Fast Convex Decomposition for Truthful Social Welfare Approximation

Approximating the optimal social welfare while preserving truthfulness is a well studied problem in algorithmic mechanism design. Assuming that the social welfare of a given mechanism design problem can be optimized by an integer program whose integrality gap is at most $\alpha$, Lavi and Swamy~\cite{Lavi11} propose a general approach to designing a randomized $\alpha$-approximation mechanism which is truthful in expectation. Their method is based on decomposing an optimal solution for the relaxed linear program into a convex combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition technique relies heavily on the ellipsoid method, which is notorious for its poor practical performance. To overcome this problem, we present an alternative decomposition technique which yields an $\alpha(1 + \epsilon)$ approximation and only requires a quadratic number of calls to an integrality gap verifier

Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design

R. Lavy and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees

Mechanism Design via Dantzig-Wolfe Decomposition

In random allocation rules, typically first an optimal fractional point is calculated via solving a linear program. The calculated point represents a fractional assignment of objects or more generally packages of objects to agents. In order to implement an expected assignment, the mechanism designer must decompose the fractional point into integer solutions, each satisfying underlying constraints. The resulting convex combination can then be viewed as a probability distribution over feasible assignments out of which a random assignment can be sampled. This approach has been successfully employed in combinatorial optimization as well as mechanism design with or without money. In this paper, we show that both finding the optimal fractional point as well as its decomposition into integer solutions can be done at once. We propose an appropriate linear program which provides the desired solution. We show that the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe decomposition is a direct implementation of the revised simplex method which is well known to be highly efficient in practice. We also show how to use the Benders decomposition as an alternative method to solve the problem. The proposed method can also find a decomposition into integer solutions when the fractional point is readily present perhaps as an outcome of other algorithms rather than linear programming. The resulting convex decomposition in this case is tight in terms of the number of integer points according to the Carath{\'e}odory's theorem

Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations

It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multi-dimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a poly-time solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings where welfare maximization can only be tractably approximated. In this paper, we extend the reduction to accommodate approximation algorithms, providing an approximation preserving reduction from (truthful) revenue maximization to (not necessarily truthful) welfare maximization. The mechanisms output by our reduction choose allocations via black-box calls to welfare approximation on randomly selected inputs, thereby generalizing also our earlier structural results on optimal multi-dimensional mechanisms to approximately optimal mechanisms. Unlike [http://arxiv.org/abs/1207.5518], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over {\em non-convex regions}

Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design

R. Lavy and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees

Application of multiplicative weights update method in algorithmic game theory

In this thesis, we apply the Multiplicative Weights Update Method (MWUM) to the design of approximation algorithms for some optimization problems in game-theoretic settings. Lavi and Swamy {LS05,LS11} introduced a randomized mechanism for combinatorial auctions that uses an approximation algorithm for the underlying optimization problem, so-called social welfare maximization and converts the approximation algorithm to a randomized mechanism that is { truthful-in-expectation}, which means each player maximizes its expected utility by telling the truth. The mechanism is powerful (e.g., see {LS05,LS11,CEF10,HKV11} for applications), but unlikely to be efficient in practice, because it uses the Ellipsoid method. In chapter 2, we follow the general scheme suggested by Lavi and Swamy and replace the Ellipsoid method with MWUM. This results in a faster and simpler approximately truthful-in-expectation mechanism. We also extend their assumption regarding the existence of an exact solution for the LP-relaxation of social welfare maximization. We assume that there exists an approximation algorithm for the LP and establish a new randomized approximation mechanism. In chapter 3, we consider the problem of computing an approximate saddle point, or equivalently equilibrium, for a convex-concave functions F: X\times Y\to \RR, where $X$ and $Y$ are convex sets of arbitrary dimensions. Our main contribution is the design of a randomized algorithm for computing an \eps-approximation saddle point for $F$. Our algorithm is based on combining a technique developed by Grigoriadis and Khachiyan {GK95}, which is a randomized variant of Brown's fictitious play {B51}, with the recent results on random sampling from convex sets (see, e.g., {LV06,V05}). The algorithm finds an \eps-approximation saddle point in an expected number of O\left(\frac{\rho^2(n+m)}{\eps^{2}}\ln\frac{R}{\eps}\right) iterations, where in each iteration two points are sampled from log-concave distributions over strategy sets. It is assumed that $X$ and $Y$ have inscribed balls of radius $1/R$ and circumscribing balls of radius $R$ and $\rho=\max_{x\in X, y\in Y} |F(x,y)|$. In particular, the algorithm requires O^*\left(\frac{\rho^2(n+m)^6}{\eps^{2}}\ln{R}\right) calls to a membership oracle, where $O^*(\cdot)$ suppresses polylogarithmic factors that depend on $n$, $m$, and \eps.In dieser Doktorarbeit verwenden wir die Multiplicative Weights Update Method (MWUM) für den Entwurf von Approximationsalgorithmen für bestimmte Optimierungsprobleme im spieltheoretischen Umfeld. Lavi und Swamy {LS05,LS11} präsentierten einen randomisierten Mechanismus für kombinatorische Auktionen. Sie verwenden dazu einen Approximationsalgorithmus für die Lösung des zugrundeliegenden Optimierungsproblem, das so genannte Social Welfare Maximization Problem, und wandeln diesen zu einem randomisierten Mechanismus um, der im Erwartungsfall anreizkompatibel ist. Dies bedeutet jeder Spieler erreicht den maximalen Gewinn, wenn er sich ehrlich verhält. Der Mechanismus ist sehr mächtig (siehe {LS05,LS11,CEF10,HKV11} für Anwendungen); trotzdem ist es unwahrscheinlich, dass er in der Praxis effizient ist, da hier die Ellipsoidmethode verwendet wird. In Kapitel 2 folgen wir dem von Lavi und Swamy vorgeschlagenem Schema und ersetzen die Ellipsoidmethode durch MWUM. Das Ergebnis ist ein schnellerer, einfacherer und im Erwartungsfall anreizkompatibler Approximationsmechanismus. Wir erweitern ihre Annahme zur Existenz einer exakten Lösung der LP-Relaxierung für das Social Welfare Maximization Problem. Wir nehmen an, dass ein Approximationsalgorithmus für das LP existiert und beschreiben darauf basierend einen neuen randomisierten Approximationsmechanismus. In Kapitel 3 betrachten wir das Problem für konvexe und konkave Funktionen $F:X\times Y\rightarrow\mathbb{R}$, wobei $X$ und $Y$ konvexe Mengen von beliebiger Dimension sind, einen Sattelpunkt zu approximieren (oder gleichbedeutend ein Equilibrium). Unser Hauptbeitrag ist der Entwurf eines randomisierten Algorithmus zur Berechnung einer $\epsilon$-Näherung eines Sattelpunktes von $F$. Unser Algorithmus beruht auf der Kombination einer Technik entwickelt durch Grigoriadis und Khachiyan {GK95}, welche eine zufallsbasierte Variation von Browns Fictitious Play {B51} ist, mit kürzlich erschienenen Resultaten im Bereich der zufälligen Stichprobennahme aus konvexen Mengen (siehe {LV06,V05}). Der Algorithmus findet eine $\epsilon$-Näherung eines Sattelpunktes im Erwartungsfall in $O(\frac{\rho^{2}(n+m)^{6}}{\epsilon^{2}}\log\frac{R}{\epsilon})$ Rechenschritten, wobei in jedem Rechenschritt zwei Punkte zufällig gemäß einer log-konkaven Verteilungen über Strategiemengen gezogen werden. Hier nehmen wir an, dass $X$ und $Y$ einbeschriebene Kugeln mit Radius $1/R$ und umschreibende Kugeln von Radius R besitzen und $\rho=\max_{x\in X,y\in Y}|F(x,y)|$. Der Algorithmus benötigt dabei $O^{*}(\frac{\rho^{2}(n+m)^{6}}{\epsilon^{2}}\log R)$ Aufrufe eines Zugehörigkeitsorakels, hier versteckt $O^{*}(\cdot)$ polylogarithmische Faktoren, die von $n,m$ und $\epsilon$ abhängen

Stability of Service under Time-of-Use Pricing

We consider "time-of-use" pricing as a technique for matching supply and demand of temporal resources with the goal of maximizing social welfare. Relevant examples include energy, computing resources on a cloud computing platform, and charging stations for electric vehicles, among many others. A client/job in this setting has a window of time during which he needs service, and a particular value for obtaining it. We assume a stochastic model for demand, where each job materializes with some probability via an independent Bernoulli trial. Given a per-time-unit pricing of resources, any realized job will first try to get served by the cheapest available resource in its window and, failing that, will try to find service at the next cheapest available resource, and so on. Thus, the natural stochastic fluctuations in demand have the potential to lead to cascading overload events. Our main result shows that setting prices so as to optimally handle the {\em expected} demand works well: with high probability, when the actual demand is instantiated, the system is stable and the expected value of the jobs served is very close to that of the optimal offline algorithm.Comment: To appear in STOC'1