博弈论实例VNMPoker4习题课件.xlsx

Start Seite 1 VNM POKER 4 r m n different cards n S 4antem 3only the 11 cells with black background can be changed on this sheet number of duplicatesr 4max betn 5 Probabilities for the given card distributions Prob1234win1234 1 0 05 0 07 0 067 0 06710 1 1 1 2 0 07 0 05 0 067 0 067210 1 1 3 0 07 0 070 05 0 0673110 1 4 0 07 0 07 0 0670 0541110 Here are Ann s payoffs for the 16 possible pure strategies of Ann and Beth FFFFFFFFCC FFFFCCCCFF FFCCFFCCFF FCFCFCFCFC FFFF FFFC FFCF FFCC FCFFFCFCFCCFFCCCCFFFCFFC CCCCCCCC0 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 CCCRCCCR0 150 000 280 13 0 28 0 13 0 42 0 27 0 28 0 13 CCRCCCRC0 550 020 40 0 13 0 68 0 15 0 53 0 00 0 68 0 15 CCRRCCRR0 700 020 680 00 0 97 0 28 0 95 0 27 0 97 0 28 CRCCCRCC0 950 420 42 0 12 0 80 0 27 0 27 0 27 1 08 0 55 CRCRCRCR1 100 420 700 02 1 08 0 40 0 68 0 00 1 37 0 68 CRRCCRRC1 500 430 82 0 25 1 48 0 42 0 80 0 27 1 77 0 70 CRRRCRRR1 650 431 10 0 12 1 77 0 55 1 22 0 00 2 05 0 83 RCCCRCCC1 350 820 820 28 0 82 0 28 0 28 0 25 1 20 0 67 RCCRRCCR1 500 821 100 42 1 10 0 42 0 70 0 02 1 48 0 80 RCRCRCRC1 900 831 220 15 1 50 0 43 0 82 0 25 1 88 0 82 RCRRRCRR2 050 831 500 28 1 78 0 57 1 23 0 02 2 17 0 95 RRCCRRCC2 301 231 230 17 1 62 0 55 0 55 0 52 2 28 1 22 RRCRRRCR2 451 231 520 30 1 90 0 68 0 97 0 25 2 57 1 35 RRRCRRRC2 851 251 630 03 2 30 0 70 1 08 0 52 2 97 1 37 RRRRRRRR3 001 251 920 17 2 58 0 83 1 50 0 25 3 25 1 50 Below you can input a behavioral strategy for Ann and one for Beth The resulting expected payoff for Ann is calculated using the matrix below Behavior Strategies Ann raises with probabilitiesBeth calls with probabilitiesExpected Payoff for Ann 0 20 50 50 50 0 8911 0 0448 for1234for1234 The probabilities of the resulting mixed strategies belonging to these two behavioral strategies are shown in the blue column and the green row below FFFFFFFFCC FFFFCCCCFF FFCCFFCCFF FCFCFCFCFC 0 000 000 000 11 0 00 0 00 0 00 0 89 0 00 0 00 CCCC0 10 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 CCCR0 10 000 000 000 00 0 00 0 00 0 00 0 02 0 00 0 00 CCRC0 10 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 CCRR0 10 000 000 000 00 0 00 0 00 0 00 0 02 0 00 0 00 CRCC0 10 000 000 000 00 0 00 0 00 0 00 0 02 0 00 0 00 Start Seite 2 CRCR0 10 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 CRRC0 10 000 000 000 00 0 00 0 00 0 00 0 02 0 00 0 00 CRRR0 10 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 RCCC0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 