Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand

Transcription

1 OPERATIONS RESEARCH Vol. 59, No. 2, March April 2011, pp issn X eissn doi /opre INFORMS Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand W. Ross Morrow Departments of Mechanical Engineering and Economics, Iowa State University, Ames, Iowa 50014, Steven J. Skerlos Department of Mechanical Engineering, University of Michigan, Ann Arhor, Michigan 48105, This article describes numerical methods that exploit fixed-point equations equivalent to the first-order condition for Bertrand-Nash equilibrium prices in a class of differentiated product market models based on the mixed-logit model of demand. One fixed-point equation is already prevalent in the literature, and one is novel. Equilibrium prices are computed for the calendar year 2005 new-vehicle market under two mixed-logit models using (i) a state-of-the-art variant of Newton s method applied to the first-order conditions as well as the two fixed-point equations and (ii) a fixed-point iteration generated by our novel fixed-point equation. A comparison of the performance of these methods for a simple model with multiple equilibria is also provided. The analysis and trials illustrate the importance of using fixed-point forms of the first-order conditions for efficient and reliable computations of equilibrium prices. Subject classifications: Bertrand-Nash equilibrium prices; differentiated product markets; mixed logit; Newton s method; GMRES-Newton hookstep; fixed-point iteration. Area of review: Marketing Science. History: Received November 2008; revisions received August 2009, March 2010; accepted April Introduction Bertrand competition has been a prominent paradigm for the empirical study of differentiated product markets for at least 20 years. Firms engaged in Bertrand competition maximize profits by choosing prices for portfolios of differentiated products, and Bertrand-Nash equilibrium prices simultaneously maximize profits for all firms. Models combining Bertrand competition with the mixed-logit discretechoice model of consumer demand have been used to study the automotive industry, electronics, entertainment, and food products and services; see Dube et al. (2002). Many applications of Bertrand competition rely on counterfactual experiments: exercises in which hypothetical market conditions are simulated with an estimated model. Such experiments have been used to study corporate mergers (Nevo 2000a), novel products and services (Petrin 2002, Goolsbee and Petrin 2004, Beresteanu and Li 2008), store locations (Thomadsen 2005), and regulatory policy changes (Goldberg 1995, 1998; Beresteanu and Li 2008). By definition, simulating market outcomes in counterfactual experiments requires computing equilibrium prices after changing the values of exogenous variables such as the number of firms or the products offered. Numerical methods for computing equilibrium prices have not yet received a thorough treatment in the literature, which currently focuses on model specification and estimation; see Knittel and Metaxoglou (2008), Dube et al. (2011), and Su and Judd (2008) for recent developments in estimation. This article fills this gap with a detailed investigation of four approaches for computing Bertrand-Nash equilibrium prices in single-period, multifirm models with mixed-logit demand. Applying Newton s method to some form of the firstorder or simultaneous stationarity condition is currently the de facto approach for computing equilibrium prices; see, for example, Nevo (1997, 2000a), Petrin (2002), Smith (2004), Doraszelski and Draganska (2006), and Jacobsen (2006). Newton s method applied directly to the first-order condition may converge when started at observed prices if changes in exogenous variables have a marginal impact on equilibrium prices. However, when the changes to exogenous variables imply significant changes in product prices, Newton s method applied directly to the first-order conditions may fail to compute equilibrium prices. Furthermore, analyses that do not have observed prices to use as an initial guess will require methods with greater reliability. This article demonstrates that solving fixed-point equations equivalent to the first-order condition for equilibrium is more reliable and efficient than solving the first-order condition itself. One fixed-point equation equivalent to the first-order conditions is the BLP-markup equation popularized by Berry et al. (1995). A second fixed-point equation, here termed the -markup equation, is a novel way to write the same condition on markups. Both markup equations lead to more-robust numerical methods than found with a simple application of Newton s method to the first-order condition. Using the fixed-point expressions in 328

2 Operations Research 59(2), pp , 2011 INFORMS 329 this way can be considered nonlinearly or analytically preconditioning the first-order condition satisfied by equilibrium prices, a technique well known in applied mathematics (Brown and Saad 1990, Cai and Keyes 2002). The existence of fixed-point equations for equilibrium suggests applying fixed-point iteration (Judd 1998) instead of Newton s method to compute equilibrium prices. The BLP-markup equation does not appear to be well suited to fixed-point iteration. Example 8 provides a case in which iterating on the BLP-markup equation is not necessarily locally convergent, whereas iterating on the -markup equation is superlinearly locally convergent. Iterating on the -markup equation also eliminates the need to solve linear systems, required to implement Newton s method and to iterate on the BLP-markup equation. This property makes fixed-point steps based on the -markup equation very inexpensive relative to Newton steps, an essential property to obtaining fast computations from generally linearly convergent fixed-point iterations. This fixed-point iteration is used to study the sensitivity of computed equilibrium prices to (i) the initial guess of equilibrium prices and (ii) the finite sample set used for simulation of a mixed-logit model (Train 2003). We find that computed equilibrium prices vary little with the initial guess, but are more sensitive to the sample set than is currently assumed. In one of our examples, computed equilibrium prices for roughly half of the products vary more than $100 even with 100,000 samples; computed equilibrium prices for 90% of products vary more than $1 even with 1,000,000 samples. Although the meaning of precision depends on the application, these results suggest that variability due to simulation should receive more attention than it currently does. Besides Newton s method and fixed-point iteration, few other practical approaches to the computation of equilibrium prices