Question

4) (20 points) Consider the following two player simultaneous move game which is another version of the Battle of the Sexes game. Bob Opera Alice 4,1 Opera Football Football 0,0 1,4 0,0 Suppose Alice plays a p - mix in which she plays Opera with probability p and Football with probability (1 – p) and Bob plays a q- mix in which he plays Opera with probability q and Football with probability (1 – 9). a) Find the mixed strategy Nash Equilibrium. Identify the equilibrium by listing the strategy of each player. b) Find and graph Alice Best Response Function. Please graph p on the vertical axis and q on the horizontal axis (Hint: you and looking for p = f(q), in other words, what p should Alice select to maximize her payoff as a function of the q that Bob selects between 0 and 1). c) Find and graph Bob's Best Response Function. Please graph q on the vertical axis and p on the horizontal axis (Hint: you and looking for q = f(p), in other words, what q should Bob select to maximize his payoff as a function of the p that Alice selects between 0 and 1). d) Combine the best response function in parts b and c to visualize the mixed strategy Nash Equilibrium you found in part a. (Hint, please reference Chapter 7, section 2 of Dixit, Skeath and Reiley, Figure 7.2 or Slide #34 from Mixed Strategies - Part 1 slides posted on Moodle)

Answer 1

Bob (ra) football opera 0,0 - b opera Alice (+b) Football 100 Alice 1,4 - (a) For Mixed Stourtegy Nash equilibrium = - consider Alice first Expected Payoff of Alice for opera Ecol= 49 For Footboll E(F) = (1-9) requires Nash mixed Strategy Ecol= E(F) 49 = 1-a =S 59 = 1 =) 9= = there Jove 19 = 4

Now Bob's expected Payoff from opera E (O) = 4P Ecoj = E(F) 3 Foom Football E(F) = u(1-P) Nash equilibrium Requines That up= u-up => 8P = 4 =) P = 4 = 1 ture fore 1-P = 1/2 Mixed Stoa tegy Nash equilibrium where Alice mix with b= 1 and CFP) = 1 Bob mix with q = 1 and (-a)=4 (b) Alice Best Responce. ECO) and E(F) Arce will play opera with EO) ? E(F) 49 > (-a) pal if

4931-a =) 58 > 1 =) q> then p=1 if qut Thein (+bj=1 =) p = 0. - BR of Alice (C). q=1 if Bobls Best des fonce ECO and ECF) Bob will pay opera with E(O) 7 E(F) up > UC1-P) 4P 4-up - 8 p > 4 p> 1 / 2

q=0 if Pat Best Res Ponce BR (Bob) = hay opera (9-1) if P> 1 Play Joot ball (9= 0) it path Play mix ( if p = 1/2 ). BR (Bob) NI- (d) By Taking b on Vertical and 9 on korrizontal Axis BRL Alice ) mixed Strategy Nash 1
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