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# A Future Friendly Digital Economy Strategy: Submission to the Government of Canada's Digital Economy Consultation — Appendix 2

## Submission (continued)

### III. The Berkman Study's Conclusions are not Supported by Quantitative Analysis

9. The Berkman Study focuses on two market characteristics associated with broadband diffusion: the intensity of competition and the rate of investment. Unfortunately, it focuses almost exclusively on intra-platform competition and public investment. It discounts the significance of platform competition and of private investment, allowing its authors to ignore the link between policies that promote private investment in competitive platforms, on the one hand, and market performance, on the other.

10. The Berkman Study claims that its attachment to intra-platform competition through mandated network unbundling of incumbent telecommunications networks at regulated prices is grounded in empirical analysis. However, it largely ignores the vast empirical literature that finds no general relationship between network unbundling and broadband penetration, while focusing on one study, by John de Ridder,Footnote 6 which appears to find such a relationship. Based on its re-estimation of de Ridder's original specifications, the Berkman Study finds that mandatory unbundling policies increase broadband penetration by one percent per year. As we explain in Section III (A) below, however, subsequent research demonstrated that de Ridder's results are spurious; and, our further econometric analysis demonstrates that the Berkman Study's efforts to resuscitate de Ridder's results are unsuccessful. Simply put, the quantitative analysis upon which the Berkman Study bases its conclusion that unbundling increases broadband penetration is invalid.

11. Given the Berkman Study's emphasis on unbundling and its mandate to "review existing literature and studies about broadband deployment and usage," we were surprised at its cursory treatment of the extensive existing literature on the effects of unbundling regulation, which on balance fails to support the proposition that unbundling enhances penetration, but does find that unbundling reduces investment incentives, especially for new technologies such as fiber to the premises.Footnote 7

12. In section III (B) below, we conduct the broader review of the existing literature we would have expected to find in the Berkman Study. Our review demonstrates that the bulk of the existing quantitative research finds no basis for concluding that unbundling leads to higher levels of broadband penetration, increased infrastructure investment, or other positive effects on significant public policy objectives.

#### A. The Berkman Study's"Do-Over" of the de Ridder Study is Fatally Flawed

13. In this section, we review the Berkman Study's econometric analyses, from which it concludes that unbundling policies increase broadband penetration. These analyses are based on the 2007 OECD de Ridder Study. Based on its econometric analysis, the Berkman Study concludes that

consistent with the findings of this recent work, and inconsistent with a recent critique of it, econometric analysis supports the proposition that unbundling contributed to broadband penetration in OECD countries. Indeed, new analyses we perform on the existing data suggest that the effect was larger than previously thought, the confidence level higher, and the finding more robust.Footnote 8

As we demonstrate below, the Berkman Study's econometric analysis does not support any of these conclusions.

14. The de Ridder Study presents least-squares regression results for a model that takes the general form:

${\mathrm{Qtot}}_{\mathrm{it}}={a}_{0}+{a}_{1}{\mathrm{GUyrs}}_{\mathrm{it}}+\underset{j=1}{\overset{n}{\Sigma }}{b}_{j}{X}_{j,\mathrm{it}}+{e}_{\mathrm{it}}$

where:

• Qtot represents total broadband penetration (as measured by the ratio of broadband connections to population in each country);
• GUyrs represents the number of years since unbundling policies were enacted in each country;
• X represents a vector of other explanatory variables (denoted by the index variable j) such as urbanization, average age, or median income;
• i is an index variable representing the country and t is an index variable representing time;
• a and b are regression coefficients; and
• e is a random and identically distributed error term that captures variance unexplained by the model.Footnote 9

15. Simply put, the thesis being tested is that, holding other factors constant, unbundling policies increase broadband penetration, with the effect increasing with the length of time unbundling policies have been in place. Thus, de Ridder estimates least squares regression parameters for various regression specifications to determine whether the coefficient on the unbundling variable (a1 in the equation above) is positive and statistically significant.

16. Specifically, the de Ridder Study estimates seven different regression equations that include the GUyrs policy variable. Four of those seven equations yield a positive coefficient for this variable that is significant at the 5 percent level. On this basis, de Ridder concludes that "unbundling (as measured by GUyrs) is currently more significant than platform competition in explaining broadband penetration."Footnote 10 Obviously, this result supports the Berkman Study's pro-unbundling position.

17. However, the de Ridder Study was effectively critiqued in a 2008 paper by Boyle, Howell and Zhang.Footnote 11 By estimating de Ridder's regressions with robust standard errors, their paper demonstrates that the statistical significance of GUyrs disappears after controlling for heteroskedasticity. In addition, Boyle, Howell, and Zhang showed that GUyrs was likely serving as a proxy for the natural diffusion of broadband into the economy – that is, rather than capturing the time elapsed since the adoption of unbundling, GUyrs was actually capturing, at least in part, the time elapsed since the introduction of DSL.Footnote 12 Indeed, when Boyle, Howell, and Zhang controlled for the number of years broadband service had been available in each country, the GUyrs variable became insignificant in both an economic and statistical sense.Footnote 13

18. The Berkman Study seeks to resuscitate the de Ridder results by making two significant modifications. First, it modifies the GUyrs variable, either by replacing it with a zero-one indicator variable for unbundling, or by replacing the values of GUyrs used by de Ridder with new values.Footnote 14 Second, it applies a more sophisticated econometric technique, utilizing a mixed-effects regression model.Footnote 15 Based on these changes, the Berkman Study concludes that unbundling, whether represented by an indicator variable or a continuous time variable, is generally significant and positive, i.e., that unbundling increases broadband penetration.Footnote 16

19. The Berkman Study's econometric analysis is flawed for three primary reasons. First, it simply fails to address the Boyle-Howell-Zhang critique of the de Ridder Study, which showed that the GUyrs variable is capturing the natural diffusion of broadband over time rather than the effect of unbundling. Second, the mixed effects regressions presented in the Berkman Study fail a Hausman Test for specification, implying that the mixed effects model is inappropriate for this data set and does not produce reliable results. Third, its decision to simply replace the values of the key variable (GUyrs) with values more consistent with its prior beliefs is simply unjustifiable as a matter of econometric technique.

20. In what follows, we present the results of our own regression analysis of the basic de Ridder/Berkman model. Our regressions utilize additional years of data (thus significantly increasing the number of observations, as well as utilizing more recent observations), and apply more appropriate regression specifications and techniques. As we demonstrate, when additional years of data and additional relevant explanatory variables are added – that is, when the model is more fully and correctly specified, and applied to a more complete data set – the effect of unbundling not only disappears but reverses: Rather than increasing broadband penetration, our results demonstrate that unbundling reduces it.

##### 1. The Inclusion of Relevant Data and Explanatory Variables Reverses the Effect of Unbundling on Penetration

21. The primary reason for the de Ridder Study's decision to limit observations to these two years was its use of regressions that include the price of DSL service, for which data is only available for 2002 and 2005. Perhaps because it initially focuses on replicating the de Ridder regressions, the Berkman Study also uses only data from 2002 and 2005.

22. However, as the Berkman Study explains, because the price of DSL service is correlated with the unbundling variable,Footnote 17 it is not appropriate to use price as a separate explanatory ("right-hand-side") variable. Accordingly, in most of its regression specifications, the Berkman Study drops the DSL price variable, which would have allowed it to utilize additional years of data. As we explain further below, our regression analyses rely on data from 2001-2006, for a total of 168 observations.

###### a. Least-Squares Regression Results

23. To examine the effect of both increases in sample size and the natural diffusion of broadband into the economy, we perform regression analysis on fourth quarter OECD data from 2001 through 2006. Table 1 lists the variables used in the analysis and presents summary statistics. The regression data reflects a 28-country (rather than 30-country) sample, because we were unable to determine the date of DSL availability in Greece and the Slovak Republic.Footnote 18

Table 1: Summary Statistics for Regression Data of 28 OECD Countries (168 Observations)
Variable Description Mean Standard Deviation Minimum Maximum
Qtot Broadband Penetration 9.976 8.497 0.010 31.788
GUyrs Years since unbundling enacted 3.940 2.942 0.000 11.000
unbundled =1 if guyrs > 0
= 0 otherwise
0.839 0.368 0.000 1.000
DSLyears Time since DSL was available 4.238 1.813 0.166 7.667
Pop dens Population density 139.838 133.943 2.527 489.183
Pops Population in millions 40.719 58.814 0.287 299.715
Gdp Gross Domestic Product per capita in US PPP 28720.140 11688.920 6178.182 80471.400
State owned = 1 if network was state owned
= 0 otherwise
0.071 0.258 0.000 1.000

It should be noted that the simple correlation coefficient between variables GUyrs and DSLyears is 0.58. This indicates that, as expected, the two variables share a moderate to strong positive correlation.

