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112
The price of stability for network design with fair cost allocation
 In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS
, 2004
"... Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite differ ..."
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Cited by 205 (26 self)
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Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context — it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widelystudied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fairdivision scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via bestresponse dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form nearoptimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of bestresponse dynamics, and discuss extensions to a weighted game.
Network Coding with a Cost Criterion
 in Proc. 2004 International Symposium on Information Theory and its Applications (ISITA 2004
, 2004
"... We consider applying network coding in settings where there is a cost associated with network use. ..."
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Cited by 61 (16 self)
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We consider applying network coding in settings where there is a cost associated with network use.
A PriceAnticipating Resource Allocation Mechanism for Distributed Shared
 Clusters”, 6th ACM Conference on Electronic Commerce
, 2005
"... In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed marketbased resource allocation system. Multiple users decide how to distribute their budget (bids) among multiple machines according to their individual preferences to maximize their ..."
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Cited by 35 (6 self)
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In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed marketbased resource allocation system. Multiple users decide how to distribute their budget (bids) among multiple machines according to their individual preferences to maximize their individual utility. We look at both the efficiency and the fairness of the allocation at the equilibrium, where fairness is evaluated through the measures of utility uniformity and envyfreeness. We show analytically and through simulations that despite being highly decentralized, such a system converges quickly to an equilibrium and unlike the social optimum that achieves high efficiency but poor fairness, the proposed allocation scheme achieves a nice balance of high degrees of efficiency and fairness at the equilibrium. 1.
NonCooperative Multicast and Facility Location Games
"... We consider a multicast game with selfish noncooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in ..."
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Cited by 35 (2 self)
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We consider a multicast game with selfish noncooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NPhard. We focus on the price of anarchy of a Nash equilibrium resulting from the bestresponse dynamics of a game course, where the players join the game sequentially. For a game with n players, we establish an upper bound of O ( √ n log 2 n) on the price of anarchy, and a lower bound of Ω(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.
Optimal Allocation of a Divisible Good to Strategic Buyers
 Proceedings of the 43d IEEE conference on Decision and Control
, 2004
"... We address the problem of allocating a divisible resource to buyers who value the quantity they receive, but strategize to maximize their net payoff (value minus payment). An allocation mechanism is used to allocate the resource based on bids declared by the buyers. The bids are equal to the payment ..."
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Cited by 32 (1 self)
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We address the problem of allocating a divisible resource to buyers who value the quantity they receive, but strategize to maximize their net payoff (value minus payment). An allocation mechanism is used to allocate the resource based on bids declared by the buyers. The bids are equal to the payments, and the buyers are assumed to be in Nash equilibrium. For two buyers such an allocation mechanism is found that guarantees that the aggregate value is always greater than of the maximum possible, and it is shown that no other mechanism achieves a larger ratio. For a general finite number of buyers an allocation mechanism is given and an expression is given for its worst case efficiency. For three buyers the expression evaluates to 0.8737, for four buyers to 0.8735 and numerical computations suggest that the numerical value does not decrease when the number of buyers is increased beyond four. A potential application of this work is the allocation of communication bandwidth on a single link.
Equilibrium of Heterogeneous Congestion Control: Existence and Uniqueness
 IEEE/ACM Transactions on Networking
, 2007
"... member, IEEE Abstract—When heterogeneous congestion control protocols that react to different pricing signals (They could be different types of signals such as packet loss, queueing delay etc. or different values of the same type of signal such as different ECN marking values based on the same actua ..."
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Cited by 29 (8 self)
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member, IEEE Abstract—When heterogeneous congestion control protocols that react to different pricing signals (They could be different types of signals such as packet loss, queueing delay etc. or different values of the same type of signal such as different ECN marking values based on the same actual link congestion level) share the same network, the current theory based on utility maximization fails to predict the network behavior. Unlike in a homogeneous network, the bandwidth allocation now depends on router parameters and flow arrival patterns. It can be nonunique, suboptimal and unstable. In [36], existence and uniqueness of equilibrium of heterogeneous protocols are investigated. This paper extends the study with two objectives: analyze the optimality and stability of such networks and design control schemes to improve them. First, we demonstrate the intricate behavior of a heterogeneous network through simulations and present a framework to help understand its equilibrium properties. Second, we propose a simple sourcebased algorithm to decouple bandwidth allocation from router parameters and flow arrival patterns by only updating a linear parameter in the sources ’ algorithms on a slow timescale. It is used to steer a network to the unique optimal equilibrium. The scheme can be deployed incrementally as the existing protocol needs no change and only the new protocols need to adopt the slow timescale adaption. I.
