Discrete Strategies in Keyword Auctions and Their Inefficiency for Locally Aware Bidders

We study formally discrete bidding strategies for the game induced by the Generalized Second Price keyword auction mechanism. Such strategies have seen experimental evaluation in the recent literature as parts of iterative best response procedures, which have been shown not to converge. We give a detailed definition of iterative best response under these strategies and, under appropriate discretization of the players' strategy spaces we find that the discretized configurations space {\em contains} socially optimal pure Nash equilibria. We cast the strategies under a new light, by studying their performance for bidders that act based on local information; we prove bounds for the worst-case ratio of the social welfare of locally stable configurations, relative to the socially optimum welfare

Labeled Traveling Salesman Problems: Complexity and approximation

We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a $1/2$-approximation algorithm based on local improvements and show that the problem is APX-hard. For the minimization version, we show that it is not approximable within $n^{1-\epsilon}$ for any fixed $\epsilon>0$. When every color appears in the graph at most $r$ times and $r$ is an increasing function of $n$, the problem is shown not to be approximable within factor $O(r^{1-\epsilon})$. For fixed constant $r$ we analyze a polynomial-time $(r +H_r)/2$ approximation algorithm, where $H_r$ is the $r$-th harmonic number, and prove APX-hardness for $r = 2$. For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances

On the inefficiency of equilibria in linear bottleneck congestion games

We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments

Inefficiency of Standard Multi-unit Auctions

We study two standard multi-unit auction formats for allocating multiple units of a single good to multi-demand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to be paid per unit. Variants of both formats find applications ranging from the allocation of state bonds to investors, to online sales over the internet. For these formats, we consider two bidding interfaces: (i) standard bidding, which is most prevalent in the scientific literature, and (ii) uniform bidding, which is more popular in practice. In this work, we evaluate the economic inefficiency of both multi-unit auction formats for both bidding interfaces, by means of upper and lower bounds on the Price of Anarchy for pure Nash equilibria and mixed Bayes-Nash equilibria. Our developments improve significantly upon bounds that have been obtained recently for submodular valuation functions. Also, for the first time, we consider bidders with subadditive valuation functions under these auction formats. Our results signify near-efficiency of these auctions, which provides further justification for their use in practice