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19
How bad is selfish routing?
 JOURNAL OF THE ACM
, 2002
"... We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route t ..."
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Cited by 678 (27 self)
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We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times—the total latency—is minimized. In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimumlatency path available to it, given the network congestion caused by the other users. In general such a “selfishly motivated ” assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total
Stackelberg scheduling strategies
 In Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing
, 2001
"... AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with loaddependent latency functions specifying the length of time necessary to complete a job; we measure ..."
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Cited by 127 (7 self)
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AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with loaddependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of &quot;selfishly controlled &quot; and &quot;centrally controlled &quot; jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a Stackelberg strategy) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system. In this paper, we prove that it is NPhard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible. We also prove stronger performance guarantees in the special case where every machine latency function is linear in the machine load.
Coordination mechanisms for selfish scheduling
 THEORETICAL COMPUTER SCIENCE
, 2009
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(Almost) optimal coordination mechanisms for unrelated maching scheduling
 IN 18TH ACMSIAM SYMP. ON DISCRETE ALGORITHMS (SODA
, 2008
"... We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and ..."
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Cited by 35 (6 self)
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We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those jobs. A strongly local policy only uses the processing time of jobs on the the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortestfirst algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem, and answers an open question raised by Ibarra and Kim [16], and Davis and Jaffe [10]. We then design a local ordering policy with the approximation ratio of Θ(log m) in equilibria, and prove that this policy is optimal among all local ordering policies. This policy orders the jobs in the nondecreasing order of their inefficiency, i.e, the ratio between the processing time on that machine over the minimum processing time. Finally, we show that best responses of players for the inefficiencybased policy may not converge to a pure Nash equilibrium, and present a Θ(log² m) policy for which we can prove fast convergence of best responses to pure Nash equilibria.
2002): “Mutual Fund Portfolio Choice in the Presence of Dynamic Flows,” working paper
"... We analyze the implications of the widely used fixed fraction of funds fees on a mutual fund manager’s portfolio decisions. In our model, a log utility investor is allowed to dynamically allocate capital between an actively managed mutual fund and a locally riskless bond. The optimal fund portfolio ..."
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Cited by 23 (2 self)
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We analyze the implications of the widely used fixed fraction of funds fees on a mutual fund manager’s portfolio decisions. In our model, a log utility investor is allowed to dynamically allocate capital between an actively managed mutual fund and a locally riskless bond. The optimal fund portfolio is shown to be the one that maximizes the market value of the fees received, and is independent of the manager’s utility function. The presence of dynamic flows induces “flow hedging ” portfolio distortions on the part of the fund, even though the investor is myopic. We predict a positive relationship between a fund’s proportional fee rate and its volatility. This is a consequence of higher fee funds holding more extreme equity positions. However, the overall dollar amount of equity held by a fund can be independent of the fee rate. While both the fund portfolio and the investor’s trading strategy depend on the proportional fee rate, the equilibrium value functions do not. Implications related to the measured performancefundflow relationship and its dependence on the fee rate are derived. Finally, we show that our results hold even if in addition to trading the fund and the bond the investor is allowed to directly trade some of the risky securities, but not all.
Dynamics Pricing: A Learning Approach
 Mathematical and Computational Models for Congestion Charging
, 2006
"... We present an optimization approach for jointly learning the demand as a function of price, and dynamically setting prices of products in an oligopoly environment in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, b ..."
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Cited by 22 (1 self)
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We present an optimization approach for jointly learning the demand as a function of price, and dynamically setting prices of products in an oligopoly environment in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. We first consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. We then extend our analysis in a competitive environment with two firms. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm’s own demand, its competitor’s demand, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following. 2 1
Inner product spaces for minsum coordination mechanisms
 In STOC
, 2011
"... We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obta ..."
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Cited by 13 (3 self)
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We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain the approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith’s Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic nonpreemptive strongly local policies. Since Smith’s Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed.
A survey of stackelberg differential game models in supply and marketing channels
 Journal of Systems Science and Systems Engineering
, 2007
"... Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature ..."
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Cited by 8 (3 self)
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Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure: a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classical advertising capital models or salesadvertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.
Stackelberg Equilibria for DiscreteTime Dynamic Games Part II: Stochastic Games with Deterministic Information Structure
"... Stackelberg equilibria for discretetime ..."
Coordination Mechanisms for Selfish Scheduling
"... In machine scheduling, a set of jobs must be scheduled on a set of machines so as to minimize some global objective function, such as the makespan considered in this paper. In practice, jobs are often controlled by independent, selfishly acting agents, each of which selects a machine for processing ..."
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In machine scheduling, a set of jobs must be scheduled on a set of machines so as to minimize some global objective function, such as the makespan considered in this paper. In practice, jobs are often controlled by independent, selfishly acting agents, each of which selects a machine for processing that minimizes the (expected) completion time of its job. This scenario can be formalized as a game in which the players are job owners, the strategies are machines, and the disutility to each player in a strategy profile is the completion time of its job in the corresponding schedule. The equilibria of these games may result in largerthanoptimal overall makespan. The price of anarchy is the ratio of the worstcase equilibrium makespan to the optimal makespan. In this paper, we design and analyze scheduling policies, or coordination mechanisms, for machines which aim to minimize the price of anarchy (restricted to pure Nash equilibria) of the corresponding game. We study coordination mechanisms for four classes of multiprocessor machine scheduling problems and derive upper and lower bounds for the price of anarchy of these mechanisms. For several of the proposed mechanisms, we are also able to prove that the system converges to a purestrategy Nash equilibrium in a linear number of rounds. Finally, we note that our results are applicable to several practical problems arising in communication networks.