American Institute of Mathematical Sciences

October  2014, 1(4): 621-638. doi: 10.3934/jdg.2014.1.621

Payoff performance of fictitious play

 1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom 2 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  August 2013 Revised  November 2014 Published  November 2014

We investigate how well continuous-time fictitious play in two-player games performs in terms of average payoff, particularly compared to Nash equilibrium payoff. We show that in many games, fictitious play outperforms Nash equilibrium on average or even at all times, and moreover that any game is linearly equivalent to one in which this is the case. Conversely, we provide conditions under which Nash equilibrium payoff dominates fictitious play payoff. A key step in our analysis is to show that fictitious play dynamics asymptotically converges to the set of coarse correlated equilibria (a fact which is implicit in the literature).
Citation: Georg Ostrovski, Sebastian van Strien. Payoff performance of fictitious play. Journal of Dynamics & Games, 2014, 1 (4) : 621-638. doi: 10.3934/jdg.2014.1.621
References:
 [1] R. J. Aumann, Subjectivity and correlation in randomized strategies, J. Math. Econom., 1 (1974), 67-96. doi: 10.1016/0304-4068(74)90037-8.  Google Scholar [2] R. J. Aumann, Correlated equilibrium as an expression of Bayesian rationality, Econometrica, 55 (1987), 1-18. doi: 10.2307/1911154.  Google Scholar [3] U. Berger, Fictitious play in $2 \times n$ games, J. Econ. Theory, 120 (2005), 139-154. doi: 10.1016/j.jet.2004.02.003.  Google Scholar [4] U. Berger, Two more classes of games with the continuous-time fictitious play property, Game. Econ. Behav., 60 (2007), 247-261. doi: 10.1016/j.geb.2006.10.008.  Google Scholar [5] U. Berger, Learning in games with strategic complementarities revisited, J. Econ. Theory, 143 (2008), 292-301. doi: 10.1016/j.jet.2008.01.007.  Google Scholar [6] D. Blackwell, Controlled random walks, In Proceedings of the International Congress of Mathematicians, 3 (1954), 336-338.  Google Scholar [7] G. W. Brown, Some notes on computation of games solutions, Technical report, Report P-78, The Rand Corporation, 1949. Google Scholar [8] G. W. Brown, Iterative solution of games by fictitious play, In Activity Analysis of Production and Allocation, volume 13, pages 374-376. John Wiley & Sons, New York, 1951.  Google Scholar [9] D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Game. Econ. Behav., 21 (1997), 40-55. doi: 10.1006/game.1997.0595.  Google Scholar [10] D. Fudenberg and D. K. Levine, Consistency and cautious fictitious play, J. Econ. Dyn. Control, 19 (1995), 1065-1089. doi: 10.1016/0165-1889(94)00819-4.  Google Scholar [11] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press Series on Economic Learning and Social Evolution, 2. MIT Press, Cambridge, MA, 1998.  Google Scholar [12] A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Game. Econ. Behav., 11 (1995), 279-303. doi: 10.1006/game.1995.1052.  