January  2019, 6(1): 65-85. doi: 10.3934/jdg.2019005

A differential game on Wasserstein space. Application to weak approachability with partial monitoring

1. 

CMLA, ENS Paris-Saclay (UMR 8536), 61 avenue du Président Wilson, 94230 Cachan

2. 

Criteo AI Lab, 32 rue Blanche, 75009 Paris, France

3. 

Laboratoire de Mathématiques de Bretagne Atlantique (CNRS UMR 6205), 6, Avenue Victor Le Gorgeu, 29200 Brest, France

* Corresponding author: vianney.perchet@normalesup.org

Received  January 2018 Revised  December 2018 Published  January 2019

Fund Project: This research was partially by research contract AFOSR-FA9550-18-1-0254. The first author also benefited from the support of the FMJH Program Gaspard Monge in optimization and operations research (supported in part by EDF) and the CNRS through the PEPS 3IA program

Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamics, we derive results on weak approachability with partial monitoring: we prove that any set satisfying a suitable compatibility condition is either weakly approachable or weakly excludable. We also obtain that the value for differential games with nonanticipative strategies is the same that those defined with a new concept of strategies very suitable to make links with repeated games.

Citation: Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
References:
[1]

J. AbernethyL. P. Bartlett and E. Hazan, Blackwell approachability and low-regret learning are equivalent, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 27-46. Google Scholar

[2]

S. As SoulaimaniM. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM J. Control Optim., 48 (2009), 2461-2479. doi: 10.1137/090749098. Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2005. Google Scholar

[4] R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, 1995. Google Scholar
[5]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[6]

P. BettiolP. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Internat. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. Google Scholar

[7]

D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1. Google Scholar

[8]

D. Blackwell, Controlled random walks, in: Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, 3 (1956), 336-338. Google Scholar

[9]

R. BuckdahnP. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games and Applications, 1 (2011), 74-114. doi: 10.1007/s13235-010-0005-0. Google Scholar

[10]

P. CardaliaguetM. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints, SIAM J. Control Optim., 39 (2000), 1615-1632. doi: 10.1137/S0363012998349327. Google Scholar

[11]

P. Cardaliaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, International Game Theory Review, 10 (2008), 1-16. doi: 10.1142/S021919890800173X. Google Scholar

[12]

P. CardaliaguetR. Laraki and S. Sorin, A continuous time approach for the asymptotic value in two-person zero-sum repeated games, SIAM J. on Control and Optimization, 50 (2012), 1573-1596. doi: 10.1137/110839473. Google Scholar

[13]

N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511546921. Google Scholar

[14]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of Hamilton Jacobi Equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X. Google Scholar

[15]

A. P. Dawid, Self-calibrating priors do not exist: Comment, J. Amer. Statist. Assoc., 80 (1985), 340-341. doi: 10.2307/2287892. Google Scholar

[16]

R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347. Google Scholar

[17]

L. Evans and T. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi Equations, ndiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040. Google Scholar

[18]

W. Fleming, The convergence problem for differential games, J. Math. Anal. Appl., 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9. Google Scholar

[19]

W. Fleming, The convergence problem for differential games, Ⅱ, in Advances in Game Theory, Ann. of Math. Studies, 52 (1964), Princeton Univ. Press, Princeton, NJ, 195-210. Google Scholar

[20]

D. P. Foster and R. Vohra, Calibrated learning and correlated equilibrium, Games and Economic Behavior, 21 (1997), 40-55. doi: 10.1006/game.1997.0595. Google Scholar

[21]

R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons, Inc., New York-London-Sydney, 1965. Google Scholar

[22]

C. Jimenez and M. Quincampoix, Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition, J. Math. Anal. Appl., 457 (2018), 1422-1451. doi: 10.1016/j.jmaa.2017.08.012. Google Scholar

[23]

E. Kohlberg, Optimal strategies in repeated games with incomplete information, Internat. J. Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399. Google Scholar

[24]

J. Kwon and V. Perchet, Online learning and blackwell approachability with partial monitoring: optimal convergence rates, PMLR Work. Conf. Proc., 54 (2014), 604-613. Google Scholar

[25]

G. LugosiS. Mannor and G. Stoltz, Strategies for prediction under imperfect monitoring, Math. Oper. Res., 33 (2008), 513-528. doi: 10.1287/moor.1080.0312. Google Scholar

[26]

S. MannorV. Perchet and G. Stoltz, Robust approachability and regret minimization in games with partial monitoring, JMLR Work. Conf. Proc., 19 (2011), 515-536. Google Scholar

[27]

S. MannorV. Perchet and G. Stoltz, Approachability in unknown games: Online learning meets multi-objective optimization, JMLR Work. Conf. Proc., 35 (2014), 1-17. Google Scholar