RCCR0 0250 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 RCRC0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 RCRR0 0250 000 000 000 00 0 00 0 00 0 00 0 00 0 00 0 00 RRCC0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 RRCR0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 RRRC0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 RRRR0 0250 000 000 000 00 0 00 0 00 0 00 0 01 0 00 0 00 1 Start Seite 3 CCCCCCComputations since the formulas would have been too long otherwise FFCCCC CCFFCC FCFCFC CFCFCFCCCCFFCCFCCCCFCCCC 0 00 0 00 0 00 0 00 0 00 0 00 The formulas just consider what would happen based on the pure strategies on the players and probabilities for the different card distributions given above Details to the right 0 42 0 27 0 42 0 27 0 55 0 40 0 53 0 00 0 82 0 28 0 67 0 13CCC 0 95 0 27 1 23 0 55 1 22 0 53CCC 0 55 0 02 0 93 0 40 0 40 0 13CCR 0 97 0 28 1 35 0 67 0 95 0 27CCR 1 08 0 02 1 75 0 68 1 07 0 00CRC 1 50 0 28 2 17 0 95 1 62 0 40CRC 0 67 0 13 0 67 0 13 0 13 0 40CRR 1 08 0 40 1 08 0 40 0 68 0 00CRR 1 20 0 13 1 48 0 42 0 80 0 27RCC 1 62 0 40 1 90 0 68 1 35 0 13RCC 1 22 0 15 1 60 0 53 0 53 0 53RCR 1 63 0 42 2 02 0 80 1 08 0 13RCR 1 75 0 15 2 42 0 82 1 20 0 40RRC 2 17 0 42 2 83 1 08 1 75 0 00RRC RRR RRR CCCCCC FFCCCC CCFFCCCCC FCFCFCCCC 0 00 0 00 0 00 0 00 0 00 0 001CCR 0 00 0 00 0 00 0 00 0 00 0 00CCR 0 00 0 00 0 00 0 00 0 00 0 00 the sum of these 256 values is the expected payoff for Ann CRC 0 00 0 00 0 00 0 00 0 00 0 00CRC 0 00 0 00 0 00 0 00 0 00 0 00CRR 0 00 0 00 0 00 0 00 0 00 0 00CRR only the 11 cells with black background can be changed on this sheet Expected Payoff for Ann Start Seite 4 0 00 0 00 0 00 0 00 0 00 0 00RCC 0 00 0 00 0 00 0 00 0 00 0 00RCC 0 00 0 00 0 00 0 00 0 00 0 00RCR 0 00 0 00 0 00 0 00 0 00 0 00RCR 0 00 0 00 0 00 0 00 0 00 0 00RRC 0 00 0 00 0 00 0 00 0 00 0 00RRC 0 00 0 00 0 00 0 00 0 00 0 00RRR 0 00 0 00 0 00 0 00 0 00 0 00RRR 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 Start Seite 5 FFFFFFFFCCCCCC FFFFCCCCFFFFCC FFCCFFCCFFCCFF FCFCFCFCFCFCFC FFFF FFFCFFCFFFCCFCFFFCFCFCCFFCCCCFFFCFFCCFCFCFCCCCFFCCFC CCCCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RCCCR 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 CCCRC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RCCRR 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 CCRCC 0 15 0 38 0 38 0 92 0 00 0 53 0 53 1 07 0 28 0 25 0 25 0 78 0 13 0 40 RCRCR 0 15 0 38 0 38 0 92 0 00 0 53 0 53 1 07 0 28 0 25 