Mixed Strategies Levent Ko¸ckesen - Koç Hastanesi

1 TheoryMixed StrategiesLevent Ko ckesenKo c UniversityLevent Ko ckesen (Ko c University) Mixed Strategies1 / PenniesPlayer 1 Player 2 HTH 1,11, 1T1, 1 1,1 How would you play?KickerGoalieLef t RightLef t 1,11, 1 Right1, 1 1,1No solution?You should try to be unpredictableChoose randomlyLevent Ko ckesen (Ko c University) Mixed Strategies2 / DrivingChief of police in istanbul concerned about drunk can set up an alcohol checkpoint or not a checkpoint always catches drunk drivers but costscYou decide whether to drink wine or cola before driving. Value of wine over cola isr Cost of drunk driving isato you andfto the city incurred only if not caught if you get caught you paydYouPoliceCheckNoWiner d, cr a, fCola0, c0,0 Assume:f > c >0;d > r > a 0 Levent Ko ckesen (Ko c University) Mixed Strategies3 / DrivingLet s work with numbers:f= 2, c= 1, d= 4, r= 2, a= 1So, the game becomes:YouPoliceCheckNoWine 2, 11, 2 Cola0, 10,0 What is the set of Nash equilibria?

2 Levent Ko ckesen (Ko c University) Mixed Strategies4 / Strategy EquilibriumAmixed strategyis a probability distribution over the set of police chooses to set up checkpoints with probability 1/3 What should you do? If you drink cola you get0 If you drink wine you get 2with What is the value of this to you? We assume the value is theexpected payoff:13 ( 2) +23 1 = 0 You are indifferent between Wine and Cola You are also indifferent between drinking Wine and Cola with anyprobabilityLevent Ko ckesen (Ko c University) Mixed Strategies5 / Strategy EquilibriumYou drink wine with probability 1/2 What should the police do? If he sets up checkpoints he gets expected payoff of 1 If he does not12 ( 2) +12 0 = 1 The police is indifferent between setting up checkpoints andnot, aswell as any Mixed strategyYour strategy is a best response to that of the police and converselyWe have aMixed Strategy EquilibriumLevent Ko ckesen (Ko c University) Mixed Strategies6 / Strategy EquilibriumIn a Mixed strategy equilibrium every action played with positiveprobability must be a best response to other players Mixed strategiesIn particular players must be indifferent between actions playedwithpositive probabilityYour probability of drinking winepThe police s probability of setting up checkpointsqYour expected payoff to Wine isq ( 2) + (1 q) 1 = 1 3q Cola is0 Indifference condition0 = 1 3qimpliesq= 1/3 Levent Ko ckesen (Ko c University) Mixed Strategies7 / Strategy EquilibriumThe police s expected payoff to Checkpoint is 1 Not isp ( 2) + (1 p)

3 0 = 2pIndifference condition 1 = 2pimpliesp= 1/2(p= 1/2, q= 1/3)is a Mixed strategy equilibriumSince there is nopure strategy equilibrium, this is also the unique NashequilibriumLevent Ko ckesen (Ko c University) Mixed Strategies8 / 1 Player 2 HDH0,06,1D1,63,3 How would you play?What could be the stable population composition?Nash equilibria? (H, D) (D, H)How about3/4hawkish and1/4dovish? On average a dovish player gets(3/4) 1 + (1/4) 3 = 3/2 A hawkish player gets(3/4) 0 + (1/4) 6 = 3/2 No type has an evolutionary advantageThis is a Mixed strategy equilibriumLevent Ko ckesen (Ko c University) Mixed Strategies9 / and Pure Strategy EquilibriaHow do you find the set of all (pure and Mixed ) Nash equilibria?In2 2games we can use the best response correspondences interms of the Mixed Strategies and plot themConsider the Battle of the Sexes gamePlayer 1 Player 2mom2,10,0o0,01,2 Denote Player 1 s strategy aspand that of Player 2 asq(probabilityof choosingm) Levent Ko ckesen (Ko c University) Mixed Strategies10 / ,10,0o0,01,2 What is Player 1 s best response?

