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.

scholarly journals Price of anarchy for Mean Field Games

2019 ◽  
Vol 65 ◽  
pp. 349-383 ◽  
Author(s):  
René Carmona ◽  
Christy V. Graves ◽  
Zongjun Tan
Keyword(s):  
Social Cost ◽  
Price Of Anarchy ◽  
Mean Field ◽  
Mean Field Games ◽  
Necessary Condition ◽  
Linear Quadratic ◽  
Selfish Behavior ◽  
Worst Case ◽  
Game Equilibrium ◽  
Routing Games

The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. A sufficient and necessary condition to have no price of anarchy is presented. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are presented.

scholarly journals Selfishness Level of Strategic Games

2014 ◽  
Vol 49 ◽  
pp. 207-240 ◽  
Author(s):  
K. R. Apt ◽  
G. Schaefer
Keyword(s):  
Nash Equilibrium ◽  
Social Welfare ◽  
Price Of Anarchy ◽  
Congestion Games ◽  
Price Of Stability ◽  
Social Optimum ◽  
Potential Game ◽  
Strategic Games ◽  
The Social ◽  
The Impact

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games. We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate. In particular, the selfishness level of finite ordinal potential games is finite, while that of weakly acyclic games can be infinite. We derive explicit bounds on the selfishness level of fair cost sharing games and linear congestion games, which depend on specific parameters of the underlying game but are independent of the number of players. Further, we show that the selfishness level of the $n$-players Prisoner's Dilemma is c/(b(n-1)-c), where b and c are the benefit and cost for cooperation, respectively, that of the n-players public goods game is (1-c/n)/(c-1), where c is the public good multiplier, and that of the Traveler's Dilemma game is (b-1)/2, where b is the bonus. Finally, the selfishness level of Cournot competition (an example of an infinite ordinal potential game), Tragedy of the Commons, and Bertrand competition is infinite.

Balancing Load via Small Coalitions in Selfish Ring Routing Games

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.

scholarly journals Selfishness Need Not Be Bad

2021 ◽  
Author(s):  
Zijun Wu ◽  
Rolf H. Möhring ◽  
Yanyan Chen ◽  
Dachuan Xu
Keyword(s):  
Price Of Anarchy ◽  
Empirical Studies ◽  
Growth Patterns ◽  
Congestion Games ◽  
Cost Functions ◽  
Recent Analysis ◽  
Selfish Routing ◽  
Public Road ◽  
Routing Games ◽  
New Framework

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.

scholarly journals Stackelberg pricing games with congestion effects

2021 ◽  
Author(s):  
Tobias Harks ◽  
Anja Schedel
Keyword(s):  
Cost Function ◽  
Stackelberg Game ◽  
Pure Strategy ◽  
Closed Form Expression ◽  
Social Optimum ◽  
Service Capacity ◽  
Worst Case ◽  
Congestion Cost ◽  
Pricing Games ◽  
Congestion Effects

AbstractWe study a Stackelberg game with multiple leaders and a continuum of followers that are coupled via congestion effects. The followers’ problem constitutes a nonatomic congestion game, where a population of infinitesimal players is given and each player chooses a resource. Each resource has a linear cost function which depends on the congestion of this resource. The leaders of the Stackelberg game each control a resource and determine a price per unit as well as a service capacity for the resource influencing the slope of the linear congestion cost function. As our main result, we establish existence of pure-strategy Nash–Stackelberg equilibria for this multi-leader Stackelberg game. The existence result requires a completely new proof approach compared to previous approaches, since the leaders’ objective functions are discontinuous in our game. As a consequence, best responses of leaders do not always exist, and thus standard fixed-point arguments á la Kakutani (Duke Math J 8(3):457–458, 1941) are not directly applicable. We show that the game is C-secure (a concept introduced by Reny (Econometrica 67(5):1029–1056, 1999) and refined by McLennan et al. (Econometrica 79(5):1643–1664, 2011), which leads to the existence of an equilibrium. We furthermore show that the equilibrium is essentially unique, and analyze its efficiency compared to a social optimum. We prove that the worst-case quality is unbounded. For identical leaders, we derive a closed-form expression for the efficiency of the equilibrium.

scholarly journals Evaluation and Analysis of CFI Schemes with Different Length of Displaced Left-Turn Lanes with Entropy Method

2021 ◽  
Vol 13 (12) ◽  
pp. 6917
Author(s):  
Binghong Pan ◽  
Shasha Luo ◽  
Jinfeng Ying ◽  
Yang Shao ◽  
Shangru Liu ◽  
...  
Keyword(s):  
Fuel Consumption ◽  
Heavy Traffic ◽  
Efficiency Index ◽  
Entropy Method ◽  
Sustainable Transportation ◽  
Assessment Model ◽  
Left Turn ◽  
Traffic Demand ◽  
Traffic Efficiency ◽  
Multi Attribute Decision Making

As an unconventional design to alleviate the conflict between left-turn and through vehicles, Continuous Flow Intersection (CFI) has obvious advantages in improving the sustainability of roadway. So far, the design manuals and guidelines for CFI are not enough sufficient, especially for the displaced left-turn lane length of CFI. And the results of existing research studies are not operational, making it difficult to put CFI into application. To address this issue, this paper presents a methodological procedure for determination and evaluation of displaced left-turn lane length based on the entropy method considering multiple performance measures for sustainable transportation, including traffic efficiency index, environment effect index and fuel consumption. VISSIM and the surrogate safety assessment model (SSAM) were used to simulate the operational and safety performance of CFI. The multi-attribute decision-making method (MADM) based on an entropy method was adopted to determine the suitability of the CFI schemes under different traffic demand patterns. Finally, the procedure was applied to a typical congested intersection of the arterial road with heavy traffic volume and high left-turn ratio in Xi’an, China, the results showed the methodological procedure is reasonable and practical. According to the results, for the studied intersection, when the Volume-to-Capacity ratio (V/C) in the westbound and eastbound lanes is less than 0.5, the length of the displaced left-turn lanes can be selected in the range of 80 to 170 m. Otherwise, other solutions should be considered to improve the traffic efficiency. The simulation results of the case showed CFI can significantly improve the traffic efficiency. In the best case, compared with the conventional intersection, the number of vehicles increases by 13%, delay, travel time, number of stops, CO emission, and fuel consumption decrease by 41%, 29%, 25%, 17%, and 17%, respectively.

Sign in / Sign up

Export Citation Format

Share Document