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"Test two, choose the better" leads to high cooperation in the Centipede game
1. | BioEcoUva and Department of Industrial Organization, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain |
2. | Department of Management Engineering, Universidad de Burgos, Avda. Cantabria s/n, 09006, Burgos, Spain |
Explaining cooperative experimental evidence in the Centipede game constitutes a challenge for rational game theory. Traditional analyses of Centipede based on backward induction predict uncooperative behavior. Furthermore, analyses based on learning or adaptation under the assumption that those strategies that are more successful in a population tend to spread at a higher rate usually make the same prediction. In this paper we consider an adaptation model in which agents in a finite population do adopt those strategies that turn out to be most successful, according to their own experience. However, this behavior leads to an equilibrium with high levels of cooperation and whose qualitative features are consistent with experimental evidence.
References:
[1] |
E. Ben-Porath,
Rationality, Nash equilibrium and backwards induction in perfect- information games, Rev. Econom. Stud., 64 (1997), 23-46.
doi: 10.2307/2971739. |
[2] |
K. Binmore,
Modeling rational players: Part Ⅰ, Economics & Philosophy, 3 (1987), 179-214.
doi: 10.1017/S0266267100002893. |
[3] |
K. Binmore, Natural Justice, Oxford University Press, 2005.
doi: 10.1093/acprof:oso/9780195178111.001.0001.![]() ![]() |
[4] |
K. Binmore and L. Samuelson, An economist’s perspective on the evolution of norms, J. Institutional Theoretical Economics (JITE), 150 (1994), 45–63. Available from: https://www.jstor.org/stable/40753015. |
[5] |
K. G. Binmore, L. Samuelson and R. Vaughan,
Musical chairs: Modeling noisy evolution, Games Econom. Behav., 11 (1995), 1-35.
doi: 10.1006/game.1995.1039. |
[6] |
G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Automata Theory and Formal Languages, Lecture Notes in Comput. Sci., 33, Springer, Berlin, 1975,134–183.
doi: 10.1007/3-540-07407-4_17. |
[7] |
J. C. Cox and D. James, On replication and perturbation of the McKelvey and Palfrey centipede game experiment, in Replication in Experimental Economics (Research in Experimental Economics, Volume 18), Emerald Group Publishing Ltd., 2015, 53–94.
doi: 10.1108/S0193-230620150000018003. |
[8] |
R. Cressman and K. H. Schlag,
The dynamic (in)stability of backwards induction, J. Econom. Theory, 83 (1998), 260-285.
doi: 10.1006/jeth.1996.2465. |
[9] |
M. Embrey, G. R. Fréchette and S. Yuksel,
Cooperation in the finitely repeated prisoner's dilemma, Quarterly J. Economics, 133 (2018), 509-551.
doi: 10.1093/qje/qjx033. |
[10] |
I. Gilboa and A. Matsui,
Social stability and equilibrium, Econometrica, 59 (1991), 859-867.
doi: 10.2307/2938230. |
[11] |
J. Y. Halpern,
Substantive rationality and backward induction, Games Econom. Behav., 37 (2001), 425-435.
doi: 10.1006/game.2000.0838. |
[12] |
J. Hofbauer, Stability for the Best Response Dynamics, University of Vienna, 1995, unpublished manuscript. |
[13] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.
![]() ![]() |
[14] |
L. R. Izquierdo, S. S. Izquierdo and W. H. Sandholm,
An introduction to $ABED$: Agent-based simulation of evolutionary game dynamics, Games Econom. Behav., 118 (2019), 434-462.
doi: 10.1016/j.geb.2019.09.014. |
[15] |
L. R. Izquierdo, S. S. Izquierdo and W. H. Sandholm,
EvoDyn-3s: A Mathematica computable document to analyze evolutionary dynamics in 3-strategy games, SoftwareX, 7 (2018), 226-233.
doi: 10.1016/j.softx.2018.07.006. |
[16] |
A. Kelley,
Stability of the center-stable manifold, J. Math. Anal. Appl., 18 (1967), 336-344.
doi: 10.1016/0022-247X(67)90061-3. |
[17] |
A. Kelley,
The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.
