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.

翻译：一笔可分宝藏位于网络节点$H$处。从给定的起始节点出发，由$n$名搜索者组成的团队各自竞相率先到达$H$，并与同时到达者平分宝藏。这种搜索博弈被称为竞争搜索，其冲突不在于藏匿者与搜索者之间，而在于搜索者之间。例如，寻找石油矿床或搜寻坠入敌境的飞行员。在我们的模型中，搜索者共享一台指向$H$的卫星导航，在每一个分支节点处以已知概率$p<1$指示方向，而每位搜索者选择以概率$q$遵循该指示。我们考虑一类星形图，其中搜索者从中心出发，$H$位于$k$个叶节点之一。我们证明，对所有参数值$n,k,p$，存在唯一的信任概率$q$构成对称均衡。均衡$q$随$p$递增、随$n$递减、随$k$递增。此外，对于固定的$k$和$p$，当$n$趋于无穷时，$q$等于$p$。这一结果提供了一个新实例，表明行为科学中所谓的概率匹配在此情境下实际上是理性的。