# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2021018
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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

* Corresponding author: Segismundo S. Izquierdo

Received  October 2020 Revised  April 2021 Early access May 2021

Fund Project: Financial support from the Spanish State Research Agency (PID2020-118906GB-I00/AEI/10.13039/501100011033), from "Junta de Castilla y León - Consejería de Educación" through BDNS 425389, from the Spanish Ministry of Science, Innovation and Universities (PRX18-00182, PRX19/00113), and from the Fulbright Program (PRX19/00113), is gratefully acknowledged

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.

Citation: Segismundo S. Izquierdo, Luis R. Izquierdo. "Test two, choose the better" leads to high cooperation in the Centipede game. Journal of Dynamics and Games, doi: 10.3934/jdg.2021018
##### 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.
A Centipede game with four decision nodes, each labeled with the deciding player. Payoffs for player 1 (P1) appear above those for player 2 (P2)
The stable rest point of Centipede under dynamic (1) for game lengths $d = 2, \ldots, 10$ and $d = 20$. Stacked bars, from the bottom to the top, represent weights on strategy [0] (continue at all decision nodes), [1] (stop at the last node), [2] (stop at the second-to-last node), etc. The dashed line separates exact ($d \leq 8$) and numerical ($d \geq 9$) results
Stable cycles in Centipede of length $d = 4$ under dynamics (2) for $\kappa = 50$ and $100$. Lighter shading represents faster motion. Shapes synchronize positions along the cycle
Expected duration of play at the stable rest point (for $\kappa \le 30$) and integrated over the stable cycle (for $\kappa \ge 40$) in Centipede of length $d = 4$, for various numbers of trials $\kappa$
Plot of the empirical distribution on population states, in each population, in a four-node Centipede with $N = 10$ and $\kappa = 1$
Smoothed 3D histogram of the empirical distribution on population states, in each population, for $N = 100$ and $\kappa = 1$
Expected fraction of matches that reach each terminal node $i \in \{1,...,5\}$ in a Centipede of length $d = 4$ with $\kappa = 1$. For $N = 10$ and $N = 100$, the height of each column corresponds to the average value over the empirical distribution on population states. The vertical lines correspond to the average $\pm$ one standard deviation. MD: mean dynamics
Expected duration of play, averaged over the empirical distribution on population states, in Centipede with $d = 4$, for various numbers of trials $\kappa$ and population sizes $N$
Sample paths of the expected duration of play in a 4-node Centipede played in populations of size $N = 100$ for various choices of $\kappa$
Expected duration of play over solution trajectories of the mean dynamic (2) in a 4-node Centipede for various choices of $\kappa$
Smoothed 3D histogram of the number of visits to states in each population for a 4-node Centipede played in populations of size $N = 100$ with $\kappa = 50$ trials. Cyclical behavior leads the empirical distribution to take a crater-like form
Expected fraction of matches that reach each terminal node $i \in \{1,...,11\}$. Centipede with 10 decision nodes ($d = 10$) and $\kappa = 1$. For $N = 10$ and $N = 100$, the height of each column corresponds to the average value over the empirical distribution on population states. The vertical lines correspond to the average $\pm$ one standard deviation
Expected duration of play in a Centipede game with 10 decision nodes, averaged over the empirical distribution on population states, for different number of trials $\kappa$
Evolution of the expected duration of play in a Centipede game with 10 decision nodes played in populations of size $N = 100$, for different number of trials $\kappa$
Expected duration of play in two 10-node Centipede games with different cost/gain ratios $c/b$, averaged over the empirical distribution on population states, for different number of trials $\kappa$. $N = 50$
Possible payoffs obtained by each strategy, with their probabilities, at a state next to the bottom of a cycle
 Str # 0.32 0.66 0.02 Str # 0.76 0.22 0.02 $1_x$ 0 0 0 $1_y$ 297 297 297 $2_x$ 198 198 198 $2_y$ 32 495 495 495 $3_x$ 76 303 300 300 $3_y$ 66 468 465 465 $4_x$ 22 241 239 236 $4_y$ 2 450 448 445 $5_x$ 2 239 237 235 $5_y$ 448 446 444 $6_x$ 239 237 235 $6_y$ 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 $1_x$ 0 0 0 $1_y$ 297 297 297 $2_x$ 198 198 198 $2_y$ 32 495 495 495 $3_x$ 76 303 300 300 $3_y$ 66 468 465 465 $4_x$ 22 241 239 236 $4_y$ 2 450 448 445 $5_x$ 2 239 237 235 $5_y$ 448 446 444 $6_x$ 239 237 235 $6_y$ 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.
