We consider the well-studied game-theoretic version of machine scheduling in which jobs correspond to self-interested users and machines correspond to resources. Here each user chooses a machine trying to minimize her own cost, and such selfish behavior typically results in some equilibrium which is not globally optimal: An equilibrium is an allocation where no user can reduce her own cost by moving to another machine, which in general need not minimize the makespan, i.e., the maximum load over the machines. We provide tight bounds on two well-studied notions in algorithmic game theory, namely, the price of anarchy and the strong price of anarchy on machine scheduling setting which lies in between the related and the unrelated machine case. Both notions study the social cost (makespan) of the worst equilibrium compared to the optimum, with the strong price of anarchy restricting to a stronger form of equilibria. Our results extend a prior study comparing the price of anarchy to the strong price of anarchy for two related machines (Epstein, Acta Informatica 2010), thus providing further insights on the relation between these concepts. Our exact bounds give a qualitative and quantitative comparison between the two models. The bounds also show that the setting is indeed easier than the two unrelated machines: In the latter, the strong price of anarchy is @math , while in ours it is strictly smaller.
The price of anarchy is for , and grows otherwise. Specifically, @math and @math @cite_16 , while @math @cite_14 . The case of a is of particular interest. For and machines, @math and @math @cite_15 , respectively. For machines, exact bounds as a function of the @math on both @math and @math are given in @cite_19 . The price of anarchy bounds are similar to related machines: @math and @math @cite_0 , where the analysis of @math is also in @cite_17 . The price of anarchy and the strong price of anarchy for pure equilibria are and bounded by a : @math @cite_12 , where the upper and lower bounds on @math can be deduced from @cite_5 and @cite_9 , respectively. Finally, @math @cite_6 @cite_16 @cite_4 . For further results on other problems and variants of these equilibrium concepts we refer the reader to e.g. @cite_8 @cite_20 @cite_1 @cite_10 and references therein.
@cite_14 @cite_4 @cite_8 @cite_10 @cite_9 @cite_1 @cite_6 @cite_0 @cite_19 @cite_5 @cite_15 @cite_16 @cite_20 @cite_12 @cite_17
2294025081 2133243407 1967821428 2123036301
The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, and to sequences of outcomes generated by natural experimentation strategies, such as successive best responses and simultaneous regret-minimization. We establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a “canonical sufficient condition” for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument. Second, we prove an “extension theorem”: every bound on the price of anarchy that is 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 every outcome sequence generated by no-regret learners. Smoothness arguments also have automatic implications for the inefficiency of approximate equilibria, for bicriteria bounds, and, under additional assumptions, for polynomial-length best-response sequences. Third, we prove that in congestion games, smoothness arguments are “complete” in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce optimal worst-case upper bounds on the price of anarchy. 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. In sponsored search auctions, advertisers compete for a number of available advertisement slots of different quality. The auctioneer decides the allocation of advertisers to slots using bids provided by them. Since the advertisers may act strategically and submit their bids in order to maximize their individual objectives, such an auction naturally defines a strategic game among the advertisers. In order to quantify the efficiency of outcomes in generalized second price auctions, we study the corresponding games and present new bounds on their price of anarchy, improving the recent results of Paes Leme and Tardos [16] and Lucier and Paes Leme [13]. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria. Given the existing lower bounds, this bound denotes that the number of advertisers has almost no impact on the price of anarchy. The proof exploits the equilibrium conditions developed in [16] and follows by a detailed reasoning about the structure of equilibria and a novel relation of the price of anarchy to the objective value of a compact mathematical program. For more general equilibrium classes (i.e., mixed Nash, correlated, and coarse correlated equilibria), we present an upper bound of 2.310 on the price of anarchy. We also consider the setting where advertisers have incomplete information about their competitors and prove a price of anarchy upper bound of 3.037 over Bayes-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments. In situations without central coordination, the price of anarchy relates the quality of any Nash equilibrium to the quality of a global optimum. Instead of assuming that all players choose their actions simultaneously, we consider games where players choose their actions sequentially. The sequential price of anarchy, recently introduced by Paes Leme, Syrgkanis, and Tardos, relates the quality of any subgame perfect equilibrium to the quality of a global optimum. The effect of sequential decision making on the quality of equilibria, depends on the specific game under consideration. We analyze the sequential price of anarchy for atomic congestion games with affine cost functions. We derive several lower and upper bounds, showing that sequential decisions mitigate the worst case outcomes known for the classical price of anarchy. Next to tight bounds on the sequential price of anarchy, a methodological contribution of our work is, among other things, a “factor revealing” linear programming approach we use to solve the case of three players.