In the U.S. GAO report (2001), there is a survey question: “About how much do you pay per month to access the Internet from your home?” Although this question does not provide an exact willingness-to-pay for Internet access but we can use this answer as a proxy for customer’s demand. The following is the distribution from this question.
[Table 7-1] Distribution of Household Expenditure for Internet Access (per Month) Sub-Range $0 ~$5 ~$10 ~$15 ~$20 ~$30 ~$40 ~$50 $50~
% 8.9 1.4 3.8 8.3 21.0 31.7 11.1 8.7 5.1
* Source: GAO-01-345, Characteristics and Choices of Internet Users, p46.
We assume price range for BE is from $0 to $50, which is the same as in the uniform distribution and the data within a sub-range is uniformly distributed. We modified the above table into an equal sub-range:
[$0-$10], [$10-$20], [$20-$30], [$30-$40], and [$40-$50]. The following is a modified piecewise uniform distribution table according to the above assumptions.
[Table 7-2] Piecewise Uniform Distribution
Sub-Range $0-$10 $10-$20 $20-$30 $30-$40 $40-$50
% 14.1 29.3 31.7 11.1 13.8
We use a two-stage RNG: (1) first, we use the above empirical distribution table (table 7-1) to match a sub-range and (2) then we use the piecewise uniform distribution (table 7-2) for a specific value. For example, if a random number (α) generated from the empirical distribution is between 0.0 and 0.141 (the 1st sub-range), then this customer’s willingness-to-pay (β) is determined by a function of uniform [$0,
$10]. Table 7-3 shows this two-stage RNG method.
[Table 7-3] Two-Stage RNG
α = RNG-1 Sub-Range β = RNG-2
0.0000 < α < 0.1410 $0 ~ $10 β = Uniform [$0, $10]
0.1411< α < 0.4340 $10 ~ $20 β = Uniform [$10, $20]
0.4341< α < 0.7510 $20 ~ $30 β = Uniform [$20, $30]
0.7511< α < 0.8620 $30 ~ $40 β = Uniform [$30, $40]
0.8621< α < 1.0000 $40 ~ $50 β = Uniform [$40, $50]
Based on these assumptions and methods, we conducted 30 times simulations and linear regressions, the results of which are presented in Appendix C. The estimated demand function for BE is presented in figure 7-1. The kinked line represents a demand curve from the simulation and the straight line represents a demand curve from the linear regression, which is “P = 44.55 – 0.00855*Q.” We estimate a QoS demand function by applying the product differentiation theory as we did in the uniform distribution case.
Subscribers
5000 4000
3000 2000
1000 0
$ 60
50
40
30
20
10
0
Empirical Estimated
[Figure 7-1] Demand Curves for Empirical Distribution
Based on this empirically estimated demand function, we analyze the 10 cases and the overall model as we did in the uniform distribution. We do not explain each case of empirical distribution in detail as we did in the uniform distribution. The detailed equilibrium quantities, prices, and profits at the equilibrium points of each case are presented in Appendix D and the payoff matrixes of 10 cases are provided in
Appendix E. The following table shows the basic demand functions of uniform and empirical distributions.
[Table 7-4] Demand Function Comparison
Uniform Empirical BE PBE = 50 – 0.01*QBE PBE = 44.55 – 0.00855*QBE
QoS PQoS = PBE + 50 - 0.01*QQoS PQoS = PBE + 44.55 - 0.00855*QQoS
If we compare two demand functions, the demand function of uniform distribution has a higher intercept (50 > 44.55) and a steeper slope (0.01 > 0.00855) than that of empirical distribution. For the most of parts, the demand curve of the uniform distribution is above the demand curve of the empirical distribution.
7.2 Progressive Market Equilibrium Paths
The empirical distribution has the same equilibrium network capacity as in the uniform distribution except for Case 773. We try to find progressive market equilibrium paths from the equilibrium points of Case 1 as we did in the uniform distribution. There are identical, two progressive market equilibrium paths as in the uniform distribution. Tables 7-5a and 7-5b show these two paths. In conclusion, there is no significant difference in the results of equilibrium analyses between two distributions, thus uniform distribution assumption is reasonable in this game model.
[Table 7-5a] Progressive Market Equilibrium Paths
Transit Peering BE QoS BE QoS
Flat Flat Two-Part Flat Flat Two-Part Empirical
73 Case 7’s equilibrium network capacity: [2K, 2K] in the uniform distribution and [1K, 2K] and [2K, 1K] in the empirical distribution.
[Table 7-5b] Progressive Market Equilibrium Paths
7.3 Overall Analysis of the Game Model
In this section, we make all 10 payoff matrixes into one combined matrix and then try to find global equilibria in the game model. The following two tables show optimal strategy sets of IAP1 and IAP2 in the empirical distribution.