0 25 0 78 0 13 0 40 CCRRC 0 15 0 38 0 38 0 92 0 00 0 53 0 53 1 07 0 28 0 25 0 25 0 78 0 13 0 40 RCRRR 0 15 0 38 0 38 0 92 0 00 0 53 0 53 1 07 0 28 0 25 0 25 0 78 0 13 0 40 CRCCC 0 55 0 02 0 02 0 52 0 02 0 52 0 52 1 05 0 40 0 13 0 13 0 67 0 13 0 67 RRCCR 0 55 0 02 0 02 0 52 0 02 0 52 0 52 1 05 0 40 0 13 0 13 0 67 0 13 0 67 CRCRC 0 55 0 02 0 02 0 52 0 02 0 52 0 52 1 05 0 40 0 13 0 13 0 67 0 13 0 67 RRCRR 0 55 0 02 0 02 0 52 0 02 0 52 0 52 1 05 0 40 0 13 0 13 0 67 0 13 0 67 CRRCC 1 50 0 43 0 43 0 63 0 82 0 25 0 25 1 32 1 48 0 42 0 42 0 65 0 80 0 27 RRRCR 1 50 0 43 0 43 0 63 0 82 0 25 0 25 1 32 1 48 0 42 0 42 0 65 0 80 0 27 CRRRC 1 50 0 43 0 43 0 63 0 82 0 25 0 25 1 32 1 48 0 42 0 42 0 65 0 80 0 27 RRRRR 1 50 0 43 0 43 0 63 0 82 0 25 0 25 1 32 1 48 0 42 0 42 0 65 0 80 0 27 FFFFFFFFCCCCCC FFFFCCCCFFFFCC FFCCFFCCFFCCFF FCFCFCFCFCFCFC FFFF FFFCFFCFFFCCFCFFFCFCFCCFFCCCCFFFCFFCCFCFCFCCCCFFCCFC CCCCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RCCCR 0 95 0 80 1 08 0 93 1 08 0 93 1 22 1 07 1 08 0 93 1 22 1 07 1 22 1 07 CCCRC 1 35 0 82 1 20 0 67 1 48 0 95 1 33 0 80 1 48 0 95 1 33 0 80 1 62 1 08 RCCRR 1 50 0 82 1 48 0 80 1 77 1 08 1 75 1 07 1 77 1 08 1 75 1 07 2 03 1 35 CCRCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RCRCR 0 95 0 80 1 08 0 93 1 08 0 93 1 22 1 07 1 08 0 93 1 22 1 07 1 22 1 07 CCRRC 1 35 0 82 1 20 0 67 1 48 0 95 1 33 0 80 1 48 0 95 1 33 0 80 1 62 1 08 RCRRR 1 50 0 82 1 48 0 80 1 77 1 08 1 75 1 07 1 77 1 08 1 75 1 07 2 03 1 35 Computations since the formulas would have been too long otherwise Start Seite 6 CRCCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RRCCR 0 95 0 80 1 08 0 93 1 08 0 93 1 22 1 07 1 08 0 93 1 22 1 07 1 22 1 07 CRCRC 1 35 0 82 1 20 0 67 1 48 0 95 1 33 0 80 1 48 0 95 1 33 0 80 1 62 1 08 RRCRR 1 50 0 82 1 48 0 80 1 77 1 08 1 75 1 07 1 77 1 08 1 75 1 07 2 03 1 35 CRRCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 RRRCR 0 95 0 80 1 08 0 93 1 08 0 93 1 22 1 07 1 08 0 93 1 22 1 07 1 22 1 07 CRRRC 1 35 0 82 1 20 0 67 1 48 0 95 1 33 0 80 1 48 0 95 1 33 0 80 1 62 1 08 RRRRR 1 50 0 82 1 48 0 80 1 77 1 08 1 75 1 07 1 77 1 08 1 75 1 07 2 03 1 35 Start Seite 7 CC CC CC FC CCCFCCCC 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 40 0 93 0 40 0 93 0 40 0 93 0 40 0 93 0 67 1 20 0 67 1 20 0 67 1 20 0 67 1 20 0 27 1 33 0 27 1 33 0 27 1 33 0 27 1 33 CC CC CC FC CCCFCCCC 0 80 0 80 1 35 1 20 1 47 0 93 2 02 1 33 0 80 0 80 1 35 1 20 1 47 0 93 2 02 1 33 Start Seite 8 0 80 0 80 1 35 1 20 1 47 0 93 2 02 1 33 0 