doi: 10.1016/0022-0396(67)90016-2. |
[18] |
D. M. Kreps, P. Milgrom, J. Roberts and R. Wilson,
Rational cooperation in the finitely repeated prisoners' dilemma, J. Econom. Theory, 27 (1982), 245-252.
doi: 10.1016/0022-0531(82)90029-1. |
[19] |
R. D. McKelvey and T. R. Palfrey,
An experimental study of the centipede game, Econometrica, 60 (1992), 803-836.
doi: 10.2307/2951567. |
[20] |
R. D. McKelvey and T. R. Palfrey,
Quantal response equilibria for extensive form games, Experimental Economics, 1 (1998), 9-41.
doi: 10.1023/A:1009905800005. |
[21] |
R. Nagel and F. F. Tang,
Experimental results on the centipede game in normal form: An investigation on learning, J. Mathematical Psychology, 42 (1998), 356-384.
doi: 10.1006/jmps.1998.1225. |
[22] |
M. J. Osborne and A. Rubinstein, Games with procedurally rational players, Amer. Economic Rev., 88 (1998), 834–847. Available from: https://www.jstor.org/stable/117008. |
[23] |
I. Palacios-Huerta and O. Volij,
Field centipedes, Amer. Economic Rev., 99 (2009), 1619-1635.
doi: 10.1257/aer.99.4.1619. |
[24] |
A. Perea,
Belief in the opponents' future rationality, Games Econom. Behav., 83 (2014), 231-254.
doi: 10.1016/j.geb.2013.11.008. |
[25] |
L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[26] |
P. Pettit and R. Sugden,
The backward induction paradox, J. Philos., 86 (1989), 169-182.
doi: 10.2307/2026960. |
[27] |
G. Ponti,
Cycles of learning in the centipede game, Games Econom. Behav., 30 (2000), 115-141.
doi: 10.1006/game.1998.0707. |
[28] |
B. D. Pulford, E. M. Krockow, A. M. Colman and C. L. Lawrence, Social value induction and cooperation in the centipede game, PLoS ONE, 11 (2016).
doi: 10.1371/journal.pone.0152352. |
[29] |
P. J. Reny,
Backward induction, normal form perfection and explicable equilibria, Econometrica, 60 (1992), 627-649.
doi: 10.2307/2951586. |
[30] |
R. W. Rosenthal,
Games of perfect information, predatory pricing and the chain-store paradox, J. Econom. Theory, 25 (1981), 92-100.
doi: 10.1016/0022-0531(81)90018-1. |
[31] |
W. H. Sandholm,
Evolution and equilibrium under inexact information, Games Econom. Behav., 44 (2003), 343-378.
doi: 10.1016/S0899-8256(03)00026-5. |
[32] |
W. H. Sandholm, Population Games and Evolutionary Dynamics, Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 2010. |
[33] |
W. H. Sandholm, S. S. Izquierdo and L. R. Izquierdo,
Best experience payoff dynamics and cooperation in the centipede game, Theor. Econ., 14 (2019), 1347-1385.
doi: 10.3982/TE3565. |
[34] |
W. H. Sandholm, S. S. Izquierdo and L. R. Izquierdo, Stability for best experienced payoff dynamics, J. Econom. Theory, 185 (2020), 35pp.
doi: 10.1016/j.jet.2019.104957. |
[35] |
R. Selten,
Reexamination of the perfectness concept for equilibrium points in extensive games, Internat. J. Game Theory, 4 (1975), 25-55.
doi: 10.1007/BF01766400. |
[36] |
R. Selten, Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit, Zeitschrift für die Gesamte Staatswissenschaft, 121 (1965), 301–324. Available from: http://www.jstor.org/stable/40748884. |
[37] |
R. Sethi,
Stability of equilibria in games with procedurally rational players, Games Econom. Behav., 32 (2000), 85-104.
doi: 10.1006/game.1999.0753. |
[38] |
J. Sijbrand,
Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469.
doi: 10.1090/S0002-9947-1985-0783998-8. |
[39] |
R. Smead,
The evolution of cooperation in the centipede game with finite populations, Philos. Sci., 75 (2008), 157-177.
doi: 10.1086/590197. |
[40] |
R. Stalnaker,
Knowledge, belief and counterfactual reasoning in games, Economics & Philosophy, 12 (1996), 133-163.
