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(ɛ²) equivalent to the outcome of the corresponding homogeneous model, where ɛ 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).

Mathematical 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 μ₁, … , μ_k. We want to calculate analytically the expected number of customers in the system, which we denote* by F(μ₁, … , μ_k). Although an explicit expression for F(μ₁, … , μ_k) is not available, there is a well-known explicit expression in the case of k homogeneous servers, which we denote by . A natural approximation for the expected number of customers in the system is

where is the average of {μ₁, … , μ_k}.

More generally, let F(μ₁, … , μ_k) denote the “outcome” of a heterogeneous model, let

denote the level of heterogeneity of {μ₁, … , μ_k}, and let F_homog.(μ) denote the outcome of the corresponding homogeneous model. If the function F(μ₁, … , μ_k) is differentiable, then it immediately follows that

Therefore, for a 10% heterogeneity level, the error of approximating F(μ₁, … , μ_k) with 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 ɛ. 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(ɛ²) 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 10% heterogeneity level is only O(1%).

The averaging principle of this study is unrelated to averaging principles that originate from laws of large numbers for k ≫ 1, such as mean-field approximations or approximations of a continuous population with a large discrete population (see, e.g., refs. 1 and 2). Thus, for example, this principle holds when there are only few servers in a queuing system or few bidders in an asymmetric auction. The averaging principle can be classified as a perturbation method in analysis. Whereas it can also be viewed as an approximation rule, we note that approximation theory is usually more interested in how well a family of simple functions approximates a given complicated function (see, e.g., ref. 3).

Averaging Principle

Let F(μ₁, … , μ_k) be the outcome of a model with a heterogeneous property, captured by the k parameters μ₁, … , μ_k, that satisfies the following two properties:

• i) Differentiability: F is twice continuously differentiable at and near the diagonal μ₁ = … = μ_k.

• ii) Symmetry: For every (μ₁, … μ_k) ∈ ℝ^k and every i ≠ j, F(…, μ_i, …, μ_j, …) = F(…, μ_j, …, μ_i, …). Thus, the outcome F is independent of the order in which we list the heterogeneous parameters.^†

Then, we have the following result:^‡

Theorem 1 (the Averaging Principle).

Let F be symmetric in its arguments and twice continuously differentiable. Then, there exist two positive functions δ(x) and C(x) such that the following holds. Let μ = (μ₁, … , μ_k) be sufficiently close to the diagonal; i.e.,

where , is the arithmetic average, and ‖⋅‖ is a vector norm on ℝ^k. Then,

where .

Theorem 1 remains valid if μ₁, … , μ_k are functions and not scalars; see Game Theory Application: Assymetric Auctions below.

Using the Geometric and Harmonic Averages.

In Theorem 1, we averaged μ₁, … , μ_k using the arithmetic mean. It is well known in homogenization theory that in some cases the correct homogenization is provided by the geometric or the harmonic mean. To address the question of the “correct” averaging, we recall the following result.

Lemma 1.

Let μ > 0, and let h₁, … , h_k ∈ ℝ. Then, as ɛ → 0, the arithmetic, geometric, and harmonic means of μ + ɛh₁, … , μ + ɛh_k are O(ɛ²) asymptotically equivalent.

Proof:

We can prove this result, using the averaging principle. Let denote the arithmetic mean of μ + ɛh₁, … , μ + ɛh_k. The geometric mean satisfies the symmetry and differentiability properties. Therefore, application of Theorem 1 gives

The proof for the harmonic mean is similar.

From Lemma 1 and the differentiability of F_homog.(μ) it follows that

Corollary 2.

The averaging principle (Theorem 1) remains valid if we replace the arithmetic mean with the geometric mean or the harmonic mean. In the former case, we add the condition that μ has positive coordinates.

A natural question is which of the three averages is “optimal,” in the sense that it minimizes the constant C in Eq. 4. The answer to this question is model specific. It can be pursued by calculating explicitly the O(ɛ²) term, as we do later on.