24. Table 2 below presents least squares regression results of "reduced form" specificationsFootnote 19 that include DSLyears, and estimate the effect of unbundling using either the GUyrs variable or, in some specifications, Unbundled, which is a one-zero indicator variable.

Table 2: Least-Squares Estimates with Qtot as the Dependent Variable
Variable Coefficient Robust t-Stat Coefficient Robust t-Stat
Specification 1 Specification 2

Note 1: For both specifications, a White test for heteroskedasticity rejects the null hypothesis of homoskedasticity. We therefore present the regression results with t-Statistics generated from White-Huber standard errors, which account for correlation between the right hand side variables and the error terms in the regression.

Note 2:
*** Denotes significance at 1 percent;
** denotes significance at 5 percent.

GUyrs 0.339 1.46
Unbundled -2.232 ** -1.98
DSLyears 2.900 *** 10.21 3.253 *** 16.85
pop dens 0.012 *** 3.19 0.013 *** 3.48
Pops -0.017 *** -4.03 -0.013 *** -3.56
Gdp 0.000 *** 5.06 0.000 *** 7.71
state owned -7.400 *** -4.26 -9.602 *** -6.17
Constant -10.042 *** -7.43 -10.544 *** -9.17
N = 168 N = 168
R-squared = 0.72 R-squared = 0.72
F (zero slopes) = 83.94 F (zero slopes) = 76.41

25. To begin, the regression results in Table 2 first show that the explanatory variables in the model are, in general, statistically significant at a 1 percent significance level. Specifically, the coefficient on DSLyears is positive and significant in both regression specifications, and positive coefficients on population density and GDP indicate that broadband penetration tends to be higher in wealthier economies and in countries with higher population densities, but lower in larger countries (those with large populations) and in countries where the telecommunications network is state-owned. All of these effects are of the anticipated sign.

26. Regarding unbundling, the Berkman Study found that the effect of GUyrs was positive and statistically significant at 1 percent. But when DSLyears is inserted into the equation, as in Specification 1, the effect disappears, indicating that the number of years unbundling policies have been in place has no independent effect on broadband penetration, and confirming the result in Boyle-Howell-Zhang that GUyrs is simply a proxy for the number of years since DSL was first introduced.Footnote 20

27. As explained above, the correct question with respect to unbundling is whether it shifts the technology diffusion curve, so that, other things equal, broadband penetration increases more rapidly in countries that adopt unbundling. To test this proposition, we introduce the indicator variable Unbundled in a regression that also controls for the time that DSL has been available in each country. The second regression specification in Table 2 provides the results of this test: as the table shows, the coefficient on Unbundled is negative and statistically significant.Footnote 21 This result indicates that unbundling has slowed the pace of broadband adoption in the sample countries, a result which directly contradicts the Berkman Study, but, as we note below, is consistent with prior empirical research.

###### b. Fixed Effects, Random Effects, and Generalized Least Squares

28. We next extend the above regressions to the estimation of fixed effects, random effects, and generalized least-squares models. A fixed effects regression essentially involves the inclusion of country-specific indicator variables in the model, and is a common practice in panel datasets. Random effects regressions take advantage of variation across countries without the inclusion of indicator variables for each country in an attempt to estimate the right hand side variables in a more efficient manner than fixed effects. Applying generalized least squares to panel data allows one to control for both heteroskedasticity (a cross-sectional data problem) and autocorrelation (a time-series problem).

Table 3: Fixed Effects, Random Effect, and GLS Regression Results with Qtot as the Dependent Variable
Variable Coefficient t-Stat Coefficient Robust
t-Stat
Coefficient Z-Stat
Fixed Effects Random Effects GLS
Unbundled -3.7421 *** -3.8 -3.8146 *** -3.91 -1.2038 * -1.93
DSLyears 2.5213 *** 8.62 3.0633 *** 19.24 2.2488 *** 13.54
pop dens -0.1093 -0.68 0.0138 * 1.94 0.0009 0.08
Pops -0.5699 *** -2.89 -0.0137 -0.85 -0.0296 ** -2.34
Gdp 0.0009 *** 5.24 0.0004 *** 4.7 0.0007 *** 6.31
Constant 14.199 0.6 -11.351 *** -4.64 -17.321 *** -5.06
N = 168 N = 168 N = 168
R-squared (within) = 0.88 R-squared (within) = 0.86 Chi-Squared = 1367.86
R-squared (between) = 0 R-squared (between) = 0.33
R-Squared (overall) = 0.01 R-Squared (overall) = 0.61

29. The results reported in Table 3 show that the Unbundled variable has a negative coefficient when any of these panel data techniques is used. In the case of fixed and random effects, the Unbundled variable is negative and statistically significant at 1 percent. In the GLS model controlling for both autocorrelationFootnote 22 and heteroskedasticity, Unbundled is again negative, and significant at 5.4 percent. Therefore, panel data analysis supports the least-squares results presented in Table 2 above – namely, that unbundling reduces broadband penetration.

30. It is also worth noting that in the fixed effects regression, the effect for the United States is both positive and statistically significant. That is, the fixed effect regression indicates that the United States has broadband penetration above what the model predicts based on the other control variables in the model, such as wealth and population density. Moreover, the United States fixed effect is statistically significant at 1 percent both in regressions that apply traditional standard errors and in regressions that use robust heteroskedastic-consistent standard errors. This finding indicates that a characteristic not specifically controlled for in the model is causing higher broadband penetration in the U.S. than the model itself predicts. Plausible candidates for such a characteristic include tastes (i.e. consumer preferences), the presence of robust infrastructure competition, or policy variables (other than unbundling) not captured in the model.

##### 2. The Mixed Effects Regressions in the Berkman Study Fail a Hausman Specification Test

31. As noted above, in addition to least squares models, the Berkman Study estimated a mixed effects model,Footnote 23 which contains both fixed and random effects. In a mixed effects model, fixed effects are estimated directly and the random effects are calculated indirectly via an estimated variance-covariance matrix.Footnote 24 As a result, a mixed effects estimator attempts to take advantage of information contained in variation between countries, whereas the fixed effects estimator includes controls for each specific country. Both techniques allow one to estimate a regression coefficient on a variable such as GDP or GUyrs, but the two estimation techniques analyze the data differently. A benefit of the random effects model is that it has greater degrees of freedom, because the estimation of many indicator variables is unnecessary.Footnote 25 Random effects estimation, however, relies on an assumption that the random effect itself is uncorrelated with the right hand side variables in the regression, which may be unlikely.

32. The key factor in selecting between fixed effects and random effects models is whether the assumptions underlying the random effects approach holds. If the effect is uncorrelated with other right hand side variables, then random effects is consistent and efficient relative to fixed effects. If this assumption fails, however, random effects models are inconsistent, and the fixed effects approach is preferred.

33. The appropriate test for determining which approach is preferred is the Hausman Specification Test,Footnote 26 which we applied to the data used in the de Ridder and Berkman analyses. The test is constructed as follows. First, both models are estimated in accordance with the original de Ridder specification, with price on the right-hand side. Next, the coefficients and variance-covariance matrices are statistically compared between the two models. If significant differences exist, then a mixed (or random) effects approach is superior. Otherwise, the fixed effects regression is preferred. We found that the Berkman Study's mixed-effects model failed the Hausman Test: That is, contrary to the Berkman Study's contentions, the mixed effects model does not offer statistical efficiency above that of the fixed effects model. Moreover, when we estimated the fixed effects model using the de Ridder data, the positive effects of unbundling highlighted in the Berkman Study disappear.

34. To perform the Hausman Test, we first estimated a fixed effects model on the following variables:Footnote 27

• log of DSL price (lnpdsl)
• Urbanization (uurb)
• Facilities competition (Cfac)
• Years of unbundling (GUyrs)
• Indicator variable for 2005 (Dummy)

We found that the coefficient on GUyrs is statistically insignificant.

35. Next, we estimated both a random effects model and a mixed effects model using identical regressors to the fixed effects regression outlined above. The Hausman Test generated a Chi-Squared statistic of 3.33 when comparing random to fixed effects, and a statistic of 4.04 when comparing the mixed and fixed effects estimates. Neither of these statistics was sufficiently large to reject the null hypothesis that fixed effects is the more efficient estimator.Footnote 28 Consequently, the regression results presented in the Berkman Study are questionable not only because of the limited data sample and the exclusion of a relevant variable, but also because they result from use of an inappropriate regression technique.