Google Scholar [13] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230.  Google Scholar [14] J. Hannan, Approximation to Bayes risk in repeated play, Contributions to the Theory of Games, 3 (1957), 97-139.  Google Scholar [15] C. Harris, On the rate of convergence of continuous-time fictitious play, Game. Econ. Behav., 22 (1998), 238-259. doi: 10.1006/game.1997.0582.  Google Scholar [16] S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430. doi: 10.1111/j.1468-0262.2005.00625.x.  Google Scholar [17] S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153.  Google Scholar [18] S. Hart and A. Mas-Colell, A general class of adaptive strategies, J. Econ. Theory, 98 (2001), 26-54. doi: 10.1006/jeth.2000.2746.  Google Scholar [19] J. Hofbauer, Stability for the Best Response Dynamics, Mimeo, 1995. Google Scholar [20] V. Krishna and T. Sjöström, On the convergence of fictitious play, Math. Oper. Res., 23 (1998), 479-511. doi: 10.1287/moor.23.2.479.  Google Scholar [21] A. Matsui, Best response dynamics and socially stable strategies, J. Econ. Theory, 57 (1992), 343-362. doi: 10.1016/0022-0531(92)90040-O.  Google Scholar [22] D. Monderer, D. Samet and A. Sela, Belief affirming in learning processes, J. Econ. Theory, 73 (1997), 438-452. doi: 10.1006/jeth.1996.2245.  Google Scholar [23] S. Morris and T. Ui, Best response equivalence, Game. Econ. Behav., 49 (2004), 260-287. doi: 10.1016/j.geb.2003.12.004.  Google Scholar [24] H. J. Moulin and J.-P. Vial, Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon, Internat. J. Game Theory, 7 (1978), 201-221. doi: 10.1007/BF01769190.  Google Scholar [25] G. Ostrovski and S. van Strien, Piecewise linear Hamiltonian flows associated to zero-sum games: Transition combinatorics and questions on ergodicity, Regul. Chaotic Dyn., 16 (2011), 128-153. doi: 10.1134/S1560354711010059.  Google Scholar [26] J. Rosenmüller, Über Periodizitätseigenschaften spieltheoretischer Lernprozesse, Z. Wahrscheinlichkeit., 17 (1971), 259-308. doi: 10.1007/BF00536300.  Google Scholar [27] L. S. Shapley, Some topics in two-person games, Advances in Game Theory, 52 (1964), 1-28.  Google Scholar [28] C. Sparrow, S. van Strien and C. Harris, Fictitious play in $3\times 3$ games: The transition between periodic and chaotic behaviour, Game. Econ. Behav., 63 (2008), 259-291. doi: 10.1016/j.geb.2007.08.005.  Google Scholar [29] K. Sydsaeter and P. Hammond, Essential Mathematics for Economic Analysis, Prentice Hall, 3rd edition, 2008. Google Scholar [30] S. van Strien and C. Sparrow, Fictitious play in $3 \times 3$ games: Chaos and dithering behaviour, Game. Econ. Behav., 73 (2011), 262-286. doi: 10.1016/j.geb.2010.12.004.  Google Scholar [31] S. van Strien, A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play, Nonlinearity, 24 (2011), 1715-1742. doi: 10.1088/0951-7715/24/6/002.  Google Scholar [32] H. P. Young, Strategic Learning and Its Limits (Arne Ryde Memorial Lectures Series), Oxford University Press, USA, 2005. Google Scholar