[28]

A. Marigonda and M. Quincampoix, Mayer control problem with probabilistic uncertainty on initial positions, J. Differential Equations, 264 (2018), 3212-3252. doi: 10.1016/j.jde.2017.11.014. Google Scholar

[29]

J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, With a foreword by Robert J. Aumann. Econometric Society Monographs, 55. Cambridge University Press, New York, 2015. doi: 10.1017/CBO9781139343275. Google Scholar

[30]

V. Perchet, Approachability of convex sets with partial monitoring, J. Optim. Theory. Appl., 149 (2011), 665-677. doi: 10.1007/s10957-011-9797-3. Google Scholar

[31]

V. Perchet, Internal regret with partial monitoring: calibration-based optimal algorithms, J. Mach. Learn. Res., 12 (2011), 1893-1921. Google Scholar

[32]

V. Perchet, Approachability, regret and calibration: Implications and equivalences, J. Dyn. Games, 1 (2014), 181-254. doi: 10.3934/jdg.2014.1.181. Google Scholar

[33]

V. Perchet, Exponential weight approachability, applications to calibration and regret minimization, Dynamic Games And Applications, 5 (2015), 136-153. doi: 10.1007/s13235-014-0119-x. Google Scholar

[34]

V. Perchet and M. Quincampoix, On a unified framework for approachability with full or partial monitoring, Mathematics of Operations Research, 40 (2014), 596-610. doi: 10.1287/moor.2014.0686. Google Scholar

[35]

S. Plaskacz and M. Quincampoix, Value-functions for differential games and control systems with discontinuous terminal cost, SIAM J. Control Optim., 39 (2000), 1485-1498. doi: 10.1137/S0363012998340387. Google Scholar

[36]

L. Pontrjagin, Linear differential games. Ⅰ, Ⅱ, Dokl. Akad. Nauk SSSR, 174 (1967), 1278-1280. Google Scholar

[37]

A. Rustichini, Minimizing regret: The general case, Games Econom. Behav., 29 (1999), 224-243. doi: 10.1006/game.1998.0690. Google Scholar

[38]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhauser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. Google Scholar

[39]

M. Sion, On general minimax theorems, Pacific J. Math, 8 (1958), 171-176. doi: 10.2140/pjm.1958.8.171. Google Scholar

[40]

X. Spinat, A necessary and sufficient condition for approachability, Math. Oper. Res., 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333. Google Scholar

[41]

T. Tomala, Belief-free Communication Equilibria, Math. Oper. Res., 38 (2013), 617-637. doi: 10.1287/moor.2013.0594. Google Scholar

[42]

N. Vieille, Weak approachability, Math. Op. Res., 17 (1992), 781-791. doi: 10.1287/moor.17.4.781. Google Scholar

[43]

C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, AMS, Vol. 58, 2003. doi: 10.1007/b12016. Google Scholar

show all references

References:
[1]

J. AbernethyL. P. Bartlett and E. Hazan, Blackwell approachability and low-regret learning are equivalent, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 27-46. Google Scholar

[2]

S. As SoulaimaniM. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM J. Control Optim., 48 (2009), 2461-2479. doi: 10.1137/090749098. Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2005. Google Scholar

[4] R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, 1995. Google Scholar
[5]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[6]

P. BettiolP. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Internat. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. Google Scholar

[7]

D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1. Google Scholar

[8]

D. Blackwell, Controlled random walks, in: Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, 3 (1956), 336-338. Google Scholar

[9]

R. BuckdahnP. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games and Applications, 1 (2011), 74-114. doi: 10.1007/s13235-010-0005-0. Google Scholar

[10]

P. CardaliaguetM. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints, SIAM J. Control Optim., 39 (2000), 1615-1632. doi: 10.1137/S0363012998349327. Google Scholar

[11]

P. Cardaliaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, International Game Theory Review, 10 (2008), 1-16. doi: 10.1142/S021919890800173X. Google Scholar

[12]

P. CardaliaguetR. Laraki and S. Sorin, A continuous time approach for the asymptotic value in two-person zero-sum repeated games, SIAM J. on Control and Optimization, 50 (2012), 1573-1596. doi: 10.1137/110839473. Google Scholar

[13]

N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511546921. Google Scholar

[14]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of Hamilton Jacobi Equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X. Google Scholar

[15]

A. P. Dawid, Self-calibrating priors do not exist: Comment, J. Amer. Statist. Assoc., 80 (1985), 340-341. doi: 10.2307/2287892. Google Scholar

[16]

R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347. Google Scholar

[17]

L. Evans and T. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi Equations, ndiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040. Google Scholar