##### 3. The Berkman Study Makes Inappropriate Modifications to the Data

36. In several of its regression specifications, the Berkman Study changes the underlying data used by de Ridder. Specifically, as shown in Table 4.8 of the Berkman Study, the authors altered 17 of the 30 values of GUyrs from the values used in the original de Ridder analysis. Using the altered data, the Berkman Study finds that GUyrs "seems to have larger effects" and "is much more significant."Footnote 29 Given the nature of the data alterations, these results are hardly surprising.

37. As the Berkman Study notes, its data alterations result "in many more countries defined to have GUyrs = 0 than before."Footnote 30 More specifically, the countries for which GUyrs were set to zero were Belgium, United States, United Kingdom, Luxembourg, Germany, Ireland, Poland, and Greece. The average value of Qtot in 2005 for these eight countries was 11.22, significantly below the average of 14.14 for all 30 countries. (The lower average for the eight "zeroed-out" countries is driven primarily by Ireland, Poland, and Greece, which have particularly low broadband penetration.) Thus, the positive effect of GUyrs in the regression specifications utilizing the altered data is the result of the exclusion of three countries that had very low broadband penetration but had adopted unbundling. In addition, the United States is assigned zero years of unbundling in 2005 despite the fact that it began to require unbundling in 1996 and, for a time, even required line sharing. And Germany is also assigned a zero, despite the fact that ECTA reported that it had 2.5 million unbundled lines devoted to broadband out of a national total of 10.7 million broadband lines!Footnote 31 In other words, many of the regression results reported by the Berkman Study are the result of simply replacing the values for selected observations with data points that are more favorable to its conclusions. Obviously, the Commission should not base its policies on analyses of what amounts to manufactured data.

#### B. The Evidence from Other Quantitative Studies Does Not Support Unbundling

38. In addition to the de Ridder Study, there is a significant body of quantitative research into the effects of mandatory unbundling on broadband penetration. Despite its mandate to conduct a "review of existing literature and studies about broadband deployment and usage," the 231-page Berkman Study devotes only half a paragraph to reviewing this literature, briefly mentioning only five studies, two of which it concedes do not find unbundling to have a significant effect on penetration. Nevertheless, based on its highly selective literature review, the Berkman Study concludes that "unbundling had a positive and significant effect on levels of penetration."Footnote 32 In this section, we conduct a more complete review of the existing literature and show, contrary to the Berkman Study's conclusion, that the vast majority of studies find either no relationship or a negative relationship between unbundling and broadband penetration. We also note that most studies also find evidence that platform competition – i.e., the U.S. model, which is rejected by the Berkman Study – does have a positive and significant effect. In short, the quantitative evidence is directly contrary to the Berkman Study's policy recommendations.

##### 1. Summary of Prior Empirical Studies

39. Table 4 summarizes the pre-existing empirical literature on broadband penetration, availability, and mandated unbundling. As the table shows, the bulk of the studies surveyed do not support the proposition that mandated unbundling, with its focus on intraplatform competition, increases broadband penetration or deployment. Most studies find the relationship to be either a negative or insignificant. The few studies reporting positive effects fail to provide persuasive evidence, owing to various biases and data deficiencies. On the other hand, the majority of the studies examining the role of inter-platform competition (all but one) find that competition across platforms leads to increased broadband penetration. Thus, the existing literature suggests that mandatory unbundling is either ineffective or counterproductive in increasing broadband penetration.

Study Data UnbundlingIncreasesBroadbandPenetration/Availability? * Distaso, Lupi & Mantenti (2005) find that inter-platform competition is a substantial driver of broadband penetration, and that competition within the market for DSL services does not play a significant role. Somewhat paradoxically, the researchers also find that a decrease in the local loop unbundling price has a positive and significant effect on penetration. However, as noted above, their econometric analysis assumes that mandated access prices are exogenous, implying that this effect may reflect reverse causality. Cross-Section, 46 US States N Y Cross-Section, 30 OECD Countries N N/A Panel, 50 U.S. States N Y Panel, 14 European Countries N* Y Panel, 30 OECD Countries N Y Panel, 30 OECD Countries N N/A Panel, 12 European Countries N N/A Panel, 30 OECD Countries N N Panel, 27 European Countries N N/A Cross-Section, 18 Countries Y N/A Panel, 30 OECD Countries Y Y Panel, 30 OECD Countries Y N/A

40. In addition to the studies summarized in Table 4, there is a substantial empirical literature on the relationship between unbundling and investment. As we discuss below, the evidence from these studies strongly supports the hypothesis that unbundling regulation reduces infrastructure investment

##### 2. Studies Finding Negative and/or Insignificant Effects of Unbundling on Penetration

41. As shown in Table 4, we identified nine studies that find either a negative or an insignificant effect of unbundling on broadband penetration.

42. First, using a cross-section of 46 U.S. states from the year 2000, Aron and Burnstein (2003) estimate the effect of intermodal competition on broadband penetration, relative to the effect of simple broadband availability, while controlling for various demand and cost drivers.Footnote 33 The authors measure head-to-head intermodal competition as the percentage of the population in a given state residing in cities where both cable modem and DSL are deployed; broadband availability is measured by the percentage of the population of a given state residing in cities where cable modem or DSL have been deployed. The demand controls include the percentage of households with internet access and an education metric; the cost controls include a metric for teledensity (measuring the number of switched access lines per mile), the average length of a local switched access line, and the regulated price of an unbundled network element.Footnote 34

43. The regression results reveal that, although broadband penetration is positively correlated with broadband availability, this effect disappears after controlling for intermodal competition. Thus, for a given level of demand and cost drivers, an increase in broadband availability in areas without intermodal competition does not stimulate additional adoption of broadband. Furthermore, the relationship between unbundled access prices and penetration is not statistically significant.Footnote 35

44. Analyzing a cross-section of 30 OECD countries from the year 2001, Bauer, Kim, and Wildman (2003) examine the effect of various policy variables on broadband penetration, including unbundling, cable-telco cross ownership, and government funding for broadband.Footnote 36 The researchers allocated the countries in their sample into one of three clusters, depending on the extent to which each of these policies was in place. The authors find that two factors, the population density and the "preparedness" of a given country (as captured by an index measuring attitudes towards advanced information technologies and the availability of complementary technologies, such as computers) are consistently significant in explaining broadband penetration. The analysis fails to detect any statistically significant relationship between membership in a given policy cluster and broadband penetration.Footnote 37

45. Denni and Gruber (2005)Footnote 38 analyze biannual state-level panel data from 1999 to 2004 in an effort to determine the extent to which intra- versus inter-platform competition affects broadband penetration, using a logistic model of technology diffusion. The dependent variable in their model is the ratio of broadband subscribers to the population of a given state. Inter-platform competition is measured with a modified version of the traditional Herfindahl index, using technologies' market shares instead of firms' market shares.Footnote 39 Intra-platform competition is measured somewhat differently, using a special case of the Herfindahl index that applies when all firms have symmetric shares.Footnote 40 Due to the potential endogeneity of these competition indices, the authors use lagged values of the endogenous variables as instruments.

46. The authors find that inter-platform completion plays a far more important role than intra-platform competition in determining the rate of diffusion of broadband infrastructure: Inter-platform competition is shown to have a substantial positive effect on diffusion in the long run, whereas intra-platform competition has only a small initial effect that rapidly dissipates. Furthermore, the authors find that mandatory unbundling actually inhibits broadband penetration. Specifically, the results indicate that the share of central offices upgraded for equal access has a negative and statistically significant effect on the rate of broadband diffusion.Footnote 41

47. Distaso, Lupi, and Manenti (2005) develop and estimate a model of oligopolistic competition between differentiated products to analyze the relative importance of intra-platform competition and inter-platform competition in driving broadband adoption.Footnote 42 A key implication of their theoretical model is that inter-platform competition (between alternative platform providers such as cable companies and fiber-optic providers) should be more effective than intra-platform competition (competition among incumbents and DSL providers using unbundled loops) in increasing broadband penetration.Footnote 43

48. When estimating their model, the authors employ a panel data set of 14 European countries running from the fourth quarter of 2000 through the second quarter of 2004. The dependent variable in each of their econometric specifications is broadband penetration, as measured by the percentage of all access lines (copper, cable, fiber, and satellite) that have been upgraded to transmit high-speed data.Footnote 44 The authors measure the degree of intra-platform competition as the level of concentration in the DSL market, using a standard Herfindahl index. The degree of inter-platform competition is modeled using a modified Herfindahl index, computed using technologies' market shares instead of firms' market shares, as in Denni and Gruber (2005). To control for potential endogeneity of the competition metrics, the two Herfindahl indices are instrumented using their lagged values.

