American Institute of Mathematical Sciences

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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

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