Selfishness Need Not Be Bad

The price of anarchy (PoA) is a standard measure for the inefficiency of selfish routing in the static Wardrop traffic model. Empirical studies and a recent analysis reveal a surprising property that the PoA tends to one when the total demand T gets large. These results are extended by a new framework for the limit analysis of the PoA in arbitrary nonatomic congestion games that apply to arbitrary growth patterns of T and all regularly varying cost functions. For routing games with Bureau of Public Road (BPR) cost functions, the convergence follows a power law determined by the degree of the BPR functions, and a related conjecture need not hold. These findings are confirmed by an empirical analysis of traffic in Beijing.

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The Inefficiency of Nash and Subgame Perfect Equilibria for Network Routing

Mathematics of Operations Research ◽

10.1287/moor.2018.0968 ◽

2019 ◽

Vol 44 (4) ◽

pp. 1286-1303 ◽

Cited By ~ 1

Author(s):

José Correa ◽

Jasper de Jong ◽

Bart de Keijzer ◽

Marc Uetz

Keyword(s):

Price Of Anarchy ◽

Subgame Perfect Equilibrium ◽

Network Routing ◽

Congestion Games ◽

Cost Functions ◽

Worst Case ◽

Subgame Perfection ◽

Subgame Perfect Equilibria ◽

Routing Games ◽

Perfect Equilibria

This paper provides new bounds on the quality of equilibria in finite congestion games with affine cost functions, specifically for atomic network routing games. It is well known that the price of anarchy equals exactly 5/2 in general. For symmetric network routing games, it is at most (5n − 2)/(2n + 1), where n is the number of players. This paper answers to two open questions for congestion games. First, we show that the price of anarchy bound (5n − 2)/(2n + 1) is tight for symmetric network routing games, thereby answering a decade-old open question. Second, we ask whether sequential play and subgame perfection allows to evade worst-case Nash equilibria, and thereby reduces the price of anarchy. This is motivated by positive results for congestion games with a small number of players, as well as recent results for other resource allocation problems. Our main result is the perhaps surprising proof that subgame perfect equilibria of sequential symmetric network routing games with linear cost functions can have an unbounded price of anarchy. We complete the picture by analyzing the case with two players: we show that the sequential price of anarchy equals 7/5 and that computing the outcome of a subgame perfect equilibrium is NP-hard.

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The price of anarchy in routing games as a function of the demand

Mathematical Programming ◽

10.1007/s10107-021-01701-7 ◽

2021 ◽

Author(s):

Roberto Cominetti ◽

Valerio Dose ◽

Marco Scarsini

Keyword(s):

Price Of Anarchy ◽

Heavy Traffic ◽

Scaling Law ◽

Congestion Games ◽

Social Optimum ◽

Asymptotic Results ◽

Worst Case ◽

Traffic Demand ◽

Break Points ◽

Routing Games

AbstractThe price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.

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Balancing Load via Small Coalitions in Selfish Ring Routing Games

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595915400035 ◽

2015 ◽

Vol 32 (01) ◽

pp. 1540003

Author(s):

Xujin Chen ◽

Xiaodong Hu ◽

Weidong Ma

Keyword(s):

Network Performance ◽

Price Of Anarchy ◽

Maximum Load ◽

Network Routing ◽

Global Optimum ◽

Ring Networks ◽

Selfish Routing ◽

Worst Case ◽

Worst Case Ratio ◽

Routing Games

This paper concerns the asymmetric atomic selfish routing game for load balancing in ring networks. In the selfish routing, each player selects a path in the ring network to route one unit traffic between its source and destination nodes, aiming at a minimum maximum link load along its own path. The selfish path selections by individuals ignore the system objective of minimizing the maximum load over all network links. This selfish ring load (SRL) game arises in a wide variety of applications in decentralized network routing, where network performance is often measured by the price of anarchy (PoA), the worst-case ratio between the maximum link loads in an equilibrium routing and an optimal routing. It has been known that the PoA of SRL with respect to classical Nash Equilibrium (NE) cannot be upper bounded by any constant, showing large loss of efficiency at some NE outcome. In an effort to improve the network performance in the SRL game, we generalize the model to so-called SRL with collusion (SRLC) which allows coordination within any coalition of up to k selfish players on the condition that every player of the coalition benefits from the coordination. We prove that, for m-player game on n-node ring, the PoA of SRLC is n - 1 when k ≤ 2, drops to 2 when k = 3 and is at least 1 + 2/m for k ≥ 4. Our study shows that on one hand, the performance of ring networks, in terms of maximum load, benefits significantly from coordination of self-interested players within small-sized coalitions; on the other hand, the equilibrium routing in SRL might not reach global optimum even if any number of players can coordinate.