49. Consistent with their theoretical predictions, the authors find that, although interplatform competition is a substantial driver of broadband adoption, competition within the market for DSL services – the type of intra-platform competition that mandatory unbundling of broadband is designed to stimulate – does not play a significant role.Footnote 45

50. Cava-Ferreruela and Alabau-Munoz (2006),Footnote 46 employing data from a panel of 30 OECD countries from 2000 to 2002, explore the determinants of wireline broadband coverage, (defined as the percentage of local loops that are DSL-enabled), as well as cable broadband coverage (defined as the percentage of homes passed by cable television networks).Footnote 47 The analysis indicates that gross national income per capita is the single most important determinant of both DSL coverage and cable coverage. The authors also find evidence that the share of DSL-enabled local loops is dramatically higher in countries where inter-platform competition is more robust, whether measured by the presence of cable infrastructures or by the number of competitors operating competing broadband platforms. In contrast, neither the existence of unbundling regulations nor the number of unbundled loops is found to be significantly correlated with DSL coverage.Footnote 48

51. Using a panel of 30 OECD countries from 1999-2003, Wallsten (2006) investigates the determinants of broadband subscribers per capita. Wallsten's regression analysis includes dummy variables for (1) various types of unbundling regulation (full unbundling, bitstream unbundling, and subloop unbundling); (2) collocation regulations; and, (3) access price regulation. The regressions also control for GDP per capita, and the number of fixed telephone lines per capita.Footnote 49

52. In his fully specified model, which includes time and country fixed effects, Wallsten finds no consistent evidence that full unbundling has a positive effect on penetration. The effect of full unbundling is positive and significant in some specifications, negative and significant in others, and statistically insignificant in his full specifications that include all covariates. With respect to bitstream unbundling, the estimated coefficients are positive, but statistically insignificant in the full specification. Furthermore, with respect to sub-loop unbundling, the effect on penetration is consistently negative and statistically significant. Wallsten does find evidence that on-site collocation requirements are positively and significantly correlated with penetration. However, he also finds a negative and statistically significant relationship between regulation of collocation charges and broadband penetration.Footnote 50

53. Waverman, Meschi, Reillier, and Dasgupta (2007) employ a panel of 12 European countries from 2002-2006 to estimate the effect of unbundling on broadband penetration.Footnote 51 The researchers' control variables include GDP, a lagged Herfindahl index (computed using technology platform shares) and the share of internet-ready cable plant.Footnote 52 Mandatory unbundling is captured with a variable measuring the price of a fully unbundled local loop as well as the number of years since unbundling was implemented. The coefficient on the unbundled price is also allowed to vary depending on whether bitstream access is available. The authors report that the number of years since adoption of unbundling has no statistically significant effect on penetration, and that the coefficients on the unbundled price variables are negative and statistically significant. This implies that that lower unbundling rates induced substitution away from alternative platforms and toward copper platforms, and that the net effect over the sample period was to steeply reduce the number of broadband consumers.Footnote 53

54. As noted above, Boyle, Howell, and Zhang (2008) analyze a panel of 30 OECD countries from 2002 – 2005 to estimate the determinants of broadband penetration under two sets of specifications.Footnote 54 In one set of regressions, unbundling is measured with a simple indicator variable equal to one if the country has implemented local loop unbundling in the year in question, and zero else. In a second set of specifications, unbundling is measured as the number of years since local loop unbundling was first implemented. The authors control for the retail price of DSL, the urban percentage of the population, the age of the population, and the number of non-DSL connections as a percentage of total broadband connections (to control for the presence of competing platforms).

55. The second set of regressions also includes a variable equal to the number of years for which broadband technology has been available in each country, to control for diffusion of broadband over time. (In the absence of this control variable, the unbundling variable employed in the second set of regressions might simply reflect spurious, diffusiondriven correlations). In all specifications, the authors find that the relationship between unbundling and penetration is statistically insignificant.Footnote 55 Finally, the authors do not find a statistically significant relationship between platform competition and penetration, although, unlike most researchers, they do not attempt to correct for endogeneity in the competition metric.

56. Finally, using a biannual panel of 27 European countries from 2002 to 2007, Wallsten and Haulsaden (2009) investigate the effects of unbundling on the penetration of nextgeneration broadband technology.Footnote 56 The authors estimate a regression in which the number of fiber broadband connections per capita is specified as a function of GDP per capita and the number of unbundled lines per capita (defined as the number of per-capita DLS connections offered over either unbundled loops or through bitstream unbundling). The authors also include country and time fixed effects. Estimating separate equations for incumbents and entrants, the authors find, in both cases, that the relationship between unbundling and fiber per capita is negative and statistically significant.Footnote 57

##### 3. Studies Finding Positive Effects of Unbundling on Penetration

57. Garcia-Murillo (2005) analyzes a cross-section of countries from 2001 to investigate the factors determining broadband availability and penetration, using two types of specifications.Footnote 58 The first set of results is of limited interest in the current context, because the dependent variable is a simple, binary indicator of whether broadband has been deployed at all in a given country. Such an analysis is incapable of assessing the degree to which unbundling does (or does not) increase penetration or investment in countries where broadband has already been deployed.

58. The dependent variable employed by Garcia-Murillo in the second set of regressions, the percentage of internet users subscribing to broadband, is more relevant. Explanatory variables in these regressions include GDP per capita, the retail price of broadband, the number of broadband providers, the percentage of internet users, and an indicator for unbundling, which is found to be positive and statistically significant in one specification. However, the sample size is quite small (less than 20), and many of the econometric results are quite anomalous, which calls the overall reliability of the model into question. For example, GDP per capita is found to have no statistically significant effect on penetration, while the estimated relationship between broadband penetration and the retail price of broadband is positive and statistically significant.Footnote 59

59. Grosso (2006) analyzes a panel of 30 OECD countries from 2001-2004, and estimates an econometric model in which broadband penetration is a function of several variables, including a dummy variable for local loop unbundling.Footnote 60 Additional independent variables in the model include GDP per capita and lagged broadband penetration, as well as a variant of the Herfindahl index computed using technology (platform) shares to measure crossplatform competition. The econometric results indicate a negative and statistically significant relationship between the Herfindahl index and broadband penetration, indicating that crossplatform competition increases broadband penetration. In addition, the relationship between unbundling and penetration is positive and statistically significant. However, due in part to data limitations, several key explanatory variables are omitted from the analysis, including demand and cost drivers such as the price of broadband and population density.Footnote 61

60. Importantly, both Garcia-Murillo (2005) and Grosso (2006), along with nearly all empirical researchers, assume that unbundling policies are exogenous when estimating the effect of unbundling on broadband penetration. This ignores the fact that regulatory outcomes such as mandated access prices and the adoption of unbundling policy regimes are subject to political and administrative processes, which implies that they are endogenous.Footnote 62 Endogeneity bias is driven by the fact that regulators may respond to incumbents' infrastructure investments by providing easier access to entrants (by mandating unbundling and/or lowering access prices). Thus, unbundling may appear to drive increased investment and broadband penetration, when in fact the causation runs in the opposite direction. Only a handful of studies have attempted to assess the empirical magnitude of this source of bias. Those that have find substantial evidence of endogenous regulation.Footnote 63

61. Moreover, as we discussed above in the context of the de Ridder Study, few empirical studies control for the fact that the passage of time should have a substantial effect on penetration, due to technology diffusion. So-called general purpose technologies, such as broadband, tend to follow a well-known "S-shaped" curve as they mature.Footnote 64 While initially only early adopters find it worthwhile to purchase the technology, eventually the technology achieves mass-market acceptance and adoption accelerates. Finally, the rate of adoption begins to level off as saturation approaches, and the few remaining non-adopters tend to be those who place a relatively low value on the technology. As noted above, Boyle et. al. (2008) have shown that failure to control for diffusion over time can lead to spurious correlations between unbundling and penetration. Furthermore, this effect will be exacerbated by endogenous regulatory outcomes, to the extent that regulatory commitment problems become more acute with movements along the diffusion curve, as broadband adoption – and the infrastructure investments that make it possible – begin to accelerate.

##### 4. Studies Relating Unbundling to Network Investment

62. There is also a large body of empirical work focusing on the relationship between mandatory unbundling and network investment, which has demonstrated that mandatory unbundling discourages investment by both incumbents and entrantsFootnote 65 and thus calls into question the "stepping stone" hypothesis,Footnote 66 which posits that mandatory unbundling can create a set of "rungs" on a "ladder of investment" allowing entrants to invest gradually in their own facilities. Indeed, the primary author of the ladder of investment thesis, Dr. Martin Cave, has acknowledged that it "remains no more than a hypothesis, as scientific testing of an imprecise proposition of this kind remains problematic."Footnote 67 The most recent authoritative review of the literature on unbundling and investment examines more than 20 empirical studies of access regulation and investment incentives, and concludes that while additional research could be useful, "most of the evidence shows that local loop unbundling…discourages both ILECs and CLECs from investing in networks."Footnote 68 The summary table from that study is reproduced as Exhibit E to this Declaration.

