2博弈论第二讲-Mixed-Strategies-复旦大学-王永钦

1、Static (or Simultaneous-Move) Games of Complete Information -Mixed Strategy NEYongqin Wang,CCES, FudanFall, 2007, FudanMatching penniesnHead is Player 1s best response to Player 2s strategy TailnTail is Player 2s best response to Player 1s strategy TailnTail is Player 1s best response to Player 2s s

2、trategy HeadnHead is Player 2s best response to Player 1s strategy HeadHence, NO Nash equilibrium-1 , 1 1 , -1 1 , -1-1 , 1Player 1Player 2TailHeadTailHeadFall, 2007, FudanSolving matching penniesnRandomize your strategiesPlayer 1 chooses Head and Tail with probabilities r and 1-r, respectively. Pla

3、yer 2 chooses Head and Tail with probabilities q and 1-q, respectively.nMixed Strategy:Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities.Player 2HeadTailPlayer 1Head-1 , 1 1 , -1Tail 1 , -1-1 , 1q1-qr1-rFall, 2007, FudanMixed strategynA

4、 mixed strategy of a player is a probability distribution over players (pure) strategies.A mixed strategy for Chris is a probability distribution (p, 1-p), where p is the probability of playing Opera, and 1-p is that probability of playing Prize Fight.If p=1 then Chris actually plays Opera. If p=0 t

5、hen Chris actually plays Prize Fight.Battle of sexesPatOperaPrize FightChrisOpera (p)2 , 10 , 0Prize Fight (1-p)0 , 01 , 2Fall, 2007, FudanSolving matching penniesnPlayer 1s expected payoffsIf Player 1 chooses Head, -q+(1-q)=1-2qIf Player 1 chooses Tail, q-(1-q)=2q-1Player 2HeadTailPlayer 1Head-1 ,

6、1 1 , -1Tail 1 , -1-1 , 1q1-q1-2q2q-1Expected payoffsr1-rFall, 2007, Fudan1qr11/21/2Solving matching penniesnPlayer 1s best responseB1(q):For q0.5, Tail (r=0)For q=0.5, indifferent (0r1)Player 2HeadTailPlayer 1Head-1 , 1 1 , -1Tail 1 , -1-1 , 1q1-q1-2q2q-1Expected payoffsr1-rFall, 2007, FudanSolving

7、 matching penniesnPlayer 2s expected payoffsIf Player 2 chooses Head, r-(1-r)=2r-1If Player 2 chooses Tail, -r+(1-r)=1-2rPlayer 2HeadTailPlayer 1Head-1 , 1 1 , -1Tail 1 , -1-1 , 11-2q2q-1Expected payoffsr1-rq1-qExpected payoffs2r-11-2rFall, 2007, FudanSolving matching penniesnPlayer 2s best response

8、B2(r):For r0.5, Head (q=1)For r=0.5, indifferent (0q1)Player 2HeadTailPlayer 1Head-1 , 1 1 , -1Tail 1 , -1-1 , 1q1-q1-2q2q-1Expected payoffsr1-rExpected payoffs2r-11-2r1qr11/21/2Fall, 2007, Fudan1qr11/21/2Solving matching penniesnPlayer 1s best responseB1(q):For q0.5, Tail (r=0)For q=0.5, indifferen

9、t (0 r 1)nPlayer 2s best responseB2(r):For r0.5, Head (q=1)For r=0.5, indifferent (0 q 1)Check r = 0.5 B1(0.5)q = 0.5 B2(0.5)Player 2HeadTailPlayer 1Head-1 , 1 1 , -1Tail 1 , -1-1 , 1r1-rq1-qMixed strategy Nash equilibriumFall, 2007, FudanMixed strategy: examplenMatching penniesnPlayer 1 has two pur

10、e strategies: H and T( 1(H)=0.5, 1(T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively.( 1(H)=0.3, 1(T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively.Fall, 2007, FudanMixed strategy