The interior rest point $\xi^\ast = \xi^\ast( d)$ of the dynamic for Centipede of lengths $d \in \{2, \ldots , 20\}$. $p$ denotes the penultimate player, $q$ the last player. The dashed lines separated exact ($d \leq 8$) from numerical ($d \geq 9$) results. The numbers shown are approximations, since the exact values are algebraic numbers that do not admit an exact rational representation
 $p$ [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 - $4.19\times 10^{-6}$ .000147 .003330 .043603 .225586 .370697 .356633 13 - $4.32\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 14 $9.04\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 15 $9.24\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 16 $9.27\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 17 $9.28\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ 20 $9.28\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 $q$ [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 - $2.42\times 10^{-5}$ .000701 .012321 .106559 .296944 .314931 .268520 12 - $2.51\times 10^{-5}$ .000703 .012326 .106566 .296938 .314925 .268518 13 $6.17\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296937 .314923 .268517 14 $6.33\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 15 $6.35\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 16 $6.36\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ 20 $6.36\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517
 $p$ [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 - $4.19\times 10^{-6}$ .000147 .003330 .043603 .225586 .370697 .356633 13 - $4.32\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 14 $9.04\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 15 $9.24\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 16 $9.27\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 17 $9.28\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ 20 $9.28\times 10^{-8}$ $4.34\times 10^{-6}$ .000147 .003331 .043604 .225586 .370695 .356632 $q$ [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 - $2.42\times 10^{-5}$ .000701 .012321 .106559 .296944 .314931 .268520 12 - $2.51\times 10^{-5}$ .000703 .012326 .106566 .296938 .314925 .268518 13 $6.17\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296937 .314923 .268517 14 $6.33\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 15 $6.35\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 16 $6.36\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517 $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ 20 $6.36\times 10^{-7}$ $2.53\times 10^{-5}$ .000703 .012327 .106567 .296936 .314923 .268517
Approximate eigenvalues of $DV( \xi^\ast)$ for the dynamic