80 0 80 1 35 1 20 1 47 0 93 2 02 1 33 FFFFFFFCFFCFFFCCFCFFFCFCFCCFFCCCCFFF 4b CCCC000000000 1 CCCR0 150 0 2833 0 1333 0 2833 0 1333 0 4167 0 2667 0 2833 5 CCRC0 55 0 01670 4 0 133 0 68330 15 0 53330 0 6833 2 CCRR0 7 0 0167 0 68330 0 9667 0 28330 95 0 2667 0 9667 6 CRCC0 95 0 4167 0 4167 0 1170 8 0 2667 0 2667 0 267 1 0833 3 CRCR1 1 0 41670 7 0 0167 1 08330 4 0 68330 1 3667 7 CRRC1 5 0 4333 0 8167 0 25 1 4833 0 41670 8 0 267 1 7667 4 CRRR1 65 0 43331 1 0 117 1 76670 55 1 216702 05 4b RCCC1 35 0 8167 0 8167 0 2833 0 8167 0 2833 0 2833 0 251 2 1 RCCR1 5 0 81671 1 0 41671 1 0 41670 7 0 0167 1 4833 5 RCRC1 9 0 8333 1 21670 151 5 0 4333 0 8167 0 25 1 8833 2 RCRR2 05 0 83331 5 0 2833 1 7833 0 5667 1 2333 0 0167 2 1667 6 RRCC2 3 1 2333 1 2333 0 1667 1 61670 550 55 0 517 2 2833 3 RRCR2 45 1 2333 1 51670 31 9 0 6833 0 9667 0 25 2 5667 7 RRRC2 851 25 1 6333 0 03332 30 7 1 0833 0 517 2 9667 4 RRRR31 25 1 9167 0 1667 2 5833 0 83331 5 0 253 25 3 4 CCCR and 1 4 RCCRMIX0 4875 0 2042 0 4875 0 2042 0 4875 0 2042 0 4875 0 2042 0 5833 00 050 10 150 20 234375 0 3 0 25 0 2 0 15 0 1 0 05 0 0 05 0 1 0 15 0 2 0 25 CCCC and RCCC CCCR and RCCR CCRC and RCRC CCRR and RCRR CRCC and RRCC CRCR and RRCR CRRC and RRRC CRRR and RRRR CFFCCFCFCFCCCCFFCCFCCCCFCCCC 000000000000 0 1333 0 4167 0 2667 0 4167 0 26670 550 40 2042 0 2042 0 2042 0 2042 0 2042 0 15 0 53330 0 8167 0 2833 0 6667 0 1333 0 062 0 048 0 034 0 02 0 006 0 28330 95 0 2667 1 23330 55 1 2167 0 53330 1417 0 15580 17 0 1842 0 1983 0 550 55 0 0167 0 93330 40 4 0 133 0 196 0 177 0 158 0 139 0 12 0 6833 0 9667 0 28331 35 0 66670 95 0 26670 00780 027 0 0461 0 0653 0 0845 0 7 1 0833 0 01671 75 0 6833 1 06670 0 259 0 226 0 192 0 159 0 126 0 83331 5 0 2833 2 16670 95 1 61670 4 0 055 0 0210 012 0 0453 0 0786 0 6667 0 6667 0 1333 0 6667 0 1333 0 1333 0 400000 0 8 1 08330 4 1 08330 4 0 683300 2042 0 2042 0 2042 0 2042 0 2042 0 81671 2 0 1333 1 4833 0 41670 8 0 267 0 063 0 048 0 034 0 02 0 006 0 95 1 61670 41 9 0 68331 35 0 13330 1417 0 15580 17 0 1842 0 1983 1 2167 1 21670 151 6 0 5333 0 5333 0 533 0 196 0 177 0 158 0 139 0 12 1 35 1 6333 0 4167 2 01670 8 1 0833 0 1330 00780 027 0 0461 0 0653 0 0845 1 36671 750 15 2 4167 0 81671 2 0 4 0 259 0 226 0 192 0 159 0 126 1 5 2 1667 0 4167 2 8333 1 08331 750 0 055 0 0210 012 0 0453 0 0786 0 3 0 58330 3 0 58330 3 0 58330 3 q100 050 10 150 2 q20 4688 0 3688 0 2688 0 1688 0 0687 q30 5313 0 5813 0 6313 0 6813 0 7313 11111 these are some Nash mixes for Beth mixing FFFC FFCC and FCCC wit