11、: examplenPlayer 1: (3/4, 0, ) is a mixed strategy. That is, 1(T)=3/4, 1(M)=0 and 1(B)=1/4.nPlayer 2: (0, 1/3, 2/3) is a mixed strategy. That is, 2(L)=0, 2(C)=1/3 and 2(R)=2/3.Player 2L (0)C (1/3)R (2/3)Player 1T (3/4)0 , 23 , 31 , 1M (0)4 , 00 , 42 , 3B (1/4)3 , 45 , 10 , 7Fall, 2007, FudanExpected

12、 payoffs: 2 players each with two pure strategiesnPlayer 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ).Player 1s expected payoff of playing s11: EU1(s11, (q, 1-q)=qu1(s11, s21)+(1-q)u1(s11, s22)Player 1s expected payoff of playing s12: EU1(s12, (q, 1-q)= qu1(s12, s

13、21)+(1-q)u1(s12, s22)nPlayer 1s expected payoff from her mixed strategy:v1(r, 1-r), (q, 1-q)=r EU1(s11, (q, 1-q)+(1-r) EU1(s12, (q, 1-q)Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12, s22), u2(s12, s22)Fal

14、l, 2007, FudanExpected payoffs: 2 players each with two pure strategiesnPlayer 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ).Player 2s expected payoff of playing s21: EU2(s21, (r, 1-r)=ru2(s11, s21)+(1-r)u2(s12, s21)Player 2s expected payoff of playing s22: EU2(s22

15、, (r, 1-r)= ru2(s11, s22)+(1-r)u2(s12, s22)nPlayer 2s expected payoff from her mixed strategy:v2(r, 1-r),(q, 1-q)=q EU2(s21, (r, 1-r)+(1-q) EU2(s22, (r, 1-r)Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12,

16、s22), u2(s12, s22)Fall, 2007, FudanExpected payoffs: examplenPlayer 1:EU1(H, (0.3, 0.7) = 0.3(-1) + 0.71=0.4EU1(T, (0.3, 0.7) = 0.31 + 0.7(-1)=-0.4v1(0.4, 0.6), (0.3, 0.7)=0.4 0.4+0.6 (-0.4)=-0.08nPlayer 2:EU2(H, (0.4, 0.6) = 0.41+0.6(-1) = -0.2EU2(T, (0.4, 0.6) = 0.4(-1)+0.61 = 0.2v2(0.4, 0.6), (0.

17、3, 0.7)=0.3(-0.2)+0.70.2=0.08Player 2H (0.3)T (0.7)Player 1H (0.4)-1 , 1 1 , -1T (0.6) 1 , -1-1 , 1Fall, 2007, FudanExpected payoffs: examplenMixed strategies: p1=( 3/4, 0, ); p2=( 0, 1/3, 2/3 ).nPlayer 1: EU1(T, p2)=3(1/3)+1(2/3)=5/3, EU1(M, p2)=0(1/3)+2(2/3)=4/3EU1(B, p2)=5(1/3)+0(2/3)=5/3. v1(p1,

18、 p2) = 5/3nPlayer 2: EU2(L, p1)=2(3/4)+4(1/4)=5/2, EU2(C, p1)=3(3/4)+3(1/4)=5/2,EU2(R, p1)=1(3/4)+7(1/4)=5/2. v1(p1, p2) = 5/2Player 2L (0)C (1/3)R (2/3)Player 1T (3/4)0 , 23 , 31 , 1M (0)4 , 00 , 42 , 3B (1/4)3 , 45 , 10 , 7Fall, 2007, FudanMixed strategy equilibriumnMixed strategy equilibriumA pro

19、bability distribution for each playerThe distributions are mutual best responses to one another in the sense of expected payoffsIt is a stochastic steady state17Mixed strategy vs. Correlated strategyFall, 2007, FudanMixed strategy equilibrium: 2-player each with two pure strategiesnMixed strategy Na

20、sh equilibrium:nA pair of mixed strategies (r*, 1-r*), (q*, 1-q*)is a Nash equilibrium if (r*,1-r*) is a best response to (q*, 1-q*), and (q*, 1-q*) is a best response to (r*,1-r*). That is,v1(r*, 1-r*), (q*, 1-q*) v1(r, 1-r), (q*, 1-q*), for all 0 r 1v2(r*, 1-r*), (q*, 1-q*) v2(r*, 1-r*), (q, 1-q),

