Krishnendu Chatterjee, Thomas A. Henzinger, and Marcin Jurdzinski
"Games with Secure Equilibria"
In 2-player non-zero-sum games, Nash equilibria capture the options
for rational behavior if each player attempts to maximize her payoff.
In contrast to classical game theory, we consider lexicographic
objectives: first, each player tries to maximize her own payoff, and
then, the player tries to minimize the opponent's payoff. Such
objectives arise naturally in the verification of systems with
multiple components. There, instead of proving that each component
satisfies its specification no matter how the other components behave,
it often suffices to prove that each component satisfies its
specification provided that the other components satisfy their
specifications. We say that a Nash equilibrium is {\em secure\/} if
it is an equilibrium with respect to the lexicographic objectives of
both players.
We prove that in graph games with Borel winning conditions, which
include the games that arise in verification, there may be several
Nash equilibria, but there is always a unique maximal payoff profile
of a secure equilibrium. We show how this equilibrium can be
computed in the case of $\omega$-regular winning conditions, and we
characterize the memory requirements of strategies that
achieve the equilibrium.