exist. Variational formulations, widely applied in economic and engineering problems (Ferris and Pang 1997), contain many solutions that need not be equilibria of the original problem. Explicit least-square minimization or Gauss-Newton methods can also be implemented but are computational disadvantages relative to applications of standard Newton-type methods for nonlinear systems. Some authors apply tattonement iterating on a game s best-response correspondence to compute equilibrium in prices or other strategic variables, including product mix (Choi et al. 1990), product characteristics (CBO 2003, Austin and Dinan 2005, Bento et al. 2005), and engineering variables (Michalek et al. 2004). Tattonement, however, has three issues: it requires the iterative computation of profitoptimal prices (a special case of the problem discussed in this article), it should be inefficient relative to direct methods whenever optimal strategies are coupled, and it lacks the global convergence guarantees of contemporary Newton solvers. Morrow and Skerlos (2010) review these conclusions in more detail. This article proceeds as follows: 2 describes Bertrand competition under mixed-logit models of demand. It derives three representations of equilibrium prices: (i) the first-order or simultaneous stationarity condition, (ii) the BLP-markup equation, and (iii) the -markup equation. Section 3 presents and motivates four numerical approaches for computing equilibrium prices based on these representations of the first-order condition. Section 4 compares the four approaches in a numerical example with 993 vehicles sold during 2005 and provides a sensitivity study using an expanded set of 5,298 vehicles. Section 5 addresses the performance of these methods in a simple example where there are multiple equilibria. Many of the technical details are provided by Morrow and Skerlos (2010). 2. Fixed-Point Equations for Equilibrium Prices Under Mixed-Logit Models This section derives the BLP- and -markup equations for (local) equilibrium prices in differentiated product market models with mixed-logit demand. Mixed-logit models are popular in empirical applications and dense in the class of random utility models (RUM) (McFadden and Train 2000). Our description of the mixed-logit model in 2.1 follows Train (2003) and is more general than the description now standard in the empirical differentiated product markets literature. This generality captures a variety of models currently used in practice, as well as the finite-sample simulators used both in estimation and computations of equilibrium prices (see Example 4). Sections 2.2 and 2.3 derive firms profits and the first-order conditions for local equilibrium given this choice model; see also Nevo (2000b) or Dube et al. (2002). Sections 2.4 and 2.5 derive the BLPand -markup equations from the first-order conditions. Tables 1 and 2 gives our nomenclature Consumers, Products, and Choice Probabilities A collection of firms offer a total of J products to a population of individuals (or households). Each product j = 1 J is defined by a price, p j = 0, and a vector of K product characteristics x j K. Individuals are identified by a vector of characteristics from some set. These individual characteristics can include both observed demographics and random coefficients (Berry et al. 1995, Nevo 2000b, Train 2003) that Table 1. Symbol = 1 2 = = 0 = 1 J K L Important sets. Description Natural numbers Real numbers Nonnegative real numbers Set of product indices Set of product characteristics Set of individual characteristics

3 330 Operations Research 59(2), pp , 2011 INFORMS Table 2. Summary of important symbols. Symbol Description Defined in Products (see 2.1) J Number of products K Number of nonprice product characteristics x j Nonprice characteristics of product j p j Price of product j p J Vector of all product prices Individual characteristics (see 2.1) Individual characteristics, including observed demographics and random coefficients Distribution of individual characteristics Choice probabilities (see 2.1) u j p j Utility of product j Utility of the outside good Pj L p 0 1 Logit choice probability for product j Equation (1) P j p 0 1 Mixed-logit choice probability for product j P p 0 1 J Vector of mixed-logit choice probabilities for all products Firms, costs, profits, and stationarity (see 2.2, 2.3) F Number of firms f Indices of the products offered by firm f c j (fixed) unit cost of product j c J Vector of all (fixed) unit costs f p Expected profits for firm f Equation (2) D k f p Derivative of firm f s profits, with respect to the Equation (3) price of product k p J Combined gradient of profits Prop. 1, Equation (4) Choice probability derivatives (see 2.3, 2.5) D k P j p Derivative of product j s choice probability Morrow and Skerlos (2010) with respect to the price of product k DP p J J Intrafirm Jacobian matrix of the choice Equation (5) probability vector p, p J J Matrices appearing in our decomposition of DP p Equation (7), Morrow and Skerlos (2010) Fixed-point equations (see 2.4, 2.5) p J The BLP-markup function (Berry et al. 1995) Equation (6) p J Our -markup function Equation (8) characterize unobserved individual-specific heterogeneity with respect to preference for product characteristics. The relative density of individual characteristic vectors in the population is described by a probability distribution over. An individual identified by receives the (random) utility U j x j p j = u x j p j + E j from purchasing product j, and U 0 = + E 0 for forgoing purchase of any of these products or purchasing the outside good. Here u is a systematic utility function, is a valuation of the no-purchase option or outside good, and E = E j J j=0 is a random vector of i.i.d. standard extreme value variables. Morrow and Skerlos (2010) gives a general specification of utility functions appropriate for equilibrium pricing. The basic requirements are that u is continuously differentiable and strictly decreasing in price and without lower bound as prices increase. Demand for each product j is characterized by choice probabilities P j J 0 1 derived from (random) utility maximization. The choice probabilities for an individual characterized by are those of the logit model (Train 2003, Chapter 3): P L j p = e u j p j e + (1) J k=1 eu k p k The vector p J denotes the vector of all product prices. Product-specific utility functions u j for all j, defined by u j p = u x j p for all p, are used in Equation (1) and in the following sections. The mixed-logit choice probabilities P j p = P L j p d follow from integrating over the distribution of individual characteristics (Train 2003, Chapter 6). The vector of mixed-logit choice probabilities for all products is denoted by P p 0 1 J.

