April  2015, 2(2): 117-140. doi: 10.3934/jdg.2015.2.117

Learning in monotone bayesian games

1. 

Wadham College, University of Oxford, Oxford, OX1 3PN, United Kingdom

Received  December 2014 Revised  October 2015 Published  December 2015

This paper studies learning in monotone Bayesian games with one-dimensional types and finitely many actions. Players switch between actions at a set of thresholds. A learning algorithm under which players adjust their strategies in the direction of better ones using payoffs received at similar signals to their current thresholds is examined. Convergence to equilibrium is shown in the case of supermodular games and potential games.
Citation: Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics and Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
References:
[1]

S. Athey, Characterizing Properties of Stochastic Objective Functions, mimeo, MIT, 1996.

[2]

S. Athey, Single crossing properties and the existence of pure strategy equilibria in games of incomplete information, Econometrica, 69 (2001), 861-889. doi: 10.1111/1468-0262.00223.

[3]

A. Beggs, Learning in bayesian games with binary actions, The B.E. Journal of Theoretical Economics: Advances in Theoretical Economics, 9 (2009), Article 33, 30pp. doi: 10.2202/1935-1704.1452.

[4]

A. Beggs, Regularity and Stability in Monotone Bayesian Games, Discussion paper, University of Oxford, 2011.

[5]

A. Beggs, Regularity and robustness in monotone bayesian games, Journal of Mathematical Economics, 60 (2015), 145-158. doi: 10.1016/j.jmateco.2015.07.002.

[6]

M. Benaïm, Dynamics of stochastic approximation algorithms, in Seminaire de Probabilités, XXXIII, vol. 1709 of Lecture Notes in Mathematics, Springer Verlag, Berlin, 1999, 1-68. doi: 10.1007/BFb0096509.

[7]

M. Benaïm, Convergence with probability one of stochastic approximation algorithms whose averageis cooperative, Nonlinearity, 13 (2000), 601-616. doi: 10.1088/0951-7715/13/3/305.

[8]

M. Benaïm and M. Faure, Stochastic approximation, cooperative dynamics and supermodular games, Annals of Applied Probability, 22 (2012), 2133-2164. doi: 10.1214/11-AAP816.

[9]

M. Benaïm and M. Hirsch, Mixed equilibria and dynamical systems arising from fictitious play in perturbed games, Games and Economic Behavior, 29 (1999), 36-72. doi: 10.1006/game.1999.0717.

[10]

U. Berger, Learning in game with strategic complementarities revisited, Journal of Economic Theory, 143 (2008), 292-301. doi: 10.1016/j.jet.2008.01.007.

[11]

P. Bianchi and J. Jakubowicz, Convergence of a multi-agent projected stochastic gradient algorithm for non-convex optimization, IEEE Transactions on Automatic Control, 58 (2013), 391-405. doi: 10.1109/TAC.2012.2209984.

[12]

J. Borwein and A. Lewis, Convex Analysis and Nonlinear Optimization, 2nd edition, Springer Verlag, New York, 2006. doi: 10.1007/978-0-387-31256-9.

[13]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[14]

Y. Chow and H. Teicher, Probablity Theory: Independence, Exchangeability and Martingales, 3rd edition, Springer Verlag, New York, 1998.

[15]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalties, Annals of Operations Research, 44 (1993), 9-42. doi: 10.1007/BF02073589.

[16]

F. Facchinei and J.-S. Pang, Finite Dimensional Variational Inequalities and Complementarity Problems, Two Volumes, Springer Verlag, New York, 2003.

[17]

D. Fudenberg and D. Kreps, Learning mixed equilibria, Games and Economic Behavior, 5 (1993), 320-367. doi: 10.1006/game.1993.1021.

[18]

I. Gilboa and D. Schmeidler, Inductive inference: An axiomatic approach, Econometrica, 71 (2003), 1-26. doi: 10.1111/1468-0262.00388.

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York, 1980.

[20]

W. Härdle and R. Nixdorf, Nonparametric sequential estimation of zeroes and extrema of regression functions, IEEE Transactions in Information Theory, 33 (1987), 367-372.

[21]

C. Henry, An existence theorem for a class of differential equations with multivalued right-hand sides, Journal of Mathematical Analysis and Applications, 41 (1973), 179-186. doi: 10.1016/0022-247X(73)90192-3.

[22]

J. Jiang, Attractors in strongly monotone flows, Journal of Mathematical Analysis and Applications, 162 (1991), 210-222. doi: 10.1016/0022-247X(91)90188-6.

[23]

V. Krishna, Learning in Games with Strategic Complementarities, Technical report, Harvard University, 1992.

[24]

H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer Verlag, New York, 1997. doi: 10.1007/978-1-4899-2696-8.

[25]

P. Milgrom and C. Shannon, Monotone comparative statics, Econometrica, 62 (1994), 157-180. doi: 10.2307/2951479.

[26]

P. Milgrom and R. Weber, A theory of auctions and competitive bidding, Econometrica, 50 (1982), 1089-1122. doi: 10.2307/1911865.

