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 & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
References:
[1]

S. Athey, Characterizing Properties of Stochastic Objective Functions,, mimeo, (1996).   Google Scholar

[2]

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

[3]

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

[4]

A. Beggs, Regularity and Stability in Monotone Bayesian Games,, Discussion paper, (2011).   Google Scholar

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

[6]

M. Benaïm, Dynamics of stochastic approximation algorithms,, in Seminaire de Probabilités, (1709), 1.  doi: 10.1007/BFb0096509.  Google Scholar

[7]

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

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

[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.  doi: 10.1006/game.1999.0717.  Google Scholar

[10]

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

[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.  doi: 10.1109/TAC.2012.2209984.  Google Scholar

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J. Borwein and A. Lewis, Convex Analysis and Nonlinear Optimization,, 2nd edition, (2006).  doi: 10.1007/978-0-387-31256-9.  Google Scholar

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

[14]

Y. Chow and H. Teicher, Probablity Theory: Independence, Exchangeability and Martingales,, 3rd edition, (1998).   Google Scholar

[15]

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

[16]

F. Facchinei and J.-S. Pang, Finite Dimensional Variational Inequalities and Complementarity Problems,, Two Volumes, (2003).   Google Scholar

[17]

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

[18]

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

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications,, Academic Press, (1980).   Google Scholar

[20]

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

[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.  doi: 10.1016/0022-247X(73)90192-3.  Google Scholar

[22]

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

[23]

V. Krishna, Learning in Games with Strategic Complementarities,, Technical report, (1992).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

S. Morris and H. Shin, Global games: Theory and applications,, in Advances in Economics and Econometrics: Theory and Applications, (2003), 56.   Google Scholar

[28]

A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks,, Wiley, (2002).   Google Scholar

[29]

R. Nelsen, An Introduction to Copulas,, 2nd edition, (2006).   Google Scholar

[30]

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

[31]

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

[32]

A. Rusczyński, Nonlinear Optimization,, Princeton University Press, (2006).   Google Scholar

[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.  doi: 10.1214/aoms/1177692703.  Google Scholar

[34]

R. Selten and J. Buchta, Experimental sealed bid first price auction with directly observed bid functions,, in Games and Human Behavior, (1998), 79.   Google Scholar

[35]

J. Steiner and C. Stewart, Learning by Similarity in Coordination Games,, Technical report, (2007).   Google Scholar

[36]

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

show all references

References:
[1]

S. Athey, Characterizing Properties of Stochastic Objective Functions,, mimeo, (1996).   Google Scholar

[2]

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

[3]

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

[4]

A. Beggs, Regularity and Stability in Monotone Bayesian Games,, Discussion paper, (2011).   Google Scholar

[5]

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

[6]

M. Benaïm, Dynamics of stochastic approximation algorithms,, in Seminaire de Probabilités, (1709), 1.  doi: 10.1007/BFb0096509.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1006/game.1999.0717.  Google Scholar

[10]

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

[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.  doi: 10.1109/TAC.2012.2209984.  Google Scholar

[12]

J. Borwein and A. Lewis, Convex Analysis and Nonlinear Optimization,, 2nd edition, (2006).  doi: 10.1007/978-0-387-31256-9.  Google Scholar

[13]

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

[14]

Y. Chow and H. Teicher, Probablity Theory: Independence, Exchangeability and Martingales,, 3rd edition, (1998).   Google Scholar

[15]

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

[16]

F. Facchinei and J.-S. Pang, Finite Dimensional Variational Inequalities and Complementarity Problems,, Two Volumes, (2003).   Google Scholar

[17]

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

[18]

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

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications,, Academic Press, (1980).   Google Scholar

[20]

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

[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.  doi: 10.1016/0022-247X(73)90192-3.  Google Scholar

[22]

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

[23]

V. Krishna, Learning in Games with Strategic Complementarities,, Technical report, (1992).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

S. Morris and H. Shin, Global games: Theory and applications,, in Advances in Economics and Econometrics: Theory and Applications, (2003), 56.   Google Scholar

[28]

A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks,, Wiley, (2002).   Google Scholar

[29]

R. Nelsen, An Introduction to Copulas,, 2nd edition, (2006).   Google Scholar

[30]

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

[31]

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

[32]

A. Rusczyński, Nonlinear Optimization,, Princeton University Press, (2006).   Google Scholar

[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.  doi: 10.1214/aoms/1177692703.  Google Scholar

[34]

R. Selten and J. Buchta, Experimental sealed bid first price auction with directly observed bid functions,, in Games and Human Behavior, (1998), 79.   Google Scholar

[35]

J. Steiner and C. Stewart, Learning by Similarity in Coordination Games,, Technical report, (2007).   Google Scholar

[36]

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

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