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Optimal Mixed Strategy for Zero-Sum Differential Games

Tao Xu, Wang Xi, Jianping He

TL;DR

The paper tackles solving zero-sum differential games (ZSDGs) under mixed strategies without requiring vanishing commitment delays. It introduces a SDG-based weak-approximation framework that maps the mixed-strategy game to a pure-strategy SDG, ensuring close agreement in state distributions and costs and enabling certified bounds on the mixed-strategy value and suboptimality. The authors prove the existence of game value under the proposed mixed-strategy definition, establish order-$n$ weak approximations, and present a five-step procedure to obtain near-optimal mixed strategies with explicit error bounds. They validate the approach on a class of control-affine dynamics with quadratic costs, showing $O(\barπ)$ scaling for both value approximation error and strategy suboptimality, and provide numerical simulations that confirm distributional closeness and practical improvements from mixed strategies.

Abstract

Solving zero-sum differential games (ZSDGs) under mixed strategies has been challenging for decades. Existing research mainly focuses on characterizing the value function, while the problem of solving optimal mixed strategies remains open. To address this issue, we propose a novel weak-approximation-based method to solve ZSDGs under mixed strategies. The key idea is to design an SDG under pure strategies that closely approximates the original game under mixed strategies, ensuring that both the state distributions and cost expectations remain nearly identical over the entire time horizon. Based on the solution of this SDG, the value function under mixed strategies can be approximated with a certified approximation error. In addition, near-optimal mixed strategies can be designed with certified suboptimality gaps. We further apply this method to a class of ZSDGs with control-affine dynamics and quadratic costs, demonstrating that the value approximation error is of order $O(\barπ)$ and the strategy suboptimality gap is of order $O(\barπ)$ with respect to the maximum commitment delay $\barπ$. Numerical examples are provided to illustrate and validate our results.

Optimal Mixed Strategy for Zero-Sum Differential Games

TL;DR

The paper tackles solving zero-sum differential games (ZSDGs) under mixed strategies without requiring vanishing commitment delays. It introduces a SDG-based weak-approximation framework that maps the mixed-strategy game to a pure-strategy SDG, ensuring close agreement in state distributions and costs and enabling certified bounds on the mixed-strategy value and suboptimality. The authors prove the existence of game value under the proposed mixed-strategy definition, establish order- weak approximations, and present a five-step procedure to obtain near-optimal mixed strategies with explicit error bounds. They validate the approach on a class of control-affine dynamics with quadratic costs, showing scaling for both value approximation error and strategy suboptimality, and provide numerical simulations that confirm distributional closeness and practical improvements from mixed strategies.

Abstract

Solving zero-sum differential games (ZSDGs) under mixed strategies has been challenging for decades. Existing research mainly focuses on characterizing the value function, while the problem of solving optimal mixed strategies remains open. To address this issue, we propose a novel weak-approximation-based method to solve ZSDGs under mixed strategies. The key idea is to design an SDG under pure strategies that closely approximates the original game under mixed strategies, ensuring that both the state distributions and cost expectations remain nearly identical over the entire time horizon. Based on the solution of this SDG, the value function under mixed strategies can be approximated with a certified approximation error. In addition, near-optimal mixed strategies can be designed with certified suboptimality gaps. We further apply this method to a class of ZSDGs with control-affine dynamics and quadratic costs, demonstrating that the value approximation error is of order and the strategy suboptimality gap is of order with respect to the maximum commitment delay . Numerical examples are provided to illustrate and validate our results.
Paper Structure (35 sections, 12 theorems, 101 equations, 2 figures, 1 algorithm)

This paper contains 35 sections, 12 theorems, 101 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivations
  3. Challenges
  4. Contributions
  5. Related Works
  6. Preliminaries and Problem Formulation
  7. Preliminaries
  8. Zero-Sum Differential Game
  9. Problems of Interest
  10. Admissible mixed strategy: definition, properties, and existence of game value
  11. Definition
  12. Properties
  13. Existence of Game Value
  14. Main Methodology
  15. Step 1: Design an SDG
  16. ...and 20 more sections

Key Result

Proposition 1

Let $\alpha, \beta$ be nonanticipative strategies for player $1$ and $2$, resp. If at least one of these strategies is NAD, then there exists a unique pair of control functions $(u,v)\in L(t_0,T;U)\times L(t_0,T;V)$ such that $u = \alpha(v)$ and $v =\beta(u)$, a.e. on $[t_0,T]$.

Figures (2)

  • Figure 1: An illustration of the main methodology.
  • Figure 2: Simulation results of $\mathbf{G}_{lq}$ under the mixed strategy spaces $\mathcal{A}_m^\pi \times \mathcal{B}_m^\pi$ and $\mathbf{\tilde{G}}_{lq}$ under the pure strategy spaces $\tilde{\mathcal{A}}_p^\pi \times \tilde{\mathcal{B}}_p^\pi$.

Theorems & Definitions (39)

  • Definition 1: NAD basarHandbookDynamicGame2018
  • Proposition 1: normal form of strategy basarHandbookDynamicGame2018
  • Definition 2: polynomial growth function
  • Definition 3: weak approximation
  • Definition 4: value functions
  • Definition 5: admissible control
  • Remark 1
  • Definition 6: admissible mixed strategy
  • Remark 2
  • Proposition 2
  • ...and 29 more