63. The Berkman Study does not express disagreement with the above results. Rather, it simply ignores the effect of policy on private network investment. Instead it focuses approvingly on public investment in broadband network infrastructure. It should be noted that many countries whose regulatory policies have failed to induce investment by incumbents or entrants are now turning to public investment to remedy their policy errors.Footnote 69 We doubt that U.S. authorities would welcome such an outcome.

## Footnotes

1. 6 John de Ridder, "Catching-up in Broadband – What Will it Take?" Organisation for Economic Co-Operation and Development (July 25, 2007) (hereafter, de Ridder Study).
2. 7 The United States has been a leader in network investment, in large part as a result of its rejection of overly broad unbundling policies, but the Berkman Study chooses to ignore this encouraging result, perhaps because it conflicts with its policy recommendations. See, e.g., Jeffrey A. Eisenach, "Broadband in the U.S. – Myths and Facts," in Australia's Broadband Future: Four Doors to Greater Competition (Melbourne: Committee for Economic Development of Australia, 2008) 48-59.
3. 8 Berkman Study at 75.
4. 9 For ease of exposition, we utilize the same system of notation as that used by both the de Ridder Study and the Berkman Study.
5. 10 de Ridder Study at 20.
6. 11 Glenn Boyle, Bronwyn Howell and Wei Zhang, "Catching up in Broadband: Does Local Loop Unbundling Really Lead to Material Increases in OECD Broadband Uptake?" New Zealand Institute for the Study of Competition and Regulation Working Paper (July 2008) (hereafter Boyle, Bronwyn and Zhang).
7. 12 The process by which new technologies are adopted has been studied by economists for many years. See, e.g., Zvi Griliches, "Hybrid Corn: An Exploration in the Economics of Technological Change," 25 Econometrica, Oct. 1957 (showing the standard technology diffusion curves in the case of the introduction of hybrid corn into farming).
8. 13 Boyle, Howell and Zhang at 7-9. Boyle, Howell and Zhang also corrected various other problems in the de Ridder Study, including de Ridder's misspecification of the GUyrs variable, which he allowed to take negative values in the years prior to the adoption of unbundling.
9. 14 Like Boyle, Howell and Zhang, the Berkman Study also eliminates negative values for GUyrs.
10. 15 The de Ridder Study performs least-squares regressions on a 30 country sample from 2005, and a pooled 54 observation sample by adding 24 data points from 2002; it also differenced the two data sets and performed regressions on the resulting 24-country sample. The regressions on the differenced data, however, did not include the GUyrs variable, as differencing it would result in a constant of 3 across all observations, which would result in the dropping of the variable from the regression. That is, one cannot estimate a regression parameter for a "variable" that does not vary.
11. 16 Berkman Study at 117. The Berkman Study presents regression results for approximately two dozen different model specifications, some of which simply replicate the results in the de Ridder Study, and the remainder of which estimate least-squares and mixed effects models using different regression specifications and different formulations of the GUyrs variable.
12. 17 Berkman Study at 115, 141.
13. 18 Our regression data are presented in Exhibit D, so that others can replicate our results and also perform original analysis using the data. We also note that more recent OECD reports have updated certain data series. To be clear, we have used broadband penetration data from Table 1G of the OECD's December 2008 Broadband Report. These data are available from the OECD Broadband Portal, at http://www.oecd.org/document/54/0,3343,en_2649_34225_38690102_1_1_1_1,00.html. Prior series are available in the December reports for various years, which are available at http://www.oecd.org/statisticsdata/0,3381,en_2649_34225_1_119656_1_1_1,00.html.
14. 19 The exclusion of price in the equations is appropriate because price is an endogenous variable that is likely correlated with the disturbance term in the Qtot regression.. Under a linear two equation model with Qtot and Price as dependent variables, the Price equation would serve as the cost equation, with the right hand side variables that affect the build out and operation of the broadband network. Population density, for example, would be a natural cost shifter that would serve this purpose. Substituting for price in the Qtot equation then yields the reduced form equation, which excludes Price, but still controls for its affect. An alternative approach would be to instrument for Price using two-stage least squares. Additional data on price over time would be required to perform this correction.
15. 20 Our confidence in this conclusion is further strengthened by the fact that, if we replicate the Berkman Study's misspecification, we get the same result: When DSLyears is removed from our regression, GUyrs becomes positive and statistically significant at 1 percent.
16. 22 An AR1 process was estimated, yielding a constant autocorrelation coefficient of 0.99 for all countries. There were 28 separate estimated covariances for the heteroskedasticity correction. Relaxing the heteroskedasticity correction yields similar results. The population variable, however, is insignificant, and the p-value for the Unbundled variable is 0.08.
17. 23 Berkman Study at 139.
18. 24 For a more complete explanation of mixed effects models, see Jose C. Pinheiro and Douglas M. Bates, Mixed-Effects Models in S and S-PLUS (Springer 2000).
19. 25 See, e.g., William H. Greene, Econometric Analysis 287-88 (Prentice Hall 5th ed., 2003) (introducing the fixed effects estimator) (hereafter Greene).
20. 26 Greene at 301-302 (discussing the Hausman Specification Test for fixed versus random effects models).
21. 27 We include the variable names, as listed in both the de Ridder Study and the Berkman Study for convenience.
22. 28 This result was also robust to the use of the Berkman Study's modified GUyrs variable, in which they altered the values in 2005 for 17 of 30 countries (we discuss this in more detail below). In particular, substituting that formulation of GUyrs still results in a failure of both random and mixed effects relative to fixed effects. Moreover, the GUyrs variable in the fixed effects model is not statistically significant.
23. 29 Berkman Study at 146.
24. 30 Id. at 145.
25. 31 ECTA, Broadband Scorecard, 4th quarter, 2005.
26. 32 Berkman Study at 115.
27. 33 Debra Aron & David Burnstein, "Broadband Adoption in the United States: An Empirical Analysis," Working Paper, LECG Ltd. (March 2003).
28. 34 Id. at 11-12.
29. 35 Id. at Table 3.
30. 36 Johannes Bauer, Jung Kim, & Steven Wildman, "Broadband Uptake in OECD Countries: Policy Lessons and Unexplained Patterns," Paper prepared for the European Regional Conference of the International Telecommunications Society (August 2003), at 14.
31. 37 Id. at 17-18.
32. 38 38Mario Denni & Harald Gruber, "The Diffusion Of Broadband Telecommunications: The Role Of Competition," Paper presented at the International Communications Society Conference (2005).
33. 39 Id. at 10. For example, if half of all broadband connections were served by DSL, and the other half by cable, then the index would be computed as 0.52 + 0.52 = 0.5
34. 40 Id. at 11. For example, if there are three DSL providers, the competition index for that market is equal to 1/3.
35. 41 Id. at 13.
36. 42 Walter Distaso, Paolo Lupi, & Fabio Manenti, "Platform Competition And Broadband Uptake: Theory And Empirical Evidence From The European Union," Paper presented at the joint PURC - University of Florida and LBS 2005 telecommunications conference (April 2005).
37. 43 Id. Corollary 1, at 12.
38. 44 Id. at 13.
39. 45 The authors do find evidence that lower unbundling prices are significantly associated with higher levels of broadband penetration. However, their analysis assumes that the regulated price of a local loop is determined exogenously. Id. at 16. As discussed in more detail below, to the extent that opportunistic regulators choose to lower access prices in response to increased levels of broadband investments by incumbents, this assumption is invalid, and results in a spurious negative correlation between penetration and unbundling prices.
40. 46 Inmaculada Cava-Ferreruela & Antonio Alabau-Munoz, "Broadband Policy Assessment: A Cross- National Empirical Analysis," Telecommunications Policy 30 (2006).
41. 47 The authors are unable to distinguish between cable networks that have or have not been upgraded to provide broadband service. Id. at 449.
42. 48 Id. at 455.
43. 49 Scott Wallsten, "Broadband and Unbundling Regulations in OECD Countries," AEI-Brookings Joint Center for Regulatory Studies, Working Paper 06-16 (June 2006).
44. 50 Id. at Table 2.
45. 51 Leonard Waverman, Meloria Meschi, Benoit Reillier, & Kalyan Dasgupta, "Access Regulation and Infrastructure Investment in the Telecommunications Sector: An Empirical Investigation," Working Paper, LECG Ltd. (Sept. 2007).
46. 52 Unlike other studies, the authors do not use the lagged Hirfindhal index to instrument for the contemporaneous index, and instead estimate an equation in which the lagged index enters directly as an independent variable. As a consequence, the authors do not directly estimate the role of inter-platform competition.
47. 53 Id. at Table 4.
48. 54 Glenn Boyle, Bronwyn Howell, & Wei Zhang, "Catching Up in Broadband Regressions: Does Local Loop Unbundling Really Lead to Material Increases in OECD Broadband Uptake?," New Zealand Institute For The Study Of Competition And Regulation (July 2008).
49. 55 Id. at Table 2.
50. 56 Scott Wallsten and Stephanie Hausladen, "Net Neutrality, Unbundling, and their Effects on International Investment in Next-Generation Networks," Review of Network Economics 8(1) (March 2009).
51. 57 As a practical matter, the architecture of the next-generation networks that have deployed in the United States would make unbundling either extremely costly or flatly infeasible. See Robert Crandall, Jeffrey Eisenach, & Robert Litan, "Vertical Separation of Telecommunications Networks: Evidence from Five Countries," Federal Communications Law Journal (forthcoming).
52. 58 Martha Garcia-Murillo, "International Broadband Deployment: The Impact of Unbundling," Communications & Strategies 57 (2005).
53. 59 Id. at Table 8.
54. 60 Marcelo Grosso, "Determinants of Broadband Penetration in OECD Nations," Working Paper, Regulatory Development Branch, Australian Competition and Consumer Commission (2006).
55. 61 Id. at Table 1.
56. 62 See, e.g., J. Gregory Sidak and Daniel F. Spulber, "Deregulatory Takings and Breach of the Regulatory Contract," New York University Law Review 71(4) (1996), discussing circumstances under which mandatory unbundling can lead to "deregulatory takings" by opportunistic regulatory agencies; see also David Newbery, Privatization, Restructuring, and Regulation of Network Utilities (MIT Press 2002); see also Jeffrey A. Eisenach and Hal J. Singer, "Irrational Expectations: Can a Regulator Credibly Commit to Removing an Unbundling Obligation?" AEI-Brookings Joint Center Working Paper No. 07-28 (December 2007), available at: http://ssrn.com/abstract=106516.
57. 63 See, e.g., Robert Crandall, Competition and Chaos (Brookings Institution Press 2005), at 71, providing evidence that regulators lower access prices in response to investments by incumbents; see also Tomaso Duso & Lars-Hendrik Röller, "Endogenous Deregulation: Evidence from OECD Countries," Economic Letters 81(1) (2003), showing that political indicators explain the degree of deregulation in the mobile telecommunications industry; see also Michal Grajek and Lars-Hendrik Röller, "Regulation and Investment in Network Industries: Evidence from European Telecoms," ESMT Working Paper No. 09-004 (2009). Using political and geographic variables as instruments, as well as lagged endogenous variables, Grajek and Röller provide empirical evidence that access regulation is determined endogenously, and that failure to control for this bias distorts the statistical relationship between regulation and telecommunications investment decisions.
58. 64 See, e.g., Elhanan Helpman, ed., General Purpose Technologies and Economic Growth (MIT Press 1998).
59. 65 See, e.g., Robert W. Crandall, Allan T. Ingraham and Hal J. Singer, "Do Unbundling Policies Discourage CLEC Facilities-Based Investment?" Topics in Economic Analysis and Policy 4 (2004). The authors use cross-state and within-state variation in the price of constructing local phone lines relative to leasing unbundled loops to identify the sensitivity of CLEC investment in local lines to the LLU rate. They show that mandatory unbundling encourages a CLEC to delay facilities-based investment by altering its relative net present value of investment between time periods. See also Jeffrey A. Eisenach, Paul Lowengrub and James C. Miller III, "An Event Analysis Study of the Economic Implications of the FCC's UNE Decision: Backdrop For Current Network Sharing Proposals," Commlaw Conspectus 17;1 (2008). An even larger body of research has examined competition and liberalization in regulated industries more generally. See Mark Armstrong and David Sappington, "Regulation, Competition, and Liberalization," Journal of Economic Literature 44:2 (2006)
60. 66 Martin Cave, "Encouraging Infrastructure via the Ladder of Investment," Telecommunications Policy 30 (2006).
61. 67 Martin Cave, "Applying the Ladder of Investment in Australia," (December 17, 2007), at 1.
62. 68 Carlo Cambini and Yanyan Jiang, "Broadband Investment and Regulation: A Literature Review," Telecommunications Policy (2009) (in press), at 11-14. (Although the article's title might suggest an exclusive focus on broadband, in reality the authors provide an extensive survey of the literature examining the relationship between regulation and investment in telecommunications infrastructure generally). The International Telecommunications Union (ITU) echoed this finding in a recent report, noting that "the reason that there was so little investment in (local loop) infrastructure by new entrants [in the United States] was that they could not deploy new infrastructure at the regulated local service prices, which were too low and acted as a disincentive to investment. " See International Telecommunications Union, Trends in Regulatory Reform 2008 (November 2008), at 52. Note also that Armstrong and Sappington, supra, at 360, observe that "providing entrants with long-term subsidized access to the incumbent's infrastructure…generally [is] not recommended."
63. 69 Two of the authors of this Declaration, Crandall and Eisenach, discuss this phenomenon in a separate paper. See Robert W. Crandall, Jeffrey Eisenach, and Robert Litan. (2009), "Vertical Separation of Telecommunications Networks: Evidence from Five Countries," Federal Communications Law Journal (forthcoming) (available at http://ssrn.com/abstract=1471960) (hereafter Crandall, Eisenach and Litan).


