We consider a general class of Bayesian Games where each players utility depends on his type (possibly multidimensional) and on the strategy profile and where players' types are distributed independently. We show that if their full information version for any fixed instance of the type profile is a smooth game then the Price of Anarchy bound implied by the smoothness property, carries over to the Bayes-Nash Price of Anarchy. We show how some proofs from the literature (item bidding auctions, greedy auctions) can be cast as smoothness proofs or be simplified using smoothness. For first price item bidding with fractionally subadditive bidders we actually manage to improve by much the existing result Hassidim2011a from 4 to @math . This also shows a very interesting separation between first and second price item bidding since second price item bidding has PoA at least 2 even under complete information. For a larger class of Bayesian Games where the strategy space of a player also changes with his type we are able to show that a slightly stronger definition of smoothness also implies a Bayes-Nash PoA bound. We show how weighted congestion games actually satisfy this stronger definition of smoothness. This allows us to show that the inefficiency bounds of weighted congestion games known in the literature carry over to incomplete versions where the weights of the players are private information. We also show how an incomplete version of a natural class of monotone valid utility games, called effort market games are universally @math -smooth. Hence, we show that incomplete versions of effort market games where the abilities of the players and their budgets are private information has Bayes-Nash PoA at most 2.
There has been a long line of research on quantifying inefficiency of equilibria starting from @cite_4 who introduced the notion of the price of anarchy. A recent work by Roughgarden @cite_5 managed to unify several of these results under a proof framework called smoothness and also showed that such inefficiency proofs also carry over to inefficiency of coarse correlated equilibria. Moreover, he showed that such techniques give tight results for the well-studied class of congestion games. Later, @cite_0 also showed that it produces tight results for the larger class of weighted congestion games. Another recent work by Schoppman and Roughgarden @cite_6 copes with games with continuous strategy spaces and shows how the smoothness framework should be adapted for such games to produce tighter results. The introduce the new notion of local smoothness for such games and showed that if an inefficiency upper bound proof lies in this framework then it also carries over to correlated equilibria.
@cite_0 @cite_5 @cite_4 @cite_6
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We characterize the price of anarchy in weighted congestion games, as a function of the allowable resource cost functions. Our results provide as thorough an understanding of this quantity as is already known for nonatomic and unweighted congestion games, and take the form of universal (cost function-independent) worst-case examples. One noteworthy byproduct of our proofs is the fact that weighted congestion games are "tight", which implies that the worst-case price of anarchy with respect to pure Nash, mixed Nash, correlated, and coarse correlated equilibria are always equal (under mild conditions on the allowable cost functions). Another is the fact that, like nonatomic but unlike atomic (unweighted) congestion games, weighted congestion games with trivial structure already realize the worst-case POA, at least for polynomial cost functions. We also prove a new result about unweighted congestion games: the worst-case price of anarchy in symmetric games is, as the number of players goes to infinity, as large as in their more general asymmetric counterparts. The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA. We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of "local smoothness", which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games.