e-companion ONLY AVAILABLE IN ELECTRONIC FORM

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1 OPERATIONS RESEARCH doi 0.87/opre ec e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 0 INFORMS Electronic Companion Fied-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mied Logit Demand by W. Ross Morrow and Steven J. Skerlos, Operations Research, doi 0.87/opre

2 ec Supplemental Material EC.. Simple Eample In this section we elaborate on Eample 6. Recall that F : R N R N is given by F = /+, has a unique finite zero at 0, and vanishes as. EC... Pure Newton We first note that DF = + [ I + ]. DF is nonsingular only if. By the Sherman-Morrison-Woodbury formula Ortega and Rheinboldt 970, Dennis and Schnabel 996, DF = + [ ] I + so long as. Thus the Newton step, s N, is s N = DF F = + = The Newton point + s N is thus given by + s N =. + We consider convergence of the pure Newton iteration to 0. Note that + s N 0 = 0 and thus the Newton step reduces the -norm error in only if <. This inequality is satisfied only if < / 3 as can be easily checked. Thus < 0 if < / 3 + s N 0 = 0 if = / 3 > 0 if > / 3, Furthermore n + s N n = k and thus the Newton step increases the absolute value of every nonzero component of when > / 3. When / 3,, the Newton steps also enforce an alternation in the signs of the components of. If = / 3 the Newton point is. The Newton iterates diverge if 0 > / 3 simply because s N n cannot vanish; indeed, if n γ / {0, } then + γ n+ n = s N n γ 0. γ

3 ec3 Note also that = + s N = 3 requires / 3,, because otherwise + s N / 3 or + s N > >. Now ψa = a 3 / a is an increasing function on [/ 3, with ψ/ 3 = / 3 and ψa as a. Thus for any γ / 3, ] there is a unique ψ γ / 3, such that ψψ γ = γ. Define a sequence of radii {r n } n=0 by r 0 = and r n = ψ r n = ψ n, n, where ψ n is the n th is the inverse map ψ iterated n times. Some iterate in the Newton sequence falls on the unit sphere if, and only if, 0 = r n for some n {0,,,..., }. EC... Line Search Line search takes the update + λs N / s N for some step length λ. We make two observations based on the formula { } s N + = sign. s N First, there is always an eact line search step to 0. Specifically, if { } + λ = sign then + λs N / s N = 0. Second, if > then any line search step with positive step length increases the distance to 0. This is because + λs N / s N = + λ. If [/ 3, the line search may enforce convergence that did not occur with the pure Newton method. Unfortunately most step length finding methods use only positive step lengths generally for good reason. For eample, a simple backtracking line search sets λ = m > 0 where m = inf { F + m s N < α m F }. Another popular choice forms a quadratic model of F + λs N and computes its minimizer, safeguarding the steplength to be some positive bounded fraction of the previous steplength. This method also generally computes only positive steplengths, and thus will diverge for 0 >. EC..3. Trust Region Methods Trust region methods try to minimize ϕ = / F + s subject to the constraint that s δ. This is itself a difficult nonlinear optimization problem, and thus practical trust region methods are based on minimizing the following quadratic local model of ϕ instead: m s = F + F DFs + s DF DFs EC..3.. Levenburg-Marquardt Method or Hookstep Minimizing this quadratic model subject to the constraint s δ is conceptually straightforward: either i s N δ and then s L = s N, or ii s N > δ and we take s L = sλ where sλ solves DF DF + λi sλ = DF F and λ > 0 is the unique solution to sλ = δ. Now DF F = + +

