The de Finetti problem with unknown competition

TL;DR

This paper extends the classical de Finetti resource extraction problem to include unknown competition modeled as a stochastic game with incomplete information, providing a Nash equilibrium characterized by singular strategies and a reflected process.

Contribution

It introduces a novel stochastic game framework for resource extraction with hidden competition and derives a Nash equilibrium using singular strategies and filtering techniques.

Findings

Nash equilibrium explicitly characterized in terms of the single-player de Finetti problem.

Equilibrium strategies involve randomised stopping times for the hidden competitor.

The model captures the dynamics of resource levels and hidden competition through a reflected process.

Abstract

We consider a resource extraction problem which extends the classical de Finetti problem for a Wiener process to include the case when a competitor, who is equipped with the possibility to extract all the remaining resources in one piece, may exist; we interpret this unknown competition as the agent being subject to possible fraud. This situation is modelled as a controller-and-stopper non-zero-sum stochastic game with incomplete information. In order to allow the fraudster to hide his existence, we consider strategies where his action time is randomised. Under these conditions, we provide a Nash equilibrium which is fully described in terms of the corresponding single-player de Finetti problem. In this equilibrium, the agent and the fraudster use singular strategies in such a way that a two-dimensional process, which represents available resources and the filtering estimate of active competition, reflects in a specific direction along a given boundary.