\sc lf}}}(n)$ corresponds to a Nash equilibrium of the non-failure Hotelling game $H(n)$ (Theorem \[thm:equiv\]).

The second variant we consider assumes a reliable environment, but failure prone players. Each player (independently) has some probability of being removed from the game. This version of the game (with $n$ players) is referred to as the [*Player Failure Hotelling game*]{}, denoted $H_{{\mbox{\small \sc pf}}}(n)$. We show that for $n\ge 3$ players, if there are at least three players, then the game admits no Nash equilibrium (Theorem \[thm:crash\]).

#### Related Work.

Fault tolerant facility location problems have been studied extensively from an optimization perspective in the operations research and computer science communities [@chechik2014robust; @chechik2015fault; @khuller2000fault; @snyder2016or; @sviridenko2002improved; @swamy2008fault]. However, relatively little work has been done from a game theoretic approach. Two recent papers do consider fault tolerant location games. Wang and Ouyang studied a failure prone competitive location game in a two dimensional environment [@wang2013continuum]. However, they studied a variant in which two players position several facilities each, and did not extend the model to a greater number of players. Zhang et al. considered a discrete competitive location model in which there are finitely many clients and finitely many potential facility locations [@zhang2016competitive]. Their 