arxiv: 2603.27286 · v2 · submitted 2026-03-28 · 📡 eess.SY · cs.SY

Prospect Theoretic Approach to Pursuit-evasion Differential Games with Risk Aversion and Probability Sensitivity

Zili Wang , Hao Yang , Xiangxiang Wang , Bin Jiang , Long Wang , Marios M. Polycarpou This is my paper

Pith reviewed 2026-05-14 22:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords pursuit-evasion gamescumulative prospect theoryrisk aversionprobability weightingNash equilibriadifferential gamescapturabilityirrational behavior

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The pith

Irrational risk perceptions in pursuit-evasion games enable captures impossible under rational play.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper incorporates cumulative prospect theory to model how both pursuer and evader irrationally assess risks and probabilities within a Bayesian pursuit-evasion differential game. It derives sufficient conditions for capturability that depend on the system dynamics together with the players' risk-aversion and probability-sensitivity parameters. The analysis demonstrates that these irrational behaviors can favor the pursuer in some settings and the evader in others, producing equilibrium outcomes where capture occurs even though rational expected-utility play would prevent it. The framework therefore supplies a rigorous way to predict behavior in human-machine control systems that involve environmental uncertainty.

Core claim

By modeling risk aversion and probability sensitivity through cumulative prospect theory and embedding the resulting value and weighting functions into a Bayesian differential-game payoff structure, the authors obtain capturability conditions and prove existence of CPT-Nash equilibria via Brouwer's fixed-point theorem. The equilibria show that irrational perceptions can reverse the outcome relative to classical rational play and can make capture feasible when rational strategies cannot achieve it.

What carries the argument

Cumulative prospect theory value function and probability weighting function applied to the players' expected payoffs, yielding modified strategy spaces whose fixed points are the CPT-Nash equilibria.

Load-bearing premise

The chosen cumulative prospect theory parameters correctly describe how real decision makers weigh risk and probability in dynamic pursuit-evasion settings, and the strategy spaces meet the compactness and convexity requirements for Brouwer's fixed-point theorem.

What would settle it

Human-subject experiments or high-fidelity simulations in which measured capture probabilities deviate systematically from the predictions obtained by solving the CPT-Nash equilibria for the fitted irrationality parameters.

Figures

Figures reproduced from arXiv: 2603.27286 by Bin Jiang, Hao Yang, Long Wang, Marios M. Polycarpou, Xiangxiang Wang, Zili Wang.

**Figure 1.** Figure 1: Risk aversion of pursuer true probabilities, but rather low probabilities are generally overweighted while high probabilities are underweighted. Hence, C + 1 represents the expected gain that makes the performance of the pursuer better than the optimal one in the rational case. True probability Probability of irrational perception The real probability The perceived probability [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 2.** Figure 2: Probability sensitivity of pursuer b. The negative prospect C − 1 contains the loss U − 1 (J) and the corresponding perceived probability w − 1 (p1). As shown by the red line in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Proof idea of Theorem 1. of the variance σ, the following result is presented. At this point, the reference point corresponds to the expected value of the performance index J. Proposition 2 : Consider the CPT-based performance index (19) subject to (2),(3) and prospect function (12), for any given prospect parameters αi , βi , γi , ϵi , and w +,− i , the equivalent form of the prospect function (19) is in… view at source ↗

**Figure 5.** Figure 5: Scenario 1: The distance under different evader’s gain sensitivity Loss sensitivity βi: Next, the effect of loss sensitivity βi is studied. The remaining CPT parameters are fixed at the rational case with α1 = α2 = β2 = ϵ1 = ϵ2 = γ1 = γ2 = 1, and β1 ∈ {0.11, 0.4, 0.72, 1} is set. The simulation is shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Scenario 1: The distance under different pursuer’s loss sensitivity On the opposite side, with the pursuer’s loss sensitivity β1 = 1 fixed, all capture can be achieved in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 4.** Figure 4: Scenario 1: The distance under different pursuer’s gain sensitivity Whereas, the pursuer’s gain sensitivity α1 = 1 and the other parameters are fixed in the rational case, α2 ∈ {0.35, 0.55, 0.8, 1} is set [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 10.** Figure 10: shows that successful capture cannot be achieved in the irrational case. The pursuer’s gain sensitivity α1 = 1 and the other parameters are fixed in the rational case, α2 ∈ {0.35, 0.55, 0.8, 1} is set [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 11.** Figure 11: Scenario 2: The distance under different evader’s gain sensitivity conditions (32)-(35). As expected, the smaller β1 leads to a successful capture. On the other hand, with the pursuer’s loss sensitivity β1 = 1 fixed, capture cannot be achieved in [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗