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The game of Pig is a simple two player dice game in which the first player to reach 100 or more points wins. Players take turns. On each turn a player rolls a six-sided die. After each roll, If the player rolls a 2-6 then he can either: ROLL AGAIN or HOLD. At this point the sum of all rolls made this turn is added to the player's total score and it becomes the other player's turn. If the player rolls a 1 then the player loses his turn. He gets no new points and it becomes the opponent's turn.If a player reaches 100 or more points after holding then the player wins. Write a program that plays the game of Pig, where one player is a human and the other is the computer. Allow the human to input r' (or R') to roll again or h' (or H') to hold. The computer program should play according to the following rule: Keep rolling on the computer's turn until it has accumulated 20 or more points, then hold. Of course, if the computer wins or rolls a 1 then the turn ends immediately. Allow the human to roll first. So here is my code:
// this game is played with a six sided die, on each player's turn he may roll
// until he gets a 1 from the die, at which point, all points accumulated to
// that point are wiped, he may hold and add the accumulation to his score at
// any time.
#include<fstream>
#include<iostream>
#include<cstdlib>
#include<stdlib.h>

using namespace std;

const int score_limit = 100;

int humanTurn(int& humanTotalScore);
int computerTurn(int& computerTotalScore);

int main ()
{
bool continue_game = 1;
int humanTotalScore = 0, computerTotalScore = 0;
cout << "This is the Game of Pig." << endl;
cout << "It is your turn. " << endl << endl;

// Determine whether or not the game is over (whether or not a score of 100 has been reached)
if ((humanTotalScore < score_limit) && (computerTotalScore < score_limit))
{
continue_game = 1;
}
else
{
continue_game = 0;
}

// Play Game
do
{
humanTurn(humanTotalScore);
computerTurn(computerTotalScore);
}
while (continue_game = 1);

// Determine who won the game
if (continue_game = 0 && (humanTotalScore > computerTotalScore))
{
cout << "Congratulations! You Win!";
}
else
{
cout << "You Lose, Computer Wins .";
}

getchar();
getchar();

return 0;
}

int humanTurn(int& humanTotalScore)
{
int accumulatedScore = 0;            //score from this turn
char ans;                            //does player want to roll again or hold
int lastRoll;                        //value of dice roll
cout << "Your current score is " << humanTotalScore << "." << endl;
cout << "Would you like to roll the dice?" << endl;
cout << "R(r) to Roll, H(h) to hold your score" << endl;
cout << "Accumulation from this set of rolls is " << accumulatedScore << endl << endl;
cin >> ans;

//If player decides to roll again
while (ans == 'R' || ans == 'r')
{
lastRoll = ((rand()%6) + 1);
if (lastRoll == 1)
{
cout << "You rolled a 1. Your score is 0 and your turn is over." << endl;
break;
}
else
{
accumulatedScore += lastRoll;
cout << "You rolled a " << lastRoll << endl;
cout << "R(r) to roll Again. H(h) to hold. Current Accumulation:" << accumulatedScore << endl;
cin >> ans;
}
}

//If player decides to hold
while (ans == 'H' || ans == 'h')
{
humanTotalScore += accumulatedScore;
cout << "Your score is " << humanTotalScore << endl;
}