4 Operations Research 59(2), pp , 2011 INFORMS 331 The examples below review several instances of this choice model. Examples 1 and 2 are also used in 4 below. Example 3 illustrates the type general specifications used in estimation. Example 4 describes one kind of simulation of a mixed-logit model (Train 2003). Example 1 (Boyd and Mellman 1980). Take = K, denoting = for and K. Set u x p = p + x and = for all K. is defined by specifying that and are independently lognormally distributed (with appropriately chosen signs, means, and variances). Example 2 (Berry et al. 1995). Take = K, denoting = 0 for, K, and 0. Set u x p = 0 = log + 0 { log p + x if p < otherwise and for some fixed coefficient > 0. represents income and is given a lognormal distribution, whereas the random coefficients 0 are independently normally distributed with some mean and variance. Note that income serves as an upper bound on the price an individual can pay for any product. Example 3 (Nevo 2000b). Take = D K+2, denoting = d for, d D, and K+2. Again, represents income; d D represents a vector of D observed demographic variables (which may include income); K+2 represents a vector of K + 2 random coefficients: one for each product characteristic, one for price, and one for the outside good. Set u d x p = + p d + p p + + d + x d = + p d + p + 0 d + 0 where, K, p 0 D, K D, p 0 K+2, and K K+2 are coefficients. The distribution of d is estimated from available data (e.g., census data) and is assumed to be standard independent multivariate normal. When + p d + p, the coefficient on price, is positive, an individual prefers higher prices. Petrin (2002) and Berry et al. (2004) adopt similar specifications that eliminate this counterintuitive property. Petrin (2002) takes the price component of utility to be log p, where is a step function. Berry et al. (2004) take the price component of utility to be p, but define = e + p d+ p. Example 4 (Simulation). Take any of the examples above, and draw S vectors s according to the distribution. Let = s S s=1, and define a probability measure over by s = 1/S for all s. Then u defines a simulator of the full mixedlogit model with u ; see Train (2003). These approximations, discussed further in 3.5, are essential in estimation of mixed-logit models and in computations of equilibrium prices Firms, Costs, and Profits Differentiated product market models are built from combining a model of choice, e.g., a mixed-logit model, with a model of firm behavior. There are F firms, each of which offers some subset of the J products offered to individuals; f = 1 J indexes the products offered by firm f. In Bertrand competition, each firm f chooses the prices of the products they offer prior to some purchasing period in which consumers opt to purchase one, or none, of the products offered by all firms. All prices remain fixed during this purchasing period, and firms satisfy all demand (Baye and Kovenock 2008). Firms choose the prices of the products they offer to maximize their expected profits max f p = j f P j p p j c j with respect to p j for all j f (2) where p J denotes the vector of prices for all products and c j denotes the constant unit cost of product j. The stationarity condition D k f p = D k P j p p j c j + P k p = 0 j f for all k f (3) is necessary for the prices of firm f s products to locally maximize firm f s profits, for fixed competitor prices. This condition requires that the mixed-logit choice probabilities are continuously differentiable in prices Local Equilibrium and the Simultaneous Stationarity Conditions Combining the stationarity condition for each firm we obtain the simultaneous stationarity condition, a first-order (necessary) condition for local equilibrium prices. Proposition 1 (Simultaneous Stationarity Condition). Suppose P is continuously differentiable. Let p J denote the combined gradient with components p j = D j f j p where f j denotes the index of the firm offering product j. If p is a local equilibrium, then p = DP p p c + P p = 0 (4) where DP p J J is the intrafirm Jacobian matrix of price derivatives of the choice probabilities defined by DP p j k D k P j p if products j and k are offered = by the same firm 0 otherwise (5)

5 332 Operations Research 59(2), pp , 2011 INFORMS Prices p satisfying Equation (4) are called simultaneously stationary. The matrix DP p has previously been denoted by (Berry et al. 1995, Petrin 2002, Beresteanu and Li 2008), (Nevo 2000a), and (Dube et al. 2002). We prefer the D notation to emphasize the relationship of DP p to the Jacobian matrix of the choice probabilities P, while using the superscript to denote the intrafirm sparsity structure. A set of simultaneously stationary prices are a local equilibrium only if every firm s profits are locally maximized at those prices; this can be verified by confirming that every firm s profits are locally concave. Note that there is no convenient condition to verify that every firm s profits are globally maximized at a particular local equilibrium. That is, there is no convenient condition to ensure that certain prices are a proper equilibrium The Markup Equation A prominent form of the first-order conditions Equation (4) is the BLP-markup equation: p = c + p where p = DP p P p (6) Note that is defined for any continuously differentiable choice probabilities with nonsingular DP p, such as certain mixed-logit models (Morrow and Skerlos 2010). Traditionally, the BLP-markup equation (6) has been used to estimate costs assuming that observed prices are in equilibrium via the formula c = p p ; see, e.g., Berry et al. (1995) or Nevo (2000a). These cost estimates form the basis of counterfactual experiments with an estimated demand model. Beresteanu and Li (2008) have recently suggested that the BLP-markup equation is also useful for computing equilibrium prices, a suggestion we explore further in 3.3 below. Note that the BLP-markup equation must be interpreted as a nonlinear fixed-point equation when applied to compute equilibrium prices Another Fixed-Point Equation In this section we derive our -markup equation for equilibrium using a particular