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).

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications,, Academic Press, (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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[32]

A. Rusczyński, Nonlinear Optimization,, Princeton University Press, (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. 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, (1998), 79.

[35]

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

[36]

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

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. Hall and C. Heyde, Martingale Limit Theory and its Applications,, Academic Press, (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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[32]

A. Rusczyński, Nonlinear Optimization,, Princeton University Press, (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. 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, (1998), 79.

[35]

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

[36]

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

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