Finite-State Contract Theory with a Principal and a Field of Agents

We use the recently developed probabilistic analysis of mean field games with finitely many states in the weak formulation, to set-up a principal / agent contract theory model where the principal faces a large population of agents interacting in a mean field manner. We reduce the problem to the optimal control of dynamics of the McKean-Vlasov type, and we solve this problem explicitly in a special case reminiscent of the linear - quadratic mean field game models. The paper concludes with a numerical example demonstrating the power of the results when applied to a simple example of epidemic containment.

[1]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[2]  Jaksa Cvitanic,et al.  Dynamic programming approach to principal–agent problems , 2015, Finance Stochastics.

[3]  Alain Bensoussan,et al.  Mean-Field-Game Model for Botnet Defense in Cyber-Security , 2015, 1511.06642.

[4]  Yuliy Sannikov.  A Continuous-Time Version of the Principal-Agent , 2005 .

[5]  Jaksa Cvitanic,et al.  Moral Hazard in Dynamic Risk Management , 2014, Manag. Sci..

[6]  Holger M. Müller.  The First-Best Sharing Rule in the Continuous-Time Principal-Agent Problem with Exponential Utility , 1998 .

[7]  Huyên Pham,et al.  Dynamic Programming for Optimal Control of Stochastic McKean-Vlasov Dynamics , 2016, SIAM J. Control. Optim..

[8]  Peiqi Wang,et al.  A Probabilistic Approach to Extended Finite State Mean Field Games , 2018, Math. Oper. Res..

[9]  Niels Richard Hansen,et al.  Exponential Martingales and Changes of Measure for Counting Processes , 2012, 1205.2244.

[10]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Robert J. Elliott,et al.  Solutions of Backward Stochastic Differential Equations on Markov Chains , 2008 .

[12]  Alain Bensoussan,et al.  Linear-Quadratic Mean Field Games , 2014, Journal of Optimization Theory and Applications.

[13]  Paul R. Milgrom,et al.  Multitask Principal–Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design , 1991 .

[14]  R. Carmona,et al.  A probabilistic weak formulation of mean field games and applications , 2013, 1307.1152.

[15]  Romuald Elie,et al.  A Tale of a Principal and Many, Many Agents , 2016, Math. Oper. Res..

[16]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[17]  Samuel N. Cohen,et al.  Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions , 2008, 0810.0055.

[18]  R. Carmona,et al.  Finite State Mean Field Games with Major and Minor Players , 2016 .

[19]  J. Sung,et al.  Linearity with Project Selection and Controllable Diffusion Rate in Continuous-Time Principal-Agent Problems , 1995 .

[20]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[21]  D. Gomes,et al.  Continuous Time Finite State Mean Field Games , 2012, 1203.3173.

[22]  Olivier Guéant.  Existence and Uniqueness Result for Mean Field Games with Congestion Effect on Graphs , 2011, 1110.3442.

[23]  Heinz Schättler,et al.  The First-Order Approach to the Continuous-Time Principal-Agent Problem with Exponential Utility , 1993 .

[24]  Klaus M. Schmidt,et al.  Discrete-time approximations of the Holmström-Milgrom brownian-motion model of intertemporal incentive provision , 2002 .