21、 for all 0 q 1Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12, s22), u2(s12, s22)Fall, 2007, FudanFind mixed strategy equilibrium in 2-player each with two pure strategiesnFind the best response corresponde

22、nce for player 1, given player 2s mixed strategynFind the best response correspondence for player 2, given player 1s mixed strategynUse the best response correspondences to determine mixed strategy Nash equilibria.20Theorem (Nash, 1950)Fall, 2007, FudanEmployee MonitoringnEmployees can work hard or

23、shirknSalary: $100K unless caught shirking nCost of effort: $50KnManagers can monitor or notnValue of employee output: $200KnProfit if employee doesnt work: $0nCost of monitoring: $10KFall, 2007, FudannEmployees best response B1(q):Shirk (r=0) if q0.5Any mixed strategy (0 r 1) if q=0.5Employee Monit

24、oringManagerMonitor ( q )Not Monitor (1-q)EmployeeWork ( r )50 , 9050 , 100Shirk (1-r )0 , -10100 , -10050100(1-q)Expected payoffsExpected payoffs100r-10200r-100Fall, 2007, FudannManagers best response B2(r):Monitor (q=1) if r0.9 Any mixed strategy (0 q 1) if r=0.9Employee MonitoringManagerMonitor (

25、 q )Not Monitor (1-q)EmployeeWork ( r )50 , 9050 , 100Shirk (1-r )0 , -10100 , -10050100(1-q)Expected payoffsExpected payoffs100r-10200r-100Fall, 2007, Fudan1qr10.5nEmployees best response B1(q):Shirk (r=0) if q0.5 Any mixed strategy (0 r 1) if q=0.5nManagers best response B2(r):Monitor (q=1) if r0.

26、9 Any mixed strategy (0 q 1) if r=0.9 Employee Monitoring0.9Mixed strategy Nash equilibrium(0.9,0.1),(0.5,0.5)Fall, 2007, FudannChris expected payoff of playing Opera: 2qnChris expected payoff of playing Prize Fight: 1-qnChris best response B1(q):Prize Fight (r=0) if q1/3 Any mixed strategy (0r1) if

27、 q=1/3Battle of sexesPatOpera (q)Prize Fight (1-q)ChrisOpera ( r )2 , 10 , 0Prize Fight (1-r)0 , 01 , 2Fall, 2007, FudannPats expected payoff of playing Opera: rnPats expected payoff of playing Prize Fight: 2(1-r)nPats best response B2(r):Prize Fight (q=0) if r2/3Any mixed strategy (0q1) if r=2/3, B

28、attle of sexesPatOpera (q)Prize Fight (1-q)ChrisOpera ( r )2 , 10 , 0Prize Fight (1-r)0 , 01 , 2Fall, 2007, Fudan1qr1nChris best response B1(q):Prize Fight (r=0) if q1/3 Any mixed strategy (0r1) if q=1/3nPats best response B2(r):Prize Fight (q=0) if r2/3 Any mixed strategy (0q1) if r=2/3Battle of se

29、xes2/3Three Nash equilibria:(1, 0), (1, 0)(0, 1), (0, 1)(2/3, 1/3), (1/3, 2/3)1/3Fall, 2007, FudanBattle of sexesnThe limitation of best-response approachnCan only deal with simple gamesnComments of mixed strategiesnRandomizationnvNM expected utilitynChoice under uncertaintyFall, 2007, Fudan2-player

30、 each with two strategiesnTheorem 1 (property of mixed Nash equilibrium)nA pair of mixed strategies (r*, 1-r*), (q*, 1-q*) is a Nash equilibrium if and only if v1(r*, 1-r*), (q*, 1-q*) EU1(s11, (q*, 1-q*)v1(r*, 1-r*), (q*, 1-q*) EU1(s12, (q*, 1-q*) v2(r*, 1-r*), (q*, 1-q*) EU2(s21, (r*, 1-r*)v2(r*,