[27]

S. Morris and H. Shin, Global games: Theory and applications, in Advances in Economics and Econometrics: Theory and Applications, Proceedings of the Eighth World Congress of the Econometric Society (eds. M. Dewatripont, L. Hansen and S. Turnovsky), Cambridge University Press, Cambridge, 2003, 56-114.

[28]

A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, Chichester, 2002.

[29]

R. Nelsen, An Introduction to Copulas, 2nd edition, Springer, New York, 2006.

[30]

R. Pemantle, Nonconvergence to unstable points in urn models and stochastic approximations, The Annals of Probability, 18 (1990), 698-712. doi: 10.1214/aop/1176990853.

[31]

R. T. Rockafellar and R. Wets, Variational Analysis, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[32]

A. Rusczyński, Nonlinear Optimization, Princeton University Press, Princeton, NJ, 2006.

[33]

E. Schuster, Joint asymptotic distribution of the estimated regression function at a finite number of distinct points, Annals of Mathematical Statistics, 43 (1972), 84-88. doi: 10.1214/aoms/1177692703.

[34]

R. Selten and J. Buchta, Experimental sealed bid first price auction with directly observed bid functions, in Games and Human Behavior, Essays in Honor of Amnon Rapoport (eds. I. Budescu, I. Erev and R. Zwick), Lawrence Erlbaum Associates, Mahwah, NJ, 1998, 79-102.

[35]

J. Steiner and C. Stewart, Learning by Similarity in Coordination Games, Technical report, CERGE, 2007.

[36]

J. Steiner and C. Stewart, Contagion through learning, Theoretical Economics, 3 (2008), 431-458.

show all references

References:
[1]

S. Athey, Characterizing Properties of Stochastic Objective Functions, mimeo, MIT, 1996.

[2]

S. Athey, Single crossing properties and the existence of pure strategy equilibria in games of incomplete information, Econometrica, 69 (2001), 861-889. doi: 10.1111/1468-0262.00223.

[3]

A. Beggs, Learning in bayesian games with binary actions, The B.E. Journal of Theoretical Economics: Advances in Theoretical Economics, 9 (2009), Article 33, 30pp. doi: 10.2202/1935-1704.1452.

[4]

A. Beggs, Regularity and Stability in Monotone Bayesian Games, Discussion paper, University of Oxford, 2011.

[5]

A. Beggs, Regularity and robustness in monotone bayesian games, Journal of Mathematical Economics, 60 (2015), 145-158. doi: 10.1016/j.jmateco.2015.07.002.

[6]

M. Benaïm, Dynamics of stochastic approximation algorithms, in Seminaire de Probabilités, XXXIII, vol. 1709 of Lecture Notes in Mathematics, Springer Verlag, Berlin, 1999, 1-68. doi: 10.1007/BFb0096509.

[7]

M. Benaïm, Convergence with probability one of stochastic approximation algorithms whose averageis cooperative, Nonlinearity, 13 (2000), 601-616. doi: 10.1088/0951-7715/13/3/305.

[8]

M. Benaïm and M. Faure, Stochastic approximation, cooperative dynamics and supermodular games, Annals of Applied Probability, 22 (2012), 2133-2164. doi: 10.1214/11-AAP816.

[9]

M. Benaïm and M. Hirsch, Mixed equilibria and dynamical systems arising from fictitious play in perturbed games, Games and Economic Behavior, 29 (1999), 36-72. doi: 10.1006/game.1999.0717.

[10]

U. Berger, Learning in game with strategic complementarities revisited, Journal of Economic Theory, 143 (2008), 292-301. doi: 10.1016/j.jet.2008.01.007.

[11]

P. Bianchi and J. Jakubowicz, Convergence of a multi-agent projected stochastic gradient algorithm for non-convex optimization, IEEE Transactions on Automatic Control, 58 (2013), 391-405. doi: 10.1109/TAC.2012.2209984.

[12]

J. Borwein and A. Lewis, Convex Analysis and Nonlinear Optimization, 2nd edition, Springer Verlag, New York, 2006. doi: 10.1007/978-0-387-31256-9.

[13]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[14]

Y. Chow and H. Teicher, Probablity Theory: Independence, Exchangeability and Martingales, 3rd edition, Springer Verlag, New York, 1998.

[15]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalties, Annals of Operations Research, 44 (1993), 9-42. doi: 10.1007/BF02073589.

[16]

F. Facchinei and J.-S. Pang, Finite Dimensional Variational Inequalities and Complementarity Problems, Two Volumes, Springer Verlag, New York, 2003.

[17]

D. Fudenberg and D. Kreps, Learning mixed equilibria, Games and Economic Behavior, 5 (1993), 320-367. doi: 10.1006/game.1993.1021.

[18]

I. Gilboa and D. Schmeidler, Inductive inference: An axiomatic approach, Econometrica, 71 (2003), 1-26. doi: 10.1111/1468-0262.00388.