doi: 10.1017/S0266267100004132. |
[41] |
U. Wilensky, Netlogo software, Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL, 1999. Available from: http://ccl.northwestern.edu/netlogo/. |
[42] |
Z. Xu,
Convergence of best-response dynamics in extensive-form games, J. Econom. Theory, 162 (2016), 21-54.
doi: 10.1016/j.jet.2015.12.001. |
[43] |
D. Zusai, Gains in evolutionary dynamics: A unifying and intuitive approach to linking static and dynamic stability, preprint, arXiv: 1805.04898. |
show all references
References:
[1] |
E. Ben-Porath,
Rationality, Nash equilibrium and backwards induction in perfect- information games, Rev. Econom. Stud., 64 (1997), 23-46.
doi: 10.2307/2971739. |
[2] |
K. Binmore,
Modeling rational players: Part Ⅰ, Economics & Philosophy, 3 (1987), 179-214.
doi: 10.1017/S0266267100002893. |
[3] |
K. Binmore, Natural Justice, Oxford University Press, 2005.
doi: 10.1093/acprof:oso/9780195178111.001.0001.![]() ![]() |
[4] |
K. Binmore and L. Samuelson, An economist’s perspective on the evolution of norms, J. Institutional Theoretical Economics (JITE), 150 (1994), 45–63. Available from: https://www.jstor.org/stable/40753015. |
[5] |
K. G. Binmore, L. Samuelson and R. Vaughan,
Musical chairs: Modeling noisy evolution, Games Econom. Behav., 11 (1995), 1-35.
doi: 10.1006/game.1995.1039. |
[6] |
G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Automata Theory and Formal Languages, Lecture Notes in Comput. Sci., 33, Springer, Berlin, 1975,134–183.
doi: 10.1007/3-540-07407-4_17. |
[7] |
J. C. Cox and D. James, On replication and perturbation of the McKelvey and Palfrey centipede game experiment, in Replication in Experimental Economics (Research in Experimental Economics, Volume 18), Emerald Group Publishing Ltd., 2015, 53–94.
doi: 10.1108/S0193-230620150000018003. |
[8] |
R. Cressman and K. H. Schlag,
The dynamic (in)stability of backwards induction, J. Econom. Theory, 83 (1998), 260-285.
doi: 10.1006/jeth.1996.2465. |
[9] |
M. Embrey, G. R. Fréchette and S. Yuksel,
Cooperation in the finitely repeated prisoner's dilemma, Quarterly J. Economics, 133 (2018), 509-551.
doi: 10.1093/qje/qjx033. |
[10] |
I. Gilboa and A. Matsui,
Social stability and equilibrium, Econometrica, 59 (1991), 859-867.
doi: 10.2307/2938230. |
[11] |
J. Y. Halpern,
Substantive rationality and backward induction, Games Econom. Behav., 37 (2001), 425-435.
doi: 10.1006/game.2000.0838. |
[12] |
J. Hofbauer, Stability for the Best Response Dynamics, University of Vienna, 1995, unpublished manuscript. |
[13] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.
![]() ![]() |
[14] |
L. R. Izquierdo, S. S. Izquierdo and W. H. Sandholm,
An introduction to $ABED$: Agent-based simulation of evolutionary game dynamics, Games Econom. Behav., 118 (2019), 434-462.
doi: 10.1016/j.geb.2019.09.014. |
[15] |
L. R. Izquierdo, S. S. Izquierdo and W. H. Sandholm,
EvoDyn-3s: A Mathematica computable document to analyze evolutionary dynamics in 3-strategy games, SoftwareX, 7 (2018), 226-233.
doi: 10.1016/j.softx.2018.07.006. |
[16] |
A. Kelley,
Stability of the center-stable manifold, J. Math. Anal. Appl., 18 (1967), 336-344.
doi: 10.1016/0022-247X(67)90061-3. |
[17] |
A. Kelley,
The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.
doi: 10.1016/0022-0396(67)90016-2. |
[18] |
D. M. Kreps, P. Milgrom, J. Roberts and R. Wilson,
Rational cooperation in the finitely repeated prisoners' dilemma, J. Econom. Theory, 27 (1982), 245-252.
