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37
Routing without regret: On convergence to nash equilibria of regretminimizing algorithms in routing games
 In PODC
, 2006
"... Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging envi ..."
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Cited by 47 (6 self)
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Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a noregret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games havesubstantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimalagents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that dependspolynomially on the players ' regret bounds and the maximum slope of any latency function. We also show that priceofanarchy results may be applied to these approximate equilibria, and alsoconsider the finitesize (noninfinitesimal) loadbalancing model of Azar [2].
Joint strategy fictitious play with inertia for potential games
 in Proceedings of the 44th IEEE Conference on Decision and Control
, 2005
"... We consider multiplayer repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these “largescale ” games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in lar ..."
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Cited by 41 (22 self)
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We consider multiplayer repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these “largescale ” games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in largescale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as “Fictitious Play ” (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion. 1
Distributed selfish load balancing
 In Proc. 17th Ann. ACM–SIAM Symp. on Discrete Algorithms (SODA
, 2006
"... 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 n ..."
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Cited by 31 (3 self)
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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 ǫNash 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
REPLEX — Dynamic traffic engineering based on Wardrop routing policies
 In Proc. 2nd Conference on Future Networking Technologies (CoNext
, 2006
"... One major challenge in communication networks is the problem of dynamically distributing load in the presence of bursty and hard to predict changes in traffic demands. Current traffic engineering operates on time scales of several hours which is too slow to react to phenomena like flash crowds or BG ..."
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Cited by 20 (4 self)
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One major challenge in communication networks is the problem of dynamically distributing load in the presence of bursty and hard to predict changes in traffic demands. Current traffic engineering operates on time scales of several hours which is too slow to react to phenomena like flash crowds or BGP reroutes. One possible solution is to use load sensitive routing. Yet, interacting routing decisions at short time scales can lead to oscillations, which has prevented load sensitive routing from being deployed since the early experiences in Arpanet. However, recent theoretical results have devised a game theoretical rerouting policy that provably avoids such oscillation and in addition can be shown to converge quickly. In this paper we present REPLEX, a distributed dynamic traffic engineering algorithm based on this policy. Exploiting the fact that most underlying routing protocols support multiple equalcost routes to a destination, it dynamically changes the proportion of traffic that is routed along each path. These proportions are carefully adapted utilising information from periodic measurements and, optionally, information exchanged between the routers about the traffic condition along the path. We evaluate the algorithm via simulations employing traffic loads that mimic actual Web traffic, i. e., bursty TCP traffic, and whose characteristics are consistent with selfsimilarity. The simulations quickly converge and do not exhibit significant oscillations on both artificial as well as real topologies, as can be expected from the theoretical results.
Multiplicative Updates Outperform Generic NoRegret . . .
, 2009
"... We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the wellknown multiplicativeweights learning algorithm results in convergence to ..."
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Cited by 15 (4 self)
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We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the wellknown multiplicativeweights learning algorithm results in convergence to pure equilibria. Our results show that natural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the loadbalancing game introduced by Koutsoupias and Papadimitriou, which has superconstant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined gametheoretically, but we show that this property holds whenever a stability criterion from the theory of dynamical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribution). We further extend our results to show that players can use algorithms with different (sufficiently small) learning rates, i.e. they can trade off convergence speed and long term average regret differently.
Distributed algorithms for multicommodity flow problems via approximate steepest descent framework
 IN PROCEEDINGS OF THE ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2007
"... We consider solutions for distributed multicommodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We show first distributed solutions that allow 1 + ǫ approximation and whose convergence time is essentially linear in the maximal path length, ..."
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Cited by 12 (5 self)
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We consider solutions for distributed multicommodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We show first distributed solutions that allow 1 + ǫ approximation and whose convergence time is essentially linear in the maximal path length, and is independent of the number of commodities and the size of the graph. Our algorithms use a very natural approximate steepest descent framework, combined with a blocking flow technique to speed up the convergence in distributed and parallel environment. Previously known solutions that achieved comparable convergence time and approximation ratio required exponential computational and space overhead per agent.
Greedy distributed optimization of multicommodity flows
 In PODC
, 2007
"... The multicommodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connectionoriented network architectures. We consider solutions for distributed multicommodity flow problems, which a ..."
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Cited by 10 (4 self)
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The multicommodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connectionoriented network architectures. We consider solutions for distributed multicommodity 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 multicommodity flow problem with polylogarithmic 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 flowpaths in the network. No prior results exist for our model. Our algorithm is a reasonable alternative to existing polynomial sequential approximation algorithms, such as GargKönemann [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 flowcontrol 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.
Fastconverging tatonnement algorithms for onetime and ongoing market problems
 In Symposium on Theory of Computing (STOC 2008
, 2008
"... 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 OneTime Markets. The Ongoing Market allows ..."
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Cited by 9 (2 self)
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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 OneTime Markets. The Ongoing Market allows trade at nonequilibrium 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
The price of stochastic anarchy
 In SAGT ’08: Proceedings of the First Annual International Symposium on Algorithmic Game Theory
, 2008
"... Abstract. We consider the solution concept of stochastic stability, and propose the price of stochastic anarchy as an alternative to the price of (Nash) anarchy for quantifying the cost of selfishness and lack of coordination in games. As a solution concept, the Nash equilibrium has disadvantages th ..."
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Cited by 7 (2 self)
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Abstract. We consider the solution concept of stochastic stability, and propose the price of stochastic anarchy as an alternative to the price of (Nash) anarchy for quantifying the cost of selfishness and lack of coordination in games. As a solution concept, the Nash equilibrium has disadvantages that the set of stochastically stable states of a game avoid: unlike Nash equilibria, stochastically stable states are the result of natural dynamics of computationally bounded and decentralized agents, and are resilient to small perturbations from ideal play. The price of stochastic anarchy can be viewed as a smoothed analysis of the price of anarchy, distinguishing equilibria that are resilient to noise from those that are not. To illustrate the utility of stochastic stability, we study the load balancing game on unrelated machines. This game has an unboundedly large price of Nash anarchy even when restricted to two players and two machines. We show that in the two player case, the price of stochastic anarchy is 2, and that even in the general case, the price of stochastic anarchy is bounded. We conjecture that the price of stochastic anarchy is O(m), matching the price of strong Nash anarchy without requiring player coordination. We expect that stochastic stability will be useful in understanding the relative stability of Nash equilibria in other games where the worst equilibria seem to be inherently brittle.