Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games

In this paper we study the inefficiency ratio of stable equilibria in load balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower and upper bounds of 7/6 and 4/3, respectively. This improves over the best known bounds in problem (19/18 and 3/2, respectively). Equivalently, the results apply to the question of how well the optimum for the $L_2$ -norm can approximate the $L_{\infty}$-norm (makespan) in identical machines scheduling

Scheduling Games with Machine-Dependent Priority Lists

We consider a scheduling game in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We characterize four classes of instances in which a pure Nash equilibrium (NE) is guaranteed to exist, and show, by means of an example, that none of these characterizations can be relaxed. We then bound the performance of Nash equilibria for each of these classes with respect to the makespan of the schedule and the sum of completion times. We also analyze the computational complexity of several problems arising in this model. For instance, we prove that it is NP-hard to decide whether a NE exists, and that even for instances with identical machines, for which a NE is guaranteed to exist, it is NP-hard to approximate the best NE within a factor of $2-\frac{1}{m}-\epsilon$ for all $\epsilon>0$. In addition, we study a generalized model in which players' strategies are subsets of resources, each having its own priority list over the players. We show that in this general model, even unweighted symmetric games may not have a pure NE, and we bound the price of anarchy with respect to the total players' costs.Comment: 19 pages, 2 figure

Congestion Games with Multisets of Resources and Applications in Synthesis

In classical congestion games, players\u27 strategies are subsets of resources. We introduce and study multiset congestion games, where players\u27 strategies are multisets of resources. Thus, in each strategy a player may need to use each resource a different number of times, and his cost for using the resource depends on the load that he and the other players generate on the resource. Beyond the theoretical interest in examining the effect of a repeated use of resources, our study enables better understanding of non-cooperative systems and environments whose behavior is not covered by previously studied models. Indeed, congestion games with multiset-strategies arise, for example, in production planing and network formation with tasks that are more involved than reachability. We study in detail the application of synthesis from component libraries: different users synthesize systems by gluing together components from a component library. A component may be used in several systems and may be used several times in a system. The performance of a component and hence the system\u27s quality depends on the load on it. Our results reveal how the richer setting of multisets congestion games affects the stability and equilibrium efficiency compared to standard congestion games. In particular, while we present very simple instances with no pure Nash equilibrium and prove tighter and simpler lower bounds for equilibrium inefficiency, we are also able to show that some of the positive results known for affine and weighted congestion games apply to the richer setting of multisets

Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective

Many packing, scheduling and covering problems that were previously considered by computer science literature in the context of various transportation and production problems, appear also suitable for describing and modeling various fundamental aspects in networks optimization such as routing, resource allocation, congestion control, etc. Various combinatorial problems were already studied from the game theoretic standpoint, and we attempt to complement to this body of research. Specifically, we consider the bin packing problem both in the classic and parametric versions, the job scheduling problem and the machine covering problem in various machine models. We suggest new interpretations of such problems in the context of modern networks and study these problems from a game theoretic perspective by modeling them as games, and then concerning various game theoretic concepts in these games by combining tools from game theory and the traditional combinatorial optimization. In the framework of this research we introduce and study models that were not considered before, and also improve upon previously known results.Comment: PhD thesi

Sampling from the {G}ibbs Distribution in Congestion Games

Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution? We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated $k$-uniform congestion games. To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution

Dynamic resource allocation games

In resource allocation games, selfish players share resources that are needed in order to fulfill their objectives. The cost of using a resource depends on the load on it. In the traditional setting, the players make their choices concurrently and in one-shot. That is, a strategy for a player is a subset of the resources. We introduce and study dynamic resource allocation games. In this setting, the game proceeds in phases. In each phase each player chooses one resource. A scheduler dictates the order in which the players proceed in a phase, possibly scheduling several players to proceed concurrently. The game ends when each player has collected a set of resources that fulfills his objective. The cost for each player then depends on this set as well as on the load on the resources in it – we consider both congestion and cost-sharing games. We argue that the dynamic setting is the suitable setting for many applications in practice. We study the stability of dynamic resource allocation games, where the appropriate notion of stability is that of subgame perfect equilibrium, study the inefficiency incurred due to selfish behavior, and also study problems that are particular to the dynamic setting, like constraints on the order in which resources can be chosen or the problem of finding a scheduler that achieves stability

