2. Static Games with Complete Information 2.3 Simultaneous-move game (SMG) with Mixed Strategies – Zero-sum Games There was no NE in some of the games we investigated. To predict an outcome in such games, we need to expand out definition of strategy and equilibrium concepts.  introduce the concept of random strategy. Pure Strategy (단순전략) 1/1 3개 중 항상 1개만 선택하는 전략 Ex) 가위를 100%로 언제나 선택하는 전략 Expand the search. To do so, strategy and equilibrium concepts need to be expanded as well If you can’t find a NE in here, Mixed Strategy (혼합전략) 3개 중 아무거나 1개 선택하되, 확률 배정해 무작위 선택하는 전략, 실제 선택되는 전략은 매번 변화 될 수 있음 Ex) 1/3, 1/3, 1/3 1/3 1/3 1/3

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.1 On Mixed Strategy A random strategy is needed when strategy by coincidence is preferred as in the tennis match(어떤 전략도 우위에 있지 않음. 따라서 무작위로 전략을 선택하게됨). This random(무작위적, 임의적) mix of pure strategy is called the ‘mixed strategy (혼합전략).’ Pure Strategy(단순전략) is a special case of Mixed Strategy; In a tennis match, if DL is chosen 100% (probability=1), DL is selected as a pure strategy (or CC is not selected as a pure strategy). Ex) 단순전략 DL을 선택한다는 것 = DL 선택확률 100% + CC 선택확률 0% If DL is selected 0%, DL is not selected as a pure strategy (or CC is selected as a pure strategy). Thus, Pure Strategy ⊂ Mixed Strategy choose scissors(pure strategy) = choose scissors with probability of 1 (mixed strategy)

Example of mixed strategy, extending Pure Strategy concept 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.1 On Mixed Strategy (cont’d) Example of mixed strategy, extending Pure Strategy concept Pure Strategy of playing DL DL과 CC를 50:50으로 섞는 전략 Pure Strategy of playing CC Thus, Pure Strategy ⊂ Mixed Strategy

Extending Payoff: Expected Payoff If Head  100, if Tail  0 Expected payoff = Prob. of Head*Payoff + Prob. of Tail*Payoff = 0.5*100 + 0.5*0 = 50 Head Tail

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.1 On Mixed Strategy (cont’d) There are infinitely many possible combinations. Ex) DL 1, CC 0; DL 0.75, CC 0.25; DL 0.5, CC 0.5; DL 0.25, CC 0.75; ...; DL 0, CC 1 If the probability of DL selected is p, the prob. of CC selected is (1- p). The payoffs are represented as a weighted sum Ex) The chance for Evert to win, when Evert uses DL (offense) and Nav uses DL (defense), is 50%. The chance for Evert to win, when Evert uses CC (offense) and Nav uses DL (defense), is 90%.  What is Evert’s expected winning chance when Evert uses DL 75% and CC 25%? 4-17-19

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.1 On Mixed Strategy (cont’d) Evert’s mixed strategy in reaction to Nav’s DL strategy is (DL 75%, CC 25%). Then, the payoff for Evert = 50*0.75 + 90*0.25 = 37.4 + 22.5 = 60. That is, when Nav takes DL and Evert uses a mixed strategy of DL 75% and CC 25%, the expected payoff (winning chance) for Evert is 60%. The mixed strategy is a special case of the continuous strategy. Thus, Pure strategy⊂Mixed strategy⊂Cont.Strategy

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.1 On Mixed Strategy (cont’d) The concept of NE can be extended easily. Definition of NE with Mixed Strategy: Given other players’ mixed strategy, the list of mixed strategy that maximizes the expected payoffs simultaneously. In other words, given p-mix(probability mix; 확률혼합) of the other player(s), NE is defined as q-mix (probability mix) that maximizes own expected payoff. In the famous theorem of Nash, he proves that such equilibrium always exists in any game.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis Mixed Strategy is a special case of Continuous strategy; Tennis match will be used as an example. In the tennis game, Chapter 2, No NE (click, see table below). Assume Evert and Nav use mixed strategy. Let the probability for Evert to choose DL be p, the probability for Nav to choose DL be q. Then, the prob. for Evert to choose CC is (1- p) and the prob. for Nav to choose CC is (1- q). Modifying the table below, we get a new table in the next slide. Nav(Defense) DL CC Evert (Offense) 50, 50 80, 10 90, 20 20, 80 Jump to the Figure 2 Pure Strategies

