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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

Abstract: We present FlightPath, a novel peer-to-peer streaming application that provides a highly reliable data stream to a dynamic set of peers. We demonstrate that FlightPath reduces jitter compared to previous works by several orders of magnitude. Furthermore, FlightPath uses a number of run-time adaptations to maintain low jitter despite 10 % of the population behaving maliciously and the remaining peers acting selfishly. At the core of FlightPath’s success are approximate equilibria. These equilibria allow us to design incentives to limit selfish behavior rigorously, yet they provide sufficient flexibility to build practical systems. We show how to use an ε-Nash equilibrium, instead of a strict Nash, to engineer a live streaming system that uses bandwidth efficiently, absorbs flash crowds, adapts to sudden peer departures, handles churn, and tolerates malicious activity. 1

We propose weakening the assumption made when studying the price of anarchy: Rather than assume that self-interested players will play according to a Nash equilibrium (which may even be computationally hard to find), we assume only that selfish players play so as to minimize their own regret. Regret minimization can be done via simple, efficient algorithms even in many settings where the number of action choices for each player is exponential in the natural parameters of the problem. We prove that despite our weakened assumptions, in several broad classes of games, this “price of total anarchy ” matches the Nash price of anarchy, even though play may never converge to Nash equilibrium. In contrast to the price of anarchy and the recently introduced price of sinking [15], which require all players to behave in a prescribed manner, we show that the price of total anarchy is in many cases resilient to the presence of Byzantine players, about whom we make no assumptions. Finally, because the price of total anarchy is an upper bound on the price of anarchy even in mixed strategies, for some games our results yield as corollaries previously unknown bounds on the price of anarchy in mixed strategies. 1

Abstract. Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For m ≫ n, the system becomes approximately balanced (an ǫ-Nash equilibrium) in expected time O(log log m). We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time O(log log m + n 4). We also give a lower bound of Ω(max{log log m, n}) for the convergence time. 1. Introduction. Suppose

We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s 2 λ), where s is the number of strategies and λ is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an ɛ-approximate Nash equilibrium when the number of strategies is two. 1

In this paper, we study the price of anarchy of traffic routing, under the assumption that users are partially altruistic or spiteful. We model such behavior by positing that the “cost ” perceived by a user is a linear combination of the actual latency of the route chosen (selfish component), and the increase in latency the user causes for others (altruistic component). We show that if all users have a coefficient of at least β> 0 for the altruistic component, then the price of anarchy is bounded by 1/β, for all network topologies, arbitrary commodities, and arbitrary semi-convex latency functions. We extend this result to give more precise bounds on the price of anarchy for specific classes of latency functions, even for β < 0 modeling spiteful behavior. In particular, we show that if all latency functions are linear, the price of anarchy is bounded by 4/(3 + 2β − β 2). We next study non-uniform altruism distributions, where different users may have different coefficients β. We prove that all such games, even with infinitely many types of players, have a Nash Equilibrium. We show that if the average of the coefficients for the altruistic components of all users is ¯ β, then the price of anarchy is bounded by 1 / ¯ β, for single commodity parallel link networks, and arbitrary convex latency functions. In particular, this result generalizes, albeit non-constructively, the Stackelberg routing results of Roughgarden and of Swamy. More generally, we bound the price of anarchy based on the class of allowable latency functions, and as a corollary obtain tighter bounds for Stackelberg routing than a recent result of Swamy.

The multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for distributed multi-commodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We provide the first stateless greedy distributed algorithm for the concurrent multi-commodity flow problem with poly-logarithmic convergence. More precisely, our algorithm achieves 1+ɛ approximation, with running time O(log P·log O(1) m·(1/ɛ) O(1)) where P is the number of flow-paths in the network. No prior results exist for our model. Our algorithm is a reasonable alternative to existing polynomial sequential approximation algorithms, such as Garg-Könemann [17]. The algorithm is simple and can be easily implemented or taught in a classroom. Remarkably, our algorithm requires that the increase in the flow rate on a link is more aggressive than the decrease in the rate. Essentially all of the existing flow-control heuristics are variations of TCP, which uses a conservative cap on the increase (e.g., additive), and a rather liberal cap on the decrease (e.g., multiplicative). In contrast, our algorithm requires the increase to be multiplicative, and that this increase is dramatically more aggressive than the decrease.

Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For