A Tale of a Principal and Many, Many Agents

Élie, Romuald; Mastrolia, Thibaut; Possamaï, Dylan

doi:10.1287/moor.2018.0931

Cited by 75 publications

(97 citation statements)

References 55 publications

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“…A different but related recent literature studies mean field games of timing where players directly choose stopping times; see Carmona and Lacker [10], Bertucci [5] and Nutz [28]. Continuous-time principal-agent problems with multiple agents have been studied by Koo, Shim and Sung [21] and Elie and Possamaï [13], and extended to the mean field setting by Elie, Mastrolia and Possamaï [12] and Bensoussan, Chau and Yam [3]. While these works have not considered rank-based rewards, a common feature is the Stackelberg equilibrium: the principal designs a reward scheme which the agents take as an external input to form a Nash equilibrium among themselves.…”

Section: Literaturementioning

confidence: 99%

A Mean Field Competition

Nutz

Zhang

2019

Mathematics of OR

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We introduce a mean field game with rank-based reward: competing agents optimize their effort to achieve a goal, are ranked according to their completion time, and paid a reward based on their relative rank. First, we propose a tractable Poissonian model in which we can describe the optimal effort for a given reward scheme. Second, we study the principal-agent problem of designing an optimal reward scheme. A surprising, explicit design is found to minimize the time until a given fraction of the population has reached the goal.

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Section: Literaturementioning

confidence: 99%

A Mean Field Competition

Nutz

Zhang

2019

Mathematics of OR

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“…A future research topic is to consider the discrete-time time-inconsistent leader-follower Stackelberg game that can be viewed as an extension of [30]. Another topic would be the existence and uniqueness of the solution to (17) and (32) and their numerical computation approaches. Note that under some assumptions on the coeffi-cients in (17) and (32), the Picard fixed point argument can be used to obtain both existence and uniqueness.…”

Section: Discussionmentioning

confidence: 99%

“…COUPLED RDES IN (32) In this appendix, we provide the explicit expressions of nonsymmetric coupled RDEs in (32). where the explicit expression of Λ 3 can be obtained from (C.1) and (C.2).…”

Section: Appendix C Explicit Expressions Of the Nonsymmetricmentioning

confidence: 99%

“…The motivation to include mean-field variables in the state equation (e.g. SDE) and the objective functional is to analyze macroscopic behavior of large-scale interacting particle systems in engineering, biology and economics, and to consider mean-variance portfolio optimization in various mathematical finance applications [1], [3], [8], [16], [20]- [32]. There are various definitions for quantifying "optimality" in (deterministic and stochastic) time-inconsistent optimal control problems.…”

Section: Introductionmentioning

confidence: 99%

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Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

Moon

Yang

2021

IEEE Trans. Automat. Contr.

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In this paper, we consider the linear-quadratic timeinconsistent mean-field leader-follower Stackelberg stochastic differential game with an adapted open-loop information structure. Given a controlled linear stochastic differential equation (SDE), the quadratic objective functionals of the leader and the follower include conditional expectations of state and control (mean field) variables. In addition, the cost parameters could be general nonexponential discounting depending on the initial time. As stated in the existing literature, these two general settings of the objective functionals induce time inconsistency in the optimal solutions. Given an arbitrary control of the leader, we first obtain the follower's (time-consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled Riccati differential equations (RDEs) and the backward SDE. This provides the rational behavior of the follower, characterized by the forward-backward SDE (FBSDE). We then obtain the leader's explicit (time-consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled RDEs, where the constraint of the leader's problem is the FBSDE induced by the follower's rational behavior. With the solvability of the nonsymmetric coupled RDEs, the equilibrium controls of the leader and the follower constitute the time-consistent Stackelberg equilibrium of the paper. Finally, the numerical examples are provided to check the solvability of the nonsymmetric coupled RDEs of the leader and the follower.Index Terms-time-inconsistent stochastic control problem, equilibrium control, Stackelberg differential games, mean-field stochastic systems. 4 This implies that the magnitude of the stochastic noise is controlled by the leader and the follower, which can be viewed as multiplicative noise of the system [51], [52]. 5 As mentioned, an equilibrium control is time consistent; see Definition 2 or [3, Definition 4.1] and [26, Definition 2.1].

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“…Cardadiaguet, Cirant, and Porretta (2018) prove the convergence of the Nash equilibria by use of the master equations when the number of minor players tends to infinity. A mean field principal-agent model is formulated by Elie, Mastrolia, and Possamai (2019). For major player models with discrete states, see (Huang 2012; Carmona and Wang, 2017;Kolokoltsov, 2017).…”

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confidence: 99%

Linear quadratic mean field games with a major player: The multi-scale approach

Huang

2020

Automatica

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This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming equations and applies a re-scaling technique introduced in Huang and Zhou (2018a, 2018b) to derive a set of Riccati equations in lower dimensions, the solvability of which determines the necessary and sufficient condition for asymptotic solvability. We next derive the mean field limit of the strategies and the value functions. Finally, we show that the two decentralized strategies can be interpreted as the best responses of a major player and a representative minor player embedded in an infinite population, which have the property of consistent mean field approximations.

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A Tale of a Principal and Many, Many Agents

Cited by 75 publications

References 55 publications

A Mean Field Competition

A Mean Field Competition

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

Linear quadratic mean field games with a major player: The multi-scale approach

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