doi: 10.1016/0022-0531(82)90029-1. |
[19] |
R. D. McKelvey and T. R. Palfrey,
An experimental study of the centipede game, Econometrica, 60 (1992), 803-836.
doi: 10.2307/2951567. |
[20] |
R. D. McKelvey and T. R. Palfrey,
Quantal response equilibria for extensive form games, Experimental Economics, 1 (1998), 9-41.
doi: 10.1023/A:1009905800005. |
[21] |
R. Nagel and F. F. Tang,
Experimental results on the centipede game in normal form: An investigation on learning, J. Mathematical Psychology, 42 (1998), 356-384.
doi: 10.1006/jmps.1998.1225. |
[22] |
M. J. Osborne and A. Rubinstein, Games with procedurally rational players, Amer. Economic Rev., 88 (1998), 834–847. Available from: https://www.jstor.org/stable/117008. |
[23] |
I. Palacios-Huerta and O. Volij,
Field centipedes, Amer. Economic Rev., 99 (2009), 1619-1635.
doi: 10.1257/aer.99.4.1619. |
[24] |
A. Perea,
Belief in the opponents' future rationality, Games Econom. Behav., 83 (2014), 231-254.
doi: 10.1016/j.geb.2013.11.008. |
[25] |
L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[26] |
P. Pettit and R. Sugden,
The backward induction paradox, J. Philos., 86 (1989), 169-182.
doi: 10.2307/2026960. |
[27] |
G. Ponti,
Cycles of learning in the centipede game, Games Econom. Behav., 30 (2000), 115-141.
doi: 10.1006/game.1998.0707. |
[28] |
B. D. Pulford, E. M. Krockow, A. M. Colman and C. L. Lawrence, Social value induction and cooperation in the centipede game, PLoS ONE, 11 (2016).
doi: 10.1371/journal.pone.0152352. |
[29] |
P. J. Reny,
Backward induction, normal form perfection and explicable equilibria, Econometrica, 60 (1992), 627-649.
doi: 10.2307/2951586. |
[30] |
R. W. Rosenthal,
Games of perfect information, predatory pricing and the chain-store paradox, J. Econom. Theory, 25 (1981), 92-100.
doi: 10.1016/0022-0531(81)90018-1. |
[31] |
W. H. Sandholm,
Evolution and equilibrium under inexact information, Games Econom. Behav., 44 (2003), 343-378.
doi: 10.1016/S0899-8256(03)00026-5. |
[32] |
W. H. Sandholm, Population Games and Evolutionary Dynamics, Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 2010. |
[33] |
W. H. Sandholm, S. S. Izquierdo and L. R. Izquierdo,
Best experience payoff dynamics and cooperation in the centipede game, Theor. Econ., 14 (2019), 1347-1385.
doi: 10.3982/TE3565. |
[34] |
W. H. Sandholm, S. S. Izquierdo and L. R. Izquierdo, Stability for best experienced payoff dynamics, J. Econom. Theory, 185 (2020), 35pp.
doi: 10.1016/j.jet.2019.104957. |
[35] |
R. Selten,
Reexamination of the perfectness concept for equilibrium points in extensive games, Internat. J. Game Theory, 4 (1975), 25-55.
doi: 10.1007/BF01766400. |
[36] |
R. Selten, Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit, Zeitschrift für die Gesamte Staatswissenschaft, 121 (1965), 301–324. Available from: http://www.jstor.org/stable/40748884. |
[37] |
R. Sethi,
Stability of equilibria in games with procedurally rational players, Games Econom. Behav., 32 (2000), 85-104.
doi: 10.1006/game.1999.0753. |
[38] |
J. Sijbrand,
Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469.
doi: 10.1090/S0002-9947-1985-0783998-8. |
[39] |
R. Smead,
The evolution of cooperation in the centipede game with finite populations, Philos. Sci., 75 (2008), 157-177.
doi: 10.1086/590197. |
[40] |
R. Stalnaker,
Knowledge, belief and counterfactual reasoning in games, Economics & Philosophy, 12 (1996), 133-163.
doi: 10.1017/S0266267100004132. |
[41] |
U. Wilensky, Netlogo software, Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL, 1999. Available from: http://ccl.northwestern.edu/netlogo/. |
[42] |
Z. Xu,
Convergence of best-response dynamics in extensive-form games, J. Econom. Theory, 162 (2016), 21-54.