[18]

W. Fleming, The convergence problem for differential games, J. Math. Anal. Appl., 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9. Google Scholar

[19]

W. Fleming, The convergence problem for differential games, Ⅱ, in Advances in Game Theory, Ann. of Math. Studies, 52 (1964), Princeton Univ. Press, Princeton, NJ, 195-210. Google Scholar

[20]

D. P. Foster and R. Vohra, Calibrated learning and correlated equilibrium, Games and Economic Behavior, 21 (1997), 40-55. doi: 10.1006/game.1997.0595. Google Scholar

[21]

R. Isaacs, Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons, Inc., New York-London-Sydney, 1965. Google Scholar

[22]

C. Jimenez and M. Quincampoix, Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition, J. Math. Anal. Appl., 457 (2018), 1422-1451. doi: 10.1016/j.jmaa.2017.08.012. Google Scholar

[23]

E. Kohlberg, Optimal strategies in repeated games with incomplete information, Internat. J. Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399. Google Scholar

[24]

J. Kwon and V. Perchet, Online learning and blackwell approachability with partial monitoring: optimal convergence rates, PMLR Work. Conf. Proc., 54 (2014), 604-613. Google Scholar

[25]

G. LugosiS. Mannor and G. Stoltz, Strategies for prediction under imperfect monitoring, Math. Oper. Res., 33 (2008), 513-528. doi: 10.1287/moor.1080.0312. Google Scholar

[26]

S. MannorV. Perchet and G. Stoltz, Robust approachability and regret minimization in games with partial monitoring, JMLR Work. Conf. Proc., 19 (2011), 515-536. Google Scholar

[27]

S. MannorV. Perchet and G. Stoltz, Approachability in unknown games: Online learning meets multi-objective optimization, JMLR Work. Conf. Proc., 35 (2014), 1-17. Google Scholar

[28]

A. Marigonda and M. Quincampoix, Mayer control problem with probabilistic uncertainty on initial positions, J. Differential Equations, 264 (2018), 3212-3252. doi: 10.1016/j.jde.2017.11.014. Google Scholar

[29]

J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, With a foreword by Robert J. Aumann. Econometric Society Monographs, 55. Cambridge University Press, New York, 2015. doi: 10.1017/CBO9781139343275. Google Scholar

[30]

V. Perchet, Approachability of convex sets with partial monitoring, J. Optim. Theory. Appl., 149 (2011), 665-677. doi: 10.1007/s10957-011-9797-3. Google Scholar

[31]

V. Perchet, Internal regret with partial monitoring: calibration-based optimal algorithms, J. Mach. Learn. Res., 12 (2011), 1893-1921. Google Scholar

[32]

V. Perchet, Approachability, regret and calibration: Implications and equivalences, J. Dyn. Games, 1 (2014), 181-254. doi: 10.3934/jdg.2014.1.181. Google Scholar

[33]

V. Perchet, Exponential weight approachability, applications to calibration and regret minimization, Dynamic Games And Applications, 5 (2015), 136-153. doi: 10.1007/s13235-014-0119-x. Google Scholar

[34]

V. Perchet and M. Quincampoix, On a unified framework for approachability with full or partial monitoring, Mathematics of Operations Research, 40 (2014), 596-610. doi: 10.1287/moor.2014.0686. Google Scholar

[35]

S. Plaskacz and M. Quincampoix, Value-functions for differential games and control systems with discontinuous terminal cost, SIAM J. Control Optim., 39 (2000), 1485-1498. doi: 10.1137/S0363012998340387. Google Scholar

[36]

L. Pontrjagin, Linear differential games. Ⅰ, Ⅱ, Dokl. Akad. Nauk SSSR, 174 (1967), 1278-1280. Google Scholar

[37]

A. Rustichini, Minimizing regret: The general case, Games Econom. Behav., 29 (1999), 224-243. doi: 10.1006/game.1998.0690. Google Scholar

[38]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhauser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. Google Scholar

[39]

M. Sion, On general minimax theorems, Pacific J. Math, 8 (1958), 171-176. doi: 10.2140/pjm.1958.8.171. Google Scholar

[40]

X. Spinat, A necessary and sufficient condition for approachability, Math. Oper. Res., 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333. Google Scholar

[41]

T. Tomala, Belief-free Communication Equilibria, Math. Oper. Res., 38 (2013), 617-637. doi: 10.1287/moor.2013.0594. Google Scholar

[42]

N. Vieille, Weak approachability, Math. Op. Res., 17 (1992), 781-791. doi: 10.1287/moor.17.4.781. Google Scholar

[43]

C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, AMS, Vol. 58, 2003. doi: 10.1007/b12016. Google Scholar

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