  This paper gives a complete analysis of worst-case equilibria for various
versions of weighted congestion games with two players and affine cost
functions. The results are exact price of anarchy bounds which are parametric
in the weights of the two players, and establish exactly how the primitives of
the game enter into the quality of equilibria. Interestingly, some of the
worst-cases are attained when the players' weights only differ slightly. Our
findings also show that sequential play improves the price of anarchy in all
cases, however, this effect vanishes with an increasing difference in the
players' weights. Methodologically, we obtain exact price of anarchy bounds by
a duality based proof mechanism, based on a compact linear programming
formulation that computes worst-case instances. This mechanism yields
duality-based optimality certificates which can eventually be turned into
purely algebraic proofs.