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) When Nav uses DL 100% (q=1) and Evert uses the mixed strategy of DL (prob. p) and CC (prob. (1- p)), the expected payoff for Evert is A=50*p + 90*(1-p). Similarly, when Nav uses DL 100% and Evert uses the mixed strategy of DL (prob. p) and CC (prob. (1- p)), the expected payoff for Nav is B = 50*p + 10*(1-p). Navratilova (Nav) DL (q=1) CC (1-q=1) q-mix Evert DL (p=1) 50, 50 80, 20 E, F CC (1-p=1) 90, 10 20, 80 G, H p-mix Ex) p=0.7, 1-p=0.3 A, B C, D DL: p, CC:1- p

The number in ( ) is the prob. Of the strategy 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) When Nav uses CC 100% and Evert uses the mixed strategy of DL (prob. p) and CC (prob. (1- p)), the expected payoff for Evert is C = 80*p + 20*(1-p). Similarly, when Nav uses CC 100% and Evert uses the mixed strategy of DL (prob. p) and CC (prob. (1- p)), the expected payoff for Nav is D = 20*p + 80*(1-p). The number in ( ) is the prob. Of the strategy Nav DL(q=1) CC(1-q=1) q-mix Evert DL(p=1) 50, 50 80, 20 E, F CC(1-p=1) 90, 10 20, 80 G, H p-mix 50*p + 90*(1-p), 50*p + 10*(1-p) C, D pure strategy mixed strategy 2 Pure Strategies + 1 Mixed Strategy

The number in ( ) is the prob. Of the strategy 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) When Evert uses DL 100% and Nav uses the mixed strategy of DL (prob. q) and CC (prob. (1- q)), the expected payoff for Evert is E = 50*q + 80*(1-q). Similarly, when Evert uses DL 100% and Nav uses the mixed strategy of DL (prob. q) and CC (prob. (1- q)), the expected payoff for Nav is F = 50*q + 20*(1-q). The number in ( ) is the prob. Of the strategy Nav DL(q) CC(1-q) q-mix Evert DL(p=1) 50, 50 80, 20 E, F CC(1-p=1) 90, 10 20, 80 G, H p-mix 50*p + 90*(1-p), 50*p + 10*(1-p) 80*p + 20*(1-p), 20*p + 80*(1-p)

The number in ( ) is the prob. Of the strategy 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Similarly, G = 90*q + 20*(1-q) H = 10*q + 80*(1-q) The number in ( ) is the prob. Of the strategy Nav DL(q) CC(1-q) q-mix Evert DL(p) 50, 50 80, 20 50*q + 80*(1-q), 50*q + 20*(1-q) CC(1-p) 90, 10 20, 80 G = 90*q + 20*(1-q), H = 10*q + 80*(1-q) p-mix 50*p + 90*(1-p), 50*p + 10*(1-p) 80*p + 20*(1-p), 20*p + 80*(1-p) NEXT

p[50q+80(1-q)]+(1-p)[90q+20(1-q)] p[50q+20(1-q)]+(1-p)[10q+80(1-q)] 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) When both use mixed strategies, Evert’s payoff is determined not only by her own p-mix but also by Nav’s q-mix. Ex) When Evert chooses DL and Nav chooses q-mix, Evert’s expected payoff is 50*q + 80*(1-q), which is a function of Nav’s q-mix (mix of q and 1-q). The number in ( ) is the prob. Of the strategy Nav DL(q) CC(1-q) q-mix Evert DL(p) 50, 50 80, 20 50*q + 80*(1-q), 50*q + 20*(1-q) CC(1-p) 90, 10 20, 80 G = 90*q + 20*(1-q), H = 10*q + 80*(1-q) p-mix 50*p + 90*(1-p), 50*p + 10*(1-p) 80*p + 20*(1-p), 20*p + 80*(1-p) p[50q+80(1-q)]+(1-p)[90q+20(1-q)] p[50q+20(1-q)]+(1-p)[10q+80(1-q)]