//If player enters a key that is not R, r, H, or h
while ( (ans != 'H') && (ans != 'h') && (ans != 'R') && (ans != 'r') ) //not sure why this is not working
{
cout << "Invalid input, please enter either R(r) or H(h)" << endl;
cin >> ans;
}

return humanTotalScore;
}

int computerTurn (int& computerTotalScore)
{
int accumulatedScore = 0;                                 //score from this turn
int lastRoll;                                             //value of dice roll

lastRoll = ((rand()%6) + 1);
if (lastRoll == 1)
{
cout << "The computer rolled a 1. It's score is 0 and your turn is over." << endl;
}
else
{
accumulatedScore += lastRoll;
cout << "The computer rolled a " << lastRoll << endl;
while (accumulatedScore < 20)
{
lastRoll = ((rand()%6) + 1);
accumulatedScore += lastRoll;
computerTotalScore += accumulatedScore;
}

while (accumulatedScore >= 20)
{
accumulatedScore += lastRoll;
cout << "Computer's score is " << computerTotalScore << endl;
cout << "It is your turn again.\n";
}
}

return computerTotalScore;
}

[Read the FAQ and learn to format your code properly for posting - Zahlman] I'm having problems with the Hold. Whenever I input h or H as the answer, it outputs a very high number repeatedly and wont stop. Can anyone tell me what i'm doing wrong?!?! [Edited by - Zahlman on April 3, 2010 10:53:26 AM]

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This loop will never terminate...

while (continue_game = 1);

##### Share on other sites
Neither will this one, if it's ever entered:
//If player decides to holdwhile (ans == 'H' || ans == 'h'){humanTotalScore += accumulatedScore;cout << "Your score is " << humanTotalScore << endl;}

Notice that `ans' doesn't change value inside the loop, so if the condition was true the first time around, it will always be true.

format: plain


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# Homework Help: Inverse Trig Function Problem

1. Jan 27, 2008

### NightSky72

1. The problem statement, all variables and given/known data
The problem here involves inverse trig functions. The question here is why doesn't arccos(cos x) yield the equation y = x?

2. Relevant equations
y = arccos(cos x)

3. The attempt at a solution
I assume the reason is due to restrictions on domain and/or range of the trig functions. I graphed it out from -4pi to 4pi, and noticed that from 0 to pi, 2pi to 3pi, etc., it does display a function similar y = x (0 to pi it IS y = x). However, on intervals such as -pi to 0, it is the same as the function y = -x. Now, I looked at the unit circle and can see that, if it is the arccos (cos x), the cos of radian measurements (essentially the sine when reading on the unit circle) is strictly positive. However, between -pi and 0, the values would be negative. They increase and decrease graphically, respectively. Anyone have any input or guiding suggestions here? Thanks.

2. Jan 27, 2008

### jambaugh

Start by looking at the graph of y=cos(x) and asking if it is an invertible function.
As such it is not! (Horizontal line test!)
What then does arccos mean?

3. Jan 27, 2008

### NightSky72

That means arccos x can only exist under certain restrictions, since an equation does not have an inverse unless its graph passes the horizontal line test. The domain of arccos x is [-1, 1] and the range is [0, pi]. Ok, so if the range can only be 0 to pi, then the graph of arccos (cos x) cannot have a range outside of that. And since the cos x only ranges between -1 and 1, then arccos (cos x) can be defined for any value of x, since the cos x is between -1 and 1, which is the domain of arccos x?


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## Linear Algebra and Its Applications, Exercise 3.4.2

Exercise 3.4.2. Given two orthonormal vectors $a_1 = (\frac{2}{3}, \frac{2}{3}, -\frac{1}{3})$ and $a_2 = (-\frac{1}{3}, \frac{2}{3}, \frac{2}{3})$ and the vector $b = (0, 3, 0)$, project $b$ onto $a_1$ and $a_2$. Also find the projection $p$ of $b$ onto the plane formed by $a_1$ and $a_2$.

Answer: Since $a_1$ is orthonormal, the projection of $b$ onto $a_1$ is given by

$a_1^Tba_1 = \begin{bmatrix} \frac{2}{3}&\frac{2}{3}&-\frac{1}{3} \end{bmatrix} \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix} \begin{bmatrix} \frac{2}{3} \\ \frac{2}{3} \\ -\frac{1}{3} \end{bmatrix} = 2 \cdot \begin{bmatrix} \frac{2}{3} \\ \frac{2}{3} \\ -\frac{1}{3} \end{bmatrix} = \begin{bmatrix} \frac{4}{3} \\ \frac{4}{3} \\ -\frac{2}{3} \end{bmatrix}$

Similarly the projection of $b$ onto $a_2$ is given by

$a_2^Tba_2 = \begin{bmatrix} -\frac{1}{3}&\frac{2}{3}&\frac{2}{3} \end{bmatrix} \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix} \begin{bmatrix} -\frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix} = 2 \cdot \begin{bmatrix} -\frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix} = \begin{bmatrix} -\frac{2}{3} \\ \frac{4}{3} \\ \frac{4}{3} \end{bmatrix}$

The projection $p$ of $b$ onto the plane formed by $a_1$ and $a_2$ is simply the sum of the projections above:

$p = a_1^Tba_1 + a_2^Tba_2 = \begin{bmatrix} \frac{4}{3} \\ \frac{4}{3} \\ -\frac{2}{3} \end{bmatrix} + \begin{bmatrix} -\frac{2}{3} \\ \frac{4}{3} \\ \frac{4}{3} \end{bmatrix} = \begin{bmatrix} \frac{2}{3} \\ \frac{8}{3} \\ \frac{2}{3} \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.

This entry was posted in linear algebra and tagged , . Bookmark the permalink.


text:
## A family of derivative-free methods with high order of convergence and its application to nonsmooth equations.(English)Zbl 1246.65079

Summary: A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to $$1.6266$$. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

### MSC:

 65H05 Numerical computation of solutions to single equations
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### References:

 [1] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. · Zbl 0241.65046 [2] A. M. Ostrowski, Solution of Equations and Systems of Equations, Pure and Applied Mathematics, Vol. 9, Academic Press, New York, NY, USA, 1966. · Zbl 0222.65070 [3] H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643-651, 1974. · Zbl 0289.65023 [4] Z. Liu, Q. Zheng, and P. Zhao, “A variant of Steffensen’s method of fourth-order convergence and its applications,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 1978-1983, 2010. · Zbl 1208.65064 [5] M. Dehghan and M. Hajarian, “Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations,” Computational & Applied Mathematics, vol. 29, no. 1, pp. 19-30, 2010. · Zbl 1189.65091 [6] A. Cordero and J. R. Torregrosa, “A class of Steffensen type methods with optimal order of convergence,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7653-7659, 2011. · Zbl 1216.65055 [7] Y. Hu and L. Fang, “A seventh-order convergence Newton-type method for solving nonlinear equations,” in 2nd International Conference on Computational Intelligence and Natural Computing, 2010. [8] M. A. Noor, W. A. Khan, K. I. Noor, and E. Al-said, “High-order iterative method free from second derivative for solving nonlinear equations,” International Journal of the Physical Science, vol. 6, no. 8, pp. 1887-1893, 2011. [9] F. Soleymani and S. K. Khattri, “Finding simple roots by seventh and eighthorder derivative-free methods,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 1, no. 6, pp. 45-52, 2012. [10] S. Amat and S. Busquier, “On a Steffensen’s type method and its behavior for semismooth equations,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 819-823, 2006. · Zbl 1096.65047 [11] A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A modified Newton-Jarratt’s composition,” Numerical Algorithms, vol. 55, no. 1, pp. 87-99, 2010. · Zbl 1251.65074 [12] A. Cordero and J. R. Torregrosa, “Variants of Newton’s method using fifth-order quadrature formulas,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 686-698, 2007. · Zbl 1122.65350 [13] S. Amat and S. Busquier, “On a higher order secant method,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 321-329, 2003. · Zbl 1035.65057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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Summer Deals - hours only!Up to 80% off on all courses and bundles.-Close
Introduction
ST_Equals
2. Get to know sf_restaurants
Relationships Between Geometries
ST_Distance and ST_Dwithin
Spatial joins practice
Summary

## Instruction

Before we begin, there's one more table we'd like you to know: sf_restaurants. Let's take a look at this table now.

## Exercise

Select all information from the sf_restaurants table.

As you can see, this table has the following 7 columns:

• id — the id of a given restaurant,
• name — the name of a given restaurant,
• food_type — the type of food served at a given restaurant,
• smoking_area — denotes whether there is an area for smokers in a given restaurant,
• rating — the rating of a given restaurant,
• avg_price — the average price that a customer pays per visit,
• coordinates — a geometry column with location of a given restaurant.


