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

2026-05-19

AI Summary

This study addresses a competitive search game on a star network, where multiple searchers simultaneously depart from the central node to locate a treasure hidden at one of the leaf nodes. The searchers share an unreliable navigation device that points toward the correct leaf with probability $ p $, and each independently chooses a trust probability $ q $ to follow its guidance. Through game-theoretic modeling and analysis of symmetric Nash equilibria, the paper establishes that for any number of searchers $ n $, leaf nodes $ k $, and device reliability $ p $, there exists a unique symmetric equilibrium trust probability $ q $. This equilibrium $ q $ increases with both $ p $ and $ k $, decreases with $ n $, and converges to $ p $ as $ n \to \infty $. These findings demonstrate that probabilistic matching emerges as a rational equilibrium strategy in the large-population limit, offering new insights into competitive search and decision-making under uncertainty.

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.

Problem

• competitive search
• faulty GPS
• probability matching
• search game
• symmetric equilibrium

Innovation

• competitive search
• probability matching
• faulty GPS
• symmetric equilibrium
• search game