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

1. Problem Statement

The paper addresses Pursuit-Evasion (PE) differential games within a Bayesian framework where environmental uncertainty affects the performance index. While traditional research assumes "rational" agents who optimize based on accurate probability distributions, this work investigates scenarios where agents (pursuer and evader) exhibit irrational perceptions.

Specifically, the problem focuses on how risk aversion and probability sensitivity (where agents overweight small probabilities and underweight large ones) alter the game dynamics. The core challenges are:

1. Modeling: How to mathematically incorporate these irrational behaviors into the continuous-time differential game framework.
2. Capturability: Under what conditions can the pursuer guarantee capture ($x \to 0$) when agents are irrational?
3. Equilibrium: Does a CPT-Nash equilibrium exist, and how does it differ from the rational Nash equilibrium?

2. Methodology

The authors propose a novel framework integrating Cumulative Prospect Theory (CPT) into Bayesian PE differential games.

A. System Modeling

• Dynamics: The game involves a pursuer ($u_r$) and an evader ($v_r$) with relative state dynamics $\dot{x} = u_r - v_r$.
• Uncertainty: The state reward weighting matrix $Q$ in the performance index is stochastic ($Q = Q_r + \Sigma_Q$), following a normal distribution.
• Performance Index: The traditional cost function $J_r$ is replaced by a Prospect Value function $C_i(J)$ for each player ($i=1$ for pursuer, $i=2$ for evader).

B. CPT Integration

The authors define specific CPT functions to model irrationality:

• Value Functions: Separate functions for gains ($U^+$) and losses ($U^-$) relative to a reference point (the rational optimal performance $J_r$).
  • Gain Sensitivity ($\alpha_i$): Controls concavity (diminishing sensitivity to gains).
  • Loss Sensitivity ($\beta_i$) & Loss Multiplier ($\epsilon_i$): Control convexity and the "loss aversion" effect (losses hurt more than equivalent gains).
• Probability Weighting Functions ($w_i$): Non-linear transformations of probabilities to model probability sensitivity (overweighting low probabilities, underweighting high ones).
• Objective: The pursuer minimizes $J_1 = J_r - C_1(J)$, and the evader maximizes $J_2 = J_r + C_2(J)$. This transforms the zero-sum game into a non-zero-sum differential game.

C. Theoretical Analysis

• Coupled Riccati Equations: The optimization problem leads to a set of coupled Algebraic Riccati Equations (AREs) that depend on the irrationality parameters.
• Fixed-Point Analysis: The existence of the CPT-Nash equilibrium is proven using Brouwer's Fixed-Point Theorem. The authors construct a mapping on a compact, convex set of matrices and show that a fixed point exists, guaranteeing a solution to the coupled AREs.
• Capturability Analysis: Using Lyapunov stability theory, the authors derive sufficient conditions for the relative state $x(t)$ to converge to zero exponentially.

3. Key Contributions

1. First Integration of CPT in PE Games: This is the first work to model irrational risk aversion and probability sensitivity in continuous-time PE differential games using a Bayesian framework.
2. Unified Theoretical Framework: The paper establishes a rigorous framework that bridges behavioral economics (CPT) with game-theoretic control.
3. Sufficient Conditions for Capturability: The authors derive specific, verifiable conditions (Theorem 1) for capture success across three distinct scenarios based on the relative control capabilities of the agents:
   • Scenario 1: Pursuer has superior control effort ($R^{-1} > \Pi^{-1}$).
   • Scenario 2: Evader has superior control effort ($R^{-1} < \Pi^{-1}$).
   • Scenario 3: Equal control effort ($R^{-1} = \Pi^{-1}$).
4. Existence of Equilibrium: Rigorous proof of the existence of CPT-Nash equilibria via Brouwer's theorem, extending beyond numerical approximations.

4. Key Results and Findings

The analysis reveals that irrationality fundamentally changes game outcomes compared to the rational case:

• Irrationality Can Be Advantageous:
  • In Scenario 1 (Pursuer is stronger), a pursuer with high loss aversion (large $\beta_1, \epsilon_1$) and low gain sensitivity is more likely to capture. Conversely, an evader with low loss aversion and high gain sensitivity facilitates capture (i.e., the evader makes mistakes).
  • In Scenario 2 (Evader is stronger, where rational capture is impossible), specific irrational behaviors can enable capture. For instance, if the evader exhibits high loss sensitivity and high loss aversion, they may overreact to perceived risks, allowing the pursuer to capture them despite the evader's superior control authority.
• Scenario 3 (Tie): Capture is only possible if the irrationality parameters of both players satisfy a specific balance condition ($2Q_r + (\Psi_1 - \Psi_2)qI = 0$).
• Numerical Validation: Simulations confirm that:
  • Rational agents follow standard capture conditions.
  • Irrational agents can achieve captures that are impossible under rationality (e.g., Scenario 2).
  • Conversely, irrationality can also lead to failure to capture in scenarios where rational agents would succeed.

5. Significance

• Human-Machine Systems: The framework is highly relevant for manned-unmanned teaming (e.g., drone swarms with human operators), where human operators often exhibit irrational risk perception. The model allows engineers to predict and regulate system behavior when humans are involved.
• Robust Control: It provides a theoretical basis for designing control strategies that account for the "irrational" nature of adversaries or partners, moving beyond the idealized assumption of perfect rationality.
• Theoretical Gap Filling: It bridges the gap between behavioral economics and differential game theory, offering a tractable method to analyze complex, non-linear decision-making in dynamic environments.

In conclusion, the paper demonstrates that irrational behaviors are not merely noise but strategic variables that can be modeled, analyzed, and potentially exploited to alter the outcome of pursuit-evasion games, offering new insights for the control of complex human-machine systems.