decomposition of the choice probability derivatives. Thus, this markup equation is specific to mixed-logit models, unlike the BLP-markup equation. First, Example 5 demonstrates this decomposition in a simple case. Example 5. Note that for simple logit, D k P L j p = j k D k u k p k P L k p D k u k p k P L k p P L j p where D k u k p k denotes the derivative of the kthproduct s utility function with respect to price, and j k = 1 if j = k and 0 otherwise (Anderson and de Palma 1988, 1992; Train 2003; Morrow 2008). If we define L k p = D k u k p k Pk L p and L j k p = D ku k p k Pk L p P j L p, we can write D k Pj L p = j k L k p L j k p. Allowing dependence on individual characteristics and integrating over the demographic distribution, we obtain a decomposition of the form DP p = p p (7) where p and p are both J J matrices that depend on prices; see Morrow and Skerlos (2010). p is a diagonal matrix with negative elements k p on the diagonal, and p is a matrix that has nonzero elements p j k = j k p only when the products indexed by j and k are offered by the same firm, like DP p. Substituting Equation (7) into Equation (4) yields the -markup equation p = c + p where p = p 1 p p c p 1 P p (8) when p is nonsingular. Numerical methods based on the BLP- and -markup equations can have entirely different properties because and are different functions. For simple logit models, these two functions coincide only at simultaneously stationary prices. There are examples of mixed-logit models with the same property. Example 8 below shows that there are some cases in which fixed-point iteration based on is not locally convergent, but fixed-point iteration based on is locally convergent. There may also be cases in which the opposite is true, or in which neither converges. 3. Computational Methods This section discusses several approaches for computing equilibrium prices using two classical numerical methods, Newton s method and fixed-point iteration, as summarized in Table 3. Section 3.1 briefly reviews Newton s method, followed by application of Newton s method to solve Equation (4) in 3.2. Newton s method applied directly to Equation (4) may compute spurious solutions with infinite prices because the combined gradient vanishes as prices increase without bound. Section 3.3 avoids this difficulty by applying Newton s method to the two markup equations instead of Equation (4) itself. Section 3.4 discusses fixed-point iterations based on the BLP- and -markup equations, and 3.5 reviews how simulation must be used in equilibrium price computations Newton s Method Newton s method, a classical technique to compute a zero of an arbitrary function F J J, is now a portfolio of related approaches to solve nonlinear systems (Ortega and Rheinboldt 1970, Kelley 1995, Dennis and Schnabel 1996, Judd 1998, Kelley 2003, Schmedders 2008). Generally speaking, Newton-type methods are differentiated in two relatively independent directions: (i) the technique used to approximate the Jacobian matrices DF and solve for

6 Operations Research 59(2), pp , 2011 INFORMS 333 Table 3. Summary of the numerical methods examined in this article. Abbr. Method Section Advantage Our experience a Newton methods (NM) Solve F p = p = Unreliable, slow Solve F p = p c p = Coercive Reliable, slow Solve F p = p c p = Coercive Reliable, slow Fixed-point iterations (FPI) Iterate p c + p 3.4 Easy to evaluate Reliable, fast Iterate p c + p 3.4 Not convergent a Conclusions on behavior of these methods is based on the numerical experiments described in 4, using a novel GMRES-Newton method with Levenberg-Marquardt style trust-region global convergence strategy. the Newton step, and (ii) the technique used to enforce convergence from arbitrary initial conditions. See Dennis and Schnabel (1996), Judd (1998), or Kelley (2003) for good treatments of these issues. Choosing the right variant of Newton s method determines the reliability and efficiency of equilibrium price computations. Problem formulation also determines the reliability and efficiency of equilibrium price computations using Newton s method. Scalings of the variables and function values are one prominent example of a problem transformation that improves the performance of Newton s method (Dennis and Schnabel 1996). Nonlinear problem preconditioning can also be important (Cai and Keyes 2002), as the following example demonstrates. Example 6. Let F N N be defined by F x = x/ 1 + x 2 2. Iterating Newton steps converges to the unique (finite) zero x = 0 only from initial conditions x 0 with x 0 2 < 1/ 3. Newton s method diverges or fails for all other starting points. Standard global convergence strategies for Newton s method (line search, trust region methods) cannot improve this poor global convergence behavior because F x 2 has unbounded level sets; see EC.1 in the electronic companion for details. The electronic companion is part of the online version that can be found at A simple nonlinear transformation overcomes this poor global convergence behavior. Note that F x = A x f x where A x = 1+ x I and f x = x. Because A x is nonsingular for all x, the problems F x = 0 and f x = 0 have identical solution sets. However, applying Newton s method to the problem f x = 0 trivially converges in a single step from any initial condition without a global convergence strategy. Example 6 illustrates why computing equilibrium prices based on the markup equations is more reliable and efficient than using Equation (4) directly. The following two sections echo the pattern of this example to provide the details Newton s Method on the Combined Gradient The most direct approach to compute equilibrium prices using Newton s method is to solve F p = p = 0, abbreviated in Table 3 and below. This approach works well when the initial condition is near an equilibrium, as required by theory (Ortega and Rheinboldt 1970, Kelley 1995, Dennis and Schnabel 1996). In practice, computing counterfactual equilibrium prices starting with the observed prices may exploit this local convergence if changes to exogeneous variables have a relatively small impact on equilibrium