George Pierrakos

On optimal single-item auctions

We revisit the problem of designing the profit-maximizing single-item auction, solved by Myerson in his seminal paper for the case in which bidder valuations are independently distributed. We focus on general joint distributions, seeking the optimal deterministic incentive compatible auction. We give a geometric characterization of the optimal auction through a duality theorem, resulting in an efficient algorithm for finding the optimal deterministic auction in the two-bidder case and an inapproximability result for three or more bidders.

On learning algorithms for nash equilibria

Can learning algorithms find a Nash equilibrium? This is a natural question for several reasons. Learning algorithms resemble the behavior of players in many naturally arising games, and thus results on the convergence or nonconvergence properties of such dynamics may inform our understanding of the applicability of Nash equilibria as a plausible solution concept in some settings. A second reason for asking this question is in the hope of being able to prove an impossibility result, not dependent on complexity assumptions, for computing Nash equilibria via a restricted class of reasonable algorithms. In this work, we begin to answer this question by considering the dynamics of the standard multiplicative weights update learning algorithms (which are known to converge to a Nash equilibrium for zero-sum games). We revisit a 3×3 game defined by Shapley [10] in the 1950s in order to establish that fictitious play does not converge in general games. For this simple game, we show via a potential function argument that in a variety of settings the multiplicative updates algorithm impressively fails to find the unique Nash equilibrium, in that the cumulative distributions of players produced by learning dynamics actually drift away from the equilibrium.

Efficiency-revenue trade-offs in auctions

When agents with independent priors bid for a single item, Myersons optimal auction maximizes expected revenue, whereas Vickreys second-price auction optimizes social welfare. We address the natural question of trade-offs between the two criteria, that is, auctions that optimize, say, revenue under the constraint that the welfare is above a given level. If one allows for randomized mechanisms, it is easy to see that there are polynomial-time mechanisms that achieve any point in the trade-off (the Pareto curve) between revenue and welfare. We investigate whether one can achieve the same guarantees using deterministic mechanisms. We provide a negative answer to this question by showing that this is a (weakly) NP-hard problem. On the positive side, we provide polynomial-time deterministic mechanisms that approximate with arbitrary precision any point of the trade-off between these two fundamental objectives for the case of two bidders, even when the valuations are correlated arbitrarily. The major problem left open by our work is whether there is such an algorithm for three or more bidders with independent valuation distributions.

On the Competitive Ratio of Online Sampling Auctions

We study online profit-maximizing auctions for digital goods with adversarial bid selection and uniformly random arrivals; in this sense, our model lies at the intersection of prior-free mechanism design and secretary problems. Our goal is to design auctions that are constant competitive with F(2). We give a generic reduction that transforms any offline auction to an online one with only a loss of a factor of 2 in the competitive ratio. We also present some natural auctions, both randomized and deterministic, and study their competitive ratio. Our analysis reveals some interesting connections of one of these auctions with RSOP.

On a noncooperative model for wavelength assignment in multifiber optical networks

We propose and investigate Selfish Path MultiColoring games as a natural model for noncooperative wavelength assignment in multifiber optical networks. In this setting, we view the wavelength assignment process as a strategic game in which each communication request selfishly chooses a wavelength in an effort to minimize the maximum congestion that it encounters on the chosen wavelength. We measure the cost of a certain wavelength assignment as the maximum, among all physical links, number of parallel fibers employed by this assignment. We start by settling questions related to the existence and computation of and convergence to pure Nash equilibria in these games. Our main contribution is a thorough analysis of the price of anarchy of such games, that is, the worst-case ratio between the cost of a Nash equilibrium and the optimal cost. We first provide upper bounds on the price of anarchy for games defined on general network topologies. Along the way, we obtain an upper bound of 2 for games defined on star networks. We next show that our bounds are tight even in the case of tree networks of maximum degree 3, leading to nonconstant price of anarchy for such topologies. In contrast, for network topologies of maximum degree 2, the quality of the solutions obtained by selfish wavelength assignment is much more satisfactory: We prove that the price of anarchy is bounded by 4 for a large class of practically interesting games defined on ring networks.

On the competitive ratio of online sampling auctions

We study online profit-maximizing auctions for digital goods with adversarial bid selection and uniformly random arrivals. Our goal is to design auctions that are constant competitive with F(2); in this sense our model lies at the intersection of prior-free mechanism design and secretary problems. We first give a generic reduction that transforms any offline auction to an online one, with only a loss of a factor of 2 in the competitive ratio; we then present some natural auctions, both randomized and deterministic, and study their competitive ratio; our analysis reveals some interesting connections of one of these auctions with RSOP, which we further investigate in our final section.

Optimal deterministic auctions with correlated priors

We revisit the problem of designing the profit-maximizing single-item auction, solved by Myerson in his seminal paper for the case in which bidder valuations are independently distributed. We focus on general joint distributions, seeking the optimal deterministic incentive compatible auction. We give a geometric characterization of the optimal auction, resulting in a duality theorem and an efficient algorithm for finding the optimal deterministic auction in the two-bidder case and an NP-completeness result for three or more bidders.

Biobjective Online Bipartite Matching

Online Matching has been a problem of considerable interest recently, particularly due to its applicability in Online Ad Allocation. In practice, there are usually multiple objectives which need to be simultaneously optimized, e.g., revenue and quality. We capture this motivation by introducing the problem of Biobjective Online Bipartite Matching. This is a strict generalization of the standard setting. In our problem, the graph has edges of two colors, Red and Blue. The goal is to find a single matching that contains a large number of edges of each color.

Selfish Resource Allocation in Optical Networks

We introduce Colored Resource Allocation Games as a new model for selfish routing and wavelength assignment in multifiber all-optical networks. Colored Resource Allocation Games are a generalization of congestion and bottleneck games where players have their strategies in multiple copies (colors). We focus on two main subclasses of these games depending on the player cost: in Colored Congestion Games the player cost is the sum of latencies of the resources allocated to the player, while in Colored Bottleneck Games the player cost is the maximum of these latencies. We investigate the pure price of anarchy for three different social cost functions and prove tight bounds for each separate case. We first consider a social cost function which is particularly meaningful in the setting of multifiber all-optical networks, where it captures the objective of fiber cost minimization. Additionally, we consider the two usual social cost functions (maximum and average player cost) and obtain improved bounds that could not have been derived using earlier results for the standard models for congestion and bottleneck games.

On a Non-cooperative Model for Wavelength Assignment in Multifiber Optical Networks

We study path multicoloring games that describe situations in which selfish entities possess communication requests in a multifiber all-optical network. Each player is charged according to the maximum fiber multiplicity that her color (wavelength) choice incurs and the social cost is the maximum player cost. We investigate the price of anarchy of such games and provide two different upper bounds for general graphs--namely the number of wavelengths and the minimum length of a path of maximum disutility, over all worst-case Nash Equilibria--as well as matching lower bounds which hold even for trees; as a corollary we obtain that the price of anarchy in stars is exactly 2. We also prove constant bounds for the price of anarchy in chains and rings in which the number of wavelengths is relatively small compared to the load of the network; in the opposite case we show that the price of anarchy is unbounded.