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Evert’s Payoff when Evert and Nav use mixed strategies = p[50q+80(1-q)]+(1-p)[90q+20(1-q)] Nav’s Payoff when Evert and Nav use mixed strategies = p[50q+20(1-q)]+(1-p)[10q+80(1-q)] For Evert to maximize her payoff, differentiate w.r.t. p. For Nav to maximize her payoff, differentiate w.r.t. q. ∂πEvert/ ∂p = 60 – 100q=0  q=0.6 (Evert’s BR fn.?) ∂πNav/ ∂q = -70 + 100p=0  p=0.7 (Nav’s BR fn.?) NE is Evert playing mixed strategy of DL 70%, Nav playing mixed strategy of DL 60%  Evert plays DL 70% and Nav blocks DL 60%. More intuitive version? NEXT.

p[50q+80(1-q)]+(1-p)[90q+20(1-q)] p[50q+20(1-q)]+(1-p)[10q+80(1-q)] 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) What is Nav’s pure strategy BR (DL or CC) when Evert chooses p-mix? Choose DL if B>D and CC if B<D. What if Evert always chooses CC (p=0)? The wining rate of Nav with CC is 80(=20*0+80(1-0)). With DL, it is 10(=50*0+10*(1-0)). Thus take CC (∵ 80 > 10). The number in ( ) is the prob. Of the strategy Nav DL(q) CC(1-q) q-mix Evert DL(p) 50, 50 80, 20 E=50*q + 80*(1-q), F=50*q + 20*(1-q) CC(1-p) 90, 10 20, 80 G = 90*q + 20*(1-q), H = 10*q + 80*(1-q) p-mix A=50*p + 90*(1-p), B=50*p + 10*(1-p) C=80*p + 20*(1-p), D=20*p + 80*(1-p) p[50q+80(1-q)]+(1-p)[90q+20(1-q)] p[50q+20(1-q)]+(1-p)[10q+80(1-q)]

Will be explained with figures 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) What if Evert always takes DL (p=1)? The winning rate of Nav with CC is 20(=20*1+80(1-1)) and with DL, it is 50(=50*1+10*(1-1)). Thus, take DL(20 < 50). Of course… If Evert always play DL, Nav will always cover DL. In a figure, next. The number in ( ) is the prob. Of the strategy Nav DL(q) CC(1-q) q-mix Evert DL(p) 50, 50 80, 20 E=50*q + 80*(1-q), F=50*q + 20*(1-q) CC(1-p) 90, 10 20, 80 G = 90*q + 20*(1-q), H = 10*q + 80*(1-q) p-mix A=50*p + 90*(1-p), B=50*p + 10*(1-p) C=80*p + 20*(1-p), D=20*p + 80*(1-p) Will be explained with figures

2. Static Games with Complete Information) 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Nav’s BR curve (Evert의 전략선택 확률에 따른 기대승률) Nav’s Expected Payoff In this range, DL(q=1) is the BR for Nav. (why?) Nav’s Expected Payoff In this range, CC(q=0) is the BR for Nav. (why?) 80 10 Nav’s Expected Payoff when Nav takes CC 50*p + 10*(1-p) 50 20 Nav’s Expected Payoff when Nav takes DL 20*p + 80*(1-p) Jump to Payoff Table p=0 p=1 Evert’s p-mix

2. Static Games with Complete Information) 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Nav’s BR curve? if p<0.7, choose CC (q=0); if p>0.7, choose DL (q=1); if p=0.7, choose any.) Nav’s Expected Payoff Nav’s Expected Payoff 80 38 10 50 20 Nav’s Expected Payoff when Nav takes DL Nav’s Expected Payoff when Nav takes CC p=0 p=0.7 p=1 Evert’s p-mix 50p+10(1-p)=20p+80(1-p)  p=0.7

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) At p=0.7, Nav is indifferent to either DL or CC because they lead to the same payoff to Nav. In words, if Evert’s prob. to take DL is lower than 0.7, Nav wants to take CC. Why isn’t the cross point 0.5 but 0.7? The winning rate is not symmetric.; Nav’s winning rate is 50% when Evert plays DL and Nav plays DL. But Nav’s winning rate is 80% (relatively higher than 50%) when Evert plays CC and Nav plays CC. That is, Nav plays CC better(Nav가 왼손잡이니까 CC 더 잘 막음). Therefore, the prob. that Evert plays DL (p=0.7) is higher than 0.5.

