American Institute of Mathematical Sciences

Advanced
January  2017, 4(1): 1-23. doi: 10.3934/jdg.2017001

On repeated games with imperfect public monitoring: From discrete to continuous time

Mathias Staudigl 1,, and  Jan-Henrik Steg 2,

1. 

Department of Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands
2. 

Center for Mathematical Economics, Bielefeld University, PO Box 10 01 31, 33 501 Bielefeld, Germany

* Corresponding author

Received  March 2016 Revised  November 2016 Published  November 2016

Motivated by recent path-breaking contributions in the theory of repeated games in continuous time, this paper presents a family of discrete-time games which provides a consistent discrete-time approximation of the continuous-time limit game. Using probabilistic arguments, we prove that continuous-time games can be defined as the limit of a sequence of discrete-time games. Our convergence analysis reveals various intricacies of continuous-time games. First, we demonstrate the importance of correlated strategies in continuous-time. Second, we attach a precise meaning to the statement that a sequence of discrete-time games can be used to approximate a continuous-time game.

Keywords: Continuous-time game theory, perfect-public equilibrium, weak convergence.
Mathematics Subject Classification: Primary: 91A25, 91A15; Secondary: 60G57.
Citation: Mathias Staudigl, Jan-Henrik Steg. On repeated games with imperfect public monitoring: From discrete to continuous time. Journal of Dynamics & Games, 2017, 4 (1) : 1-23. doi: 10.3934/jdg.2017001
References:
[1]

D. Abreu, D. Pearce and E. Stacchetti, Toward a theory of discounted repeated games with imperfect monitoring, Econometrica, 58 (1990), 1041-1063.  doi: 10.2307/2938299.  Google Scholar

[2]

C. Alos-Ferrer and K. Ritzberger, Trees and extensive forms, Journal of Economic Theory, 143 (2008), 216-250.  doi: 10.1016/j.jet.2007.11.002.  Google Scholar

[3]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering Springer -Stochastic Modelling and Applied Probability, 2009. doi: 10.1007/978-0-387-76896-0.  Google Scholar

[4]

B. Bernard and C. Frei, The folk theorem with imperfect public information in continuous time, Theoretical Economics, 11 (2016), 411-453, https://econtheory.org/ojs/index.php/te/article/view/20160411. doi: 10.3982/TE1687.  Google Scholar

[5]

B. Biais, T. Mariotti, G. Plantin and J. -C. Rochet, Dynamic security design: Convergence to continuous time and asset pricing implications, The Review of Economic Studies, 74 (2007), 345-390, URL http://www.jstor.org/stable/4626144. doi: 10.1111/j.1467-937X.2007.00425.x.  Google Scholar

[6]

P. Billingsley, Convergence of Probability Measures 2nd edition, Wiley Series in Probability and Statistics, 1999. doi: 10.1002/9780470316962.  Google Scholar

[7]

P. Cardaliaguet, C. Rainer, D. Rosenberg and N. Vieille, Markov games with frequent actions and incomplete information, HEC Paris Research Paper No. ECO/SCD-2013-1007, (2013), 1–37, arXiv: 1307.3365v1, [math.OC]. doi: 10.2139/ssrn.2344780.  Google Scholar

[8]

J. Cvitanić and J. Zhang, Contract Theory in Continuous-Time Models Springer Finance, 2013. doi: 10.1007/978-3-642-14200-0.  Google Scholar

[9]

N. El Karoui, D. Nguyen and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 20 (1987), 169-219.  doi: 10.1080/17442508708833443.  Google Scholar

[10]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[11]

E. Faingold, Building a reputation under frequent decisions, 2008, Unpublished manuscript, Yale University. Google Scholar

[12]

D. Fudenberg and D. K. Levine, Limit games and limit equilibria, Journal of Economic Theory, 38 (1986), 261-279.  doi: 10.1016/0022-0531(86)90118-3.  Google Scholar

[13]

D. Fudenberg and D. K. Levine, Continuous time limits of repeated games with imperfect public monitoring, A Long-Run Collaboration on Long-Run Games, (2008), 369-388.  doi: 10.1142/9789812818478_0017.  Google Scholar

[14]

D. Fudenberg and D. K. Levine, Repeated games with frequent signals, Quarterly Journal of Economics, 124 (2009), 233-265.  doi: 10.1162/qjec.2009.124.1.233.  Google Scholar

[15]

D. Fudenberg, D. K. Levine and E. Maskin, The folk theorem with imperfect public information, Econometrica, 62 (1994), 997-1039.  doi: 10.2307/2951505.  Google Scholar

[16]

F. Gensbittel, Continuous-time limit of dynamic games with incomplete information and a more informed player, International Journal of Game Theory, 45 (2016), 321-352.  doi: 10.1007/s00182-015-0507-5.  Google Scholar

[17]

I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes Ⅲ Springer-Verlag, Berlin Heidelberg New York, 1979.  Google Scholar

[18]

M. F. Hellwig and K. M. Schmidt, Discrete-time approximation of the Holmström-Milgrom model of intertemporal incentive provision, Econometrica, 70 (2002), 2225-2264.  doi: 10.1111/1468-0262.00375.  Google Scholar

