On the Existence of Nash Equilibria in Strategic Search Games

We consider a general multi-agent framework in which a set of n agents are roaming a network where m valuable and sharable goods (or resources or services or information) are hidden in m different vertices of the network. We analyze several strategic situations that arise in this setting by means of game theory. To do so we introduce a class of strategic search games. In such a game each agent has to select a simple path in the network that starts from a predetermined set of initial vertices. Depending on how the value of the retrieved goods is splitted among the agents we consider two game types: finders-share in which the agents that find a good split among them the corresponding benefit and firsts-share in which only the agents that first find a good share the corresponding benefit. We show that finders-share games always have pure Nash equilibria (pne). For obtaining this result we introduce the notion of Nash preserving reduction between strategic games. We show that finders-share games are Nash reducible to single-source network congestion games. This is done through a series of Nash preserving reductions. For firsts-share games we show the existence of games with and without pne. Furthermore we identify some graph families in which the firsts-share game has always a pne that is computable in polynomial time.