We present a new class of vertex cover and set cover games. The price of
anarchy bounds match the best known constant factor approximation guarantees
for the centralized optimization problems for linear and also for submodular
costs -- in contrast to all previously studied covering games, where the price
of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In
particular, we describe a vertex cover game with a price of anarchy of 2. The
rules of the games capture the structure of the linear programming relaxations
of the underlying optimization problems, and our bounds are established by
analyzing these relaxations. Furthermore, for linear costs we exhibit linear
time best response dynamics that converge to these almost optimal Nash
equilibria. These dynamics mimic the classical greedy approximation algorithm
of Bar-Yehuda and Even [3]