2. Static Games with Complete Information) 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Same result: Evert’s BR curve to her own p-mix? Evert’s Expected Payoff Use these pairs to get p* Evert’s Expected Payoff Use these pairs to get q* Evert’s Expected Payoff when Nav plays DL  50*p+90*(1-p) 90 62 20 Evert’s BR curve to her own p-mix Right-upward BR  If Nav plays CC, Evert can increase her expected payoff by playing DL more often (increased p). Right-downward BR  If Nav plays DL, Evert plays less DL. 80 50 Evert’s Expected Payoff when Nav plays CC  80*p+20*(1-p) Nav’s BR curve to Evert’s p-mix p=0 p=0.7 p=1 Evert’s p-mix *Jump to organized BR curves

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2. Static Games with Complete Information) 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Evert’s BR curve to Nav’s q-mix? if q<0.6, play DL (p=1); if q>0.6, play CC(p=0); if q=0.6, play any. Evert’s Expected Payoff Evert’s Expected Payoff Evert’s Expected Payoff when she plays DL  50*q + 80*(1-q) 90 50 If Nav plays DL (q=1), Evert plays CC. 80 20 If Nav plays CC (if q is low), Evert plays DL. If Evert plays DL, it is more advantageous for Nav to increase q (prob. Nav plays DL). If Evert plays CC, it is more advantageous for Nav to decrease q (prob. Nav plays DL). Evert’s wining rate is minimized at this point. Evert’s Expected Payoff when she plays CC  90*q + 20*(1-q) q=0 q=0.6 q=1 50q+80(1-q)= 90q+20(1-q)  q=0.6 4-19-19 Nav’s q-mix

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) This is what coach or consultants do in professional tennis. By instinct, we know that we should hit the ball to where the opponent does not block. However, if we study GT, we can calculate the optimal ratio of how to mix the strategies (optimal p-mix and q-mix), given wining rates of offense/defense. This is the difference btwn someone equipped with GT and others. Experts are expected to provide a quantified strategy. Q: What happens if Evert is playing optimally at p=0.7 but Nav isn’t playing optimally at q=0.6? Evert’s winning rate goes up

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) Using the BR curves, find the NE. The simplified/integrated BR curve of Nav’s q-mix to Evert’s p-mix is as follows. q 1 *E가 DL을 칠 확률이 낮으면 (p=0.7 미만), N는 항상 DL을 안 막는다 (q=0) *E가 DL을 칠 확률이 높으면 (p=0.7 초과), N는 항상 DL을 막는다 (q=1)  N이 판단하는 E의 DL 칠 확률의 높고 낮음의 기준: p=0.7 0 0.7 1 p if p<0.7, q=0 if p=0.7, 0≤q≤1 if p>0.7, q=1 ; If the prob. Evert plays DL is smaller than 0.7, Nav plays CC (q=0). ; If the prob. Evert plays DL is 0.7, Nav plays CC or DL (0≤q≤1). ; If the prob. Evert plays DL is bigger than 0.7, Nav plays DL (q=1).

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) The simplified/integrated BR curve of Evert’s p-mix to Nav’s q-mix is as follows. Or switch the axses-> p 1 q 1 0.6 *N이 DL을 막을 확률이 낮으면 (q=0.6 미만), E는 항상 DL을 친다(p=1) *N이 DL을 막을 확률이 높으면 (q=0.6 초과), E는 항상 DL을 안 친다 (q=0)  E가 판단하는 N의 DL 막을 확률의 높고 낮음의 기준: q=0.6 0 0.6 1 q 0 1 p if q<0.6, p=1 if q=0.6, 0≤p≤1 if q>0.6, p=0 ; If the prob. Nav plays DL is smaller than 0.6, Evert plays DL (p=1). ; If the prob. Nav plays DL is 0.6, Evert plays CC or DL (0≤p≤1). ; If the prob. Nav plays DL is bigger than 0.6, Evert plays CC (p=0).

