We consider the price of anarchy of pure Nash equilibria in congestion games with linear latency functions. For asymmetric games, the price of anarchy of maximum social cost is Θ( √ N ), where N is the number of players. For all other cases of symmetric or asymmetric games and for both maximum and average social cost, the price of anarchy is 5/2. We extend the results to latency functions that are polynomials of bounded degree. We also extend some of the results to mixed Nash equilibria.

We study the following Bayesian setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both m and n.Combinatorial Auctions. In a combinatorial auction m items M = {1, . . . , m} are offered for sale to n bidders N = {1, . . . , n}. Each bidder i has a valuation function (or valuation, in short) v i that assigns a non-negative real number to every subset of the items. v i expresses i's preferences over bundles of items. The value v i (S) can be thought of as specifying i's maximum willingness to pay for S. Two standard assumptions are made on each v i : v i (∅) = 0 (normalization), and v i (S) ≤ v i (T ) for every two bundles S ⊆ T (monotonicity). The objective is to find a partition of the items among the bidders S 1 , . . . , S n (where S i ∩ S j = ∅ for all i = j) such that the social welfare Σ i v i (S i ) is maximized.The interplay between selfishness and computational optimization in combinatorial auctions is well-studied. Each of these aspects alone can be handled in a satisfactory way: The celebrated VCG mechanisms [19,3,10] motivate agents to truthfully report their private information, and optimize the social-welfare. The caveat is that this may take exponential time [15,16] (in the natural parameters of the problem m and n). On the other hand, if we disregard strategic issues, it is possible to obtain good approximations to the optimal social-welfare

We introduce the notion of coordination mechanisms to improve the performance in systems with independent selfish and non-colluding agents. The quality of a coordination mechanism is measured by its price of anarchy-the worst-case performance of a Nash equilibrium over the (centrally controlled) social optimum. We give upper and lower bounds for the price of anarchy for selfish task allocation and congestion games.

We consider the price of stability for Nash and correlated equilibria of linear congestion games. The price of stability is the optimistic price of anarchy, the ratio of the cost of the best Nash or correlated equilibrium over the social optimum. We show that for the sum social cost, which corresponds to the average cost of the players, every linear congestion game has Nash and correlated price of stability at most 1.6. We also give an almost matching lower bound of 1 + √ 3/3 = 1.577. We also consider the price of anarchy of correlated equilibria. We extend some of the results in [2,4] to correlated equilibria and show that for the sum social cost, the price of anarchy is 2.5. The same bound holds for symmetric games as well. This matches the lower bounds given in [2,4] for pure Nash equilibria. We also extend the results in [2] for weighted congestion games to correlated equilibria. Specifically, we show that when the social cost is the total latency, the price of anarchy is (3 + √ 5)/2 = 2.618.

We study the mechanism design problem of scheduling tasks on n unrelated machines in which the machines are the players of the mechanism. The problem was proposed and studied in the seminal paper of Nisan and Ronen on algorithmic mechanism design, where it was shown that the approximation ratio of mechanisms is between 2 and n. We improve the lower bound to 1 + √ 2 for 3 or more machines.