31、1-r*), (q*, 1-q*) EU2(s22, (r*, 1-r*)Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12, s22), u2(s12, s22)Fall, 2007, FudanExpected payoffs: 2 players each with two pure strategiesnPlayer 1 plays a mixed stra

32、tegy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ).Player 1s expected payoff of playing s11: EU1(s11, (q, 1-q)=qu1(s11, s21)+(1-q)u1(s11, s22)Player 1s expected payoff of playing s12: EU1(s12, (q, 1-q)= qu1(s12, s21)+(1-q)u1(s12, s22)nPlayer 1s expected payoff from her mixed strategy:v1(r,

33、 1-r), (q, 1-q)=r EU1(s11, (q, 1-q)+(1-r) EU1(s12, (q, 1-q)Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12, s22), u2(s12, s22)Fall, 2007, FudanExpected payoffs: 2 players each with two pure strategiesnPlaye

34、r 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ).Player 2s expected payoff of playing s21: EU2(s21, (r, 1-r)=ru2(s11, s21)+(1-r)u2(s12, s21)Player 2s expected payoff of playing s22: EU2(s22, (r, 1-r)= ru2(s11, s22)+(1-r)u2(s12, s22)nPlayer 2s expected payoff from he

35、r mixed strategy:v2(r, 1-r),(q, 1-q)=q EU2(s21, (r, 1-r)+(1-q) EU2(s22, (r, 1-r)Player 2s21 ( q )s22 ( 1- q )Player 1s11 ( r )u1(s11, s21), u2(s11, s21)u1(s11, s22), u2(s11, s22)s12 (1- r )u1(s12, s21), u2(s12, s21)u1(s12, s22), u2(s12, s22)Fall, 2007, FudanTheorem 1: illustrationnPlayer 1:EU1(H, (0

36、.5, 0.5) = 0.5(-1) + 0.51=0EU1(T, (0.5, 0.5) = 0.51 + 0.5(-1)=0v1(0.5, 0.5), (0.5, 0.5)=0.5 0+0.5 0=0nPlayer 2:EU2(H, (0.5, 0.5) = 0.51+0.5(-1) =0EU2(T, (0.5, 0.5) = 0.5(-1)+0.51 = 0v2(0.5, 0.5), (0.5, 0.5)=0.50+0.50=0Matching penniesPlayer 2H (0.5)T (0.5)Player 1H (0.5)-1 , 1 1 , -1T (0.5) 1 , -1-1

37、 , 1Fall, 2007, FudanTheorem 1: illustrationnPlayer 1:v1(0.5, 0.5), (0.5, 0.5) EU1(H, (0.5, 0.5)v1(0.5, 0.5), (0.5, 0.5) EU1(T, (0.5, 0.5)nPlayer 2:v2(0.5, 0.5), (0.5, 0.5) EU2(H, (0.5, 0.5)v2(0.5, 0.5), (0.5, 0.5) EU2(T, (0.5, 0.5)nHence, (0.5, 0.5), (0.5, 0.5) is a mixed strategy Nash equilibrium

38、by Theorem 1.Matching penniesPlayer 2H (0.5)T (0.5)Player 1H (0.5)-1 , 1 1 , -1T (0.5) 1 , -1-1 , 1Fall, 2007, FudannEmployees expected payoff of playing “work”EU1(Work, (0.5, 0.5) = 0.550 + 0.550=50 nEmployees expected payoff of playing “shirk”EU1(Shirk, (0.5, 0.5) = 0.50 + 0.5100=50nEmployees expe

39、cted payoff of her mixed strategy v1(0.9, 0.1), (0.5, 0.5)=0.9 50+0.1 50=50Theorem 1: illustrationEmployee MonitoringManagerMonitor (0.5)Not Monitor (0.5)EmployeeWork (0.9)50 , 9050 , 100Shirk (0.1)0 , -10100 , -100Fall, 2007, FudannManagers expected payoff of playing “Monitor”EU2(Monitor, (0.9, 0.1

40、) = 0.990+0.1(-10) =80nManagers expected payoff of playing “Not”EU2(Not, (0.9, 0.1) = 0.9100+0.1(-100) = 80nManagers expected payoff of her mixed strategy v2(0.9, 0.1), (0.5, 0.5)=0.580+0.580=80Theorem 1: illustrationEmployee MonitoringManagerMonitor (0.5)Not Monitor (0.5)EmployeeWork (0.9)50 , 9050