To extend the scope of the averaging principle, we define a weaker symmetry property.^§

Weak Symmetry. For every μ and η, the outcome is independent of the value of j for 1 ≤ j ≤ k, where is the jth unit vector in ℝ^k.

Thus, F(μ₁, … , μ_k) is weakly symmetric if, whenever all but one of the parameters are identical, the outcome F is independent of the identity (coordinate) of the heterogeneous parameter.

Every symmetric function F is also weakly symmetric, but not vice versa.^¶ Nevertheless, the proof of Theorem 1 implies the following:

Corollary 3.

The averaging principle (Theorem 1) remains valid if we replace the assumption of symmetry with the assumption of weak symmetry.

Queuing Theory Application: An M/M/k Queue with Heterogeneous Service Rates

Consider a system with k servers. Server i has a random service time that is distributed according to an exponential distribution with rate μ_i. Customers arrive randomly according to a Poisson distribution with arrival rate λ. An arriving customer is randomly allocated to one of the nonbusy servers, if such a server exists. Otherwise, the customer joins a waiting queue, which is unbounded in length. Once a customer is allocated to a server, he gets the service he needs and then leaves the system. This setup is known in the Queuing literature as the M/M/k model.^║ Examples for such multiserver queuing systems are call centers, queues in banks, parallel computing, and communications in Integrated Services Digital Network (ISDN) protocols.

Let F(μ₁, … , μ_k) denote the expected number of customers in the system (i.e., waiting in the queue or receiving service) in steady state. In the case of two heterogeneous servers, F(μ₁, μ₂) can be explicitly calculated (SI Text):

Lemma 2.

Consider an M/M/2 queue with heterogeneous servers, where . Then, the expected number of customers in the system is given by

Finding an explicit solution for F(μ₁, … , μ_k) when k ≥ 3 is computationally challenging, because it involves solving a system of 2^k − 1 linear equations. In the homogeneous case μ₁ = … = μ_k = μ; however, it is well known that (e.g., ref. 4)

The function F(μ₁, …, μ_k) can be written as a sum of solutions of a system of linear equations with coefficients that depend smoothly on μ₁, … , μ_k (SI Text). Therefore, F is differentiable. Because customers are randomly allocated to the free servers, renaming the servers does not affect the expected number of customers in the system. Hence, F is also symmetric. Therefore, we can use the averaging principle to obtain an explicit O(ɛ²) approximation for F(μ₁, … , μ_k):

Theorem 4.

Consider an M/M/k queue with heterogeneous servers whose service rates are μ₁, … , μ_k. The expected number of customers in the system is given by

where is given by Eq. 6, , and ɛ is given by Eq. 2.

For example, by Theorem 4, the expected number of customers with two heterogeneous servers is

where

Indeed, substituting in Eq. 5 and expanding in ɛ gives Eq. 7.

In the case of k = 8 heterogeneous servers, even writing the system of 2⁸ − 1 = 255 equations for the 255 unknowns is a formidable task, not to mention solving it explicitly. By the averaging principle, however,

where is given by Eq. 6 with k = 8.

We ran stochastic simulations of an M/M/8 queuing system with eight heterogeneous servers, using the ARENA simulation software, and used it to calculate the expected number of customers in the system. The simulation parameters were

and ɛ varies between 0 and 1 in increments of 0.05. Because , the average service rate is . Therefore, by Theorem 4,

In addition, by Eq. 6, F_homog.(5) = 6.2314.

To illustrate the accuracy of this approximation, we plot in Fig. 1 the relative error of the averaging-principle approximation . As expected, this error scales as ɛ². Note that even when the heterogeneity is not small, the averaging-principle approximation is quite accurate. This is because the coefficient (0.594) of the O(ɛ²) term is small.** For example, when ɛ = 0.5, the relative error is ∼2%, and for ɛ = 1 it is below 10%.

Fig. 1.

