) E can choose p=0.7 to maximin to get 62 no matter N picks DL or CC

7
q N ’ s payoff E ’ s CC, N ’ s payoff=80-70q E ’ s DL, N ’ s payoff=20+30q 0.6 80 20

8
Maximin strategy (p, q)=(0.7,0.6) With N playing q=0.6, E ’ s expected payoff is 50x.6+80x.4=62 for playing DL purely 90x.6+20x.4=62 for playing CC purely With E playing p=0.7, N ’ s expected payoff is 50x.7+10x.3=38 for playing DL purely 20x.7+80x.3=38 for playing CC purely

9
N.E. in mixed strategy Theorem If a player would mix two or more strategies as the N.E strategy, the expected payoffs from playing those strategies purely (given opponents ’ equilibrium strategies) should be the same, as the equilibrium payoff under the mixed strategy.

10
DLCCq-mix DL50, 5080, 20 50q+80(1-q), 50q+20(1-q) CC90, 1020, 80 90q+20(1-q), 10q+80(1-q) p-mix 50p+90(1-p), 50p+10(1-p) 80p+20(1-p), 20p+80(1-p)

11
For Evert 50q+80(1-q)>90q+20(1-q) if q<0.6 Pure DL (p=1) when q<0.6 50q+80(1-q) 0.6 Pure CC (p=0) when q>0.6 50q+80(1-q)=90q+20(1-q) if q=0.6 Any mix (p=0~1) when q=0.6

12
For Navratilova 50p+10(1-p)>20p+80(1-p) if p>0.7 Pure DL (q=1) when p>0.7 50p+10(1-p)<20p+80(1-p) if p<0.7 Pure CC (q=0) when p<0.7 50p+10(1-p)=20p+80(1-p) if p=0.7 Any mix (q=0~1) when p=0.7

13
p q 0.6 1 0 1 E ’ s best response N ’ s best response N.E. (0.7, 0.6) 0.7

14
Follow the theorem, E would mix if 50q+80(1-q)=90q+20(1-q) or q=0.6, so that she ’ s indifferent between DL and CC (both 62). And N would mix if p=0.7. Opponent ’ s indifference property. N.E. as a system of beliefs. Consider mixed strategy, then N.E. will be the same as minimax method.

15
When one has 3 strategies DLCCq-mix DL5080 50q+80(1 -q) CC9020 90q+20(1 -q) Lob7060 70q+60(1 -q) p-mix 50p1+90p 2+70(1- p1-p2) 80p1+20p 2+60(1- p1-p2)

16
E will mix the three if there ’ s a q such that 50q+80(1-q)= 90q+20(1- q)= 70q+60(1-q). However the answer is NO. It means E will only mix the two out of the three.

17
Case1: E mixes DL & CC only. From previous argument p=0.7 and q=0.6, and payoffs are 62 for E, 38 for N. When playing Lob purely with q=0.6, expected payoff=66, thus E will deviate. (0.7, 0.3, 0) cannot be E ’ s equilibrium strategy.

18
Case2: E mixes DL & Lob only. E will mix the two when 50q+80(1- q)=70q+60(1-q) or q=0.5. For N to mix, when 50p+30(1- p)=20p+40(1-p), or p=0.25. Payoffs are 65 and 35 respectively When playing CC purely with q=0.5, expected payoff=55, thus E will NOT deviate. [(0.25, 0, 0.75), (0.5, 0.5)] is a N.E.

19
Case3: E mixes CC & Lob only. E will mix the two when 90q+20(1- q)=70q+60(1-q) or q=2/3. For N to mix, when 10p+30(1- p)=80p+40(1-p), never. Actually N will play CC purely since CC dominates DL when E mixes only CC & Lob. In equilibrium, E will not just mix CC & Lob.

20
Homework 1, Question 3 of Exercise 2. Find the N.E in the following strategic form game LMR U1, 33, 41, 0 D0, 52, 23, 6

Similar presentations

OK

Midterm Scores Total of 200 points, 40 per question. A--155-200 B—120-154 C—80-119 D—50-79 F

Midterm Scores Total of 200 points, 40 per question. A--155-200 B—120-154 C—80-119 D—50-79 F

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on democracy and equality in indian Ppt on water activity meter Ppt on shell structures architecture Ppt on low level language list Ppt on vitamin d deficiency in india Ppt on nuclear power plants in india Ppt on places in our neighbourhood clip Ppt on malignant bone tumors Ppt on fibonacci numbers and nature Ppt on unix shell scripting