Averaging principle for second-order approximation of heterogeneous models with homogeneous models

Abstract

Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(e 2 ) equivalent to the outcome of the corresponding homogeneous model, where e is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing).homogenization | perturbation methods M athematical modeling is a powerful tool in scientific research.Typically, the mathematical model is merely an approximation of the actual problem. Therefore, when choosing the model to work with, one has to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulations. This dilemma arises, for example, when the model contains a heterogeneous quantity. In such cases, a huge simplification is usually achieved by replacing the heterogeneous quantity with its average value. The natural question that arises is whether this approximation is "legitimate," i.e., whether the error that is introduced by this approximation is sufficiently small.Let us illustrate this with the following example, which is discussed in detail below. Consider a queue with k heterogeneous servers, whose expected service times are μ 1 , . . . , μ k . We want to calculate analytically the expected number of customers in the system, which we denote* by F(μ 1 , . . . , μ k ). Although an explicit expression for F(μ 1 , . . . , μ k ) is not available, there is a well-known explicit expression in the case of k homogeneous servers, which we denote by F homog:for the expected number of customers in the system iswhere μ is the average of {μ 1 , . . . , μ k }. More generally, let F(μ 1 , . . . , μ k ) denote the "outcome" of a heterogeneous model, letdenote the level of heterogeneity of {μ 1 , . . . , μ k }, and let F homog. (μ) denote the outcome of the corresponding homogeneous model. If the function F(μ 1 , . . . , μ k ) is differentiable, then it immediately follows thatTherefore, for a 10% heterogeneity level, the error of approximating F(μ 1 , . . . , μ k ) with F homog: ðμÞ is, roughly speaking, on the order of 10%. In many studies in different fields, however, researchers have noted that the error of this approximation is considerably smaller than e. Moreover, this observation seems to hold even when the level of heterogeneity is not small. In this study we show that these observations follow from a general principle, which we call the averaging principle. Specifically, we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(e 2 ) asymptotically equivalent to the outcome of the corresponding homogeneous model; i.e.,Thus, if the function F is also symmetric, the error of the approximation in Eq. 1 for a 1...

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“…The idea presented here is inspired from the works in [21,22,20] on the so-called averaging principle. These previous works are limited to static and one-shot games.…”

Nonasymptotic Mean-Field Games

“…The idea presented here is inspired from the works in [21,22,20] on the so-called averaging principle. These previous works are limited to static and one-shot games.…”