[Table 7-6] IAP1’s Optimal Strategy Set
Transit-BE-Flat Transit-QoS-Flat Transit-QoS-2P Peering-BE-Flat Peering-QoS-Flat Peering-QoS-2P IAP2’s
[Table 7-7] IAP2’s Optimal Strategy Set
Transit-BE-Flat Transit-QoS-Flat Transit-QoS-2P Peering-BE-Flat Peering-QoS-Flat Peering-QoS-2P IAP1’s
There are four global equilibrium points, ε, two φs, and γ in table 7-9, and two of them, ε ([2K, 1K] of Case 6) and γ ([2K, 2K] of Case 10), are the same positions as α and δ in the uniform distribution. These four global equilibria are also the local equilibrium points in Cases 6, 7, and 10. While in the result of uniform distribution QoS-IAPs could choose flat-rate or a two-part tariff in their global equilibrium points, in the empirical distribution QoS-IAPs could not choose flat-rate. Table 7-8 shows a collection of local equilibrium points in the empirical distribution and the shaded cells are the global equilibrium points.
[Table 7-8] Equilibrium Profits
[1K, 2K] [2K, 1K] [2K, 2K]
Case 1 [15891,31808] [31808,15891]
Case 2 [15891,35385]
Case 3 [9730,30782] [18815,18598]
Case 4 [8273,30015] [19690,18598]
Case 5 [16101,16101]
Case 6 [18598,30015] ε: [30015,18598]
Case 7 φ: [11591,28508] φ:[28508,11591]
Case 8 [10640,20965]
Case 9 [19501,19501]
Case 10 γ: [20965,20965]
[Table 7-9] Equilibrium Points in Empirical Distribution
Transit Peering BE QoS BE QoS
Flat Flat Two-Part Flat Flat Two-Part Empirical
We calculate social welfares of all 90 equilibrium points (= 9 cells * 10 cases) in the empirical distribution as we did in the uniform distribution. The detailed numbers are presented at table F-2 in Appendix F. The following table shows the highest social welfare in the empirical distribution. This point is [1K, 3K] in Case 2 (=$147,936), which is not a local equilibrium in Case 2 and which is different from the highest social welfare points in the uniform distribution. When we compare social welfares of [2K, 2K] in Case 5 (one of the highest social welfare points in the uniform distribution) between table F-1 (uniform) and table F-2 (empirical) in Appendix F, there is a big difference in the BE consumer surplus (66,390 vs. 39,145). But there is a small difference of BE consumer surplus at [1K, 3K] in Case 2 (the highest social welfare in the empirical distribution) in both distributions (53,999 vs. 51,400). The reason for different points of highest social welfares in the two distributions mainly comes from the level of
difference in the BE consumer surplus by using different estimated demand functions. However, if we consider only local equilibrium of each case, the highest social welfares are [2K, 2K] of Case 5 and Case 9 (=$127,386), which are the same points as in the uniform distribution.
[Table 7-10] Highest Social Welfare Points
Transit Peering BE QoS BE QoS
Flat Flat Two-Part Flat Flat Two-Part Uniform
If we try to compare the social welfares at the global equilibrium points from the two distributions, social welfares of uniform distribution are higher than those of empirical distribution. The reason for higher social welfare in the uniform distribution comes from the fact that the demand functions of empirical distribution are below those of uniform distribution, which causes lower consumer surplus, lower profits, and finally lower social welfare. The following two tables display the social welfare values of the global equilibrium points in both distributions. In the both distributions, social welfares of peering equilibrium points (β, χ, δ, φ, and γ) are higher than those of transit equilibrium points (α and ε), which means that peering is socially more desirable than transit. But comparing the combined profit of both IAPs at these global equilibrium points, the profit of transit is higher than that of peering, which is another obstacle to peer each other.
[Table 7-11] Social Welfare of Global Equilibrium in the Uniform Distribution CS PS Uniform
Distribution BE QoS Total Profits Fixed Cost Total SW α Case 6 [2K,1K] 38806 -14725 24081 51920 44000 95920 120,001 β Case 7 [2K,2K] 72200 N/A 72200 25600 46600 72200 144,400 χ Case 9 [2K,2K] 66390 5810 72200 45330 45200 90530 162,730 δ Case 10 [2K,2K] 49251 -14725 34526 45850 45200 91050 125,576
[Table 7-12] Social Welfare of Global Equilibrium in the Empirical Distribution CS PS Empirical
Distribution BE QoS Total Profits Fixed Cost Total SW ε Case 6 [2K,1K] 32040 -18055 13985 48613 44000 92613 106,598 φ Case 7
[1K,2K], [2K,1K] 38424 N/A 38424 40099 37600 77699 116,123 γ Case 10 [2K,2K] 46405 -18055 28350 41930 45200 87130 115,480
[CHAPTER 8] CONCLUSION