prices. On the other hand, can be unreliable when started far from equilibrium. The challenge is the tendency for the derivatives of profits to vanish as prices become large (Morrow and Skerlos 2010), as demonstrated in Example 7 below. Example 7. Consider a simple logit model with linear in price utility and an outside good: u p = p +v for some > 0 and any v, and >. The derivative of firm f s profit function with respect to the price of product k f is D k f p = P L k p p k c k + P L k p f p + P L k p Because Pk L p and P k L p p k c k both vanish as p k (as is easily checked), f p is bounded in p. Thus, D k f p 0 as p k. Note that even though the price derivatives vanish at infinity, this does not mean that infinite prices maximize profits. Nonetheless, may converge to a zero of F with some components equal to infinity that is not an equilibrium. Moreover, because the components of F p can vanish over some divergent sequences, standard global convergence strategies based on minimizing F p 2 will not be effective ways of avoiding this behavior. As in Example 6, we must reformulate the problem to obtain reliable and efficient approaches for computing equilibrium prices Newton s Method and the Markup Equations Reliable and efficient implementations of Newton s method are found by observing that the combined gradient, F, can be written as follows: F p = DP p F p where F p =p c p (9) F p = p F p where F p =p c p (10)

7 334 Operations Research 59(2), pp , 2011 INFORMS Either F or F can be used to compute simultaneously stationary prices when DP p and p, respectively, are nonsingular (Morrow and Skerlos 2010). Of course, F and F recast the first-order condition as a fixed-point problem: F is zero if and only if the BLP-markup equation holds, and F is zero if and only if the -markup equation holds. Solving F p = 0 or F p = 0, abbreviated and, respectively, in Table 3 and below, requires the solution of nontrivial nonlinear systems with Newton s method. and, however, are less likely to have the computational problems that exhibits because they exploit norm-coercivity of the maps F and F (Morrow and Skerlos 2010). A norm-coercive map has a norm that tends to infinity with the norm of its argument (Ortega and Rheinboldt 1970, Harker and Pang 1990). Globally convergent implementations of Newton s method that decrease the value of F p 2 in each step produce bounded sequences of iterates when F is norm-coercive. Thus, solving the BLPor -markup equation instead of the literal first-order condition removes the tendency for applications of Newton s method to compute spurious solutions at infinity Fixed-Point Iteration In addition to applications of Newton s method, the BLPand -markup equations suggest applying fixed-point iteration to solve for equilibrium prices. The fixed-point iteration p c + p based on the -markup equation, here abbreviated, can efficiently compute equilibrium prices for some problems. has relatively efficient steps because no linear systems need to be solved, unlike every other method listed in Table 3. Although we are not aware of a general convergence proof for, this iteration has converged reliably on test problems including the examples in 4 and 5 below. The fixed-point iteration p c + p, abbreviated, based on the BLP-markup equation need not converge. Example 8 below gives a case in which can fail to be even locally convergent. Example 8. Consider multiproduct monopoly pricing with a simple logit model having u j p = p + v j for some > 0, any v j, and >. It is well known that for a single-product firm, unique profit-maximizing prices exist (Anderson and de Palma 1988, Milgrom and Roberts 1990, Caplin and Nalebuff 1991). Morrow (2008) proves that profit-optimal prices p are unique for the multiproduct case and even so with multiple firms even though profits are not quasi-concave (Hanson and Martin 1996). In this example, is not always locally convergent near p, whereas is always superlinearly locally convergent. For an arbitrary continuously differentiable function F and p = F p, F is contractive on some neighborhood of p in some norm if DF p < 1 where A (Ortega and Rheinboldt 1970). We show that D p > 1 may hold while D p = 0, where A denotes the spectral radius of the matrix A. The components of the BLP-markup function are given by k p = 1 1 J j=1 P j L p 1 for all k. From this formula, the equation J j=1 D p = P j L p J 1 J j=1 P j L p = j=1 e u j p j can be derived. For valuations of the outside good,, sufficiently close to, D p > 1 can hold; see EC.2 in the e-companion for details. To prove the claim regarding D p, note that k p = p + 1/, and thus D l k p = D l p = 0 for all k l. Even if the BLP-markup equation does generate a convergent fixed-point iteration, evaluating involves the solution of F linear systems that grow in size with the number of products offered by the firms. The work required to evaluate using a direct method like PLU or QR factorization is max f J f 3, given values of P p, p, and p as approximated using simulation. The work to evaluate is only max f J f 2 given P p, p, and p (Morrow and Skerlos 2010). Generally speaking, function evaluations must be cheap for the linear convergence of fixed-point iterations to result in faster computations than the superlinearly or quadratically convergent variants of Newton s method Simulation In most mixed-logit models, P p, p, and p cannot be computed exactly because they involve integrals with no closed-form expression (Train 2003). Instead, they are approximated with sample averages over a finite set of samples from. This sampling technique, called simulation, is commonly applied in estimation (McFadden 1989, Train 2003, Draganska and Jain 2004). Simulation is applied to compute equilibrium prices by applying,,, and to the finitesample simulator defined in Example 4. Approximating P p, p, and p (among other quantities) in this way dominates the computational burden of each approach to computing equilibrium prices (Morrow and Skerlos 2010). Other sampling techniques such as importance or quasirandom sampling could be employed to reduce this burden (Train 2003). In 4.3 we investigate how the variation in computed equilibrium prices derived from drawing different sample sets changes with S, the sample set size. 4. A Numerical Example This section studies a differentiated product market with many products and a high degree of firm and product heterogeneity: the 2005 new-vehicle market in the United States. The new-vehicle market has played a significant role

