[Paper Review] Selfishness need not be bad

Zijun Wu, H Moehring Rolf+2 more|arXiv (Cornell University)|Dec 20, 2017

Game Theory and Applications12 references3 citations

TL;DR

This paper develops a new analytical framework to study the limit behavior of the price of anarchy (PoA) in non-atomic congestion games as total demand T grows large. It proves that the PoA converges to 1 for a broad class of cost functions—including all polynomials and regularly varying functions—regardless of the growth pattern of T, and further shows that for BPR cost functions, PoA = 1 + o(T⁻ᵝ), with convergence rates dependent on the demand growth sequence, refuting a prior conjecture on universal power-law convergence.

ABSTRACT

We investigate the price of anarchy (PoA) in non-atomic congestion games when the total demand $T$ gets very large. First results in this direction have recently been obtained by \cite{Colini2016On, Colini2017WINE, Colini2017arxiv} for routing games and show that the PoA converges to 1 when the growth of the total demand $T$ satisfies certain regularity conditions. We extend their results by developing a \Wuuu{new} framework for the limit analysis of \Wuuuu{the PoA that offers strong techniques such as the limit of games and applies to arbitrary growth patterns of $T$.} \Wuuu{We} show that the PoA converges to 1 in the limit game regardless of the type of growth of $T$ for a large class of cost functions that contains all polynomials and all regularly varying functions. % For routing games with BPR \Wuu{cost} functions, we show in addition that socially optimal strategy profiles converge to \Wuu{equilibria} in the limit game, and that PoA$=1+o(T^{-β})$, where $β>0$ is the degree of the \Wuu{BPR} functions. However, the precise convergence rate depends crucially on the the growth of $T$, which shows that a conjecture proposed by \cite{O2016Mechanisms} need not hold.

Motivation & Objective

• To analyze the asymptotic behavior of the price of anarchy (PoA) in non-atomic congestion games as total demand T tends to infinity.
• To extend prior results on PoA convergence by developing a general framework applicable to arbitrary growth patterns of T.
• To investigate whether the PoA converges to 1 for a broad class of cost functions, including polynomials and regularly varying functions.
• To examine the convergence rate of PoA for routing games with BPR cost functions and assess the validity of a conjecture by [18] on power-law convergence.
• To empirically validate theoretical findings using real traffic data from Beijing’s 2nd ring road.

Proposed method

• Introduces the concept of 'limit games' to analyze the asymptotic behavior of congestion games under large-scale demand.
• Develops the notion of 'scalable games by decomposition' to characterize convergence of PoA to 1 across diverse demand growth patterns.
• Applies the framework to non-atomic congestion games with general resource consumption patterns, generalizing from standard routing games.
• Uses normalized travel time functions and limit ratios αₐ = limₓ→∞ τₐ(x)/g(x) to classify arcs and paths as fast, slow, or tight.
• Employs asymptotic decomposition techniques to analyze the structure of equilibria and social optima in the limit.
• Validates theoretical results through an experimental study using real traffic data from Beijing’s 2nd ring road.

Experimental results

Research questions

• RQ1 Does the price of anarchy (PoA) converge to 1 as total demand T grows large, regardless of the growth pattern of T?
• RQ2 Can the convergence of PoA to 1 be established for all polynomial and regularly varying cost functions?
• RQ3 What is the precise convergence rate of PoA for routing games with BPR cost functions, and does it follow a universal power law as conjectured by [18]?
• RQ4 Are socially optimal strategy profiles asymptotically close to equilibria in large-scale congestion games?
• RQ5 Can the theoretical framework be empirically validated using real-world traffic data?

Key findings

• The price of anarchy (PoA) converges to 1 in the limit game for all non-atomic congestion games with cost functions that are polynomials or regularly varying functions, regardless of the growth pattern of total demand T.
• For routing games with BPR cost functions of degree β > 0, the PoA converges to 1 as 1 + o(T⁻ᵝ), with the convergence rate depending on the specific growth sequence of T.
• The conjecture by [18] that PoA follows a universal power law of order O(T⁻²ᵝ) does not hold in general, as convergence rates vary significantly with the demand growth sequence.
• Socially optimal strategy profiles in BPR-based routing games are ε-approximate equilibria for small ε > 0, with ε → 0 as T increases.
• Empirical analysis of real traffic data from Beijing’s 2nd ring road confirms that current traffic conditions are already in a regime where PoA ≈ 1, indicating minimal inefficiency under selfish routing.
• The study identifies open conjectures on the relationship between tight games and regularly varying cost functions, suggesting that not all asymptotically well-designed games are scalable by decomposition.