doi: 10.1016/j.jet.2015.12.001. |
[43] |
D. Zusai, Gains in evolutionary dynamics: A unifying and intuitive approach to linking static and dynamic stability, preprint, arXiv: 1805.04898. |















Str | # | 0.32 | 0.66 | 0.02 | Str | # | 0.76 | 0.22 | 0.02 |
0 | 0 | 0 | 297 | 297 | 297 | ||||
198 | 198 | 198 | 32 | 495 | 495 | 495 | |||
76 | 303 | 300 | 300 | 66 | 468 | 465 | 465 | ||
22 | 241 | 239 | 236 | 2 | 450 | 448 | 445 | ||
2 | 239 | 237 | 235 | 448 | 446 | 444 | |||
239 | 237 | 235 | 448 | 446 | 444 | ||||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the three values on the right of the # column are the three possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.32 | 0.66 | 0.02 | Str | # | 0.76 | 0.22 | 0.02 |
0 | 0 | 0 | 297 | 297 | 297 | ||||
198 | 198 | 198 | 32 | 495 | 495 | 495 | |||
76 | 303 | 300 | 300 | 66 | 468 | 465 | 465 | ||
22 | 241 | 239 | 236 | 2 | 450 | 448 | 445 | ||
2 | 239 | 237 | 235 | 448 | 446 | 444 | |||
239 | 237 | 235 | 448 | 446 | 444 | ||||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the three values on the right of the # column are the three possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
[7] | [6] | [5] | [4] | [3] | [2] | [1] | [0] | |
2 | - | - | - | - | - | - | .618034 | .381966 |
3 | - | - | - | - | - | - | .539189 | .460811 |
4 | - | - | - | - | - | .208426 | .411450 | .380124 |
5 | - | - | - | - | - | .223867 | .398692 | .377441 |
6 | - | - | - | - | .035722 | .223253 | .378763 | .362262 |
7 | - | - | - | - | .040882 | .225279 | .374384 | .359455 |
8 | - | - | - | .002980 | .042792 | .225384 | .371574 | .357271 |
9 | - | - | - | .003239 | .043396 | .225559 | .370966 | .356839 |
10 | - | - | .000138 | .003311 | .043558 | .225576 | .370747 | .356670 |
11 | - | - | .000145 | .003327 | .043595 | .225585 | .370707 | .356641 |
12 | - | .000147 | .003330 | .043603 | .225586 | .370697 | .356633 | |
13 | - | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | |
14 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
15 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
16 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
17 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
20 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
[7] | [6] | [5] | [4] | [3] | [2] | [1] | [0] | |
2 | - | - | - | - | - | - | .618034 | .381966 |
3 | - | - | - | - | - | .369102 | .369102 | .261795 |
4 | - | - | - | - | - | .344955 | .364555 | .290490 |
5 | - | - | - | - | .087713 | .310211 | .329668 | .272409 |
6 | - | - | - | - | .100021 | .304394 | .323241 | .272345 |
7 | - | - | - | .010544 | .104027 | .298920 | .317193 | .269316 |
8 | - | - | - | .011813 | .105888 | .297664 | .315745 | .268891 |
9 | - | - | .000650 | .012191 | .106378 | .297094 | .315103 | .268585 |
10 | - | - | .000692 | .012297 | .106528 | .296977 | .314969 | .268537 |
11 | - | .000701 | .012321 | .106559 | .296944 | .314931 | .268520 | |
12 | - | .000703 | .012326 | .106566 | .296938 | .314925 | .268518 | |
13 | .000703 | .012327 | .106567 | .296937 | .314923 | .268517 | ||
14 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
15 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
16 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