4 ec4 and [ ] DF DF = I + + In general the LM system is { [ ] } I + λi sλ = [ + λ + ] I sλ = + + Define Λ = + λ + and A = / +. Then [ ΛI A ] sλ = + [ ] A I sλ = Λ Λ + [ ] sλ = A I + Λ + Λ A = + Λ A If > and Λ > A, then sλ increases the distance to the origin. Now Λ > A if, and only if, λ > + + Note that if >, then this inequality is trivially satisfied for all λ 0. In other words, the hookstep also increases the length of, and thus does not enforce global convergence to 0 for 0 >. EC..3.. Powell s Hybrid Method or Dogleg Step Powell s Hybrid method or the Dogleg step s D is the Newton step if s N δ, otherwise is the unique step of -norm δ on the piecewise linear path from 0 to the Cauchy step s C defined below and then to the Newton step. This does not alleviate the poor global convergence behavior because the Cauchy point and the Newton point coincide. Thus the dogleg step is simply a scaled Newton step: s D = min{ s N, δ}s N / s N. If >, this step points increases all components of and, regardless of δ, + s D > again. If <, this step points in the right direction and the scaling could effectively enforce convergence to 0. In the remainder of this section we prove our claim that the Cauchy point and the Newton point coincide. We first compute the steepest descent direction for ϕ = / F : d = DF F = + + Note that the steepest descent steps, like the Newton step, also just re-scale. The Cauchy step s C is the minimizer of m s in the steepest descent direction. Since α m αd = F + α d + DFd

5 ec5 we must have However note that so that and DFd = + = + = + d α =. DFd DF d α = s C = αd = = + { } = sign + + { } = sign = = s N That is, the Cauchy step is the Newton step, and thus the Cauchy and Newton points coincide. The dogleg step is thus a line search method, and cannot enforce global convergence to 0 for 0 >. EC.. Eample 8. Here we develop Eample 8 more than in the tet. This eample gives a case in which η-fpi is not locally convergent while ζ-fpi is. What we need to show here is that we can choose ϑ > to ensure that ρdηp ϑ > ; note that we must include the dependence of equilibrium prices on the value of the outside good. The ζ-markup equations states that Morrow 008 ˆπp ϑ, ϑ = α PL p ϑ, ϑ P L p ϑ, ϑ = ρ Dηp ϑ. α where p ϑ is the unique vector of profit-maimizing prices for a given value of the outside good, ϑ. Below we show that ˆπp ϑ, ϑ +, and thus ρdηp ϑ +, as ϑ. This establishes that ρdηp ϑ > for sufficiently small ϑ. We want to show that for all M > 0, there eists some ϑ R such that ˆπp ϑ, ϑ M for all ϑ ϑ. Now, by definition, ˆπp ϑ, ϑ ˆπp, ϑ for any p. In particular, ˆπp ϑ, ϑ ˆπc + m, ϑ = Ec + m e ϑ + Ec + m m

6 ec6 for any m > 0 where Ep = J j= eu jp j. Furthermore ˆπc + m, ϑ M if, and only if, ϑ ϑm, m M M = log + log Ec + m. M Thus for any M > 0, ˆπp ϑ, ϑ ˆπc + M +, ϑ M if we take ϑ ϑm +, M = log M + log Ec + M +. That is, there is a ϑ negative enough to make ˆπp ϑ, ϑ as large as we want. EC.3. Additional Details Regarding the Numerical Eamples EC.3.. The Two Demand Models To characterize demand for new vehicles, we employ modified versions of two eisting models of new vehicle purchasing Boyd and Mellman 980, Berry et al Here we give a brief description of the versions of these models used in our eamples below. Random coefficient distributions are described by the parameters given in Tables 5 and 6. The utility function in the Boyd and Mellman 980 model is linear in characteristics and price with lognormally distributed unobserved demographic variables random coefficients θ, has no observed demographic variables, and does not model an outside good i.e., ϑθ. Specifically, with θ = α, β P R 3, uα, β,, p = αp + β = αp + β + β β 3 3 We include the vehicle characteristics : size length times width over height, all in inches, : acceleration 60 divided by the 0-60 acceleration time, in seconds, and 3 : fuel consumption 00 times fuel consumption, in gallons per mile. A simple function of horsepower to weight ratio approimates 0-60 acceleration, a commonly used proy in econometric models of the vehicle market. Due to a lack of data, we eclude several consumer reports ratings ride, handling, and noise used in the original model. We use 980 dollars as our monetary units. The utility function in the Berry et al. 995 model is linear in characteristics with independent and normally distributed unobserved demographics random coefficients, nonlinear in income minus price, and models an outside good. Specifically, with θ = φ, β, β 0 P R 3 R, uφ, β, β 0,, p = α logφ p + β and ϑφ, ψ, ν = α log φ + β 0 for a price coefficient α = >, income φ, and random coefficients β. Consistent with the original model, we take income, price, and cost to be in thousands of 983 dollars where income is lognormally distributed with a log-mean of 0 3 log and a log-standard deviation of CPS 007. For vehicle characteristics we include : operating cost, in 0 mile increments driven per dollar spent using a fuel price of $ = $.7 983, : horsepower to weight ratio weight is in 0 lbs., and 3 length times width both in hundreds of inches. We effectively eclude a dummy variable for standard air conditioning present in the original model by assuming standard air conditioning on all vehicles the utility value for this is absorbed into the utility of the outside good. No other characteristics from the original model differ in our version. The random coefficients β and β 0 are independently normally distributed with means and standard deviations listed in Table 6.

