How to detect a salami slicer: a stochastic controller-stopper game with unknown competition

TL;DR

This paper models a complex stochastic game involving control, stopping, and asymmetric information to detect fraud, deriving Nash equilibria and exploring strategies under uncertainty and strategic interaction.

Contribution

It introduces a novel non-zero-sum game framework combining filtering, control, stopping, and asymmetric information, with explicit equilibrium solutions.

Findings

Existence of pure strategy equilibria for certain parameters

Existence of randomized stopping equilibria for other parameters

Application to fraud detection scenarios

Abstract

We consider a stochastic game of control and stopping specified in terms of a process $X_t=-\theta \Lambda_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $\theta$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $\Lambda$ is a non-decreasing process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $\theta$ or $\Lambda$ directly, but can merely observe the path of the process $X$ and may choose a stopping rule $\tau$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is the victim of fraud and operates in this sense under unknown competition. Player 2 can observe both $\theta$ and $W$ and seeks to choose the fraud strategy $\Lambda$ that maximizes the expected discounted amount \[{\mathbb E} \left [\theta\int _0^{\tau} e^{-rs} d\Lambda_s \right ],\] whereas Player 1 seeks to choose the stopping strategy $\tau$ so as to minimize the expected discounted cost \[{\mathbb E} \left [\theta\int _0^{\tau} e^{-rs} d\Lambda_s + e^{-r\tau}M{\mathbb I}_{\{\tau<\infty\}} \right ].\] This non-zero-sum game appears to be novel and is motivated by applications in fraud detection; it combines filtering (detection), non-singular control, stopping, strategic features (games) and asymmetric information. We derive Nash equilibria for this game; for some parameter values we find an equilibrium in pure strategies, and for other parameter values we find an equilibrium by allowing for randomized stopping strategies.