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Barriers to nearoptimal equilibria
 IN PROCEEDINGS OF THE 55TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
"... This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worstcase quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each moti ..."
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This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worstcase quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games; welfaremaximization in combinatorial auctions with “blackbox” bidder valuations; and welfaremaximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worstcase price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players’ utility functions. For this reason, our lower bounds are particularly significant for problems of game design — ranging from the design of simple combinatorial auctions to the existence of effective tolls for routing networks — where the goal is to design a game that has only nearoptimal equilibria. For example, our results imply that the simultaneous firstprice auction format is optimal among all “simple combinatorial auctions” in several settings.
CS369E: Communication Complexity (for Algorithm Designers) Lecture #7: Lower Bounds in Algorithmic Game Theory∗
, 2015
"... This lecture explains some applications of communication complexity to proving lower bounds in algorithmic game theory (AGT), at the border of computer science and economics. In AGT, the natural description size of an object is often exponential in a parameter of interest, and the goal is to perfor ..."
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This lecture explains some applications of communication complexity to proving lower bounds in algorithmic game theory (AGT), at the border of computer science and economics. In AGT, the natural description size of an object is often exponential in a parameter of interest, and the goal is to perform nontrivial computations in time polynomial in the parameter (i.e., logarithmic in the description size). As we know, communication complexity is a great tool for understanding when nontrivial computations require looking at most of the input. 2 The Welfare Maximization Problem The focus of this lecture is the following optimization problem, which has been studied in AGT more than any other. 1. There are k players. 2. There is a set M of m items. 3. Each player i has a valuation vi: 2 M → R+. The number vi(T) indicates i’s value, or willingness to pay, for the items T ⊆ M. The valuation is the private input of player i — i knows vi but none of the other vj’s. We assume that vi(∅) = 0 and that the valuations are monotone, meaning vi(S) ≤ vi(T) whenever S ⊆ T. To avoid bit complexity issues, we’ll also assume that all of the vi(T)’s are integers with description length polynomial in k and m. ∗ c©2015, Tim Roughgarden.