 $d=2$ $-.8090\pm.4468 \, \mathrm{i}$ $d=3$ $-.9071$ $-.6556\pm.3376 \, \mathrm{i}$ $d=4$ $-.8715\pm.1608 \, \mathrm{i}$ $-.5238\pm.2572 \, \mathrm{i}$ $d=5$ $-.6851\pm.1242 \, \mathrm{i}$ $-.4412 \pm.1803 \, \mathrm{i}$ $-1.0845$ $d=6$ $-.5778\pm.1093 \, \mathrm{i}$ $-.3687\pm.1286\, \mathrm{i}$ $-1.1369$ $-.8289$ $d=7$ $-.4762\pm.0829 \, \mathrm{i}$ $-.3145\pm.0920 \, \mathrm{i}$ $-1.2171$ $-.8220$ $-.7632$ $d=8$ $-.4262\pm.0657 \, \mathrm{i}$ $-.2781\pm.0620 \, \mathrm{i}$ $-1.2194$ $-.9406$ $-.7827$ $-.5425$ $d=9$ $-.3647\pm.0549 \, \mathrm{i}$ $-.2471\pm.0516 \, \mathrm{i}$ $-1.2203$ $-.9703$ $-.8741$ $-.6137$ $-.5069$ $d=10$ $-.3394\pm.0404 \, \mathrm{i}$ $-.2248\pm.0383 \, \mathrm{i}$ $-1.1978$ $-.9917$ $-.9230$ $-.6928$ $-.5786$ $-.4027$ $d=11$ $-.2982\pm.0404 \, \mathrm{i}$ $-.2040\pm.0378 \, \mathrm{i}$ $-1.1856$ $-.9975$ $-.9427$ $-.7738$ $-.6353$ $-.4986$ $-.3874$ $d=12$ $-.2835\pm.0306 \, \mathrm{i}$ $-.1878\pm.0305 \, \mathrm{i}$ $-1.1666$ $-.9995$ $-.9562$ $-.8129$ $-.6930$ $-.5637$ $-.4666$ $-.3280$ $d=13$ $-.2531\pm.0331 \, \mathrm{i}$ $-.1731\pm.0312 \, \mathrm{i}$ $-1.1568$ $-.9999$ $-.9583$ $-.8390$ $-.7301$ $-.6389$ $-.5184$ $-.4212$ $-.3214$ $d=14$ $-.2431\pm.0260 \, \mathrm{i}$ $-.1610\pm.0261 \, \mathrm{i}$ $-1.1429$ $-1.0000$ $-.9637$ $-.8555$ $-.7466$ $-.6921$ $-.5724$ $-.4800$ $-.3974$ $-.2804$
 $d=2$ $-.8090\pm.4468 \, \mathrm{i}$ $d=3$ $-.9071$ $-.6556\pm.3376 \, \mathrm{i}$ $d=4$ $-.8715\pm.1608 \, \mathrm{i}$ $-.5238\pm.2572 \, \mathrm{i}$ $d=5$ $-.6851\pm.1242 \, \mathrm{i}$ $-.4412 \pm.1803 \, \mathrm{i}$ $-1.0845$ $d=6$ $-.5778\pm.1093 \, \mathrm{i}$ $-.3687\pm.1286\, \mathrm{i}$ $-1.1369$ $-.8289$ $d=7$ $-.4762\pm.0829 \, \mathrm{i}$ $-.3145\pm.0920 \, \mathrm{i}$ $-1.2171$ $-.8220$ $-.7632$ $d=8$ $-.4262\pm.0657 \, \mathrm{i}$ $-.2781\pm.0620 \, \mathrm{i}$ $-1.2194$ $-.9406$ $-.7827$ $-.5425$ $d=9$ $-.3647\pm.0549 \, \mathrm{i}$ $-.2471\pm.0516 \, \mathrm{i}$ $-1.2203$ $-.9703$ $-.8741$ $-.6137$ $-.5069$ $d=10$ $-.3394\pm.0404 \, \mathrm{i}$ $-.2248\pm.0383 \, \mathrm{i}$ $-1.1978$ $-.9917$ $-.9230$ $-.6928$ $-.5786$ $-.4027$ $d=11$ $-.2982\pm.0404 \, \mathrm{i}$ $-.2040\pm.0378 \, \mathrm{i}$ $-1.1856$ $-.9975$ $-.9427$ $-.7738$ $-.6353$ $-.4986$ $-.3874$ $d=12$ $-.2835\pm.0306 \, \mathrm{i}$ $-.1878\pm.0305 \, \mathrm{i}$ $-1.1666$ $-.9995$ $-.9562$ $-.8129$ $-.6930$ $-.5637$ $-.4666$ $-.3280$ $d=13$ $-.2531\pm.0331 \, \mathrm{i}$ $-.1731\pm.0312 \, \mathrm{i}$ $-1.1568$ $-.9999$ $-.9583$ $-.8390$ $-.7301$ $-.6389$ $-.5184$ $-.4212$ $-.3214$ $d=14$ $-.2431\pm.0260 \, \mathrm{i}$ $-.1610\pm.0261 \, \mathrm{i}$ $-1.1429$ $-1.0000$ $-.9637$ $-.8555$ $-.7466$ $-.6921$ $-.5724$ $-.4800$ $-.3974$ $-.2804$
Possible payoffs obtained by each strategy, with their probabilities, at a state near the bottom of a cycle
 Str # 0.04 0.89 0.07 Str # 0.66 0.34 $1_x$ 0 0 0 $1_y$ 4 297 297 $2_x$ 66 189 186 186 $2_y$ 89 300 297 $3_x$ 34 114 112 109 $3_y$ 7 266 264 $4_x$ 107 105 103 $4_y$ 266 264 $5_x$ 107 105 103 $5_y$ 266 264 $6_x$ 107 105 103 $6_y$ 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 $1_x$ 0 0 0 $1_y$ 4 297 297 $2_x$ 66 189 186 186 $2_y$ 89 300 297 $3_x$ 34 114 112 109 $3_y$ 7 266 264 $4_x$ 107 105 103 $4_y$ 266 264 $5_x$ 107 105 103 $5_y$ 266 264 $6_x$ 107 105 103 $6_y$ 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.