**Figure 9.** Figure 9: Scenario 1: The distance under different evader’s loss multiplier B. Scenario 2 The settings of probability variance (9), initial positions z10, z20 and solution algorithm in Scenario 2 are the same as those in Scenario 1. In this case, the weighting matrices in the performance index (7) with Qr = I3×3, R = 0.9I3×3, Π = I3×3 are considered. Gain sensitivity αi : For the pursuer, the other parameters are fi… view at source ↗

**Figure 12.** Figure 12: Scenario 2: The distance under different pursuer’s loss sensitivity [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗

**Figure 13.** Figure 13: Scenario 2: The distance under different evader’s loss sensitivity 2.2 0 1.8 1.4 0 5 5 Distance x(t) 10 Time t(s) 10 1 15 15 20 Loss multiplier ǫ1 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗

**Figure 14.** Figure 14: Scenario 2: The distance under different pursuer’s loss multiplier VI. CONCLUSION To characterize the subjective bias in perceiving the probabilistic characteristics of environmental uncertainty in practical human-machine systems, this paper develops a CPT-based PE differential game framework to model irrational behaviors, with particular emphasis on risk aversion and probability sensitivity. By bridging… view at source ↗

**Figure 15.** Figure 15: Scenario 2: The distance under different evader’s loss multiplier player scenarios. Future work will extend the proposed framework to networked multi-player differential games, focusing on how irrationality propagates and couples among interacting agents, and on developing scalable equilibrium computation and capturability analysis for such settings. APPENDIX Let Hi ≜ x T 0 Mx0 and the density function gi… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies CPT to pursuit-evasion games for the first time and derives parameter-dependent capturability conditions plus an existence claim via Brouwer, but the continuity needed for that fixed-point step is not obviously preserved under nonlinear probability weighting.

read the letter

The core contribution is straightforward: they replace the standard expected-payoff structure in a Bayesian pursuit-evasion differential game with cumulative prospect theory valuations that include both value-function curvature for risk aversion and a nonlinear weighting function for probabilities. From that they extract sufficient conditions on the dynamics and the CPT parameters that guarantee capture, and they assert existence of a CPT-Nash equilibrium by invoking Brouwer on the best-response map. The abstract also notes that certain capture sets become reachable only when one or both players are irrational in the CPT sense, which is the main qualitative takeaway. That framing is new for this class of games and the direction is reasonable if you want to model human operators who overweight low probabilities or exhibit loss aversion. The conditions themselves look like straightforward extensions of the rational case once the prospect values are substituted in. The soft spot is the fixed-point argument. Nonlinear weighting typically destroys the continuity or convexity properties that make best responses well-behaved, so the strategy sets need to be shown compact and convex and the CPT-modified payoff map needs to be continuous (or at least upper hemicontinuous) in a topology where Brouwer applies. The abstract does not indicate that this verification was done explicitly, and the stress-test concern about discontinuity is therefore live. The parameters remain free, so the results are conditional on particular irrationality levels rather than calibrated to data. No numerical examples or simulations appear in the abstract to illustrate effect sizes. This is for control theorists and game theorists who already work on differential games and are curious about behavioral extensions. A serious referee should check the continuity proof and the precise statement of the strategy spaces; the paper is worth that review rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. This paper develops a pursuit-evasion differential game framework that incorporates Cumulative Prospect Theory (CPT) to model both players' risk aversion and probability sensitivity under environmental uncertainty. It derives sufficient conditions for capturability expressed in terms of the system dynamics and the CPT parameters, and establishes existence of CPT-Nash equilibria by invoking Brouwer's fixed-point theorem. The central results claim that irrational CPT behaviors can benefit the pursuer in some regimes and the evader in others, enabling captures that are impossible under standard rational (expected-value) play.

