Records 
Author 
ColiniBaldeschi, R.; Cominetti, R.; Mertikopoulos, P.; Scarsini, M. 
Title 
When Is Selfish Routing Bad? The Price of Anarchy in Light and Heavy Traffic 
Type 

Year 
2020 
Publication 
Operations Research 
Abbreviated Journal 
Oper. Res. 
Volume 
68 
Issue 
2 
Pages 
411434 
Keywords 
nonatomic congestion games; price of anarchy; light traffic; heavy traffic; regular variation 
Abstract 
This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin/destination (O/D) pairs. Empirical studies in realworld networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the following question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple threelink networks with a single O/D pair and smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials) and inflow patterns, the price of anarchy does converge to 1 in both heavy and light traffic, irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree can be computed explicitly when the network's cost functions are polynomials. 
Address 
[ColiniBaldeschi, Riccardo] Facebook Inc, Core Data Sci Grp, London W1T 1FB, England, Email: rickuz@fb.com; 
Corporate Author 

Thesis 

Publisher 
Informs 
Place of Publication 

Editor 

Language 
English 
Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
0030364x 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000521730200006 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
1128 
Permanent link to this record 



Author 
ColiniBaldeschi, R.; Cominetti, R.; Scarsini, M. 
Title 
Price of Anarchy for Highly Congested Routing Games in Parallel Networks 
Type 

Year 
2019 
Publication 
Theory Of Computing Systems 
Abbreviated Journal 
Theor. Comput. Syst. 
Volume 
63 
Issue 
1 
Pages 
90113 
Keywords 
Nonatomic routing games; Price of Anarchy; Regularly varying functions; Wardrop equilibrium; Parallel networks; High congestion 
Abstract 
We consider nonatomic routing games with one source and one destination connected by multiple parallel edges. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we prove that under suitable conditions on the costs the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case, and that these counterexamples already occur in simple networks with only 2 parallel links. 
Address 
[ColiniBaldeschi, Riccardo; Scarsini, Marco] LUISS, Dipartimento Econ & Finanza, Viale Romania 32, I00197 Rome, Italy, Email: rcolini@luiss.it; 
Corporate Author 

Thesis 

Publisher 
Springer 
Place of Publication 

Editor 

Language 
English 
Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
14324350 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000456320200005 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
974 
Permanent link to this record 



Author 
Cominetti, R.; Dose, V.; Scarsini, M. 
Title 
The price of anarchy in routing games as a function of the demand 
Type 

Year 
2022 
Publication 
Mathematical Programming 
Abbreviated Journal 
Math. Program. 
Volume 
Early Access 
Issue 

Pages 

Keywords 
Nonatomic routing games; Price of anarchy; Affine cost functions; Variable demand 
Abstract 
The 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 worstcase 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. 
Address 

Corporate Author 

Thesis 

Publisher 

Place of Publication 

Editor 

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Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
00255610 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000693858200001 
Approved 

Call Number 
UAI @ alexi.delcanto @ 
Serial 
1468 
Permanent link to this record 



Author 
Cominetti, R.; Quattropani, M.; Scarsini, M. 
Title 
The BuckPassing Game 
Type 

Year 
2022 
Publication 
Mathematics Operations Research 
Abbreviated Journal 
Math. Oper. Res. 
Volume 
Early Access 
Issue 

Pages 

Keywords 
priorfree equilibrium; generalized ordinal potential game; finite improvement property; fairness of equilibria; price of anarchy; price of stability; Markov chain tree theorem; PageRank; PageRank game 
Abstract 
We consider two classes of games in which players are the vertices of a directed graph. Initially, nature chooses one player according to some fixed distribution and gives the player a buck. This player passes the buck to one of the player's outneighbors in the graph. The procedure is repeated indefinitely. In one class of games, each player wants to minimize the asymptotic expected frequency of times that the player receives the buck. In the other class of games, the player wants to maximize it. The PageRank game is a particular case of these maximizing games. We consider deterministic and stochastic versions of the game, depending on how players select the neighbor to which to pass the buck. In both cases, we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. If the graph on which the game is played admits a Hamiltonian cycle, then this is the outcome of priorfive Nash equilibrium in the minimizing game. For the minimizing game, we then use the price of anarchy and stability to measure fairness of these equilibria. 
Address 

Corporate Author 

Thesis 

Publisher 

Place of Publication 

Editor 

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Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
0364765X 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000731930100001 
Approved 

Call Number 
UAI @ alexi.delcanto @ 
Serial 
1502 
Permanent link to this record 



Author 
Cominetti, R.; Scarsini, M.; Schroder, M.; StierMoses, N. 
Title 
Approximation and Convergence of Large Atomic Congestion Games 
Type 

Year 
2022 
Publication 
Mathematics of Operations Research 
Abbreviated Journal 
Math. Oper. Res. 
Volume 
Early Access 
Issue 

Pages 

Keywords 
unsplittable atomic congestion games; nonatomic congestion games; Wardrop equilibrium; Poisson games; symmetric equilibrium; price of anarchy; price of stability; total variation 
Abstract 
We consider the question of whether and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider games in which each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider an increasing number of players with a unit weight that participate in the game with a decreasingly small probability. In this case, the Nash equilibrium flows converge in total variation toward Poisson random variables whose expected values are War drop equilibria of a different nonatomic game with suitably defined costs. The latter can be viewed as symmetric equilibria in a Poisson game in the sense of Myerson, establishing a plausible connection between the Wardrop model for routing games and the stochastic fluctuations observed in real traffic. In both settings, we provide explicit approximation bounds, and we study the convergence of the price of anarchy. Beyond the case of congestion games, we prove a general result on the convergence of large games with random players toward Poisson games. 
Address 

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Publisher 

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Editor 

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Summary Language 

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Series Issue 

Edition 

ISSN 
0364765X 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000850694300001 
Approved 

Call Number 
UAI @ alexi.delcanto @ 
Serial 
1647 
Permanent link to this record 