The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential Games

Bensoussan, Alain; Chen, Shaokuan; Sethi, Suresh P.

doi:10.1137/140958906

Cited by 131 publications

(79 citation statements)

References 32 publications

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“…It is important to mention that this equilibrium concept is time‐consistent, in contrast to others such as global open‐loop Stackelberg and global close‐loop Stackelberg that are in general not (Bensoussan et al. ).…”

Section: Introductionmentioning

confidence: 89%

Dynamic Pricing, Production, and Channel Coordination with Stochastic Learning

Sethi

2015

Production and Operations Management

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We consider a decentralized two‐period supply chain in which a manufacturer produces a product with benefits of cost learning, and sells it through a retailer facing a price‐dependent demand. The manufacturer's second‐period production cost declines linearly in the first‐period production, but with a random learning rate. The manufacturer may or may not have the inventory carryover option. We formulate the resulting problems as two‐period Stackelberg games and obtain their feedback equilibrium solutions explicitly. We then examine the impact of mean learning rate and learning rate variability on the pricing strategies of the channel members, on the manufacturer's production decisions, and on the retailer's procurement decisions. We show that as the mean learning rate or the learning rate variability increases, the traditional double marginalization problem becomes more severe, leading to greater efficiency loss in the channel. We obtain revenue sharing contracts that can coordinate the dynamic supply chain. In particular, when the manufacturer may hold inventory, we identify two major drivers for inventory carryover: market growth and learning rate variability. Finally, we demonstrate the robustness of our results by examining a model in which cost learning takes place continuously.

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Section: Introductionmentioning

confidence: 89%

Dynamic Pricing, Production, and Channel Coordination with Stochastic Learning

Sethi

2015

Production and Operations Management

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“…The follower then determines his optimal solution by responding to the optimal solution of the leader. Under this setting, the problem can be solved in a reverse way [40], [41]. Specifically, (S.1) solve the follower's optimal control problem with arbitrary x 0 ∈ R n and control of the leader…”

Section: Problem Statementmentioning

confidence: 99%

“…In fact, (S.1) above characterizes the rational reaction behavior of the follower, which is the optimization constraint of (S.2). Due to the hierarchy between the leader and the follower mentioned above, the problem considered in the paper can be referred to as the (adapted open-loop) linear-quadratic (LQ) mean-field Stackelberg differential game [39]- [41]. We also note that if the solutions to (S.1) and (S.2) exist, then the pair (ū,v) constitutes the (adapted open-loop) Stackelberg equilibrium [40], [41] (see also Definition 1 below).…”

Section: Problem Statementmentioning

confidence: 99%

“…The class of leader-follower differential games is also known as Stackelberg differential games [39]- [41]. The leader holds a dominating position; the leader chooses and then announces his optimal strategy by considering the rational behavior of the follower.…”

Section: Introductionmentioning

confidence: 99%

“…Note that depending on the open-loop or closed-loop information structure between the leader and the follower, the Stackelberg game has to be treated differently. 3 In particular, for the adapted open-loop case, the (stochastic) Stackelberg game can be analyzed by applying the stochastic maximum principle to the following: (i) the follower's problem given an arbitrary strategy of the leader and (ii) the leader's problem, where the constraint is the follower's rational behavior characterized by the forward-backward SDE (FBSDE) from (i) [39]- [41]. There are wide ranges of applications for Stackelberg games including engineering, economics and biology; see [29], [46]- [50] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Infinite time linear quadratic stackelberg game problem for unknown stochastic discrete‐time systems via adaptive dynamic programming approach

Liu

2020

Asian Journal of Control

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In this paper, we propose an adaptive dynamic programming (ADP) approach to solve the infinite horizon linear quadratic (LQ) Stackelberg game problem for unknown stochastic discrete‐time systems with multiple decision makers. Firstly, the stochastic LQ Stackelberg game problem is converted into the deterministic problem by system transformation. Next, a value iteration ADP approach is put forword and the convergence is given. Thirdly, in order to implement the iterative method, back propagation neural network (BPNN) is chosen to design model network, critic network and action network to approximate the unknown systems, objective functions and Stackelberg strategies. Finally, simulation results show that the algorithm is effective.

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The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential Games

Cited by 131 publications

References 32 publications

Dynamic Pricing, Production, and Channel Coordination with Stochastic Learning

Dynamic Pricing, Production, and Channel Coordination with Stochastic Learning

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

Infinite time linear quadratic stackelberg game problem for unknown stochastic discrete‐time systems via adaptive dynamic programming approach

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