## Abstract

Bounding the price of anarchy (PoA), which quantifies the degradation in the quality of outcomes in a (pure) Nash equilibrium of a game, is one of the fundamental questions in computational game theory. However, for a large class of games, a pure NE may not always exist and hence a natural question to pursue is to quantify the inefficiency for weaker notions of equilibrium such as mixed Nash equilibrium, correlated equilibrium or coarse correlated equilibrium, all of which are known to exist for finite games. Several techniques have been developed for bounding the price of anarchy, yet, only a handful of them are applicable for proving the PoA bounds for general equilibrium concepts. Most notable among such techniques is Roughgarden's elegant smoothness framework, which led to the concept of robust price of anarchy. The term refers to the inefficiency bounds applicable to general equilibrium notions such as coarse correlated equilibrium.

In this paper, we develop a new framework based on LP and Fenchel duality for bounding the robust price of anarchy for a large class of games. We use our framework to give the first PoA bounds for temporal routing games on graphs and energy minimization games in machine scheduling. Most notably, we present the first coordination mechanisms with bounded PoA for temporal routing over general graphs, show a related lowerbound result, and an improved bound on the price of stability for this game. Previously, coordination mechanisms with bounded PoA were only known for restricted classes of graphs such as trees or parallel edges. Furthermore, we demonstrate the wide applicability of our framework by giving new proofs of the PoA bounds for three classical games – weighted affine congestion games, competitive facility location games and simultaneous second price auctions. Our price anarchy bounds for these games match the ones known in the literature or obtained using the smoothness framework.

All our proofs use the following technique: we first show that for a wide class of games, one can formulate the underlying optimization problem as a linear (or convex) program such that the (Fenchel) dual of the relaxation encodes the equilibrium condition. Further, the dual program has a specific structure with variables for players and resources, which can be naturally interpreted as the cost incurred by the players and the congestion of the resource in an equilibrium outcome. This lets us argue that our definition of dual variables satisfy the dual constraints and using the weak duality theorem we establish the PoA bounds.