41、 , 100Shirk (0.1)0 , -10100 , -100Fall, 2007, FudannEmployeev1(0.9, 0.1), (0.5, 0.5) EU1(Work, (0.5, 0.5)v1(0.9, 0.1), (0.5, 0.5) EU1(Shirk, (0.5, 0.5)nManagerv2(0.9, 0.1), (0.5, 0.5) EU2(Monitor, (0.9, 0.1)v2(0.9, 0.1), (0.5, 0.5) EU2(Not, (0.9, 0.1)n Hence, (0.9, 0.1), (0.5, 0.5) is a mixed strate

42、gy Nash equilibrium by Theorem 1.Theorem 1: illustrationEmployee MonitoringManagerMonitor (0.5)No Monitor (0.5)EmployeeWork (0.9)50 , 9050 , 100Shirk (0.1)0 , -10100 , -100Fall, 2007, FudannUse Theorem 1 to check whether (2/3, 1/3), (1/3, 2/3) is a mixed strategy Nash equilibrium.Theorem 1: illustra

43、tionBattle of sexesPatOpera (1/3)Prize Fight (2/3)ChrisOpera (2/3 )2 , 10 , 0Prize Fight (1/3)0 , 01 , 2Fall, 2007, FudanMixed strategy equilibrium: 2-player each with two strategiesnTheorem 2 Let (r*, 1-r*), (q*, 1-q*) be a pair of mixed strategies, where 0 r*1, 0q*0 and p1n*0 then EU1(s1m, p2*) =

44、EU1(s1n, p2*); if p1m*0 and p1n*=0 then EU1(s1m, p2*) EU1(s1n, p2*)player 2: for any i and k, if p2i*0 and p2k*0 then EU2(s2i, p1*) = EU2(s2k, p1*); if p2i*0 and p2k*=0 then EU2(s2i, p1*) EU2(s2k, p1*) Fall, 2007, Fudan2-player each with a finite number of pure strategiesnWhat does Theorem 4 tell us

45、? nA pair of mixed strategies (p1*, p2*), wherep1*=(p11*, p12*, ., p1J* ), p2*=(p21*, p22*, ., p2K* ) is a mixed strategy Nash equilibrium if and only if they satisfies the following conditions:nGiven player 2s p2*, player 1s expected payoff of every pure strategy to which she assigns positive proba

46、bility is the same, and player 1s expected payoff of any pure strategy to which she assigns positive probability is not less than the expected payoff of any pure strategy to which she assigns zero probability.nGiven player 1s p1*, player 2s expected payoff of every pure strategy to which she assigns

47、 positive probability is the same, and player 2s expected payoff of any pure strategy to which she assigns positive probability is not less than the expected payoff of any pure strategy to which she assigns zero probability.Fall, 2007, Fudan2-player each with a finite number of pure strategiesnTheor

48、em 4 implies that we have a mixed strategy Nash equilibrium in the following situation Given player 2s mixed strategy, Player 1 is indifferent among the pure strategies to which she assigns positive probabilities. The expected payoff of any pure strategy she assigns positive probability is not less

49、than the expected payoff of any pure strategy she assigns zero probability.Given player 1s mixed strategy, Player 2 is indifferent among the pure strategies to which she assigns positive probabilities. The expected payoff of any pure strategy she assigns positive probability is not less than the exp

50、ected payoff of any pure strategy she assigns zero probability.Fall, 2007, FudanTheorem 4: illustrationnCheck whether (3/4, 0, 1/4), (0, 1/3, 2/3) is a mixed strategy Nash equilibrium nPlayer 1: EU1(T, p2) = 0 0+3 (1/3)+1 (2/3)=5/3, EU1(M, p2) = 4 0+0 (1/3)+2 (2/3)=4/3EU1(B, p2) = 3 0+5 (1/3)+0 (2/3