20 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 |
[7] | [6] | [5] | [4] | [3] | [2] | [1] | [0] | |
2 | - | - | - | - | - | - | .618034 | .381966 |
3 | - | - | - | - | - | - | .539189 | .460811 |
4 | - | - | - | - | - | .208426 | .411450 | .380124 |
5 | - | - | - | - | - | .223867 | .398692 | .377441 |
6 | - | - | - | - | .035722 | .223253 | .378763 | .362262 |
7 | - | - | - | - | .040882 | .225279 | .374384 | .359455 |
8 | - | - | - | .002980 | .042792 | .225384 | .371574 | .357271 |
9 | - | - | - | .003239 | .043396 | .225559 | .370966 | .356839 |
10 | - | - | .000138 | .003311 | .043558 | .225576 | .370747 | .356670 |
11 | - | - | .000145 | .003327 | .043595 | .225585 | .370707 | .356641 |
12 | - | .000147 | .003330 | .043603 | .225586 | .370697 | .356633 | |
13 | - | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | |
14 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
15 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
16 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
17 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
20 | .000147 | .003331 | .043604 | .225586 | .370695 | .356632 | ||
[7] | [6] | [5] | [4] | [3] | [2] | [1] | [0] | |
2 | - | - | - | - | - | - | .618034 | .381966 |
3 | - | - | - | - | - | .369102 | .369102 | .261795 |
4 | - | - | - | - | - | .344955 | .364555 | .290490 |
5 | - | - | - | - | .087713 | .310211 | .329668 | .272409 |
6 | - | - | - | - | .100021 | .304394 | .323241 | .272345 |
7 | - | - | - | .010544 | .104027 | .298920 | .317193 | .269316 |
8 | - | - | - | .011813 | .105888 | .297664 | .315745 | .268891 |
9 | - | - | .000650 | .012191 | .106378 | .297094 | .315103 | .268585 |
10 | - | - | .000692 | .012297 | .106528 | .296977 | .314969 | .268537 |
11 | - | .000701 | .012321 | .106559 | .296944 | .314931 | .268520 | |
12 | - | .000703 | .012326 | .106566 | .296938 | .314925 | .268518 | |
13 | .000703 | .012327 | .106567 | .296937 | .314923 | .268517 | ||
14 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
15 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
16 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 | ||
20 | .000703 | .012327 | .106567 | .296936 | .314923 | .268517 |
|
||||||||||||
|
||||||||||||
Str | # | 0.04 | 0.89 | 0.07 | Str | # | 0.66 | 0.34 |
0 | 0 | 0 | 4 | 297 | 297 | |||
66 | 189 | 186 | 186 | 89 | 300 | 297 | ||
34 | 114 | 112 | 109 | 7 | 266 | 264 | ||
107 | 105 | 103 | 266 | 264 | ||||
107 | 105 | 103 | 266 | 264 | ||||
107 | 105 | 103 | 266 | 264 | ||||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.04 | 0.89 | 0.07 | Str | # | 0.66 | 0.34 |
0 | 0 | 0 | 4 | 297 | 297 | |||
66 | 189 | 186 | 186 | 89 | 300 | 297 | ||
34 | 114 | 112 | 109 | 7 | 266 | 264 | ||
107 | 105 | 103 | 266 | 264 | ||||
107 | 105 | 103 | 266 | 264 | ||||
107 | 105 | 103 | 266 | 264 | ||||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.2 | 0.75 | 0.02 | 0.01 | 0.01 | 0.01 | Str | # | 0.78 | 0.15 | 0.02 | 0.02 | 0.03 |
0 | 0 | 0 | 0 | 0 | 0 | 20 | 297 | 297 | 297 | 297 | 297 | |||
78 | 141 | 138 | 138 | 138 | 138 | 138 | 75 | 264 | 261 | 261 | 261 | 261 | ||
15 | 76 | 74 | 71 | 71 | 71 | 71 | 2 | 263 | 261 | 258 | 258 | 258 | ||
2 | 80 | 78 | 76 | 73 | 73 | 73 | 1 | 271 | 269 | 267 | 264 | 264 | ||