A new model for coalition formation games

We present two broad categories of games, namely, group matching games and bottleneck routing games on grids. Borrowing ideas from coalition formation games, we introduce a new category of games which we call group matching games. We investigate how these games perform when agents are allowed to make selfish decisions that increase their individual payoffs versus when agents act towards the social benefit of the game as a whole. The Price of Anarchy (PoA) and Price of Stability (PoS) metrics are used to quantify these comparisons. We show that the PoA for a group matching game is at most kα and the PoS is at most k/α where k is the maximum size of a group and α is a switching cost. Furthermore we show that the PoA and PoS of the games do not change significantly even if we increase γ, the number of groups that an agent is allowed to join. We also consider routing games on grid network topologies. The social cost is the worst congestion in any of the network edges (bottleneck congestion). Each player\u27s objective is to find a path that minimizes the bottleneck congestion in its path. We show that the price of anarchy in bottleneck games in grids is proportional to the number of bends β that the paths are allowed to take in the grids\u27 space. We present games where the PoA is O(β). We also give a respective lower bound of Ω(β) which shows that our upper bound is within only a poly-log factor from the best achievable price of anarchy. A significant impact of our analysis is that there exist bottleneck routing games with small number of bends which give a poly-log approximation to the optimal coordinated solution that may use an arbitrary number of bends. To our knowledge, this is the first tight analysis of bottleneck games on grids

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Online Platforms in Networked Markets: Transparency, Anticipation and Demand Management

The global economy has been transformed by the introduction of online platforms in the past two decades. These companies, such as Uber and Amazon, have benefited and undergone massive growth, and are a critical part of the world economy today. Understanding these online platforms, their designs and how participation change with anticipation and uncertainty can help us identify the necessary ingredients for successful implementation of online platforms in the future, especially for those with underlying network constraints, e.g., the electricity grid. This thesis makes three main contributions. First, we identify and compare common access and allocation control designs for online platforms, and highlight their trade-offs between transparency and control. We make these comparisons under a networked Cournot competition model and consider three popular designs: (i) open access, (ii) discriminatory access, and (iii) controlled allocation. Our findings reveal that designs that control over access are more efficient than designs that control over allocations, but open access designs are susceptible to substantial search costs. Next, we study the impact of demand management in a networked Stackelberg model considering network constraints and producer anticipation. We provide insights on limiting manipulation under these constrained networked marketplaces with nodal prices, and show that demand management mechanisms that traditionally aid system stability also help plays a vital role economically. In particular, we show that demand management empower consumers and give them "market power" to counter that of producers, limiting the impact of their anticipation and their potential for manipulation. Lastly, we study how participants (e.g., drivers on Uber) make competitive real-time production (driving) decisions. To that end, we design a novel pursuit algorithm for making online optimization under limited inventory constraints. Our analysis yields an algorithm that is competitive and applicable to achieve optimal results in the well known one-way trading problem, and new variants of the original problem.</p

Atomic Routing Games on Maximum Congestion

We study atomic routing congestion games in which each player chooses a path in the network from its strategy set (a collection of paths) with the objective to minimize the maximum congestion along any edge on its selected path. The social cost is the global maximum congestion on any edge in the network. We show that for arbitrary routing games, the price of stability is 1, and the price of anarchy, PoA, is bounded by κ − 1 ≤ PoA ≤ c(κ 2 + log 2 n), where κ is the length of the longest cycle in the network, n is the size of the network and c is a constant. Further, any best response dynamic converges to a Nash equilibrium. Our bounds show that for maximum congestion games, the topology of the network, in particular the length of cycles, plays an important role in determining the quality of the Nash equilibria. A fundamental issue in the management of large scale communication networks is to route the packet traffic so as to optimize the network performance. Our measure of network performance is the worst bottleneck (most used link) in the system. The model we use for network traffic is that of finite, unsplittable packets (atomic flow), and each packet’s path is controlled independentl