The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

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Handbook of Computational Social Choice

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.

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Intrinsic robustness of the price of anarchy

We consider a bilevel model where the leader wants to maximize revenues from a taxation scheme, while the follower rationally reacts to those tax levels. We focus our attention on the special case of a toll-setting problem defined on a multicommodity transportation network. We show that the general problem is NP-complete, while particular instances are polynomially solvable. Numerical examples are given.

A Bilevel Model of Taxation and its Application to Optimal Highway Pricing

We present a cross-layer modeling and design approach for multigigabit indoor wireless personal area networks (WPANs) utilizing the unlicensed millimeter (mm) wave spectrum in the 60 GHz band. Our approach accounts for the following two characteristics that sharply distinguish mm wave networking from that at lower carrier frequencies. First, mm wave links are inherently directional: directivity is required to overcome the higher path loss at smaller wavelengths, and it is feasible with compact, low-cost circuit board antenna arrays. Second, indoor mm wave links are highly susceptible to blockage because of the limited ability to diffract around obstacles such as the human body and furniture. We develop a diffraction-based model to determine network link connectivity as a function of the locations of stationary and moving obstacles. For a centralized WPAN controlled by an access point, it is shown that multihop communication, with the introduction of a small number of relay nodes, is effective in maintaining network connectivity in scenarios where single-hop communication would suffer unacceptable outages. The proposed multihop MAC protocol accounts for the fact that every link in the WPAN is highly directional, and is shown, using packet level simulations, to maintain high network utilization with low overhead.

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Blockage and directivity in 60 GHz wireless personal area networks: from cross-layer model to multihop MAC design

Bin packing approximation algorithms: Survey and classification

We study geometric versions of the min-size k-clustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications We prove that the problem can be solved in polynomial time when the points to be clustered are located on a line For Euclidean spaces of higher dimensions, we show that the problem is NP-hard and present polynomial time approximation schemes The latter result yields an improved approximation algorithm for the related problem of k-clustering to minimize the sum of cluster diameters

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Geometric clustering to minimize the sum of cluster sizes

We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, ie, contains a Hamiltonian path For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation

Almost Envy-Free Allocations with Connected Bundles.

In the non cooperative version of the classical minimum bin packing problem, an item is charged a cost according to the percentage of the used bin space it requires. We study the game induced by the selfish behavior of the items which are interested in being packed in one of the bins so as to minimize their cost. We prove that such a game always converges to a pure Nash equilibrium starting from any initial packing of the items, estimate the number of steps needed to reach one such equilibrium, prove the hardness of computing good equilibria and give an upper and a lower bound for the price of anarchy of the game. Then, we consider a multidimensional extension of the problem in which each item can require to be packed in more than just one bin. Unfortunately, we show that in such a case the induced game may not admit a pure Nash equilibrium even under particular restrictions. The study of these games finds applications in the analysis of the bandwidth cost sharing problem in non cooperative networks.

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On the packing of selfish items

We consider broadcast network design games in undirected networks in which every player is a node wishing to receive communication from a distinguished source node s and the cost of each communication link is equally shared among the downstream receivers according to the Shapley value. We prove that the Price of Stability of such games is constant, thus closing a long-standing open problem raised in [2]. Our result is obtained by means of homogenization, a new technique that, in any intermediate state locally diverging from a given optimal solution T*, is able to restore local similarity by exploiting cost differences between nearby players in T*.

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The Price of Stability for Undirected Broadcast Network Design with Fair Cost Allocation Is Constant

Bounding the price of stability of undirected network design games with fair cost allocation is a challenging open problem in the Algorithmic Game Theory research agenda. Even though the generalization of such games in directed networks is well understood in terms of the price of stability (it is exactly H n , the n-th harmonic number, for games with n players), far less is known for network design games in undirected networks. The upper bound carries over to this case as well while the best known lower bound is 42/23?1.826. For more restricted but interesting variants of such games such as broadcast and multicast games, sublogarithmic upper bounds are known while the best known lower bound is 12/7?1.714. In the current paper, we improve the lower bounds as follows. We break the psychological barrier of 2 by showing that the price of stability of undirected network design games is at least 348/155?2.245. Our proof uses a recursive construction of a network design game with a simple gadget as the main building block. For broadcast and multicast games, we present new lower bounds of 20/11?1.818 and 1.862, respectively.

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Vittorio Bilò

Papers

Geometric clustering to minimize the sum of cluster sizes

Almost Envy-Free Allocations with Connected Bundles.

On the packing of selfish items

The Price of Stability for Undirected Broadcast Network Design with Fair Cost Allocation Is Constant

Improved Lower Bounds on the Price of Stability of Undirected Network Design Games