The relative error of the averaging-principle approximation for the steady-state number of customers in a system with eight heterogeneous servers, as a function of the heterogeneity parameter ɛ. The solid line is error = 0.594ɛ². The crosses denote the relative error of the improved approximation given by Eq. 16. The dotted line is error = 0.074ɛ³.

Remark:

We can also use the averaging principle to obtain O(ɛ²) approximations to other quantities of interest that satisfy the symmetry property, such as the average waiting time in the queue or the probability that there are exactly m customers in the queue.

Game Theory Application: Asymmetric Auctions

Consider a sealed-bid first-price auction with k bidders, in which the bidder who places the highest bid wins the object and pays his bid, and all other bidders pay nothing.^†† A common assumption in auction theory is that of independent private-value auctions, which says that each bidder knows his own valuation for the object, does not know the valuation of the other bidders, but does know the cumulative distribution functions (CDF) of the valuations of the other bidders. Bidders are also characterized by their attitude toward risk: The literature usually assumes that bidders are risk neutral, because this simplifies the analysis. More often than not, however, bidders are risk averse.

A strategy of bidder i is a function b_i that assigns a bid b_i(υ_i) to each possible valuation υ_i of that bidder. The bid that a bidder places depends on his valuation υ_i and on his beliefs about the distributions of the valuations of the other bidders and about their bidding behavior. An equilibrium in this setup is a vector of k strategies , such that no single bidder can profit by deviating from his bidding strategy, whatever his valuation might be, as long as all other bidders follow their equilibrium bidding strategies.

Most of the auction literature focuses on the symmetric (homogeneous) case, in which the beliefs of any bidder about any other bidder (e.g., about his distribution of valuations, his attitude toward risk, etc.) are the same. In this case, one can look for a symmetric equilibrium, in which all bidders adopt the same strategy. In practice, however, bidders are usually asymmetric (heterogeneous), both in their attitude toward risk and in the distribution of their valuations. Each bidder then faces a different competition. As a result, the equilibrium strategies of the bidders are not the same.

The addition of asymmetry usually leads to a huge complication in the analysis. For example, consider a first-price auction for a single object with risk-neutral bidders that have private values that are independently distributed in the unit interval [0, 1] according to a common cumulative distribution function F. Denote by b the symmetric Nash equilibrium bidding strategy. Then υ:= b⁻¹, the inverse of b, satisfies the ordinary differential equation (ODE)^‡‡

This equation can be solved explicitly, yielding

Therefore, this case is “completely understood.” From the seller’s point of view, a key property of an auction is his expected revenue. In the symmetric case, Eq. 8 can be used to calculate the seller’s expected revenue R_homog.[F], yielding

In the asymmetric case, where the value of bidder i is independently distributed in [0, 1] according to F_i, the inverse equilibrium strategies are the solutions of the system of ODEs,

for i = 1, …, k, subject to the initial conditions

and the “end condition” at some unknown ,

Thus, the addition of asymmetry leads to a huge complication of the mathematical model: Instead of a single ODE that can be explicitly integrated, the mathematical model consists of a system of coupled nonlinear ODEs with a nonstandard boundary condition. As a result, Eq. 10 cannot be explicitly solved, and it is poorly understood, compared with the symmetric case.

In ref. 6, Fibich and Gavious considered Eq. 10 in the weakly asymmetric case F_i = F + ɛH_i, i = 1, … , k. After several pages of informal perturbation-analysis calculations, they obtained O(ɛ²) asymptotic approximations of the inverse equilibrium strategies . Substituting these approximations in the expression for the seller’s expected revenue showed that it is given by

This is, in fact, a special case of the averaging principle. Indeed, symmetry holds because changing the indexes of the bidders does not affect the revenue. Therefore, assuming differentiability, the averaging principle for functions (SI Text) yields

where . Substituting in Eq. 9 and expanding in powers of ɛ gives

Hence, Eq. 11 follows.

In ref. 7, Lebrun rigorously proved that the asymmetric equilibrium bids and the expected revenue are once differentiable. Lebrun noted that this proves Eq. 11 [with o(ɛ) error instead of O(ɛ²)]. In retrospect, this can be viewed as an early application of the averaging principle.