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#### On the Dispersions of Three Network Information Theory Problems

##### Vincent Y. F. Tan, Oliver Kosut

We analyze the dispersions of distributed lossless source coding (the Slepian-Wolf problem), the multiple-access channel and the asymmetric broadcast channel. For the two-encoder Slepian-Wolf problem, we introduce a quantity known as the entropy dispersion matrix, which is analogous to the scalar dispersions that have gained interest recently. We prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability. To gain better intuition about the rate at which the non-asymptotic rate region converges to the Slepian-Wolf boundary, we define and characterize two operational dispersions: the local dispersion and the weighted sum-rate dispersion. The former represents the rate of convergence to a point on the Slepian-Wolf boundary while the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit. Interestingly, when we approach either of the two corner points, the local dispersion is characterized not by a univariate Gaussian but a bivariate one as well as a subset of off-diagonal elements of the aforementioned entropy dispersion matrix. Finally, we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple-access channel and the asymmetric broadcast channel in terms of dispersion matrices. All our proofs are unified a so-called vector rate redundancy theorem which is proved using the multidimensional Berry-Esseen theorem.

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# Henri Auguste Delannoy

### Quick Info

Born
28 September 1833
Bourbonne-les-Bains, Haute-Marne, France
Died
5 February 1915
Guéret, La Creuse, France

Summary
Henri Delannoy was a French mathematician who worked on probability and recreational mathematics. He was also intersted inthe history of mathematics.

### Biography

Henri Delannoy's parents were Omère Benjamin Joseph Delannoy, a military accountant, and Françoise Delage. In fact on his birth certificate his name appears as 'Henry' but in later life he had it officially changed to 'Henri' so in this biography we use the version of the name by which Delannoy himself wanted to be known. Omère Delannoy had fought in a number of Napoleon's military campaigns, in particular at the Battle of Waterloo, before becoming an administrator in the military. He met Françoise Delage in Guéret, in central France, and they were married there on 24 November 1830. Henri was brought up in Guéret where he attended the Collège de Guéret.

In 1849 he was awarded his baccalaureate, having received permission to take the examination when he was still officially too young. He then spent two years studying mathematics at the Lycée de Bourges, going there because his family were living in the vicinity at the time. He then attended Sainte-Barbe College in Paris, in order to prepare for the difficult entrance exams to the prestigious École Polytechnique. He was ranked 62nd in the entrance examinations and entered the École Polytechnique in 1853. The records contain the following physical description of him: dark brown hair, height 1.68m, average forehead, medium nose, blue eyes, small mouth, round chin, round face. His academic record was certainly not particularly strong since he came 91st out of a class of 106 in 1854 and he graduated in the following year coming 67th out of 94 students. He joined the military on 1 May 1855 and trained at the Artillery School in Metz, graduating 12th out of 37 students in 1856. He was given the rank of sub-lieutenant in 1856 and promoted to lieutenant on 1 May of the following year.

Delannoy saw action in 1859 in the Second Italian War of Independence. France, led by Napoleon III, was allied with the Kingdom of Piedmont-Sardinia against Austria. War between Austria and Sardinia was declared on 29 April 1859 and a large French army marched towards Piedmont to support the greatly out-numbered Sardinian forces. Delannoy was with this French army and took part in the decisive Battle of Solferino on 24 June which proved to be a tactical victory for the French-Piedmontese forces but at a high cost in terms of soldiers lost and wounded. Delannoy survived unscathed and, after spending time stationed in Brescia and on Lake Garda, returned to Guéret where he married Olympe-Marguerite Guillon, daughter of the chemist Antoine Guillon, on 10 November 1859; they had two daughters and one son.

In 1863 Delannoy was promoted to captain [2]:-
He spent three years in Africa (6 October 1866-25 October 1869). He was the governor of the military Hospital of Sidi-bel-Abes, Algeria, during the terrible typhus epidemic (he belonged the Supply Corps and they were in charge of the sanitary affairs). He translated for himself, and perhaps also for his hierarchy, several German books/notes about the Supply Corps. He took part in the 1870 war between France and Prussia. ... he was in Germany on July 26, 1870 (that is, 4 days after the declaration of war ...) and on March 7, 1871 (that is, 3 days before the Treaty of London ...).
In 1876 Delannoy's wife died after being severely burnt in a tragic accident in her own kitchen. We have not mentioned any mathematical activity by Delannoy, except as part of his schooling, up to this point and this is actually an accurate picture of his involvement in mathematics. Only in 1879 did he start to become interested in mathematics when he read articles by Édouard Lucas in La Revue Scientifique. He began corresponding with Lucas in the following year and the two began to exchange mathematical ideas. His interests were in mathematical puzzles, mathematical recreations, and probability. In all he seems to have published eleven articles but he also published many problems and solutions to problems set by others. His first publication was Emploi de l'échiquier pour la solution de problèmes arithmétiques (1886) followed by Sur la durée du jeu (1888). Both of these were published while he was still in the army, working as a military intendant in the city of Orléans. However, Delannoy decided that he wanted to retire from the army and pursue his interests in mathematics and also his interests as a historian. He retired from the army on 9 January 1889 and returned to Guéret where he had been brought up. He lived there for the rest of his life devoting himself to his mathematical and historical interests. Let us now look briefly at his mathematical contributions.

The first paper by Delannoy, mentioned above, literally means 'Using a chessboard to solve arithmetical problems' but it would be much more understandable if we translated 'échiquier' as 'array' for indeed this is what Delannoy uses. In this paper he introduces numbers which are now known as 'ballot numbers' or 'Delannoy-Segner numbers'. They count the number of lattice paths between (1, 0) and $(m, n), m > n$, which do not cross the diagonal. The 1888 paper, The length of a game, uses lattice paths to analyse the following game previously considered by Eugène Rouché and Joseph Bertrand:
Pierre and Paul play a game against each other with equal probabilities. At the start of the game each has $n$ francs; at each round, the winner gets one franc from his opponent. The game continues until one player is ruined. What is the probability $P$ that the game ends at the beginning of the round $m$.
In Emploi de l'échiquier pour la résolution de divers problèmes de probalilités (1889), Delannoy uses his lattice paths in arrays to solve seven problems already studied by mathematicians such as Ampère, Bertrand, Huygens, Laplace and Rouché. In this paper Delannoy introduces the 'Delannoy numbers' which count the lattice paths from the origin to $(m, n)$ where the path from each point can go up, right, or across the (right-up) diagonal. For example, the number of such paths from the origin to (1, 1), to (2, 2), to (3, 3), to (4, 4), ... is
3, 13, 63, 321, 1683, 8989, 48639, 265729, ...

It is interesting to note that in the 1950s a mysterious connection was spotted between these diagonal Delannoy numbers and Legendre polynomials. Only very recently has Gábor Hetyei explained this connection, which was previously thought to be a coincidence, giving a geometric interpretation of the relation between the diagonal Delannoy numbers and the Legendre polynomials in [3].

Other papers by Delannoy are: Problèmes divers concernant le jeu (1890), Formules relatives aux coefficients du binôme (1890), Sur le nombre d'isomères possibles dans une molécule carbonnée (1894), Sur les arbres géométriques et leur emploi dans la théorie des combinaisons chimiques (1894), Sur une question de probabilités traitée par d'Alembert (1895), Emploi de l'échiquier pour la résolution de certains problèmes de probabilités (1895), and Sur la probabilité des événements composés (1898).

We mentioned above that, in addition to his mathematical interests, Delannoy was an active historian, indeed he published numerous articles in the Mémoires de la Société des Sciences Naturelles et archéologiques de la Creuse. He wrote about criminal trials in La Marche, riots in Guéret in 1705, grapevines in La Creuse, local abbeys and abbots, and other similar topics. Following his retirement, he was very active in the Société des Sciences Naturelles et archéologiques de la Creuse, serving as its president from 1896 until his death in 1915. Finally let us note another of Delannoy's interests - he painted watercolours.

### References (show)

1. J-M Autebert, A-M Décaillot and S R Schwer, Henri-Auguste Delannoy et la publication des oeuvres posthumes d'Édouard Lucas, Gaz. Math. No. 95 (2003), 50-62.
2. C Banderier and S Schwer, Why Delannoy numbers?, J. Statist. Plann. Inference 135 (1) (2005), 40-54.
3. G Hetyei, Delannony orthants of Legendre polytopes, Discrete Comput. Geom. 42 (4) (2009), 705-721.
4. J Schröder, Delannoy and tetrahedral numbers, Comment. Math. Univ. Carolin. 48 (3) (2007), 389-394.
5. S R Schwer and J-M Autebert, Henri-Auguste Delannoy, une biographie. I, Math. Sci. Hum. Math. Soc. Sci. No. 174 (2006), 25-67.