+ = 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy – BR Analysis (cont’d) NE? The point where the two BR curves cross. That is, at the cross point, the both are responding the best and there is no other point where their payoffs would increase simultaneously.  Def. of NE. NE: Evert plays DL 70%, CC 30% and Nav plays DL 60%, CC 40%! Note: Nav’s best response to Evert’s playing DL 70% is not playing DL 70%. This is what we usually think. But Nav should play DL 10% less, 60%. Lesson: Do Not exactly match the rate. React more or less! q 1 Only when Nav believes that the prob. for Evert to play DL is higher than 0.7, Nav plays DL. q 1 0.6 q 1 BR curve of Nav Evert plays CC (p=0) BR curve of Evert 0.6 + = Evert plays DL (p=1) Nav plays CC 0 0.7 1 p 0 1 p 0 0.7 1 p =Hakenkreuz? Hook Cross=German Cross=Nazi’s Icon, 卍을 뒤집어 놓은 것 만자 [卍字] : 부처의 가슴이나 손발에 나타나는 만덕(萬德)의 상징.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) The same result can be acquired using Minimax method. In zero-sum games, if one player is indifferent to selection of strategies, others become indifferent as well. That is, insisting on a specific strategy (ex: play DL always) will be disadvantageous. Thus, for my predictability not to be exploited, strategies should be mixed, showing no pattern. Using this idea, Minimax will be used to solve the same game. Consider the next payoff table.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Only Evert’s payoffs are shown. Write the R’s minima and C’s maxima on the sides. If Evert chooses DL, her minimum payoff is 50. If Evert chooses CC, her minimum payoff is 20. Thus, Evert’s maximin is 50, which is the maximum payoff can be secured or reserved by playing pure strategies. This is because Nav wants Evert’s payoff as low as possible. Nav(Defense) DL CC Evert (Offense) 50 80 90 20 min=50 min=20 max=90 max=80

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Maximum in columns are as follows. If Nav plays DL, the maximum payoff for Evert is 90. If Nav plays CC, the maximum payoff for Evert is 80. Thus, Evert’s minimax is 80; the minimum payoff that is received by Evert when Nav plays pure strategies. Nav will play CC (∵ 90>80). Nav(Defense) DL CC Evert (Offense) 50 80 90 20 min=50 min=20 max=90 max=80

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) There exists no equilibrium because Evert’s maximin (50) and Nav’s minimax (80) are different; Evert and Nav will randomly mix strategies to get higher payoffs, Evert trying to get higher maximin and Nav trying to get lower minimax. The table is extended below to show this. The minimum in rows changes depending on Evert’s selection of p (p -mix). In a figure? Next. Nav(Defense) DL CC Evert 50 80 90 20 p-mix 50*p + 90*(1-p) 80*p + 20*(1-p) min=50 min=20 min=?

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Similar figure to BRA is used. Evert’s expected payoff Evert’s expected payoff when Nav plays DL 90 62 20 80 50 Evert’s expected payoff when Nav plays CC p=0 p=0.7 p=1 Evert’s p-mix 50*p + 90*(1-p)=80*p + 20*(1-p) -> p=0.7