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References:
 [1] R. J. Aumann, Subjectivity and correlation in randomized strategies, J. Math. Econom., 1 (1974), 67-96. doi: 10.1016/0304-4068(74)90037-8.  Google Scholar [2] R. J. Aumann, Correlated equilibrium as an expression of Bayesian rationality, Econometrica, 55 (1987), 1-18. doi: 10.2307/1911154.  Google Scholar [3] U. Berger, Fictitious play in $2 \times n$ games, J. Econ. Theory, 120 (2005), 139-154. doi: 10.1016/j.jet.2004.02.003.  Google Scholar [4] U. Berger, Two more classes of games with the continuous-time fictitious play property, Game. Econ. Behav., 60 (2007), 247-261. doi: 10.1016/j.geb.2006.10.008.  Google Scholar [5] U. Berger, Learning in games with strategic complementarities revisited, J. Econ. Theory, 143 (2008), 292-301. doi: 10.1016/j.jet.2008.01.007.  Google Scholar [6] D. Blackwell, Controlled random walks, In Proceedings of the International Congress of Mathematicians, 3 (1954), 336-338.  Google Scholar [7] G. W. Brown, Some notes on computation of games solutions, Technical report, Report P-78, The Rand Corporation, 1949. Google Scholar [8] G. W. Brown, Iterative solution of games by fictitious play, In Activity Analysis of Production and Allocation, volume 13, pages 374-376. John Wiley & Sons, New York, 1951.  Google Scholar [9] D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Game. Econ. Behav., 21 (1997), 40-55. doi: 10.1006/game.1997.0595.  Google Scholar [10] D. Fudenberg and D. K. Levine, Consistency and cautious fictitious play, J. Econ. Dyn. Control, 19 (1995), 1065-1089. doi: 10.1016/0165-1889(94)00819-4.  Google Scholar [11] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press Series on Economic Learning and Social Evolution, 2. MIT Press, Cambridge, MA, 1998.  Google Scholar [12] A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Game. Econ. Behav., 11 (1995), 279-303. doi: 10.1006/game.1995.1052.  Google Scholar [13] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230.  Google Scholar [14] J. Hannan, Approximation to Bayes risk in repeated play, Contributions to the Theory of Games, 3 (1957), 97-139.  Google Scholar [15] C. Harris, On the rate of convergence of continuous-time fictitious play, Game. Econ. Behav., 22 (1998), 238-259. doi: 10.1006/game.1997.0582.  Google Scholar [16] S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430. doi: 10.1111/j.1468-0262.2005.00625.x.  Google Scholar [17] S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153.  Google Scholar [18] S. Hart and A. Mas-Colell, A general class of adaptive strategies, J. Econ. Theory, 98 (2001), 26-54. doi: 10.1006/jeth.2000.2746.  Google Scholar [19] J. Hofbauer, Stability for the Best Response Dynamics, Mimeo, 1995. Google Scholar [20] V. Krishna and T. Sjöström, On the convergence of fictitious play, Math. Oper. Res., 23 (1998), 479-511. doi: 10.1287/moor.23.2.479.  Google Scholar [21] A. Matsui, Best response dynamics and socially stable strategies, J. Econ. Theory, 57 (1992), 343-362. doi: 10.1016/0022-0531(92)90040-O.  Google Scholar [22] D. Monderer, D. Samet and A. Sela, Belief affirming in learning processes, J. Econ. Theory, 73 (1997), 438-452. doi: 10.1006/jeth.1996.2245.  Google Scholar [23] S. Morris and T. Ui, Best response equivalence, Game. Econ. Behav., 49 (2004), 260-287. doi: 10.1016/j.geb.2003.12.004.  Google Scholar [24] H. J. Moulin and J.-P. Vial, Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon, Internat. J. Game Theory, 7 (1978), 201-221. doi: 10.1007/BF01769190.  Google Scholar [25] G. Ostrovski and S. van Strien, Piecewise linear Hamiltonian flows associated to zero-sum games: Transition combinatorics and questions on ergodicity, Regul. Chaotic Dyn., 16 (2011), 128-153. doi: 10.1134/S1560354711010059.  Google Scholar [26] J. Rosenmüller, Über Periodizitätseigenschaften spieltheoretischer Lernprozesse, Z. Wahrscheinlichkeit., 17 (1971), 259-308. doi: 10.1007/BF00536300.  Google Scholar [27] L. S. Shapley, Some topics in two-person games, Advances in Game Theory, 52 (1964), 1-28.  Google Scholar [28] C. Sparrow, S. van Strien and C. Harris, Fictitious play in $3\times 3$ games: The transition between periodic and chaotic behaviour, Game. Econ. Behav., 63 (2008), 259-291. doi: 10.1016/j.geb.2007.08.005.  Google Scholar [29] K. Sydsaeter and P. Hammond, Essential Mathematics for Economic Analysis, Prentice Hall, 3rd edition, 2008. Google Scholar [30] S. van Strien and C. Sparrow, Fictitious play in $3 \times 3$ games: Chaos and dithering behaviour, Game. Econ. Behav., 73 (2011), 262-286. doi: 10.1016/j.geb.2010.12.004.  Google Scholar [31] S. van Strien, A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play, Nonlinearity, 24 (2011), 1715-1742. doi: 10.1088/0951-7715/24/6/002.  Google Scholar [32] H. P. Young, Strategic Learning and Its Limits (Arne Ryde Memorial Lectures Series), Oxford University Press, USA, 2005. Google Scholar
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