  The notion of fault tolerant Nash equilibria has been introduced as a way of
studying the robustness of Nash equilibria. Under this notion, a fixed number
of players are allowed to exhibit faulty behavior in which they may deviate
arbitrarily from an equilibrium strategy. A Nash equilibrium in a game with $N$
players is said to be $\alpha$-tolerant if no non-faulty user wants to deviate
from an equilibrium strategy as long as $N-\alpha-1$ other players are playing
the equilibrium strategies, i.e., it is robust to deviations from rationality
by $\alpha$ faulty players. In prior work, $\alpha$-tolerance has been largely
viewed as a property of a given Nash equilibria. Here, instead we consider
following Nash's approach for showing the existence of equilibria, namely,
through the use of best response correspondences and fixed-point arguments. In
this manner, we provide sufficient conditions for the existence an
$\alpha$-tolerant equilibrium. This involves first defining an
$\alpha$-tolerant best response correspondence. Given a strategy profile of
non-faulty agents, this correspondence contains strategies for a non-faulty
player that are a best response given any strategy profile of the faulty
players. We prove that if this correspondence is non-empty, then it is
upper-hemi-continuous. This enables us to apply Kakutani's fixed-point theorem
and argue that if this correspondence is non-empty for every strategy profile
of the non-faulty players then there exists an $\alpha$-tolerant equilibrium.
However, we also illustrate by examples, that in many games this best response
correspondence will be empty for some strategy profiles even though
$\alpha$-tolerant equilibira still exist.
