When Smoothness is Not Enough: Toward Exact Quantification and Optimization of the Price-of-Anarchy

Today's multiagent systems have grown too complex to rely on centralized controllers, prompting increasing interest in the design of distributed algorithms. In this respect, game theory has emerged as a valuable tool to complement more traditional techniques. The fundamental idea behind this approach is the assignment of agents' local cost functions, such that their selfish minimization attains, or is provably close to, the global objective. Any algorithm capable of computing an equilibrium of the corresponding game inherits an approximation ratio that is, in the worst case, equal to its price-of-anarchy. Therefore, a successful application of the game design approach hinges on the possibility to quantify and optimize the equilibrium performance. Toward this end, we introduce the notion of generalized smoothness, and show that the resulting efficiency bounds are significantly tighter compared to those obtained using the traditional smoothness approach. Leveraging this newly-introduced notion, we quantify the equilibrium performance for the class of local resource allocation games. Finally, we show how the agents' local decision rules can be designed in order to optimize the efficiency of the corresponding equilibria, by means of a tractable linear program.Comment: 9 pages, double column, 1 figure, 1 table, to appear in the proceedings of the 2019 IEEE Conference on Decision and Contro

Optimal Price of Anarchy in Cost-Sharing Games

The design of distributed algorithms is central to the study of multiagent systems control. In this paper, we consider a class of combinatorial cost-minimization problems and propose a framework for designing distributed algorithms with a priori performance guarantees that are near-optimal. We approach this problem from a game-theoretic perspective, assigning agents cost functions such that the equilibrium efficiency (price of anarchy) is optimized. Once agents' cost functions have been specified, any algorithm capable of computing a Nash equilibrium of the system inherits a performance guarantee matching the price of anarchy. Towards this goal, we formulate the problem of computing the price of anarchy as a tractable linear program. We then present a framework for designing agents' local cost functions in order to optimize for the worst-case equilibrium efficiency. Finally, we investigate the implications of our findings when this framework is applied to systems with convex, nondecreasing costs.Comment: 8 pages, double column, 1 figure, 2 tables, submitted to 2019 American Control Conferenc

Utility Design for Distributed Resource Allocation -- Part II: Applications to Submodular, Covering, and Supermodular Problems

A fundamental component of the game theoretic approach to distributed control is the design of local utility functions. Relative to resource allocation problems that are additive over the resources, Part I showed how to design local utilities so as to maximize the associated performance guarantees [1], which we measure by the price of anarchy. The purpose of the present manuscript is to specialize these results to the case of submodular, covering, and supermodular problems. In all these cases we obtain tight expressions for the price of anarchy that often match or improve the guarantees associated to state-of-the-art approximation algorithms. Two applications and corresponding numerics are presented: the vehicle-target assignment problem and a coverage problem arising in wireless data caching.Comment: 15 pages, 10 figure

Optimal mechanisms for distributed resource-allocation

As the complexity of real-world systems continues to increase, so does the need for distributed protocols that are capable of guaranteeing a satisfactory system performance, without the reliance on centralized decision making. In this respect, game theory provides a valuable framework for the design of distributed algorithms in the form of equilibrium efficiency bounds. Arguably one of the most widespread performance metrics, the price-of-anarchy measures how the efficiency of a system degrades when moving from centralized to distributed decision making. While the smoothness framework -- introduced in Roughgarden 2009 -- has emerged as a powerful methodology for bounding the price-of-anarchy, the resulting bounds are often conservative, bringing into question the suitability of the smoothness approach for the design of distributed protocols. In this paper, we introduce the notion of generalized smoothness in order to overcome these difficulties. First, we show that generalized smoothness arguments are more widely applicable, and provide tighter price-of-anarchy bounds compared to those obtained using the existing smoothness framework. Second, we show how to leverage the notion of generalized smoothness to obtain a tight characterization of the price-of-anarchy, relative to the class of local cost-sharing games. Within this same class of games we show that the price-of-anarchy can be computed and optimized through the solution of a tractable linear program. Finally, we demonstrate that our approach subsumes and generalizes existing results for three well-studied classes of games.Comment: 27 pages, 6 figure

Distributed control and game design: From strategic agents to programmable machines

Large scale systems are forecasted to greatly impact our future lives thanks to their wide ranging applications including cooperative robotics, mobility on demand, resource allocation, supply chain management. While technological developments have paved the way for the realization of such futuristic systems, we have a limited grasp on how to coordinate the individual components to achieve the desired global objective. This thesis deals with the analysis and coordination of large scale systems without the need of a centralized authority. In the first part of this thesis, we consider non-cooperative decision making problems where each agent's objective is a function of the aggregate behavior of the population. First, we compare the performance of an equilibrium allocation with that of an optimal allocation and propose conditions under which all equilibrium allocations are efficient. Towards this goal, we prove a novel result bounding the distance between the strategies at a Nash and Wardrop equilibrium that might be of independent interest. Second, we show how to derive scalable algorithms that guide agents towards an equilibrium allocation. In the second part of this thesis, we consider large-scale cooperative problems, where a number of agents need to be allocated to a set of resources with the goal of jointly maximizing a given submodular or supermodular set function. Since this class of problems is computationally intractable, we aim at deriving tractable algorithms for attaining approximate solutions. We approach the problem from a game-theoretic perspective and ask the following: how should we design agents' utilities so that any equilibrium configuration is almost optimal? To answer this question we introduce a novel framework that allows to characterize and optimize the system performance as a function of the chosen utilities by means of a tractable linear program.Comment: PhD Thesis, ETH Zuric