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On repeated games with imperfect public monitoring: From discrete to continuous time
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Billingsley, Convergence of Probability Measures 2nd edition, Wiley Series in Probability and Statistics, 1999.
doi: 10.1002/9780470316962. |
[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. |
[8] |
J. Cvitanić and J. Zhang, Contract Theory in Continuous-Time Models Springer Finance, 2013.
doi: 10.1007/978-3-642-14200-0. |
[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. |
[10] |
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley, New York, 1986.
doi: 10.1002/9780470316658. |
[11] |
E. Faingold, Building a reputation under frequent decisions, 2008, Unpublished manuscript, Yale University. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes Ⅲ Springer-Verlag, Berlin Heidelberg New York, 1979. |
[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. |
[19] |
B. Holmström and P. Milgrom,
Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.
doi: 10.2307/1913238. |
[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. |
[21] |
O. Kallenberg, Foundations of Modern Probability 2nd edition, Springer, New York [u. a. ], 2002.
doi: 10.1007/978-1-4757-4015-8. |
[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. |
[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. |
[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. |
[25] |
G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, Oxford, 2006.
![]() |
[26] |
P. Morando, Measures aleatoires, Seminaire de Probabilites, Lecture Notes in Mathematics, Spinger-Verlag, 3 (1969), 190-229. |
[27] |
A. Neyman,
Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278.
doi: 10.1007/s13235-013-0083-x. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[34] |
M. Staudigl, On repeated games with imperfect public monitoring: Characterization of continuation payoff processes, 2014, URL http://www.mwpweb.eu/MathiasStaudigl/, Bielefeld University. |
[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. |
[36] |
J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972. |
[37] |
L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders, Philadelphia, 1969. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Billingsley, Convergence of Probability Measures 2nd edition, Wiley Series in Probability and Statistics, 1999.
doi: 10.1002/9780470316962. |
[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. |
[8] |
J. Cvitanić and J. Zhang, Contract Theory in Continuous-Time Models Springer Finance, 2013.
doi: 10.1007/978-3-642-14200-0. |
[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. |
[10] |
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley, New York, 1986.
doi: 10.1002/9780470316658. |
[11] |
E. Faingold, Building a reputation under frequent decisions, 2008, Unpublished manuscript, Yale University. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes Ⅲ Springer-Verlag, Berlin Heidelberg New York, 1979. |
[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. |
[19] |
B. Holmström and P. Milgrom,
Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.
doi: 10.2307/1913238. |
[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. |
[21] |
O. Kallenberg, Foundations of Modern Probability 2nd edition, Springer, New York [u. a. ], 2002.
doi: 10.1007/978-1-4757-4015-8. |
[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. |
[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. |
[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. |
[25] |
G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, Oxford, 2006.
![]() |
[26] |
P. Morando, Measures aleatoires, Seminaire de Probabilites, Lecture Notes in Mathematics, Spinger-Verlag, 3 (1969), 190-229. |
[27] |
A. Neyman,
Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278.
doi: 10.1007/s13235-013-0083-x. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[34] |
M. Staudigl, On repeated games with imperfect public monitoring: Characterization of continuation payoff processes, 2014, URL http://www.mwpweb.eu/MathiasStaudigl/, Bielefeld University. |
[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. |
[36] |
J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972. |
[37] |
L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders, Philadelphia, 1969. |
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