4 Expected payoff to mis2q ois1 qIf2q >1 qorq >1/3 best response ism(or equivalentlyp= 1)If2q <1 qorq <1/3 best response iso(or equivalentlyp= 0)If2q= 1 qorq= 1/3 he is indifferent best response is anyp [0,1]Player 1 s best response correspondence:B1(q) = {1},ifq >1/3[0,1],ifq= 1/3{0},ifq <1/3 Levent Ko ckesen (Ko c University) Mixed Strategies11 / ,10,0o0,01,2 What is Player 2 s best response?Expected payoff to misp ois2(1 p)Ifp >2(1 p)orp >2/3 best response ism(or equivalentlyq= 1)Ifp <2(1 p)orp <2/3 best response iso(or equivalentlyq= 0)Ifp= 2(1 p)orp= 2/3 she is indifferent best response is anyq [0,1]Player 2 s best response correspondence:B2(p) = {1},ifp >2/3[0,1],ifp= 2/3{0},ifp <2/3 Levent Ko ckesen (Ko c University) Mixed Strategies12 / (q) = {1},ifq >1/3[0,1],ifq= 1/3{0},ifq <1/3B2(p) = {1},ifp >2/3[0,1],ifp= 2/3{0},ifp <2/3 Set of Nash equilibria{(0,0),(1,1),(2/3,1/3)}qp1102/ 31/3bbbB1(q)B2(p) Levent Ko ckesen (Ko c University) Mixed Strategies13 / Actions and Mixed StrategiesUp to now we tested actions only against other actionsAn action may be undominated by any other action, yet bedominated by a Mixed strategyConsider the following gameLRT1,11,0M3,00,3B0,14,0No action dominatesTBut Mixed strategy( 1(M) = 1/2, 1(B) = 1/2)strictly dominatesTA strictly dominated action is never used with positive probability in amixed strategy equilibriumLevent Ko ckesen (Ko c University) Mixed Strategies14 / Actions and Mixed StrategiesAn easy way to figure out dominated actions is to compare expectedpayoffsLet player 2 s Mixed strategy given byq=prob(L)LRT1,11,0M3,00,3B0,14,0u1(T, q)

5 = 1u1(M, q) = 3qu1(B, q) = 4(1 q)01234014/712/7qu1(., q)u1(T, q)u1(M, q)u1(B, q)An action is anever bestresponseif there is no belief (onA i) that makes that action abest responseTis a never best responseAn action is a NBR iff it isstrictly dominatedLevent Ko ckesen (Ko c University) Mixed Strategies15 / if there are no strictly dominated actions?LRT2,02,1M3,30,0B0,13,0 Denote player 2 s Mixed strategy byq=prob(L)u1(T, q) = 2, u1(M, q) = 3q, u1(B, q) = 3(1 q)qu1(., q)1/23/22/31/31320u1(T, q)u1(M, q)u1(B, q)Pure strategy Nash eq.(M, L) Mixed strategy equilibria? Only one player mixes? Not possible Player 1 mixes over{T, M, B}? Not possible Player 1 mixes over{M, B}? Not possible Player 1 mixes over{T, B}? Letp=prob(T)q= 1/3,1 p=p p= 1/2 Player 1 mixes over{T, M}? Letp=prob(T)q= 2/3,3(1 p) =p p= 3/4 Levent Ko ckesen (Ko c University) Mixed Strategies16 / Life Examples?Ignacio Palacios-Huerta (2003): 5 years worth of penalty kicksEmpirical scoring probabilitiesLRL58,4295,5R93,770,30 Ris the natural side of the kickerWhat are the equilibrium Strategies ?

6 Levent Ko ckesen (Ko c University) Mixed Strategies17 / KickLRL58,4295,5R93,770,30 Kicker must be indifferent58p+ 95(1 p) = 93p+ 70(1 p) p= keeper must be indifferent42q+ 7(1 q) = 5q+ 30(1 q) q= see Walker and Wooders (2001): WimbledonLevent Ko ckesen (Ko c University) Mixed Strategies18 / 18