2 | 83 | 81 | 79 | 77 | 74 | 74 | 1 | 275 | 273 | 271 | 269 | 266 | ||
3 | 84 | 82 | 80 | 78 | 76 | 73 | 1 | 272 | 270 | 268 | 266 | 264 | ||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.2 | 0.75 | 0.02 | 0.01 | 0.01 | 0.01 | Str | # | 0.78 | 0.15 | 0.02 | 0.02 | 0.03 |
0 | 0 | 0 | 0 | 0 | 0 | 20 | 297 | 297 | 297 | 297 | 297 | |||
78 | 141 | 138 | 138 | 138 | 138 | 138 | 75 | 264 | 261 | 261 | 261 | 261 | ||
15 | 76 | 74 | 71 | 71 | 71 | 71 | 2 | 263 | 261 | 258 | 258 | 258 | ||
2 | 80 | 78 | 76 | 73 | 73 | 73 | 1 | 271 | 269 | 267 | 264 | 264 | ||
2 | 83 | 81 | 79 | 77 | 74 | 74 | 1 | 275 | 273 | 271 | 269 | 266 | ||
3 | 84 | 82 | 80 | 78 | 76 | 73 | 1 | 272 | 270 | 268 | 266 | 264 | ||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.24 | 0.55 | 0.03 | 0.04 | 0.08 | 0.06 | Str | # | 0.73 | 0.10 | 0.04 | 0.04 | 0.09 |
0 | 0 | 0 | 0 | 0 | 0 | 24 | 297 | 297 | 297 | 297 | 297 | |||
73 | 129 | 126 | 126 | 126 | 126 | 126 | 55 | 279 | 276 | 276 | 276 | 276 | ||
10 | 116 | 114 | 111 | 111 | 111 | 111 | 3 | 303 | 301 | 298 | 298 | 298 | ||
4 | 149 | 147 | 145 | 142 | 142 | 142 | 4 | 325 | 323 | 321 | 318 | 318 | ||
4 | 173 | 171 | 169 | 167 | 164 | 164 | 8 | 339 | 337 | 335 | 333 | 330 | ||
9 | 177 | 175 | 173 | 171 | 169 | 166 | 6 | 330 | 328 | 326 | 324 | 322 | ||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Str | # | 0.24 | 0.55 | 0.03 | 0.04 | 0.08 | 0.06 | Str | # | 0.73 | 0.10 | 0.04 | 0.04 | 0.09 |
0 | 0 | 0 | 0 | 0 | 0 | 24 | 297 | 297 | 297 | 297 | 297 | |||
73 | 129 | 126 | 126 | 126 | 126 | 126 | 55 | 279 | 276 | 276 | 276 | 276 | ||
10 | 116 | 114 | 111 | 111 | 111 | 111 | 3 | 303 | 301 | 298 | 298 | 298 | ||
4 | 149 | 147 | 145 | 142 | 142 | 142 | 4 | 325 | 323 | 321 | 318 | 318 | ||
4 | 173 | 171 | 169 | 167 | 164 | 164 | 8 | 339 | 337 | 335 | 333 | 330 | ||
9 | 177 | 175 | 173 | 171 | 169 | 166 | 6 | 330 | 328 | 326 | 324 | 322 | ||
Str: strategy; #: number of players using the corresponding strategy. For each strategy, the values on the right of the # column are the possible total payoffs obtainable at the considered state by the corresponding strategy in 99 trials. Top numbers in bold: probability of obtaining the payoffs on that column, at the considered state. |
Duration of play | |
||||||||||||
Payoff player 1 | |||||||||||||
Payoff player 2 | |||||||||||||
Stopping strategy | none |
Duration of play | |
||||||||||||
Payoff player 1 | |||||||||||||
Payoff player 2 | |||||||||||||
Stopping strategy | none |
Stopping strategy | none | |||||||||
First row: stopping strategy at the first trial. First column: stopping strategy at the second trial. |
Stopping strategy | none | |||||||||
First row: stopping strategy at the first trial. First column: stopping strategy at the second trial. |
Duration of play | |||||||||||||
Payoff player 1 | -1 | 5 | 4 | 10 | 9 | 15 | 14 | 20 | 19 | 25 | 24 | 30 | |
Payoff player 2 | 6 | 5 | 11 | 10 | 16 | 15 | 21 | 20 | 26 | 25 | 31 | 30 | |
Stopping strategy | none |
Duration of play | |||||||||||||
Payoff player 1 | -1 | 5 | 4 | 10 | 9 | 15 | 14 | 20 | 19 | 25 | 24 | 30 | |
Payoff player 2 | 6 | 5 | 11 | 10 | 16 | 15 | 21 | 20 | 26 | 25 | 31 | 30 | |
Stopping strategy | none |
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