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Leader-Follower Stochastic Differential Game with Asymmetric Information and Applications

Jingtao Shi, Guangchen Wang, Jie Xiong

arXiv:1509.03982v1math.OC

TL;DR

The paper studies leader-follower stochastic differential games in which the follower has less information than the leader. It derives partial-information optimality tools and applies them to an LQ game, obtaining a state-feedback representation of the open-loop equilibrium when the relevant Riccati equations are solvable.

  • Problem

    The paper addresses leader-follower stochastic games with asymmetric information, a setting distinguished from cited prior games by the follower's information being a sub-σ-algebra of the leader's.

    Source: §1.3 · p. 5

  • Method

    The paper derives partial-information maximum principles and verification theorems, then applies them to an LQ game using filtering and Riccati equations.

    Source: §1.3 · p. 5

  • Results

    The LQ game admits an open-loop Stackelberg equilibrium with a state-feedback representation when the relevant Riccati equations are solvable.

    Source: §1.3 · p. 5, §3.2 · p. 18

  • Takeaways & Limitations

    In the supply-chain examples, the nested-information setup represents a manufacturer choosing policies while the retailer responds with less information.

    Source: §1.1 · p. 2

    The treatment solves the LQ case; extending it to nonlinear states or nonquadratic costs and studying closed-loop equilibrium remain future work.

    Source: §4 · p. 19

Abstract

from arXiv · show

This paper is concerned with a leader-follower stochastic differential game with asymmetric information, where the information available to the follower is based on some sub-$\sigma$-algebra of that available to the leader. Such kind of game problem has wide applications in finance, economics and management engineering such as newsvendor problems, cooperative advertising and pricing problems. Stochastic maximum principles and verification theorems with partial information are obtained, to represent the Stackelberg equilibrium. As applications, a linear-quadratic leader-follower stochastic differential game with asymmetric information is studied. It is shown that the open-loop Stackelberg equilibrium admits a state feedback representation if some system of Riccati equations is solvable.

1 Introduction

The introduction motivates and formulates a stochastic Stackelberg game in which the follower has less information than the leader, then presents partial-information control tools and an LQ application.

1.1 Motivation

  • Newsvendor and cooperative advertising/pricing examples place the retailer as follower and manufacturer as leader, with the retailer's information nested within the manufacturer's.The manufacturer anticipates the retailer's response when choosing its policies.Source: §1.1 · pp. 2–3

1.2 Problem Formulation

  • The formulation lets the follower choose a control adapted to G1,t after the leader's choice, then has the leader optimize while anticipating that response, with G1,t ⊆ G2,t.Both players' information filtrations are sub-σ-algebras of the full filtration.Source: §1.2 · pp. 3–4

1.3 Literature Review and Contributions of This Paper

  • The paper introduces an LQ game with asymmetric information and characterizes its open-loop Stackelberg equilibrium through filtering equations and state-feedback representations involving new Riccati equations.The leader's problem is a conditional mean-field FBSDE control problem, while the follower's is a partially observed control problem.Source: §1.3 · p. 5
  • For the general game, the paper gives partial-information maximum principles and verification theorems for the follower and leader problems.The leader's state equation uses the stochastic Hamiltonian system corresponding to the follower's optimal response.Source: §1.3 · p. 5

2 Maximum Principle and Verification Theorem for Stackelberg Equilibrium

The section develops partial-information maximum principles and verification theorems for the follower’s and leader’s stochastic control problems. The leader’s problem treats the state as the quadruple of the original state and the follower’s adjoint processes.

2.1 The Follower’s Problem

  • The follower’s partial-information maximum principle gives a necessary condition for an optimal control, while the verification theorem supplies sufficient conditions for optimality.The verification result uses concavity and convexity assumptions.Source: §2.1 · p. 6

2.2 The Leader’s Problem

  • Substituting the follower’s response into the leader’s formulation yields a controlled conditional mean-field FBSDE as the leader’s state equation.The leader’s control problem is formulated over this controlled system.Source: §2.2 · pp. 6–7
  • The leader’s partial-information maximum principle gives a maximum condition for every admissible control comparison.The leader’s state is the quadruple of the original state and the follower’s adjoint processes.Source: §2.2 · pp. 7–8
  • The leader’s verification theorem provides sufficient conditions under which the candidate control is optimal.The proposition assumes convexity conditions, including convexity of G2.Source: §2.2 · p. 8

3 LQ Leader-Follower Stochastic Differential Game with Asymmetric Information

This section applies the paper’s theoretical results to a partially observed linear-quadratic leader-follower game, analyzing the follower’s and leader’s problems. It describes an open-loop Stackelberg equilibrium with state-feedback representations for both players when the coupled Riccati equations have a differentiable solution pair; the leader’s feedback is optimal and the follower’s is observable.

3.1 The Follower’s LQ Problem

  • The follower cannot directly observe the system state and instead observes a signal; the analysis changes measure so the signal and the other driving process form a two-dimensional standard Brownian motion.The change of measure is justified by the martingale property of Z, which follows from the Novikov condition.Source: §3.1 · pp. 8–9
  • Under assumptions (A3.1) and (A3.2), when Riccati equation (49) admits a unique differentiable solution P1(·), the follower’s LQ problem admits an optimal state-feedback control (45), whose process triple is the unique FY_t-adapted solution to FBSDFE (52).The control is generally anticipating, motivating a “real” state-feedback representation in terms of the original state.Source: §3.1 · p. 12

3.2 The Leader’s LQ Problem

  • The leader’s problem incorporates the process pair (ϕ, β) into its state, yielding a conditional mean-field FBSDE that is treated as a complete-information control problem.The stated solvability of the leader’s state equation relies on the previously obtained solvability of the follower’s FBSDFE.Source: §3.2 · p. 13
  • The coupled Riccati system is entirely new; its first equation has a unique solution by standard theory, but solvability of the second remains open and is assumed.The leader’s state-feedback derivation decouples a conditional mean-field FBSDE to obtain the coupled Riccati equations.Source: §3.2 · pp. 15–17
  • Under assumptions (A3.1)–(A3.6) and a differentiable solution pair to the Riccati equations, the open-loop Stackelberg equilibrium has state-feedback representations for both players.The leader’s feedback is optimal, and the follower’s feedback is observable; the corresponding optimal state equations are specified for each.Source: §3.2 · p. 18

4 Concluding Remarks

The paper develops a leader-follower stochastic differential game with asymmetric information and applies its results to linear-quadratic models. It derives a state-feedback representation for the open-loop Stackelberg equilibrium under appropriate assumptions, while identifying broader cases for future work.

  • The paper introduces asymmetric information in a leader-follower stochastic differential game, with the follower’s information contained in a sub-σ-algebra of the leader’s, and uses filtering for adjoint processes.The two players have distinct roles, so approaches for games with equivalent player roles do not apply.Source: §4 · p. 19
  • For the partial-observation linear-quadratic problem, the open-loop Stackelberg equilibrium has a state-feedback representation under appropriate assumptions on the state and cost coefficients.The paper introduces conditional mean-field FBSDEs and Riccati equations for the leader’s LQ problem.Source: §4 · p. 19
  • The paper leaves the non-linear-state or non-quadratic-cost case, closed-loop equilibrium, and broader partially observable cases for future research.It also identifies solvability of the system of Riccati equations (76) as a topic for further study.Source: §4 · p. 19
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