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Because this is a zero-sum game, Evert predicts that Nav will play a strategy that gives minimum payoff to her (Evert). Thus, if p<0.7, Nav plays CC (Black line); if p>0.7, Nav plays DL (Gray line). That is, the inverted V lines are the minimum that are expected for Evert to receive when Evert chooses p. Evert’s expected payoff 90 62 20 Evert’s expected payoff when Nav plays DL 80 50 Evert’s expected payoff when Nav plays CC Evert’s p-mix p=0 p=0.7 p=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) When p =1(Evert plays DL for sure); if Nav plays DL, Evert’s expected payoff is 50; if Nav plays CC, Evert’s expected payoff is 80. When p =0(Evert plays CC for sure); if Nav plays DL, Evert’s expected payoff is 90; if Nav plays CC, Evert’s expected payoff is 20. Thus, all of the first two row strategies in the table are included in the figure. What is the best strategy for Evert? Select p so that the expected minimum payoff is maximized  choose p=0.7. Evert’s expected payoff in this case (or maximin) is 50*0.7+90*(1-0.7)=62.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Conclusion: If Evert uses pure strategy, the winning probability is 50% (maximin when playing DL). But, if she plays the mixed strategy, it increases to 62%. The same goes for Nav. In a conclusion, if Nav plays q=0.6 (50q+80(1-q)=90q+20(1-q)  q=0.6), Nav’s payoff(minimax) is 50*0.6+80*0.4=62%. Thus, Evert’s maximin(=62) becomes equal to Nav’s minimax (=62). Thus, Nash Equilibrium. This brings better outcomes than payoff with pure strategies (Evert’s 90, 80 or Nav’s 10, 20). From Nav’s view point, Evert’s payoff decreases 90>80>62 or Nav’s payoff increases 10<20<38 by playing mixed strategies.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.2 Random Selection of Strategy–Minimax method (Skip) Final Conclusion: Selecting random strategy is advantageous to both of them rather than insisting on a pure strategy. That is, making the other players constantly guess what I will do is a better strategy. -> sporadic enforcement in psychology.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.3 NE as a system of Beliefs and Responses In SMG, responding to other player’s strategy is impossible. Instead, each player makes judgment on others’ decision and take the best action. In 2.1, we call this thinking the belief (所信) on other players’ strategic choice. We assume that this belief is correct and each player will make the best choice. This was the assumption behind NE. We applied this idea of NE to zero-sum games, minimax analysis, dominance, prisoners’ dilemma game, focal point, coordination game, and game of chicken.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.3 NE as a system of Beliefs and Responses (cont’d) That is, we assumed so far that all players would choose a specific pure strategy that is advantageous. However, because we introduced a more general ‘mixed strategy,’ we need to re-think the NE as a system of beliefs and responses for its applicability. In the coordination game, in which JD and JP decide where to meet, both were unsure of the probabilities for them to go to whether CE or CALES. Also, in the tennis game, Nav was unsure to which direction Evert would hit the ball. However, there should be a distinct between uncertainty and correct belief.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.3 NE as a system of Beliefs and Responses (cont’d) For example, in the tennis game, there is no way for Nav to find out for sure to which direction Evert would hit the ball. However, Nav can have a correct belief on Evert’s mixed strategy(p-mix of DL and CC). That is, having a correct belief means that one knows, calculates or guesses with what probability the other player are playing the mixed strategy among pure strategies. In the tennis game, Evert’s equilibrium mixed strategy is mixing DL 70% and CC 30%. If Nav believes this, Nav’s belief is correct.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.3 NE as a system of Beliefs and Responses (cont’d) Thus, NE can be re-defined in consideration of ‘Belief’ as the following. An equilibrium is a NE when; 1) Each player has a correct belief on others’ choices of mixed strategies and 2) each selects the best (mixed) strategy that reflects such beliefs. In the next section 2.4, we will consider mixed strategies for non-zero-sum games(JP-JD 미팅게임). In the NZG, there is no reason to believe pursuing my own interest would harm other players.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.3 NE as a system of Beliefs and Responses (cont’d) Thus, there is no reason to hide one’s intentions(I will go to CE!). In zero-sum games, revealing one’s intentions (I will play DL!) led to disadvantageous situations. However, when choices are made simultaneously, each player has uncertain belief on others’ choices. Therefore, the role of subjective and uncertain but correct belief becomes important.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.4 Mixed Strategy with more than 3 Pure Strategies So far we dealt with 2 pure strategy game and associated mixed strategy games. In reality, we play games with more than 3 pure strategy games in general. But, in that case, algorithm to solve gets complicated rapidly. Thus, we have to use computer software to solve the game. For example, let’s add one more strategy of ‘Lob’ for Evert. Down-the-line (DL) Evert Nav Lob 5-3-19 Cross-Court (CC)

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.4 Mixed Strategy with more than 3 Pure Strategies The payoff table is below. p1 : prob. that Evert plays DL p2 : prob. that Evert plays CC 1-p1-p2 : prob. that Evert plays Lob q : prob. that Nav plays DL 1-q : prob. that Nav plays CC *No blocking Lob for Nav? Why? Nav DL CC q-mix Evert 50 80 50q + 80(1-q) 90 20 90q+20(1-q) Lob 70 60 70q+60(1-q) p-mix 50*p1 + 90*p2 + 70*(1-p1-p2) 80*p1 + 20*p2 + 60*(1-p1-p2) q 1-q p1 p2 1-p1-p2 Let’s use these eqs. to draw the graph