51、)=5/3.Hence, EU1(T, p2) = EU1(B, p2) EU1(M, p2)Player 2L (0)C (1/3)R (2/3)Player 1T (3/4)0 , 23 , 31 , 1M (0)4 , 00 , 42 , 3B (1/4)3 , 45 , 10 , 7Fall, 2007, FudanTheorem 4: illustrationnPlayer 2: EU2(L, p1)=2(3/4) + 00 + 4(1/4)=5/2, EU2(C, p1)=3(3/4) + 40 + 1(1/4)=5/2,EU2(R, p1)=1(3/4) + 30 + 7(1/4

52、)=5/2.Hence, EU2(C, p1)=EU2(R, p1)EU2(L, p1)Therefore, (3/4, 0, 1/4), (0, 1/3, 2/3) is a mixed strategy Nash equilibrium by Theorem 4.Player 2L (0)C (1/3)R (2/3)Player 1T (3/4)0 , 23 , 31 , 1M (0)4 , 00 , 42 , 3B (1/4)3 , 45 , 10 , 7Fall, 2007, FudanExample: Rock, paper and scissorsnCheck whether th

53、ere is a mixed strategy Nash equilibrium in which p110, p120, p130, p210, p220, p230.Player 2Rock (p21)Paper (p22)Scissors (p23)Player 1Rock (p11) 0 , 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample: Rock, paper and scissorsnIf each player assign

54、s positive probability to every of her pure strategy, then by Theorem 4, each player is indifferent among her three pure strategies.Player 2Rock (p21)Paper (p22)Scissors (p23)Player 1Rock (p11) 0 , 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample:

55、 Rock, paper and scissorsnPlayer 1 is indifferent among her three pure strategies: EU1(Rock, p2) = 0 p21+(-1) p22+1 p23EU1(Paper, p2) = 1 p21+0 p22+(-1) p23EU1(Scissors, p2) = (-1) p21+1 p22+0 p23nEU1(Rock, p2)= EU1(Paper, p2)= EU1(Scissors, p2)nTogether with p21+ p22+ p23=1, we have three equations

56、 and three unknowns. Player 2Rock (p21)Paper (p22)Scissors (p23)Player 1Rock (p11) 0 , 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample: Rock, paper and scissorsn0 p21+(-1) p22+1 p23= 1 p21+0 p22+(-1) p23 0 p21+(-1) p22+1 p23 = (-1) p21+1 p22+0 p2

57、3 p21+ p22+ p23=1 nThe solution is p21= p22= p23=1/3Player 2Rock (p21)Paper (p22)Scissors (p23)Player 1Rock (p11) 0 , 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample: Rock, paper and scissorsnPlayer 2 is indifferent among her three pure strategie

58、s: EU2(Rock, p1)=0 p11+(-1) p12+1 p13EU2(Paper, p1)=1 p11+0 p12+(-1) p13EU2(Scissors, p1)=(-1) p11+1 p12+0 p13 nEU2(Rock, p1)= EU2(Paper, p1)=EU2(Scissors, p1)nTogether with p11+ p12+ p13=1, we have three equations and three unknowns. Player 2Rock (p21)Paper (p22)Scissors (p23)Player 1Rock (p11) 0 ,

59、 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample: Rock, paper and scissorsn0 p11+(-1) p12+1 p13=1 p11+0 p12+(-1) p130 p11+(-1) p12+1 p13=(-1) p11+1 p12+0 p13 p11+ p12+ p13=1 nThe solution is p11= p12= p13=1/3Player 2Rock (p21)Paper (p22)Scissors

60、(p23)Player 1Rock (p11) 0 , 0-1 , 1 1 , -1Paper (p12) 1 , -1 0 , 0-1 , 1Scissors (p13)-1 , 1 1 , -1 0 , 0Fall, 2007, FudanExample: Rock, paper and scissorsnPlayer 1: EU1(Rock, p2) = 0 (1/3)+(-1) (1/3)+1 (1/3)=0 EU1(Paper, p2) = 1 (1/3)+0 (1/3)+(-1) (1/3)=0 EU1(Scissors, p2) = (-1) (1/3)+1 (1/3)+0 (1