Weighted Congestion Games: Price of Anarchy, Universal Worst-Case Examples, and Tightness

2010 ◽  
pp. 17-28 ◽  
Author(s):  
Kshipra Bhawalkar ◽  
Martin Gairing ◽  
Tim Roughgarden
Keyword(s):  
Price Of Anarchy ◽  
Congestion Games ◽  
Worst Case ◽  
Case Examples

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

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.

scholarly journals The Inefficiency of Nash and Subgame Perfect Equilibria for Network Routing

2019 ◽  
Vol 44 (4) ◽  
pp. 1286-1303 ◽  
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.

Congestion Games, Load Balancing, and Price of Anarchy

2005 ◽  
pp. 13-27 ◽  
Author(s):  
Anshul Kothari ◽  
Subhash Suri ◽  
Csaba D. Tóth ◽  
Yunhong Zhou
Keyword(s):  
Load Balancing ◽  
Price Of Anarchy ◽  
Congestion Games

scholarly journals The Toxicological Effects of Heavy Fuel Oil Category Substances

2013 ◽  
Vol 33 (1_suppl) ◽  
pp. 95S-109S ◽  
Author(s):  
Richard H. McKee ◽  
Fred Reitman ◽  
Ceinwen Schreiner ◽  
Russell White ◽  
Jeffrey H. Charlap ◽  
...  
Keyword(s):  
Developmental Toxicity ◽  
Hematological Parameters ◽  
Target Organ ◽  
Fuel Oil ◽  
Heavy Fuel Oil ◽  
Worst Case ◽  
Case Examples ◽  
Toxicological Effects ◽  
Fetal Survival ◽  
Developmental Effects

Heavy fuel oil (HFO) category substances are used to manufacture HFO, a product used in industrial boilers and marine diesel engines. Commercial HFOs and blending stream components are substances of complex and variable composition, composed of C20 to >C50 hydrocarbons, although lower molecular weight material may be added to reduce viscosity and improve flow characteristics. An HFO blending stream (catalytically cracked clarified oil [CCCO]) was tested for target organ and developmental toxicity in rats following repeated dermal administration at doses of 5, 25, or 50 mg/kg/d. In the repeated dose study, there was evidence of increased liver weights, reduced thymus weights, and reductions in hematological parameters with an overall no observed adverse effect level (NOAEL) of 5 mg/kg/d. In the developmental toxicity test, there were significant reductions in fetal survival, significant increases in resorption frequency, and significantly reduced fetal weights with an overall NOAEL of 5 mg/kg/d. These target organ and developmental effects are associated with the types and levels of aromatic constituents in these substances. Among HFO blending streams, CCCOs have the highest levels of aromatics and, because they produce the characteristic toxicological effects at the lowest levels, are considered as “reasonable worst-case examples” for this group of substances. Other HFO category members with lower levels of aromatics produce similar effects but have higher NOAELs. The potential for target organ and developmental effects of other HFO category members can be predicted from information on the types and levels of the aromatic constituents present in these substances.

THE PRICE OF NASH EQUILIBRIA IN MULTICAST TRANSMISSION GAMES

2010 ◽  
Vol 11 (03n04) ◽  
pp. 97-120 ◽  
Author(s):  
VITTORIO BILÒ
Keyword(s):  
Nash Equilibrium ◽  
Cost Sharing ◽  
Price Of Anarchy ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Price Of Stability ◽  
Common Source ◽  
Cost Share ◽  
Worst Case ◽  
The Cost

We consider the problem of sharing the cost of multicast transmissions in non-cooperative undirected networks where a set of receivers R wants to be connected to a common source s. The set of choices available to each receiver r ∈ R is represented by the set of all (s, r)-paths in the network. Given the choices performed by all the receivers, a public known cost sharing method determines the cost share to be charged to each of them. Receivers are selfish agents aiming to obtain the transmission at the minimum cost share and their interactions create a non-cooperative game. Devising cost sharing methods yielding games whose price of anarchy (price of stability), defined as the worst-case (best-case) ratio between the cost of a Nash equilibrium and that of an optimal solution, is not too high is thus of fundamental importance in non-cooperative network design. Moreover, since cost sharing games naturally arise in socio-economical contests, it is convenient for a cost sharing method to meet some constraining properties. In this paper, we first define several such properties and analyze their impact on the prices of anarchy and stability. We also reconsider all the methods known so far by classifying them according to which properties they satisfy and giving the first non-trivial lower bounds on their price of stability. Finally, we propose a new method, namely the free-riders method, which admits a polynomial time algorithm for computing a pure Nash equilibrium whose cost is at most twice the optimal one. Some of the ideas characterizing our approach have been independently proposed in Ref. 10.

scholarly journals Exact Price of Anarchy for Polynomial Congestion Games

2011 ◽  
Vol 40 (5) ◽  
pp. 1211-1233 ◽  
Author(s):  
Sebastian Aland ◽  
Dominic Dumrauf ◽  
Martin Gairing ◽  
Burkhard Monien ◽  
Florian Schoppmann
Keyword(s):  
Price Of Anarchy ◽  
Congestion Games

The sequential price of anarchy for affine congestion games with few players

2019 ◽  
Vol 47 (2) ◽  
pp. 133-139
Author(s):  
Jasper de Jong ◽  
Marc Uetz
Keyword(s):  
Price Of Anarchy ◽  
Congestion Games

scholarly journals Worst case examples of an exterior point algorithm for the assignment problem

2008 ◽  
Vol 5 (3) ◽  
pp. 605-614 ◽  
Author(s):  
Charalampos Papamanthou ◽  
Konstantinos Paparrizos ◽  
Nikolaos Samaras ◽  
Konstantinos Stergiou
Keyword(s):  
Assignment Problem ◽  
Worst Case ◽  
Case Examples
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