Competitive Search with a Faulty Satnav (GPS): When Probability Matching is Rational

TL;DR

This paper models a competitive search game on star graphs with probabilistic GPS guidance, showing that probability matching can be a rational strategy in certain search scenarios.

Contribution

It introduces a game-theoretic model demonstrating that probability matching is rational, deriving a unique equilibrium trust probability in a multi-searcher setting.

Findings

Existence of a unique symmetric equilibrium trust probability q.

q increases with GPS accuracy p and number of leaves k, decreases with number of searchers n.

In the limit of large n, q equals p, confirming probability matching as rational.

Abstract

A divisible treasure is located at a node $H$ of a network. From a given start node a group of $n$ Searchers each seek to reach $H$ first, dividing the treasure equally with the other first arrivers. This type of search game is called competitive search, where the conflict is not between hider and searcher but between searchers. Examples are search for oil deposits or for a pilot downed over enemy territory. In our model, the Searchers have a common Satnav (GPS) which points to $H$ at each branch node with a known probability $p<1$ and each Searcher chooses the probability $q$ with which they follow the pointer. We consider a family of star graphs where the Searchers start at the center and $H$ lies at one of the $k$ leaf nodes. We show that for all parameter values $n,k,p,$ there is a unique trust probability $q$ which forms a symmetric equilibrium. The equilibrium $q$ is increasing in $p,$ decreasing in $n$ and increasing in $k$. Furthemore for fixed $k$ and $p$ we have $q$ equal to $p$ in the limit of $n$ tending to infinity. This last fact is a new example where what is known in behavioural science as probability matching is in fact rational.