[19]

B. Holmström and P. Milgrom, Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.  doi: 10.2307/1913238.  Google Scholar

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes Springer Verlag -Grundlehren der Mathematischen Wissenschaften, Volume 288, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.  Google Scholar

[21]

O. Kallenberg, Foundations of Modern Probability 2nd edition, Springer, New York [u. a. ], 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[22]

H. Kushner, Numerical approximations for nonzero-sum stochastic differential games, SIAM Journal on Control and Optimization, 46 (2007), 1942-1971.  doi: 10.1137/050647931.  Google Scholar

[23]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser: System & Control: Foundations & Applications, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[24]

H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time 2nd edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[25] G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, Oxford, 2006.   Google Scholar
[26]

P. Morando, Measures aleatoires, Seminaire de Probabilites, Lecture Notes in Mathematics, Spinger-Verlag, 3 (1969), 190-229.  Google Scholar

[27]

A. Neyman, Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278.  doi: 10.1007/s13235-013-0083-x.  Google Scholar

[28]

D. Rosenberg, E. Solan and N. Vieille, The MaxMin value of stochastic games with imperfect monitoring, International Journal of Game Theory, 32 (2003), 133-150.  doi: 10.1007/s001820300150.  Google Scholar

[29]

Y. Sannikov, A continuous-time version of the principal-agent problem, Review of Economic Studies, 75 (2008), 957-984.  doi: 10.1111/j.1467-937X.2008.00486.x.  Google Scholar

[30]

Y. Sannikov and A. Skrzypacz, The role of information in repeated games with frequent actions, Econometrica, 78 (2010), 847-882.  doi: 10.3982/ECTA6420.  Google Scholar

[31]

Y. Sannikov, Games with imperfectly observable actions in continuous time, Econometrica, 75 (2007), 1285-1329.  doi: 10.1111/j.1468-0262.2007.00795.x.  Google Scholar

[32]

H. Schättler and J. Sung, The first-order approach to the continuous-time principal agent problem with exponential utility, Journal of Economic Theory, 61 (1993), 331-371, URL http://www.sciencedirect.com/science/article/pii/S0022053183710720. doi: 10.1006/jeth.1993.1072.  Google Scholar

[33]

L. K. Simon and M. B. Stinchcombe, Extensive form games in continuous time: Pure strategies, Econometrica, 57 (1989), 1171-1214.  doi: 10.2307/1913627.  Google Scholar

[34]

M. Staudigl, On repeated games with imperfect public monitoring: Characterization of continuation payoff processes, 2014, URL http://www.mwpweb.eu/MathiasStaudigl/, Bielefeld University. Google Scholar

[35]

R. H. Stockbridge, Time-average control of martingale problems: Existence of a stationary solution, The Annals of Probability, 18 (1990), 190-205, URL http://www.jstor.org/stable/2244233. doi: 10.1214/aop/1176990944.  Google Scholar

[36]

J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972.  Google Scholar

[37]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders, Philadelphia, 1969.  Google Scholar

show all references

References:
[1]

D. Abreu, D. Pearce and E. Stacchetti, Toward a theory of discounted repeated games with imperfect monitoring, Econometrica, 58 (1990), 1041-1063.  doi: 10.2307/2938299.  Google Scholar

[2]

C. Alos-Ferrer and K. Ritzberger, Trees and extensive forms, Journal of Economic Theory, 143 (2008), 216-250.  doi: 10.1016/j.jet.2007.11.002.  Google Scholar

[3]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering Springer -Stochastic Modelling and Applied Probability, 2009. doi: 10.1007/978-0-387-76896-0.  Google Scholar

[4]

B. Bernard and C. Frei, The folk theorem with imperfect public information in continuous time, Theoretical Economics, 11 (2016), 411-453, https://econtheory.org/ojs/index.php/te/article/view/20160411. doi: 10.3982/TE1687.  Google Scholar

[5]

B. Biais, T. Mariotti, G. Plantin and J. -C. Rochet, Dynamic security design: Convergence to continuous time and asset pricing implications, The Review of Economic Studies, 74 (2007), 345-390, URL http://www.jstor.org/stable/4626144. doi: 10.1111/j.1467-937X.2007.00425.x.  Google Scholar

[6]

P. Billingsley, Convergence of Probability Measures 2nd edition, Wiley Series in Probability and Statistics, 1999. doi: 10.1002/9780470316962.  Google Scholar

[7]

P. Cardaliaguet, C. Rainer, D. Rosenberg and N. Vieille, Markov games with frequent actions and incomplete information, HEC Paris Research Paper No. ECO/SCD-2013-1007, (2013), 1–37, arXiv: 1307.3365v1, [math.OC]. doi: 10.2139/ssrn.2344780.  Google Scholar

[8]

J. Cvitanić and J. Zhang, Contract Theory in Continuous-Time Models Springer Finance, 2013. doi: 10.1007/978-3-642-14200-0.  Google Scholar

[9]

N. El Karoui, D. Nguyen and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 20 (1987), 169-219.  doi: 10.1080/17442508708833443.  Google Scholar