Numerical calculations (ref. 6, table 1, and ref. 8, tables 1 and 2) show that the error of the averaging-principle approximation in Eq. 12 is small (typically below 1%), even when the asymmetry level is not small (e.g., ɛ = 0.4). This provides another illustration that the averaging-principle approximation can be useful even when ɛ is not very small.

The averaging principle not only leads to a simpler derivation of Eq. 11, but also enables us to derive a more general novel result:

Theorem 5.

Consider an anonymous auction^§§ in which all k bidders have the same attitude toward risk, and all bidders follow the same “rules” when they determine their bidding strategies.^¶¶ Let F₁, … , F_k be the cumulative distribution functions of the valuations of the bidders, and let R[F₁, … , F_k] be the expected revenue of the seller. If R is twice differentiable at and near the diagonal, then

where , is the average of F₁, …, F_k, and ɛ is the level of heterogeneity.

Indeed, the assumptions of the theorem imply that F is symmetric. Therefore, if F is also differentiable, the theorem follows from the averaging principle.

Social-Networks Application: Diffusion of New Products

Diffusion of new products is a fundamental problem in marketing, which has been studied in diverse areas such as retail service; industrial technology; agriculture; and educational, pharmaceutical, and consumer-durables markets (10). Typically, the diffusion process begins when the product is first introduced into the market and progresses through a series of adoption events. An individual can adopt the product due to external influences such as mass media or commercials and/or due to internal influences by other individuals who have already adopted the product (word of mouth). The internal influences depend on the underlying social-network structure, because adopters can influence only people that they “know.” The social network is usually modeled by an undirected graph, where each vertex is an individual, and two vertices are connected by an edge if they can influence each other.

The first quantitative analysis of diffusion of new products was the Bass model (11), which inspired a huge body of theoretical and empirical research. In this model and in many of the subsequent product-diffusion models:

• i) A new product is introduced at time t = 0.

• ii) Once a consumer adopts the product, he remains an adopter at all later times.

• iii) If consumer j has not adopted before time t, the probability that he adopts the product in the time interval [t, t + s), given that the product was already adopted by n_j(t) people that are connected to j, and that no other consumer adopts the product in the time interval [t, t + s), is

as s → 0, where m_j is the total number of individuals connected to consumer j and the parameters p_j and q_j describe the likelihood of individual j to adopt the product due to external and internal influences, respectively.

We say that a social network is translation invariant if any individual sees exactly the same network structure. Therefore, in particular, m_j is independent of j. Examples of translation-invariant social networks are as follows (Fig. 2 A–D):

• A) A complete graph, in which any two individuals are connected.

• B) A one-dimensional circle, in which each individual is connected to his two nearest neighbors.

• C) A one-dimensional circle, in which each individual is connected to his four nearest neighbors.

• D) A two-dimensional torus, in which each individual is connected to his four nearest neighbors.

Fig. 2.

(A–D) Examples of translation-invariant networks.

We say that the individuals are homogeneous when all individuals share the same parameters; i.e., p_j = p and q_j = q for every individual j. Let N(t) denote the number of adopters at time t. The expected aggregate adoption curve E_homog.[N(t; p, q)] in several translation-invariant social networks with homogeneous individuals was analytically calculated in refs. 12 and 13. In these studies, the assumption that all individuals are homogeneous was essential for the analysis.

One of the fundamentals of marketing theory is that consumers are anything but homogeneous. An explicit calculation of the expected aggregate adoption curve E[N(t;{p_j},{q_j})] in the heterogeneous case, however, is much harder than in the homogeneous case. As a result, the effect of heterogeneity is not well understood.

The averaging principle allows us to approximate the heterogeneous model with the corresponding homogeneous model. Consider a translation-invariant network. Then, for t ≥ 0 the function F({p_j},{q_j}) := E[N(t;{p_j},{q_j})] is differentiable and weakly symmetric (SI Text). Therefore, by the averaging principle,

Theorem 6.