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York, 1980.

[20]

W. Härdle and R. Nixdorf, Nonparametric sequential estimation of zeroes and extrema of regression functions, IEEE Transactions in Information Theory, 33 (1987), 367-372.

[21]

C. Henry, An existence theorem for a class of differential equations with multivalued right-hand sides, Journal of Mathematical Analysis and Applications, 41 (1973), 179-186. doi: 10.1016/0022-247X(73)90192-3.

[22]

J. Jiang, Attractors in strongly monotone flows, Journal of Mathematical Analysis and Applications, 162 (1991), 210-222. doi: 10.1016/0022-247X(91)90188-6.

[23]

V. Krishna, Learning in Games with Strategic Complementarities, Technical report, Harvard University, 1992.

[24]

H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer Verlag, New York, 1997. doi: 10.1007/978-1-4899-2696-8.

[25]

P. Milgrom and C. Shannon, Monotone comparative statics, Econometrica, 62 (1994), 157-180. doi: 10.2307/2951479.

[26]

P. Milgrom and R. Weber, A theory of auctions and competitive bidding, Econometrica, 50 (1982), 1089-1122. doi: 10.2307/1911865.

[27]

S. Morris and H. Shin, Global games: Theory and applications, in Advances in Economics and Econometrics: Theory and Applications, Proceedings of the Eighth World Congress of the Econometric Society (eds. M. Dewatripont, L. Hansen and S. Turnovsky), Cambridge University Press, Cambridge, 2003, 56-114.

[28]

A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, Chichester, 2002.

[29]

R. Nelsen, An Introduction to Copulas, 2nd edition, Springer, New York, 2006.

[30]

R. Pemantle, Nonconvergence to unstable points in urn models and stochastic approximations, The Annals of Probability, 18 (1990), 698-712. doi: 10.1214/aop/1176990853.

[31]

R. T. Rockafellar and R. Wets, Variational Analysis, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[32]

A. Rusczyński, Nonlinear Optimization, Princeton University Press, Princeton, NJ, 2006.

[33]

E. Schuster, Joint asymptotic distribution of the estimated regression function at a finite number of distinct points, Annals of Mathematical Statistics, 43 (1972), 84-88. doi: 10.1214/aoms/1177692703.

[34]

R. Selten and J. Buchta, Experimental sealed bid first price auction with directly observed bid functions, in Games and Human Behavior, Essays in Honor of Amnon Rapoport (eds. I. Budescu, I. Erev and R. Zwick), Lawrence Erlbaum Associates, Mahwah, NJ, 1998, 79-102.

[35]

J. Steiner and C. Stewart, Learning by Similarity in Coordination Games, Technical report, CERGE, 2007.

[36]

J. Steiner and C. Stewart, Contagion through learning, Theoretical Economics, 3 (2008), 431-458.

[1]

Saeed Hadikhanloo, Rida Laraki, Panayotis Mertikopoulos, Sylvain Sorin. Learning in nonatomic games, part Ⅰ: Finite action spaces and population games. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022018

[2]

Kashi Behrstock, Michel Benaïm, Morris W. Hirsch. Smale strategies for network prisoner's dilemma games. Journal of Dynamics and Games, 2015, 2 (2) : 141-155. doi: 10.3934/jdg.2015.2.141

[3]

Miguel A. Dumett, Roberto Cominetti. On the stability of an adaptive learning dynamics in traffic games. Journal of Dynamics and Games, 2018, 5 (4) : 265-282. doi: 10.3934/jdg.2018017

[4]

Tigran Bakaryan, Diogo Gomes, Héctor Sánchez Morgado. Discrete approximation of stationary Mean Field Games. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022022

[5]

Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks and Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197

[6]

Weihua Ruan. Markovian strategies for piecewise deterministic differential games with continuous and impulse controls. Journal of Dynamics and Games, 2019, 6 (4) : 337-366. doi: 10.3934/jdg.2019022

[7]

Miquel Oliu-Barton. Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. Journal of Dynamics and Games, 2019, 6 (4) : 259-275. doi: 10.3934/jdg.2019018

[8]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020

[9]

Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

[10]

Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719

[11]

Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021021

[12]

Samuel Drapeau, Peng Luo, Alexander Schied, Dewen Xiong. An FBSDE approach to market impact games with stochastic parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 237-260. doi: 10.3934/puqr.2021012

[13]

Jiequn Han, Ruimeng Hu, Jihao Long. Convergence of deep fictitious play for stochastic differential games. Frontiers of Mathematical Finance, 2022, 1 (2) : 287-319. doi: 10.3934/fmf.2021011

[14]

Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial and Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795

[15]

Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics and Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103

[16]

Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics and Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003

[17]

Matt Barker. From mean field games to the best reply strategy in a stochastic framework. Journal of Dynamics and Games, 2019, 6 (4) : 291-314. doi: 10.3934/jdg.2019020

[18]

Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics and Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555

[19]

Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics and Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008

[20]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375

 Impact Factor: 

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]