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.4 Mixed Strategy with more than 3 Pure Strategies In a figure, the game is depicted below. Evert’s Expected Payoff Evert’s Expected Payoff If q<0.5, Evert DL. If 0.5<q<0.667, Evert Lob. If q>0.667, Evert CC. However, Nav will play q=0.5 (why?) Answer: Next E Plays DL E Plays Lob E Plays CC 90 70 50 90q+20(1-q) E가 CC할 때 N의 q에 따른 E의 기대보상 80 60 20 70q+60(1-q) E가 Lob할 때 N의 q에 따른 E의 기대보상 DL or Lob are indifferent to Evert 50q + 80(1-q) E가 DL할 때 N의 q에 따른 E의 기대보상 q=0 0.5 0.6 0.667 q=1 Nav’s q-mix

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.4 Mixed Strategy with more than 3 Pure Strategies (cont’d) Nav’s best is Evert’s worst. Evert’s worst  the lowest point of upper envelop of Evert’s BR curves  Thus, at q=0.5, Nav blocks DL and CC half and half, Evert plays DL or Lob only (doesn’t play CC at all(p2 or CC=0)  q=0.667 is never selected. Evert’s Expected Payoff E Plays DL E Plays Lob E Plays CC *Because Evert knows that Nav will play at q=0.5, Evert will play DL or Lob that provide payoff of 65, rather than CC that provides payoff of 55. 90 70 50 80 65 60 55 20 90q+20(1-q) 70q+60(1-q) 50q + 80(1-q) 90q+20(1-q) q=0 0.5 0.6 0.667 q=1 Nav’s q-mix

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.4 Mixed Strategy with more than 3 Pure Strategies (cont’d) 50*p1 + 90*p2 + 70*(1-p1-p2) = 80*p1 + 20*p2+ 60*(1-p1-p2), p2 or CC = 0 (why?)  p1 or DL=0.25 Because Evert discards CC, remove CC  Again, 2 pure strategies for each. Solving for 2 pure strategies game, the equilibrium is pDL=0.25 and pLob=0.75. In general, we tend to end up with unused pure strategies. This is because all the BR curves should coincide at one point for all the pure strategies to be used. This is a very rare case. These 2 pure strategies will not be used. General Case Exceptional Case

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies In the previous game, only one player has 3 pure strategies. What if both have more than 3 pure strategies? Ex) penalty kick Goalie (Kicker’s)Left Center (Kicker’s)Right Kicker Left 45 90 85 Right 95 60 p-mix 45PL+85PC+95PR 90PL+0+95PR 90PL+85PC+60PR Left Kicker Goalie Center Right

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) In a 3D figure. * In the shaded area, PL+PR>1. Thus, impossible  no concern. 95 Kicker’s expected payoff when goalie blocks left = 45PL+85PC+95PR =45PL+85(1-PL-PR) +95PR = 85-40PL+10PR 85 pR=1 55 45 = 85-40PL+10PR = 85-40*1+10*0 pL=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) Kicker’s expected payoff when goalie blocks right= 90PL+85(1-PL-PR)+60PR =85+5PL-25PR 85 60 65 pR=1 90 pL=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) 95 Kicker’s expected payoff when goalie blocks center= 90PL+95PR pR=1 90 pL=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) All 3 planes put together=upper surface=Kicker’s BR surface 95 95 85 85 60 65 90 pR=1 90 55 0.475 45 0.355 pL=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) In a simplified figure. Kicker’s expected payoff against goalie’s pure strategies (left, right, center) Thus, Kicker’s best strategy is to mix; To right 45.7% To left 35.5% To center 18.8% pR=1 pR=0.457 pL=0.355 pL=1

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) However intuitive the graphical solving algorithm is, it is complicated and not accurate. More importantly, it doesn’t work when we have more than 4 pure strategies (why?). Thus, we need to devise a method to solve these games algebraically. (Next)

