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Irrational pursuit-evasion differential games: A cumulative prospect theory approach

Zili Wang, Hao Yang, Xiangxiang Wang, Bin Jiang, Long Wang

Abstract

This paper considers for the first time pursuit-evasion (PE) differential games with irrational perceptions of both pursuer and evader on probabilistic characteristics of environmental uncertainty. Firstly, the irrational perceptions of risk aversion and probability sensitivity are modeled and incorporated within a Bayesian PE differential game framework by using Cumulative Prospect Theory (CPT) approach; Secondly, several sufficient conditions of capturability are established in terms of system dynamics and irrational parameters; Finally, the existence of CPT-Nash equilibria is rigorously analyzed by invoking Brouwer's fixed-point theorem. The new results reveal that irrational behaviors benefit the pursuer in some cases and the evader in others. Certain captures that are unachievable under rational behaviors can be achieved under irrational ones. By bridging irrational behavioral theory with game-theoretic control, this framework establishes a rigorous theoretical foundation for practical control engineering within complex human-machine systems.

Irrational pursuit-evasion differential games: A cumulative prospect theory approach

Abstract

This paper considers for the first time pursuit-evasion (PE) differential games with irrational perceptions of both pursuer and evader on probabilistic characteristics of environmental uncertainty. Firstly, the irrational perceptions of risk aversion and probability sensitivity are modeled and incorporated within a Bayesian PE differential game framework by using Cumulative Prospect Theory (CPT) approach; Secondly, several sufficient conditions of capturability are established in terms of system dynamics and irrational parameters; Finally, the existence of CPT-Nash equilibria is rigorously analyzed by invoking Brouwer's fixed-point theorem. The new results reveal that irrational behaviors benefit the pursuer in some cases and the evader in others. Certain captures that are unachievable under rational behaviors can be achieved under irrational ones. By bridging irrational behavioral theory with game-theoretic control, this framework establishes a rigorous theoretical foundation for practical control engineering within complex human-machine systems.

Paper Structure

This paper contains 23 sections, 109 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Problem Description and Contributions
  4. Model Establishment of Irrational Perceptions
  5. Notations
  6. Bayesian PE Differential Game
  7. CPT-Based PE Differential Games
  8. CPT-Based PE Capturability and Nash Equilibrium
  9. Main result
  10. Analysis of Conditions
  11. Ancillary results and proof of main result
  12. An equivalent problem for CPT-based optimal controller and AREs
  13. Solvability of CPT-based AREs and Capturability Analysis
  14. Analytic Characterization of the Scalar Fixed-point
  15. Illustrative example
  16. ...and 8 more sections

Figures (15)

  • Figure 2: Risk aversion of pursuer
  • Figure 3: Probability sensitivity of pursuer
  • Figure 4: Proof idea of Theorem 1.
  • Figure 5: Scenario 1: The distance under different pursuer's gain sensitivity
  • Figure 6: Scenario 1: The distance under different evader's gain sensitivity
  • ...and 10 more figures