Possible payoffs obtained by each strategy, with their probabilities, at a state leaving the bottom of a cycle
 Str # 0.2 0.75 0.02 0.01 0.01 0.01 Str # 0.78 0.15 0.02 0.02 0.03 $1_x$ 0 0 0 0 0 0 $1_y$ 20 297 297 297 297 297 $2_x$ 78 141 138 138 138 138 138 $2_y$ 75 264 261 261 261 261 $3_x$ 15 76 74 71 71 71 71 $3_y$ 2 263 261 258 258 258 $4_x$ 2 80 78 76 73 73 73 $4_y$ 1 271 269 267 264 264 $5_x$ 2 83 81 79 77 74 74 $5_y$ 1 275 273 271 269 266 $6_x$ 3 84 82 80 78 76 73 $6_y$ 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 $1_x$ 0 0 0 0 0 0 $1_y$ 20 297 297 297 297 297 $2_x$ 78 141 138 138 138 138 138 $2_y$ 75 264 261 261 261 261 $3_x$ 15 76 74 71 71 71 71 $3_y$ 2 263 261 258 258 258 $4_x$ 2 80 78 76 73 73 73 $4_y$ 1 271 269 267 264 264 $5_x$ 2 83 81 79 77 74 74 $5_y$ 1 275 273 271 269 266 $6_x$ 3 84 82 80 78 76 73 $6_y$ 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.
Possible payoffs obtained by each strategy, with their probabilities, at a state leaving the bottom of a cycle
 Str # 0.24 0.55 0.03 0.04 0.08 0.06 Str # 0.73 0.10 0.04 0.04 0.09 $1_x$ 0 0 0 0 0 0 $1_y$ 24 297 297 297 297 297 $2_x$ 73 129 126 126 126 126 126 $2_y$ 55 279 276 276 276 276 $3_x$ 10 116 114 111 111 111 111 $3_y$ 3 303 301 298 298 298 $4_x$ 4 149 147 145 142 142 142 $4_y$ 4 325 323 321 318 318 $5_x$ 4 173 171 169 167 164 164 $5_y$ 8 339 337 335 333 330 $6_x$ 9 177 175 173 171 169 166 $6_y$ 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 $1_x$ 0 0 0 0 0 0 $1_y$ 24 297 297 297 297 297 $2_x$ 73 129 126 126 126 126 126 $2_y$ 55 279 276 276 276 276 $3_x$ 10 116 114 111 111 111 111 $3_y$ 3 303 301 298 298 298 $4_x$ 4 149 147 145 142 142 142 $4_y$ 4 325 323 321 318 318 $5_x$ 4 173 171 169 167 164 164 $5_y$ 8 339 337 335 333 330 $6_x$ 9 177 175 173 171 169 166 $6_y$ 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.