Recall we have the following payoff table. 2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) Recall we have the following payoff table. Goalie (Kicker’s)Left Center (Kicker’s)Right Kicker Left 45 90 85 Right 95 60 pL: prob. that Kicker plays Left pR: prob. that Kicker plays Right pC: prob. that Kicker plays Center; pC=1-pL- pR qL: prob. that Goalie plays Left (blocking Kicker’s Left) qR: prob. that Goalie plays Right qC: prob. that Goalie plays Center; qC=1-qL- qR

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) Therefore the expected payoff are; Goalie (Kicker’s)Left Center (Kicker’s)Right Kicker Left 45 90 85 Right 95 60 Kicker’s expected payoff when Goalie plays Left = 45*pL+85*pC+95*pR = 45*pL+85*(1-pL- pR)+95*pR= 85-40*pL+10*pR Kicker’s expected payoff when Goalie plays Center = 90*pL+0*pC+95*pR = 90*pL+95*pR Kicker’s expected payoff when Goalie plays Right = 90*pL+85*pC+60*pR = 90*pL+85*(1-pL- pR)+60*pR= 85+5*pL-25*pR

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) In the previous figure on slide 50, NE is where 3 curves cross (where Kicker’s expected payoffs are equal). Thus, *The best strategies (rates) of the 2 players do NOT match! Left = Right : 85-40*pL+10*pR = 85+5*pL-25*pR 35*pR = 45*pL  pR = 45/35*pL  pR = 9/7*pL Center = Right : 90*pL+95*pR = 85+5*pL-25*pR 85*pL+120*pR = 85  85*pL+120*9/7pL = 85 (85+120*9/7)pL = 85  pL =0.355; pR = 9/7*0.355 = 0.457 ; pC=1-pL- pR , pC = 0.188 Similarly, qL = 0.325, qR = 0.561, qC = 0.113 32.5% Left:35.5% Center: 18.8% 11.3% Right:45.7% 56.1%

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies –Zero-sum Games 2.3.5 When Both have more than 3 Pure Strategies (cont’d) Another Example: R-P-S. How should I mix my strategies of R, P, and S? B R (q1) P (q2) S (1-q1-q2) q-mix A R (p1) 0, 0 0, 1 1, 0 1-q1-q2, q2 P (p2) q1, 1-q1-q2 S (1-p1-p2) q2, q1 p-mix p2 , 1-p1-p2 1-p1-p2, p1 p1, p2 πA=p1(1-q1-q2)+p2q1+(1-p1-p2)q2, πB=p1q2+p2(1-q1-q2)+(1-p1-p2)q1 ∂πA/ ∂p1 = 1 -q1 -2q2 = 0. ∂πA/ ∂p2 = q1 - q2 = 0. Thus, q1 = q2 = 1/3 = 1-q1-q2 ∂πB/ ∂q1 = - 2p2 + 1 - p1 = 0. ∂πB/ ∂q2 = p1 -p2 = 0. Thus, p1 = p2 = 1/3 = 1-p1-p2 Conclusion: NE is when both players put the same probabilities of 1/3 for R, P, and S. Why? Equal/symmetric winning rates.

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies – Zero-sum Games 2.3.6 Real-life Examples of Mixed Strategy in Zero-sum Games Results from laboratory experiments suggest that players don’t play mixed strategies. (Davis and Holt, 1993). However, in reality, the usage of mixed strategies has been reported. (Beresford and Peston, 1995). In late 1940’s, during the Malaya Civil War, there were frequent attacks on transported war supplies. To avoid the attack, a British army officer chose transportation route by holding a knife in one of his hands and letting a soldier decide which route to take randomly. DO NOT SHOW PATTERNS!!!

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies – Zero-sum Games 2.3.6 Real-life Examples of Mixed Strategy in Zero-sum Games (cont’d) Examples of usage of mixed strategies are more frequently found in sports.  Tennis.(Walker and Wooders, 2001). The prediction approaches the result Same for penalty kicks in soccer (Chiappori, Groseclose and Levitt, 2002).

2. Static Games with Complete Information 2.3 SMG with Mixed Strategies – Zero-sum Games 2.3.7 Using Mixed Strategies in Real Life It is important that the other player(s) don’t know which strategy I will play. That is, show no pattern. Use Random function such as RAND() in Excel or Wheel of Fortune. Excel RAND()