8 Operations Research 59(2), pp , 2011 INFORMS 335 in the development of differentiated product market models based on Bertrand competition. Section 4.1 describes two models of the vehicle market. Section 4.2 compares the methods in Table 3 using these models, and 4.3 investigates the sensitivity of computed equilibrium prices to choice of initial condition and the sample set Model Description This section describes the demand models and vehicle data used in the numerical trials. These models are not intended to provide an accurate depiction of the new-vehicle market suitable for making inferences about firm behavior or conducting policy analysis. Rather, these models only provide a realistic example of the type of problem that arises in empirical applications Two Demand Models. Modified versions of the Boyd and Mellman (1980) model of Example 1 and the Berry et al. (1995) model of Example 2 characterize demand for new vehicles. Some vehicle characteristics originally included in these models are excluded due to a lack of data, but the originally estimated coefficients are used for each characteristic included; see EC.3.1 in the e-companion for details. The modified Boyd and Mellman model uses three vehicle characteristics: a metric of size, 0 60 acceleration, and fuel consumption. The coefficients used are those reported by Boyd and Mellman and are reproduced here in Table 5. The modified Berry et al. again uses three vehicle characteristics: operating cost, horsepower-to-weight ratio (a common proxy for 0 60 acceleration), and vehicle length times vehicle width. Again the coefficients used are those reported by Berry et al. and are reproduced here in Table 6. Income has a lognormal distribution fit to current population survey data from 2005 (Berry et al. 1995, Petrin 2002, Berry et al. 2004, CPS 2007). Note that price enters the Boyd and Mellman model in 1980 dollars, whereas price enters the Berry et al. model in 1983 dollars Firms, Vehicle Characteristics, and Costs. The firms for our examples are the 38 brands or makes enumerated in Table 4. Characteristics for 5,298 modelyear plus trim vehicles (e.g., 2005 Ford Focus ZX3 S, 2005 Ford Focus ZX3 SE, etc.) are taken from Wards Automotive Yearbook (Wards ). Virtually all vehicles with a gross vehicle weight rating above 8,500 lbs. are excluded due to an absence of fuel economy data for these vehicles. Average cost to dealers, in 2005 dollars, for 993 model-year vehicles (e.g., 2005 Ford Focus ) sold during 2005 are taken from data reported to J. D. Power by dealers. Because these are costs to dealers and not automobile manufacturers, this example could be viewed as the dealers pricing problem. Furthermore, such data should not be used directly to infer unit costs to automobile manufacturers. The Consumer Price Index (BLS 2009) deflates costs in 2005 dollars to costs matching the units in which the two demand models were estimated. Table 4. Make statistics for the 2005 data set. MK ML MLY MLYV MK ML MLY MLYV Acura Lexus Audi Lincoln BMW Mazda Buick Mercedes-Benz Cadillac Mercury Chevrolet Mini Chrysler Mitsubishi Dodge Nissan Ford Oldsmobile GMC Pontiac Honda Porsche Hummer Saab Hyundai Saturn Infiniti Scion Isuzu Subaru Jaguar Suzuki Jeep Toyota Kia Volkswagen Land Rover Volvo Median Total ,298 Note. Abbreviations: MK: makes; ML: models; MLY: model-year vehicles; MLYV: model-year variants. Note that the costs from J.D. Power are not derived from an econometrically estimated cost model. Although counterfactuals in econometric applications would generally be based on an estimated cost model, equilibrium computations can also be based on direct cost data as in our case or cost models derived independently of demand estimation Method Comparisons This section compares the performance of,,, and on a 993-vehicle model. To obtain the 993-vehicle model, vehicle characteristics from the Ward s data were averaged to obtain 993 typical vehicle models matched to the J.D. Power cost data. Using this smaller model enabled detailed comparisons of the different methods Details for Newton s Method. For,, and we employ a new GMRES-Newton method with a Levenberg-Marquardt or hookstep global convergence strategy (Viswanath 2007). GMRES (Golub and Van Loan 1996, Trefethen and Bau 1997) generalizes the Table 5. Coefficients describing the (lognormal) distribution of the random coefficients mixed-logit model estimated by Boyd and Mellman (1980) Log-mean Log-std. dev

9 336 Operations Research 59(2), pp , 2011 INFORMS Table 6. Coefficients describing the (normal) distribution of the random coefficients mixed-logit model estimated by Berry et al. (1995) Mean Std. dev Note. is lognormally distributed, and thus reported values are the log-mean and log-standard deviation. popular conjugate gradient iterative method for solving symmetric linear systems (Judd 1998, Nocedal and Wright 2006) to nonlinear systems of equations in which the Jacobian DF is, in general, asymmetric. GMRES-Newton methods have been applied to large-scale nonlinear fluid dynamics and radiative transfer problems; see, for example, Brown and Saad (1990), Eisenstat and Walker (1994), Pernice and Walker (1998), Kelley (2003), Pawlowski et al. (2006, 2008), Viswanath (2009), Viswanath and Cvitanovic (2009), and Halcrow et al. (2009). Viswanath s approach combines the GMRES method for solving linear systems with the Levenberg-Marquardt hookstep globalization strategy (Levenberg 1944, Marquardt 1963, Dennis and Schnabel 1996). Line search (Dennis and Schnabel 1996) and Powell s hybrid or dogleg step (Powell 1970, Dennis and Schnabel 1996) globalization strategies have also been extended to GMRES-Newton methods; see, e.g., Brown and Saad (1990), Eisenstat and Walker (1994), Kelley (1995), Pernice and Walker (1998), Kelley (2003), Pawlowski et al. (2006, 2008). Other important details are as follows. The termination condition for all methods is p Note that this is the natural termination condition for, but not the usual termination condition for - NM or (Morrow and Skerlos 2010). The secondorder conditions are always checked at computed points. In, GMRES uses a point-dependent preconditioner (Morrow and Skerlos 2010), whereas neither or appear to require preconditioning. GMRES is given 50 steps to solve the unpreconditioned Newton system to a tight relative residual tolerance of 10 6, and GMRES is never restarted. Although loosening this relative residual tolerance (Viswanath 2007) or including adaptive tolerances (Eisenstat and Walker 1996, Pernice and Walker 1998, Kelley 2003) can improve computation speed and reliability (Tuminaro et al. 2002), we do not feel this will make a significant impact on our results. In our examples GMRES is already very fast: generally, fewer than 10 GMRES steps are required to achieve convergence to an inexact Newton step, even with the tight relative residual tolerance. Additional GMRES steps are cheap relative to the overhead requirement of computing the Jacobian matrices. For, these inclusions may have a more pronounced improvement in computational times. Finally, a primary benefit from GMRES-Newton methods is that they can be built on evaluations of F alone (Brown and Saad 1990, Pernice and Walker 1998). Using the analytic Jacobians is usually the most efficient approach, and thus the computations below use the analytic Jacobians (Morrow and Skerlos 2010) Starting at Unit Costs. Unit costs give a reasonable starting point in the absence of an educated guess of equilibrium prices. Tables 7 and EC.1 detail the results of 10 trials starting from unit costs with S = Newton s method generally fails to converge for the problem under both demand models, and thus is not included. Typical convergence curves are provided in Figures 1 and 2. The,, and methods converge reliably to essentially the same prices (Table EC.1). While and display a rapid convergence rate in terms of iterations, the relative computational intensity of taking Newton steps even with the GMRES-Newton Hookstep method makes the the fastest method. For the example illustrated in Figure 1, requires 9 iterations and 87 seconds of CPU time to compute equilibrium prices. In comparison, requires 39 iterations and 30 seconds of CPU time. For the example illustrated in Figure 2, the method requires 5 iterations and 51 seconds of CPU time. In comparison, required 20 iterations and only 25 seconds of CPU time Starting Near Equilibrium. The following test demonstrates that Newton s method performs well when started near equilibrium prices. Equilibrium prices p were computed using the for a single 1,000 sample set starting from unit costs. The results of the trials described in suggest that and would compute numerical approximations of the same equilibrium. With this equilibrium, a set of T = 10 initial conditions p t are generated by p t = p + t for all t T by drawing t from a uniform distribution on 1 1. The multiplier is given the values $1, $10, $100, and $1,000, in units appropriate for the demand model. One can think of these Table 7. Results of price equilibrium computations starting at unit costs under both demand models for sample sets. Boyd and Mellman (1980) Iterations (#) 8/10/15 10/15/20 23/39/70 CPU time (s) 71/87/123 80/118/147 17/29/53 Successful (#/#) 7/10 6/10 10/10 Berry et al. (1995) Iterations (#) 6/6/9 8/9/11 18/19/23 CPU time (s) 50/52/70 64/73/90 23/24/29 Successful (#/#) 10/10 9/10 10/10 Notes. Iterations and CPU time are again listed as minimum/median/maximum, where we only include successful trials; that is, trials in which a simultaneously stationary point was computed. The number of successful trials is also given. All successful trials resulted in prices satisfying the second-order conditions.

10 Operations Research 59(2), pp , 2011 INFORMS 337 Figure 1. Convergence curves for the fixed-point iteration and variants of Newton s method started at unit costs for a fixed 1,000-sample set under the Boyd and Mellman (1980) model Iterations (#) CPU time (s) Note. Newton s method applied to the preconditioned combined gradient did not converge and is not included. T initial conditions as guesses of equilibrium prices only known to be correct within 2 dollars. Table 8 contains a summary of results for this experiment under the Boyd and Mellman (1980) model. All methods, including, appear to be a reliable way to solve for equilibrium prices. Except for a single trial with the formulation and = $1 000, all trials converged to simultaneously stationary prices satisfying the second-order conditions. All methods converge rapidly for small, with,, and taking only one to two steps for $1 $10. The method converges in very few iterations, even with = $ Another feature to note is the speed of the method. Generally speaking, the approaches based on Newton s method take fewer iterations than (30% 50%), but more CPU time (250% 770%). Even though,, and converge in a single step with = $1, they take more than twice as much CPU time to converge as. Figure 3 illustrates a typical performance by the various approaches we consider under the Boyd and Mellman model. These plots clearly illustrate the rapid convergence of Newton s method applied to any formulation, in terms of iterations. However, when considering CPU time we again observe that a single step of Newton s method can take longer than it takes to converge. For = $100 or $1 000, using Newton s method took longer than the original started at unit costs, regardless of the formulation to which Newton s method is applied. Figure 2. Convergence curves for the fixed-point iteration and variants of Newton s method started at unit costs for a fixed 1,000-sample set under the Berry et al. (1995) model Iterations (#) CPU time (s) Note. Newton s method applied to the preconditioned combined gradient did not converge and is not included.