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SELFISH ROUTING IN THE PRESENCE OF NETWORK UNCERTAINTY

Parallel Processing Letters ◽

10.1142/s0129626409000122 ◽

2009 ◽

Vol 19 (01) ◽

pp. 141-157 ◽

Cited By ~ 8

Author(s):

CHRYSSIS GEORGIOU ◽

THEOPHANIS PAVLIDES ◽

ANNA PHILIPPOU

Keyword(s):

Nash Equilibria ◽

Price Of Anarchy ◽

Probability Distributions ◽

Congestion Games ◽

Selfish Routing ◽

Network Information ◽

Negative Results ◽

Special Cases ◽

Interesting Open Problem ◽

Incomplete Network Information

We study the problem of selfish routing in the presence of incomplete network information. Our model consists of a number of users who wish to route their traffic on a network of m parallel links with the objective of minimizing their latency. However, in doing so, they face the challenge of lack of precise information on the capacity of the network links. This uncertainty is modeled via a set of probability distributions over all the possibilities, one for each user. The resulting model is an amalgamation of the KP-model of [14] and the congestion games with user-specific functions of [22]. We embark on a study of Nash equilibria and the price of anarchy in this new model. In particular, we propose polynomial-time algorithms (w.r.t. our model's parameters) for computing some special cases of pure Nash equilibria and we show that negative results of [22], for the non-existence of pure Nash equilibria in the case of three users, do not apply to our model. Consequently, we propose an interesting open problem, that of the existence of pure Nash equilibria in the general case of our model. Furthermore, we consider appropriate notions for the social cost and the price of anarchy and obtain upper bounds for the latter. With respect to fully mixed Nash equilibria, we show that when they exist, they are unique. Finally, we prove that the fully mixed Nash equilibrium is the worst equilibrium.

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Congestion Games, Load Balancing, and Price of Anarchy

Combinatorial and Algorithmic Aspects of Networking - Lecture Notes in Computer Science ◽

10.1007/11527954_3 ◽

2005 ◽

pp. 13-27 ◽

Cited By ~ 5

Author(s):

Anshul Kothari ◽

Subhash Suri ◽

Csaba D. Tóth ◽

Yunhong Zhou

Keyword(s):

Load Balancing ◽

Price Of Anarchy ◽

Congestion Games

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Algorithm for Searching an Equilibrium in a Routing Game with Piecewise Constant Cost Functions

International Game Theory Review ◽

10.1142/s0219198918500068 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1850006 ◽

Cited By ~ 1

Author(s):

Andrey Parfenov

Keyword(s):

Function Method ◽

Piecewise Linear ◽

User Equilibrium ◽

Transportation Systems ◽

Cost Functions ◽

Hard Problem ◽

Piecewise Constant ◽

Constant Cost ◽

Routing Games ◽

Min Cost Flow

We consider nonatomic routing games which are used to model transportation systems with a large number of agents and suggest an algorithm to search for the user equilibrium in such games, which is a generalization of the notion of Nash equilibrium. In general, finding a user equilibrium in routing games is computationally a hard problem. We consider the following subclass of routing games: games with piecewise constant cost functions, and construct an algorithm finding equilibrium in such games. This algorithm is based on the potential function method and the method of piecewise linear (PWL) costs enumeration which finds min-cost flow in a network with PWL cost functions. If each cost function increases, then the complexity of our algorithm is polynomial in the parameters of the network. But if some cost functions have decreasing points, then the complexity is exponential by the number of such points.

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Price of Anarchy of Network Routing Games with Incomplete Information

Lecture Notes in Computer Science - Internet and Network Economics ◽

10.1007/11600930_107 ◽

2005 ◽

pp. 1066-1075 ◽

Cited By ~ 7

Author(s):

Dinesh Garg ◽

Yadati Narahari

Keyword(s):

Incomplete Information ◽

Price Of Anarchy ◽

Network Routing ◽

Games With Incomplete Information ◽

Routing Games

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Do Himalayan treelines respond to recent climate change? An evaluation of sensitivity indicators

Earth System Dynamics ◽

10.5194/esd-6-245-2015 ◽

2015 ◽

Vol 6 (1) ◽

pp. 245-265 ◽

Cited By ~ 59

Author(s):

U. Schickhoff ◽

M. Bobrowski ◽

J. Böhner ◽

B. Bürzle ◽

R. P. Chaudhary ◽

...

Keyword(s):

Climate Change ◽

Climate Warming ◽

Tree Growth ◽

Empirical Studies ◽

Growth Patterns ◽

Niche Modelling ◽

Ongoing Research ◽

Recent Climate Change ◽

Recent Climate ◽

Response Gradient

Abstract. Climate warming is expected to induce treelines to advance to higher elevations. Empirical studies in diverse mountain ranges, however, give evidence of both advancing alpine treelines and rather insignificant responses. The inconsistency of findings suggests distinct differences in the sensitivity of global treelines to recent climate change. It is still unclear where Himalayan treeline ecotones are located along the response gradient from rapid dynamics to apparently complete inertia. This paper reviews the current state of knowledge regarding sensitivity and response of Himalayan treelines to climate warming, based on extensive field observations, published results in the widely scattered literature, and novel data from ongoing research of the present authors. Several sensitivity indicators such as treeline type, treeline form, seed-based regeneration, and growth patterns are evaluated. Since most Himalayan treelines are anthropogenically depressed, observed advances are largely the result of land use change. Near-natural treelines are usually krummholz treelines, which are relatively unresponsive to climate change. Nevertheless, intense recruitment of treeline trees suggests a great potential for future treeline advance. Competitive abilities of seedlings within krummholz thickets and dwarf scrub heaths will be a major source of variation in treeline dynamics. Tree growth–climate relationships show mature treeline trees to be responsive to temperature change, in particular in winter and pre-monsoon seasons. High pre-monsoon temperature trends will most likely drive tree growth performance in the western and central Himalaya. Ecological niche modelling suggests that bioclimatic conditions for a range expansion of treeline trees will be created during coming decades.

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