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

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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:

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[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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