The expected aggregate adoption curve in a translation-invariant social network with heterogeneous individuals can be approximated with

where and are the averages of {p_j} and {q_j}, respectively, and ɛ is the level of heterogeneity of {p_j} and {q_j}.

Theorem 6 is consistent with previous numerical findings:

• 1) In ref. 14, simulations of an agent-based model with a complete graph showed that heterogeneity in p and q had a minor effect on the expected aggregate adoption curve.

• 2) Simulations of agent-based models with 1D and 2D translation-invariant networks (ref. 12, figure 18) showed that when the values of {p_j} and {q_j} are uniformly distributed within ±20% of the corresponding values and of the homogeneous individuals, the heterogeneous and homogeneous adoption curves are nearly indistinguishable. Even when the heterogeneity level was increased to ±50%, the two adoption curves were still very close.

Calculating the O(ɛ²) Term

The averaging principle is based on a two-term Taylor expansion of F. Therefore, the error of this approximation is given, to leading order, by the quadratic term in this expansion. When F satisfies the differentiability and symmetry properties*** and μ is the arithmetic mean, this error is given by (SI Text)

where

Therefore,

• i) The magnitude of this error is .

• ii) The sign of this error is the same as the sign of α.

A Taylor expansion in h gives

This shows that to determine the sign of α, one can compare the effect of adding h units to two parameters with the corresponding effect of adding 2h units to a single parameter.

The value of α can be calculated as follows:

Lemma 3.

Assume that F(μ₁, … , μ_k) satisfies the differentiability and symmetry properties. Then,

Therefore, this calculation requires only the explicit calculations of F in the homogeneous case μ₁ = … = μ_k and in the case that the heterogeneity is limited to a single coordinate (i.e., when μ₂ = … = μ_k). In many applications this is a considerably easier task than the explicit calculation of F in the fully heterogeneous case.

To illustrate this, consider again the M/M/k example of Fig. 1 with k = 8. Whereas the fully heterogeneous case requires solving 2^k − 1 = 255 equations, the single-coordinate heterogeneous case requires solving only 2 ⋅ k = 16 equations. Solving these 16 equations symbolically and using Lemma 3 yields (SI Text)

where the values of {c_i, b_i} are listed in Table 1.

Table 1.

Values of {c_i, b_i}

i	ci	bi
0	1	1
1	45	14
2	999	126
3	14,280	840
4	144,720	4,200
5	1,088,640	15,120
6	6,249,600	35,280
7	27,941,760	40,320
8	97,977,600
9	263,390,400
10	514,382,400
11	653,184,000
12	406,425,600

In particular, substituting and λ = 28 yields α ∼ 0.00837. This leads to the improved approximation

The error of this improved approximation scales as 0.074ɛ³ (Fig. 1), which is the next term in the Taylor expansion. In particular, the relative error of Eq. 16 is below 1.5% for 0 ≤ ɛ ≤ 1.

Final Remarks

The averaging principle is based on a simple observation: The leading-order effects of heterogeneity cancel out when the outcome is symmetric. Nevertheless, it can lead to a significant simplification of mathematical models in all branches of science.

The symmetry and the weak-symmetry properties are usually easy to check. The differentiability of F is easy to check in some applications, but can be quite a challenge in others. We note, however, that more often than not, functions that arise in mathematical models are differentiable, unless there is a “very good reason” why they are not. Although this is a very informal statement, we make it to point out that the “generic” case is that the outcome F is differentiable, rather then the other way around.

An important issue is the “level of heterogeneity” that is covered by the averaging principle. Strictly speaking, the level of heterogeneity should be “sufficiently small.” In practice, however, in many cases the averaging principle provides good approximations even when ɛ = 0.5. In other words, the coefficient of the O(ɛ²) term is O(1). Whereas this is also an informal statement, we make it to point out that one should not be “surprised” that the averaging principle holds even when ɛ is not very small.