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 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.

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.

UNIQUENESS IN RANDOM-PROPOSER MULTILATERAL BARGAINING

2009 ◽  
Vol 11 (04) ◽  
pp. 407-417 ◽  
Author(s):  
HUIBIN YAN
Keyword(s):  
Subgame Perfect Equilibrium ◽  
Characteristic Functions ◽  
Bargaining Model ◽  
Equilibrium Outcome ◽  
Multilateral Bargaining ◽  
Legislative Bargaining ◽  
Uniqueness Results ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria ◽  
Special Case

Solution uniqueness is an important property for a bargaining model. Rubinstein's (1982) seminal 2-person alternating-offer bargaining game has a unique Subgame Perfect Equilibrium outcome. Is it possible to obtain uniqueness results in the much enlarged setting of multilateral bargaining with a characteristic function? This paper investigates a random-proposer model first studied in Okada (1993) in which each period players have equal probabilities of being selected to make a proposal and bargaining ends after one coalition forms. Focusing on transferable utility environments and Stationary Subgame Perfect Equilibria (SSPE), we find ex ante SSPE payoff uniqueness for symmetric and convex characteristic functions, considerably expanding the conditions under which this model is known to exhibit SSPE payoff uniqueness. Our model includes as a special case a variant of the legislative bargaining model in Baron and Ferejohn (1989), and our results imply (unrestricted) SSPE payoff uniqueness in this case.

BARGAINING MODEL WITH SEQUENCES OF DISCOUNT RATES AND BARGAINING COSTS

2004 ◽  
Vol 06 (02) ◽  
pp. 265-280 ◽  
Author(s):  
AGNIESZKA RUSINOWSKA
Keyword(s):  
Discount Rate ◽  
Necessary Conditions ◽  
Subgame Perfect Equilibrium ◽  
Bargaining Model ◽  
Perfect Equilibrium ◽  
Discount Rates ◽  
Sufficient And Necessary Conditions ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

The paper is a kind of generalization of Rubinstein bargaining model. Rubinstein assumed that preferences of the players were constant in time, and he analyzed models in which preferences of each player were defined either by constant discount rate or by constant bargaining cost. In this paper, a bargaining model is presented, in which preferences of each player are expressed simultaneously by sequence of discount rates and sequence of bargaining costs varying in time. The results presented in the paper concern subgame perfect equilibria. There is a theorem concerning sufficient and necessary conditions for the existence of subgame perfect equilibrium of the game. Moreover, some theorems presenting forms of subgame perfect equilibria for various cases of the model analyzed have been proved here. A possibility of delay in reaching an agreement is also considered in the paper. If we analyze a class of strategies, that depend on the former history, a delay can appear for some models. The adequate examples are presented. In the paper, some applications of the bargaining model are also described.

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 Symmetric Subgame-Perfect Equilibria in Resource Allocation

2014 ◽  
Vol 49 ◽  
pp. 323-361 ◽  
Author(s):  
L. Cigler ◽  
B. Faltings
Keyword(s):  
Resource Allocation ◽  
Subgame Perfect Equilibrium ◽  
Efficient Allocation ◽  
Perfect Equilibrium ◽  
Use Of Resources ◽  
Strategy Profile ◽  
Global Coordination ◽  
Resource Allocation Games ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

We analyze symmetric protocols to rationally coordinate on an asymmetric, efficient allocation in an infinitely repeated N-agent, C-resource allocation problems, where the resources are all homogeneous. Bhaskar proposed one way to achieve this in 2-agent, 1-resource games: Agents start by symmetrically randomizing their actions, and as soon as they each choose different actions, they start to follow a potentially asymmetric "convention" that prescribes their actions from then on. We extend the concept of convention to the general case of infinitely repeated resource allocation games with N agents and C resources. We show that for any convention, there exists a symmetric subgame-perfect equilibrium which implements it. We present two conventions: bourgeois, where agents stick to the first allocation; and market, where agents pay for the use of resources, and observe a global coordination signal which allows them to alternate between different allocations. We define price of anonymity of a convention as a ratio between the maximum social payoff of any (asymmetric) strategy profile and the expected social payoff of the subgame-perfect equilibrium which implements the convention. We show that while the price of anonymity of the bourgeois convention is infinite, the market convention decreases this price by reducing the conflict between the agents.

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