[10]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[11]

E. Faingold, Building a reputation under frequent decisions, 2008, Unpublished manuscript, Yale University. Google Scholar

[12]

D. Fudenberg and D. K. Levine, Limit games and limit equilibria, Journal of Economic Theory, 38 (1986), 261-279.  doi: 10.1016/0022-0531(86)90118-3.  Google Scholar

[13]

D. Fudenberg and D. K. Levine, Continuous time limits of repeated games with imperfect public monitoring, A Long-Run Collaboration on Long-Run Games, (2008), 369-388.  doi: 10.1142/9789812818478_0017.  Google Scholar

[14]

D. Fudenberg and D. K. Levine, Repeated games with frequent signals, Quarterly Journal of Economics, 124 (2009), 233-265.  doi: 10.1162/qjec.2009.124.1.233.  Google Scholar

[15]

D. Fudenberg, D. K. Levine and E. Maskin, The folk theorem with imperfect public information, Econometrica, 62 (1994), 997-1039.  doi: 10.2307/2951505.  Google Scholar

[16]

F. Gensbittel, Continuous-time limit of dynamic games with incomplete information and a more informed player, International Journal of Game Theory, 45 (2016), 321-352.  doi: 10.1007/s00182-015-0507-5.  Google Scholar

[17]

I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes Ⅲ Springer-Verlag, Berlin Heidelberg New York, 1979.  Google Scholar

[18]

M. F. Hellwig and K. M. Schmidt, Discrete-time approximation of the Holmström-Milgrom model of intertemporal incentive provision, Econometrica, 70 (2002), 2225-2264.  doi: 10.1111/1468-0262.00375.  Google Scholar

[19]

B. Holmström and P. Milgrom, Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.  doi: 10.2307/1913238.  Google Scholar

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes Springer Verlag -Grundlehren der Mathematischen Wissenschaften, Volume 288, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.  Google Scholar

[21]

O. Kallenberg, Foundations of Modern Probability 2nd edition, Springer, New York [u. a. ], 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[22]

H. Kushner, Numerical approximations for nonzero-sum stochastic differential games, SIAM Journal on Control and Optimization, 46 (2007), 1942-1971.  doi: 10.1137/050647931.  Google Scholar

[23]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser: System & Control: Foundations & Applications, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[24]

H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time 2nd edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[25] G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, Oxford, 2006.   Google Scholar
[26]

P. Morando, Measures aleatoires, Seminaire de Probabilites, Lecture Notes in Mathematics, Spinger-Verlag, 3 (1969), 190-229.  Google Scholar

[27]

A. Neyman, Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278.  doi: 10.1007/s13235-013-0083-x.  Google Scholar

[28]

D. Rosenberg, E. Solan and N. Vieille, The MaxMin value of stochastic games with imperfect monitoring, International Journal of Game Theory, 32 (2003), 133-150.  doi: 10.1007/s001820300150.  Google Scholar

[29]

Y. Sannikov, A continuous-time version of the principal-agent problem, Review of Economic Studies, 75 (2008), 957-984.  doi: 10.1111/j.1467-937X.2008.00486.x.  Google Scholar

[30]

Y. Sannikov and A. Skrzypacz, The role of information in repeated games with frequent actions, Econometrica, 78 (2010), 847-882.  doi: 10.3982/ECTA6420.  Google Scholar

[31]

Y. Sannikov, Games with imperfectly observable actions in continuous time, Econometrica, 75 (2007), 1285-1329.  doi: 10.1111/j.1468-0262.2007.00795.x.  Google Scholar

[32]

H. Schättler and J. Sung, The first-order approach to the continuous-time principal agent problem with exponential utility, Journal of Economic Theory, 61 (1993), 331-371, URL http://www.sciencedirect.com/science/article/pii/S0022053183710720. doi: 10.1006/jeth.1993.1072.  Google Scholar

[33]

L. K. Simon and M. B. Stinchcombe, Extensive form games in continuous time: Pure strategies, Econometrica, 57 (1989), 1171-1214.  doi: 10.2307/1913627.  Google Scholar

[34]

M. Staudigl, On repeated games with imperfect public monitoring: Characterization of continuation payoff processes, 2014, URL http://www.mwpweb.eu/MathiasStaudigl/, Bielefeld University. Google Scholar

[35]

R. H. Stockbridge, Time-average control of martingale problems: Existence of a stationary solution, The Annals of Probability, 18 (1990), 190-205, URL http://www.jstor.org/stable/2244233. doi: 10.1214/aop/1176990944.  Google Scholar

[36]

J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972.  Google Scholar

[37]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders, Philadelphia, 1969.  Google Scholar

[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
[2]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164
[3]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032
[4]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019
[5]

Manuel Friedrich, Martin Kružík, Ulisse Stefanelli. Equilibrium of immersed hyperelastic solids. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021003
[6]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[7]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[8]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1
[9]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[10]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[11]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[12]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[13]

David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial & Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121
[14]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[15]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
[16]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[17]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[18]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[19]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[20]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

 Impact Factor: 

Tools

Metrics

  • PDF downloads (176)
  • HTML views (133)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences