國(guó)外博弈論課件lecture(23)-資料下載頁(yè)
2024-10-19 13:59本頁(yè)面
【正文】 cissors ? Player 2 is indifferent among her three pure strategies: EU2(Rock, p1)=0?p11+(1)? p12+1? p13 EU2(Paper, p1)=1? p11+0? p12+(1)? p13 EU2(Scissors, p1)=(1)? p11+1? p12+0? p13 ? EU2(Rock, p1)= EU2(Paper, p1)=EU2(Scissors, p1) ? Together with p11+ p12+ p13=1, we have three equations and three unknowns. Player 2 Rock (p21) Paper (p22) Scissors (p23) Player 1 Rock (p11) 0 , 0 1 , 1 1 , 1 Paper (p12) 1 , 1 0 , 0 1 , 1 Scissors (p13) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 25 Example: Rock, paper and scissors ? 0?p11+(1)? p12+1? p13=1? p11+0? p12+(1)? p13 0?p11+(1)? p12+1? p13=(1)? p11+1? p12+0? p13 p11+ p12+ p13=1 ? The solution is p11= p12= p13=1/3 Player 2 Rock (p21) Paper (p22) Scissors (p23) Player 1 Rock (p11) 0 , 0 1 , 1 1 , 1 Paper (p12) 1 , 1 0 , 0 1 , 1 Scissors (p13) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 26 Example: Rock, paper and scissors ? Player 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?(1/3)=0 ? Player 2: EU2(Rock, p1)=0?(1/3)+(1)?(1/3)+1?(1/3)=0 EU2(Paper, p1)=1?(1/3)+0?(1/3)+(1)?(1/3)=0 EU2(Scissors, p1)=(1)?(1/3)+1?(1/3)+0?(1/3)=0 ? Therefore, (p1=(1/3, 1/3, 1/3), p2=(1/3, 1/3, 1/3)) is a mixed strategy Nash equilibrium by Theorem 4. Player 2 Rock (1/3) Paper (1/3) Scissors (1/3) Player 1 Rock (1/3) 0 , 0 1 , 1 1 , 1 Paper (1/3) 1 , 1 0 , 0 1 , 1 Scissors (1/3) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 27 Example: Rock, paper and scissors ? Check whether there is a mixed strategy Nash equilibrium in which one of p11, p12, p13 is positive, and at least two of p21, p22, p23 are positive. ? The answer is No. Player 2 Rock (p21) Paper (p22) Scissors (p23) Player 1 Rock (p11) 0 , 0 1 , 1 1 , 1 Paper (p12) 1 , 1 0 , 0 1 , 1 Scissors (p13) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 28 Example: Rock, paper and scissors ? Check whether there is a mixed strategy Nash equilibrium in which two of p11, p12, p13 is positive, and at least two of p21, p22, p23 are positive. ? The answer is No. Player 2 Rock (p21) Paper (p22) Scissors (p23) Player 1 Rock (p11) 0 , 0 1 , 1 1 , 1 Paper (p12) 1 , 1 0 , 0 1 , 1 Scissors (p13) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 29 Example: Rock, paper and scissors ? Therefore, (p1=(1/3, 1/3, 1/3), p2=(1/3, 1/3, 1/3)) is the unique mixed strategy Nash equilibrium by Theorem 4. Player 2 Rock (p21) Paper (p22) Scissors (p23) Player 1 Rock (p11) 0 , 0 1 , 1 1 , 1 Paper (p12) 1 , 1 0 , 0 1 , 1 Scissors (p13) 1 , 1 1 , 1 0 , 0 June 2, 2021 73347 Game TheoryLecture 10 30 Summary ? Find mixed strategy Nash equilibrium in a 2player game each with a finite number of pure strategies ? Next time ? Review HW1 ? Reading lists ? Cha of Osborne ? Solutions to HW
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