Significance. If the continuity and compactness conditions required for Brouwer's theorem are verified under the nonlinear CPT transformations, the work supplies the first rigorous existence result for prospect-theoretic equilibria in differential games and yields explicit, parameter-dependent capturability criteria. These contributions would bridge behavioral decision theory with game-theoretic control, offering a foundation for analyzing human-machine pursuit-evasion scenarios where agents exhibit documented probability weighting and loss aversion.

major comments (1)

[Existence of CPT-Nash equilibria] Existence analysis (final section): the application of Brouwer's fixed-point theorem presupposes that the strategy sets remain compact and convex and that the CPT-modified payoff map is continuous (or upper hemicontinuous) in a suitable topology. Because the value function v(·) and weighting function w(·) are strictly concave/convex and non-linear, the best-response correspondence may lose continuity or convexity; the manuscript must supply an explicit verification that these properties survive the CPT transformation for the given linear dynamics, otherwise the equilibrium existence and the derived capturability conditions rest on an unproven step.

minor comments (2)

[Abstract] The abstract states that the framework is presented 'for the first time'; a brief comparison paragraph with existing behavioral extensions of differential games (e.g., prospect-theoretic Stackelberg or risk-sensitive games) would clarify the precise increment in novelty.
[Modeling section] Notation for the CPT parameters (risk-aversion coefficient and probability-sensitivity exponents) should be introduced once in a dedicated subsection and then used consistently; scattered re-definitions make it difficult to track how each parameter enters the capturability inequalities.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The positive assessment of the overall contribution is appreciated. We address the single major comment point by point below.

read point-by-point responses

Referee: Existence analysis (final section): the application of Brouwer's fixed-point theorem presupposes that the strategy sets remain compact and convex and that the CPT-modified payoff map is continuous (or upper hemicontinuous) in a suitable topology. Because the value function v(·) and weighting function w(·) are strictly concave/convex and non-linear, the best-response correspondence may lose continuity or convexity; the manuscript must supply an explicit verification that these properties survive the CPT transformation for the given linear dynamics, otherwise the equilibrium existence and the derived capturability conditions rest on an unproven step.

Authors: We agree that the manuscript would benefit from an explicit verification of the topological properties required by Brouwer's theorem after the CPT transformations. The original submission invoked the theorem on the basis that the underlying strategy sets (closed and bounded subsets of the admissible control spaces) remain compact and convex for the linear dynamics, and that the CPT-modified payoffs preserve continuity because both the value function and the weighting function are continuous (though nonlinear) and the underlying probability measures are fixed. However, we acknowledge that a self-contained argument showing upper hemicontinuity of the best-response correspondence under these transformations was not spelled out. In the revised version we will add a dedicated lemma (with proof) establishing that (i) the strategy sets stay compact and convex, (ii) the CPT payoff map is continuous on the product space, and (iii) the best-response correspondence is upper hemicontinuous, thereby rigorously justifying the application of Brouwer's theorem. This addition will also clarify why the capturability conditions remain valid. revision: yes

Circularity Check

0 steps flagged

No circularity: CPT-Nash existence and capturability conditions derived from standard Brouwer theorem and game dynamics

full rationale

The paper models risk aversion and probability sensitivity via standard Cumulative Prospect Theory parameters inside a Bayesian pursuit-evasion differential game, then derives sufficient capturability conditions directly from the system dynamics and those parameters. Existence of CPT-Nash equilibria is asserted by direct invocation of Brouwer's fixed-point theorem on the strategy space. No step reduces a derived quantity to a fitted input by construction, no self-citation supplies a load-bearing uniqueness result, and no ansatz is smuggled via prior work by the same authors. The derivation chain therefore remains self-contained against external mathematical facts and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical tools and the CPT modeling choice. No new physical entities are introduced. The irrational parameters function as modeling choices rather than data-fitted constants in the abstract description.

free parameters (1)

CPT risk aversion and probability sensitivity parameters
These parameters encode irrational perceptions and are incorporated into the game payoffs; their specific values are not derived from first principles in the abstract.

axioms (2)

standard math Brouwer's fixed-point theorem applies to the compact convex strategy space of the CPT-modified game
Invoked to establish existence of CPT-Nash equilibria.
domain assumption The system dynamics and uncertainty structure permit formulation as a Bayesian differential game
Required to embed CPT into the pursuit-evasion setting.

pith-pipeline@v0.9.0 · 5461 in / 1294 out tokens · 30713 ms · 2026-05-14T22:13:22.187758+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the irrational perceptions of risk aversion and probability sensitivity are modeled ... by using Cumulative Prospect Theory (CPT) approach ... existence of CPT-Nash equilibria is rigorously analyzed by invoking Brouwer’s fixed-point theorem
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

U⁺₁(J) ≜ (−(J−Jr))^{α₁} ... w⁺₁(p₁) ≜ exp{−(−log(p₁))^{γ₁}} ... Ψ_i ≜ (−1)^i (α_i ∫ ... )

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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