Payoff obtained by each player depending on the stopping strategy
 Duration of play $\cdots$ $d - 10$ $d - 9$ $d - 8$ $d - 7$ $d - 6$ $d - 5$ $d - 4$ $d - 3$ $d - 2$ $d - 1$ $d$ $d + 1$ Payoff player 1 $\cdots$ $d-13$ $d-10$ $d-11$ $d-8$ $d-9$ $d-6$ $d-7$ $d-4$ $d-5$ $d-2$ $d-3$ $d$ Payoff player 2 $\cdots$ $d-9$ $d-10$ $d-7$ $d-8$ $d-5$ $d-6$ $d-3$ $d-4$ $d-1$ $d-2$ $d +1$ $d$ Stopping strategy $\cdots$ $[6]_y$ $[5]_x$ $[5]_y$ $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none
 Duration of play $\cdots$ $d - 10$ $d - 9$ $d - 8$ $d - 7$ $d - 6$ $d - 5$ $d - 4$ $d - 3$ $d - 2$ $d - 1$ $d$ $d + 1$ Payoff player 1 $\cdots$ $d-13$ $d-10$ $d-11$ $d-8$ $d-9$ $d-6$ $d-7$ $d-4$ $d-5$ $d-2$ $d-3$ $d$ Payoff player 2 $\cdots$ $d-9$ $d-10$ $d-7$ $d-8$ $d-5$ $d-6$ $d-3$ $d-4$ $d-1$ $d-2$ $d +1$ $d$ Stopping strategy $\cdots$ $[6]_y$ $[5]_x$ $[5]_y$ $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none
Payoff obtained by player 2 depending on the stopping strategies, two trials
 Stopping strategy $1^{st}$ trial: $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none $2^{nd}$ trial: $[4]_x$ $2 d-16$ $2 d-13$ $2 d-14$ $2 d-11$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $[4]_y$ $2 d-13$ $2 d-10$ $2 d-11$ $2 d-8$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $[3]_x$ $2 d-14$ $2 d-11$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $[3]_y$ $2 d-11$ $2 d-8$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $[2]_x$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $[2]_y$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $2 d$ $2 d-1$ $[1]_x$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $2 d-1$ $2 d-2$ $[1]_y$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $2 d$ $2 d-1$ $2 d+2$ $2 d+1$ $none$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $2 d-1$ $2 d-2$ $2 d+1$ $2 d$ First row: stopping strategy at the first trial. First column: stopping strategy at the second trial.
 Stopping strategy $1^{st}$ trial: $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none $2^{nd}$ trial: $[4]_x$ $2 d-16$ $2 d-13$ $2 d-14$ $2 d-11$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $[4]_y$ $2 d-13$ $2 d-10$ $2 d-11$ $2 d-8$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $[3]_x$ $2 d-14$ $2 d-11$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $[3]_y$ $2 d-11$ $2 d-8$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $[2]_x$ $2 d-12$ $2 d-9$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $[2]_y$ $2 d-9$ $2 d-6$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $2 d$ $2 d-1$ $[1]_x$ $2 d-10$ $2 d-7$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $2 d-1$ $2 d-2$ $[1]_y$ $2 d-7$ $2 d-4$ $2 d-5$ $2 d-2$ $2 d-3$ $2 d$ $2 d-1$ $2 d+2$ $2 d+1$ $none$ $2 d-8$ $2 d-5$ $2 d-6$ $2 d-3$ $2 d-4$ $2 d-1$ $2 d-2$ $2 d+1$ $2 d$ First row: stopping strategy at the first trial. First column: stopping strategy at the second trial.
Payoff obtained by each player depending on the stopping strategy
 Duration of play $\cdots$ $d - 10$ $d - 9$ $d - 8$ $d - 7$ $d - 6$ $d - 5$ $d - 4$ $d - 3$ $d - 2$ $d - 1$ $d$ $d + 1$ Payoff player 1 $\cdots$ -1 5 4 10 9 15 14 20 19 25 24 30 Payoff player 2 $\cdots$ 6 5 11 10 16 15 21 20 26 25 31 30 Stopping strategy $\cdots$ $[6]_y$ $[5]_x$ $[5]_y$ $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none
 Duration of play $\cdots$ $d - 10$ $d - 9$ $d - 8$ $d - 7$ $d - 6$ $d - 5$ $d - 4$ $d - 3$ $d - 2$ $d - 1$ $d$ $d + 1$ Payoff player 1 $\cdots$ -1 5 4 10 9 15 14 20 19 25 24 30 Payoff player 2 $\cdots$ 6 5 11 10 16 15 21 20 26 25 31 30 Stopping strategy $\cdots$ $[6]_y$ $[5]_x$ $[5]_y$ $[4]_x$ $[4]_y$ $[3]_x$ $[3]_y$ $[2]_x$ $[2]_y$ $[1]_x$ $[1]_y$ none
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