Alexa Sharp

Computing Shapley Value in Supermodular Coalitional Games

Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.

An incremental model for combinatorial maximization problems

Many combinatorial optimization problems aim to select a subset of elements of maximum value subject to certain constraints. We consider an incremental version of such problems, in which some of the constraints rise over time. A solution is a sequence of feasible solutions, one for each time step, such that later solutions build on earlier solutions incrementally. We introduce a general model for such problems, and define incremental versions of maximum flow, bipartite matching, and knapsack. We find that imposing an incremental structure on a problem can drastically change its complexity. With this in mind, we give general yet simple techniques to adapt algorithms for optimization problems to their respective incremental versions, and discuss tightness of these adaptations with respect to the three aforementioned problems.

The price of civil society

Most work in algorithmic game theory assumes that players ignore costs incurred by their fellow players. In this paper, we consider superimposing a social network over a game, where players are concerned with minimizing not only their own costs, but also the costs of their neighbors in the network. We aim to understand how properties of the underlying game are affected by this alteration to the standard model. The new social game has its own equilibria, and the price of civil society denotes the ratio of the social cost of the worst such equilibrium relative to the worst Nash equilibrium under standard selfish play. We initiate the study of the price of civil society in the context of a simple class of games. Counterintuitively, we show that when players become less selfish (optimizing over both themselves and their friends), the resulting outcomes may be worse than they would have been in the base game. We give tight bounds on this phenomenon in a simple class of load-balancing games, over arbitrary social networks, and present some extensions.

On distance coloring

An undirected graph G =(V ,E ) is (d ,k )-colorable if there is a vertex coloring using at most k colors such that no two vertices within distance d have the same color. It is well known that (1,2)-colorability is decidable in linear time, and that (1,k )-colorability is NP -complete for k ≥3. This paper presents the complexity of (d ,k )-coloring for general d and k , and enumerates some interesting properties of (d ,k )-colorable graphs. The main result is the dichotomy between polynomial and NP -hard instances: for fixed d ≥2, the distance coloring problem is polynomial time for

New models for the CS1 course: a fifteen year retrospective

k \leq \lfloor \frac{3d}{2} \rfloor

Mediated Equilibria in Load-Balancing Games

and NP-hard for

Distance coloring

k > \lfloor \frac{3d}{2} \rfloor

Incremental algorithms: solving problems in a changing world

. We present a reduction in the latter case, as well as an algorithm in the former. The algorithm entails several innovations that may be of independent interest: a generalization of tree decompositions to overlay graphs other than trees; a general construction that obtains such decompositions from certain classes of edge partitions; and the use of homology to analyze the cycle structure of colorable graphs. This paper is both a combining and reworking of the papers of Sharp and Kozen [14, 10].

Equilibria and Efficiency Loss in Games on Networks

The year was 1994; the place, Phoenix Arizona. A panel session moderated by Barbara Boucher Owens entitled New Models for the CS1 Course: What Are They and Are They Leading to the Same Place? [1] took place before a packed audience at SIGCSE ’94. At the time, the World Wide Web was just over the horizon, and disk drives held at most a few hundred megabytes. The most popular language for CS1 was Pascal. At the SIGCSE’94 panel we heard about four diverse approaches to introductory computer science: the “breadthfirst” approach, which includes discrete mathematics, logic and problem solving; the “object-oriented” approach, which introduces objects from the beginning; the“formal”approach, which mandates correctness proofs; and the “Scheme” approach, which focuses on abstraction. For SIGCSE’09 three of the four panelists from the ’94 panel have returned to update the discussion. Joining this group is a younger faculty member who, as a student in the 1990’s, was directly affected by the teaching philosophies of the time. The veteran panelists will comment on how their approaches have evolved over the years. Have the expectations of today’s student population forced these educators to adjust their philosophies, or have they been able to adapt their core beliefs to the current environment? Looking ahead, the most important question remains, how does one craft a CS1 curriculum that can appeal to a wide audience and best prepare students for the road ahead.

Incremental flow

Mediators are third parties to whom the players in a game can delegate the task of choosing a strategy; a mediator forms a mediated equilibrium if delegating is a best response for all players. Mediated equilibria have more power to achieve outcomes with high social welfare than Nash or correlated equilibria, but less power than a fully centralized authority. Here we begin the study of the power of mediation by using the mediation analogue of the price of stability--the ratio of the social cost of the best mediated equilibrium