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.

Linear–Quadratic Time-Inconsistent Mean Field Games

2013 ◽  
Vol 3 (4) ◽  
pp. 537-552 ◽  
Author(s):  
A. Bensoussan ◽  
K. C. J. Sung ◽  
S. C. P. Yam
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Linear Quadratic ◽  
Time Inconsistent

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 Linear quadratic mean field game with control input constraint

2018 ◽  
Vol 24 (2) ◽  
pp. 901-919 ◽  
Author(s):  
Ying Hu ◽  
Jianhui Huang ◽  
Xun Li
Keyword(s):  
Backward Stochastic Differential Equation ◽  
Consistency Condition ◽  
Mean Field ◽  
Control Process ◽  
Mean Field Games ◽  
Monotonicity Condition ◽  
Projection Operators ◽  
Control Input ◽  
Linear Quadratic ◽  
The Individual

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset Γ of full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on Γ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related ϵ-Nash equilibrium property is also verified.

scholarly journals Linear-Quadratic Mean Field Games

2015 ◽  
Vol 169 (2) ◽  
pp. 496-529 ◽  
Author(s):  
A. Bensoussan ◽  
K. C. J. Sung ◽  
S. C. P. Yam ◽  
S. P. Yung
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Linear Quadratic ◽  
Quadratic Mean

scholarly journals PERFORMANCE OF NON-COOPERATIVE ROUTING OVER PARALLEL NON-OBSERVABLE QUEUES

2016 ◽  
Vol 30 (3) ◽  
pp. 455-469 ◽  
Author(s):  
Olivier Brun
Keyword(s):  
Autonomic Computing ◽  
Price Of Anarchy ◽  
Service Discipline ◽  
Case Scenario ◽  
Selfish Behavior ◽  
Fixed Amount ◽  
Worst Case ◽  
Computer Services ◽  
The Mean ◽  
The Impact

Autonomic computing is emerging as a significant new approach to the design of computer services. Its goal is the development of services that are able to manage themselves with minimal direct human intervention, and, in particular, are able to sense their environment and to tune themselves to meet end-user needs. However, the impact on performance of the interaction between multiple uncoordinated self-optimizing services is not yet well understood. We present some recent results on a non-cooperative load-balancing game which help to better understand the result of this interaction. In this game, users generate jobs of different services, and the jobs have to be processed on one of the servers of a computing platform. Each service has its own dispatcher which probabilistically routes jobs to servers so as to minimize the mean processing cost of its own jobs. We first investigate the impact of heterogeneity in the amount of incoming traffic routed by dispatchers and present a result stating that, for a fixed amount of total incoming traffic, the worst-case overall performance occurs when each dispatcher routes the same amount of traffic. Using this result we then study the so-called Price of Anarchy (PoA), an oft-used worst-case measure of the inefficiency of non-cooperative decentralized architectures. We give explicit bounds on the PoA for cost functions representing the mean delay of jobs when the service discipline is PS or SRPT. These bounds indicate that significant performance degradations can result from the selfish behavior of self-optimizing services. In practice, though, the worst-case scenario may occur rarely, if at all. Some recent results suggest that for the game under consideration the PoA is an overly pessimistic measure that does not reflect the performance obtained in most instances of the problem.

scholarly journals Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations

Games ◽  
2018 ◽  
Vol 9 (4) ◽  
pp. 88 ◽  
Author(s):  
Alexander Aurell
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Social Cost ◽  
Price Of Anarchy ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Regularity Conditions ◽  
Field Type ◽  
Cost Functionals

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

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