7 ec7 EC.3.. Etrapolation of Costs Section 4.3 considers a differentiated product market model with 5,98 vehicles derived from the Ward s data. To generate this larger problem we etrapolate the J. D. Power cost data by assuming that variation in dealer costs is reflected in the MSRP reported by Ward s. Specifically, we define a model-year plus trim s dealer cost as the average dealer cost for the model-year vehicle with which it is associated plus the deviation of the model-year variant specific MSRP from the average MSRP across all variants of that model-year. EC.3.3. Arbitrary Initial Conditions Our arbitrary initial conditions used in Section 4.3 were formally defined as follows. For the Boyd and Mellman model initial conditions are drawn uniformly from [min j NJ c j, ma j NJ c j ] J. For the Berry et al. model, the finite reservation price income makes choosing arbitrary initial conditions more difficult. We draw initial conditions uniformly from [0, 9000] J, where the 70 th percentile of income is approimately $9, Using this upper limit ensures that, loosely speaking, the upper 30 % of the sampled population can buy any vehicle at the initial prices, and does not preclude the eistence of individuals in the sampled population that have an income too low to buy any of the vehicles at their initial prices. EC.3.4. Additional Results EC Cost Trials unit costs. Table EC. provides detailed results regarding our trials starting at

8 ec8 Table EC. Results of price equilibrium computations starting at unit costs under both demand models for ten, 000-sample sets. n: iterations to termination maimum is 75; t: CPU time in seconds; FO : First-order conditions satisfied S or failed F ; SO : Second-order conditions satisfied S or failed F ; p p FPI : absolute deviation from equilibrium prices computed by ζ-fpi in 980 and 983 dollars. Boyd and Mellman 980 Berry et al. 995 η-nm p p FPI η-nm p p FPI n t FO SO min median ma n t FO SO min median ma S S.30E-06.84E E S S 8.97E E-0.6E F F 6.88E-0.84E+0 6.7E S S 8.7E E-0.44E S S 8.5E E E S S.90E-07.E-04.78E F F 7.06E E+0 7.8E S S 3.9E E E S S.6E-04 8.E E S S.80E E-04.6E S S.95E E E S S 6.8E E E S S.3E-04.06E-0.9E S S 5.36E E E S S 5.3E-04.37E-0.33E S S 9.86E E-05.7E F F 8.66E-0.77E+0 4.5E S S 8.56E-06.4E-04.6E S S.57E-05.0E E S S.07E E-04.64E min min med med 5.84 ma ma ζ-nm p p FPI ζ-nm p p FPI n t FO SO min median ma n t FO SO min median ma S S.59E-06.84E E S S 4.E E-0.E S S 4.99E-05.06E-0.E S S.03E-03.97E-0.7E S S 6.47E E E F F 3.6E E-0.48E F F 9.78E E+0 5.6E S S.36E E E F F 7.8E-0.58E+0.58E S S 3.48E-07.E E S S 8.E E E S S 7.07E E E S S 4.45E-05.07E-0.9E S S.7E E E S S 3.3E-05.35E-0.33E S S.88E E E F F.3E-03.43E+0 4.5E S S.55E E E F F.37E-0.03E E S S 9.80E E E min min med med ma 90.6 ma ζ-fpi ζ-fpi n t FO SO n t FO SO S S S S S S 8.58 S S S S 7.47 S S 9.0 S S S S S S S S S S S S S S S S S S S S S S 7.39 S S S S S S min 8.58 min med med ma ma