Network optimization and control
 Foundations and Trends in Networking
"... We study how protocol design for various functionalities within a communication network architecture can be viewed as a distributed resource allocation problem. This involves understanding what resources are, how to allocate them fairly, and perhaps most importantly, how to achieve this goal in a di ..."
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Cited by 24 (3 self)
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We study how protocol design for various functionalities within a communication network architecture can be viewed as a distributed resource allocation problem. This involves understanding what resources are, how to allocate them fairly, and perhaps most importantly, how to achieve this goal in a distributed and stable fashion. We start with ideas of a centralized optimization framework and show how congestion control, routing and scheduling in wired and wireless networks can be thought of as fair resource allocation. We then move to the study of controllers that allow a decentralized solution of this problem. These controllers are the analytical equivalent of protocols in use on the Internet today, and we describe existing protocols as realizations of such controllers. The Internet is a dynamic system with feedback delays and flows that arrive and depart, which means that stability of the system cannot be taken for granted. We show how to incorporate
Efficient, strategyproof and almost budgetbalanced assignment
, 2007
"... Call a VickreyClarkeGroves (VCG) mechanism to assign p identical objects among n agents, feasible if cash transfers yield no deficit. The efficiency loss of such a mechanism is the worst (largest) ratio of the budget surplus to the efficient surplus, over all profiles of non negative valuations. T ..."
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Cited by 23 (3 self)
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Call a VickreyClarkeGroves (VCG) mechanism to assign p identical objects among n agents, feasible if cash transfers yield no deficit. The efficiency loss of such a mechanism is the worst (largest) ratio of the budget surplus to the efficient surplus, over all profiles of non negative valuations. The optimal (smallest) efficiency loss � L(n, p) satisfies is strictly smaller or strictly �L(n, p) ≤ �L(n, { n 4
Revenue and stability of a mechanism for efficient allocation of a divisible good
, 2005
"... Abstract A class of efficient mechanisms for allocating a divisible good is studied. This class was discovered independently by Maheswaran and Ba¸sar and by us. Strategic buyers play a game by submitting onedimensional bids, or signals, to the seller. The seller allocates the good in proportion to ..."
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Cited by 19 (0 self)
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Abstract A class of efficient mechanisms for allocating a divisible good is studied. This class was discovered independently by Maheswaran and Ba¸sar and by us. Strategic buyers play a game by submitting onedimensional bids, or signals, to the seller. The seller allocates the good in proportion to the bids and charges the buyers nonuniform prices according to the mechanism. Under some mild conditions on the valuation functions of the buyers, there is a unique Nash equilibrium point (NEP) and the allocation at the NEP is efficient. The prices charged to the buyers at the NEP are bounded above by, and can be made arbitrarily close to, the uniform market clearing price for pricetaking buyers. A globally stable decentralized algorithm is given, allowing the buyers to reach the NEP. The work is motivated by the problem of rate allocation on the links of a communication network. 1
Proportional response dynamics leads to market equilibrium
 In STOC
"... One of the main reasons of the recent success of peer to peer (P2P) file sharing systems such as BitTorrent is its builtin titfortat mechanism. In this paper, we model the bandwidth allocation in a P2P system as an exchange economy and study a titfortat dynamics, namely the proportional respons ..."
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Cited by 18 (1 self)
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One of the main reasons of the recent success of peer to peer (P2P) file sharing systems such as BitTorrent is its builtin titfortat mechanism. In this paper, we model the bandwidth allocation in a P2P system as an exchange economy and study a titfortat dynamics, namely the proportional response dynamics, in this economy. In a proportional response dynamics each player distributes its good to its neighbors proportional to the utility it received from them in the last period. We show that this dynamics not only converges but converges to a market equilibrium, a standard economic characterization of efficient exchanges in a competitive market. In addition, for some classes of utility functions we consider, it converges much faster than the classical tâtonnement process and any existing algorithms for computing market equilibria. As a part of our proof we study the double normalization of a matrix, an operation that linearly scales the rows of a matrix so that each row sums to a prescribed positive number, followed by a similar scaling of the columns. We show that the double normalization process of any nonnegative matrix always converges. This complements the previous studies in matrix scaling that has focused on the convergence condition of the process when the row and column normalizations are considered as separate steps. 1