11 338 Operations Research 59(2), pp , 2011 INFORMS Table 8. Summary of results for 10 perturbation trials under the Boyd and Mellman (1980) model for a fixed set of 1,000 samples. Boyd and Mellman (1980) 1 USD Iterations (#) 1/1/1 1/1/1 1/1/1 5/7/8 CPU time (s) 15/15/16 18/18/18 18/18/18 4/5/6 10 USD Iterations (#) 2/2/2 2/2/2 2/2/2 8/11/11 CPU time (s) 23/23/24 27/27/28 26/26/27 6/8/9 100 USD Iterations (#) 2/3/4 2/3/3 2/3/4 12/14/15 CPU time (s) 25/31/36 30/33/36 30/35/41 9/10/ USD Iterations (#) 4/6/14 a 3/4/5 6/7/10 15/17/19 CPU time (s) 39/47/94 a 41/48/53 57/68/82 11/12/14 Notes. Both iterations and CPU time are given in the form minimum/median/maximum, each taken over the 10 trials. a The results of two trials failed to satisfy the second order conditions. Figure 3. Typical convergence curves for perturbation trials under the Boyd and Mellman (1980) model. $1 $1 Original FPI $ $ $100 ı $1,000 ı $ $1, Iterations ( ) CPU time (s) Notes. The fixed-point iteration and the GMRES-Hookstep Newton s method are starting from random points in p USD, where p is a vector of equilibrium prices computed using (convergence curve given as dotted black line). Results for a fixed sample set S = under the Boyd and Mellman (1980) model.

12 Operations Research 59(2), pp , 2011 INFORMS 339 Table 9. Summary of results for 10 perturbation trials under the Berry et al. (1995) model for a fixed set of 1,000 samples. Both iterations and CPU time are given in the form minimum/median/maximum, each taken over the 10 trials. Berry et al. (1995) 1 USD Iterations (#) 2/2/3 2/2/3 2/2/3 8/9/10 CPU time (s) 18/18/24 20/20/26 19/19/25 10/11/12 10 USD Iterations (#) 3/3/3 3/3/3 3/3/3 10/12/13 CPU time (s) 24/24/24 24/26/26 25/25/27 12/14/ USD Iterations (#) 4/4/5 3/4/4 3/4/5 12/14/17 CPU time (s) 31/31/37 27/34/34 27/33/39 14/17/ USD Iterations (#) 8/10/50 a 5/5/6 5/6/26 b 17/18/22 CPU time (s) 58/79/320 a 46/46/52 45/52/248 b 20/21/26 a failed to compute simultaneously stationary prices in one trial and failed to compute prices satisfying the second order conditions in another. b failed to compute simultaneously stationary prices in one trial. Table 9 details a similar comparison under the Berry et al. (1995) model. Again, all methods are fairly robust near equilibrium, with only three exceptions. In one trial, failed to compute simultaneously stationary prices. In two separate trials, failed to compute simultaneously stationary prices and to compute prices satisfying the second-order condition. Again,,, or take fewer iterations than (30% 60%), but more CPU time (150% 360%). has the significant advantage of requiring, on average, only 1.2 seconds per iteration regardless of. Newton s method appears to requires six seconds per iteration for small and seven for large. This increase in average time per iteration is related to relatively more expensive hooksteps becoming required as the initial prices deviate from equilibrium. Such a small time increase due to the globalization strategy is a testament to the efficiency of Viswanath s GMRES-Newton hookstep. Figure 4 illustrates typical convergence curves under the Berry et al. (1995) model. Again, these plots clearly illustrate the rapid convergence of Newton s method applied to any formulation, in terms of iterations. However the again appears to be the fastest method for computing equilibrium prices. For = $100 and $1,000, using Newton s method again takes longer than the original run used to generate the initial conditions near equilibrium. Under both models, the prices computed by the and the variants of Newton s method compare quite well to the computed equilibrium prices used to generate the perturbed initial conditions. Using any convergent method, roughly 97% of vehicles have prices deviating from the originally computed equilibrium prices by less than $1 (in the appropriately deflated units). This should be interpreted as reflecting local convergence properties of the different iterations Sensitivity This section examines the sensitivity of computed equilibrium prices to the initial condition and to the finitesample simulator with a 5,298-vehicle model. To generate this larger problem, the J. D. Power cost data is extrapolated to model year plus trim vehicles by assuming that variation in dealer costs is reflected in the MSRP reported by Wards; see the e-companion, EC.3.2, for details. These extrapolated costs are neither intended nor expected to be an accurate reflection of real unit costs to either dealers or automotive firms. In fact, this technique is a problematic way of obtaining cost estimates. This extrapolation is employed only to generate a large-scale numerical example with the features of empirical applications. The results for this section were obtained using only, based on the results of 4.2 and the size of the problem. In all of the trials reported on below, the combined gradient norm p consistently decreased to 10 6 in fewer than 40 fixed-point steps, and the computed prices always satisfied the second-order conditions. Computation times are provided in Table Variability Due to Initial Conditions. To test the variability in computed equilibrium prices over different initial conditions, was started at 10 arbitrary initial conditions for a fixed sample set; see the e-companion, EC.3.3, for details. Figure 5 illustrates that variability in equilibrium prices computed from these different initial conditions is negligible. Each curve plots the cumulative distribution of standard deviation in computed equilibrium prices for the 10 initial conditions with a fixed sample set of a certain size. Under the Boyd and Mellman model all of the 5,298 vehicles have standard deviations in computed equilibrium prices less than $1 (1980) over these different initial conditions and sample set sizes. With 100,000 samples, 97% of these vehicles have standard deviations less than $0.01 (1980). Under the Berry et al. model, more than 99% of the 5,298 vehicles have standard deviations in computed equilibrium prices less than $0.01 (1983). For the larger sample set sizes of 50,000 and 100,000 all computed equilibrium prices vary less than $0.01 (1983). Thus it appears that for fixed sample sets, regardless of size, computed equilibrium prices are very stable over arbitrary initial conditions when using.