Utility-based choice models are often used to determine a consumer’s purchase decision among a list of available products; to provide an estimate of product demands; and, when data on purchase decisions or market shares are available, to infer consumers’ preferences over observed product characteristics. These models also serve as a building block in modeling firms’ pricing and assortment optimization problems. We consider a firm’s multiproduct pricing problem, in which product demands are determined by a pure characteristics model. A sample average approximation (SAA) method is used to approximate the expected market share of products and the firm profit. We propose an SAA-regularized method for the multiproduct price optimization problem. We present convergence analysis and numerical examples to show the efficiency and the effectiveness of the proposed method.

We undertake an empirical study of the impact of delay announcements on callers’ abandonment behavior and the performance of a call center with two priority classes. A Cox regression analysis reveals that in this call center, callers’ abandonment behavior is affected by the announcement messages heard. To account for this, we formulate a structural estimation model of callers’ (endogenous) abandonment decisions. In this model, callers are forward-looking utility maximizers and make their abandonment decisions by solving an optimal stopping problem. Each caller receives a reward from service and incurs a linear cost of waiting. The reward and per-period waiting cost constitute the structural parameters that we estimate from the data of callers’ abandonment decisions as well as the announcement messages heard. The call center performance is modeled by a Markovian approximation. The main methodological contribution is the definition of an equilibrium in steady state as one where callers’ expectation of their waiting time, which affects their (rational) abandonment behavior, matches their actual waiting time in the call center, as well as the characterization of such an equilibrium as the solution of a set of nonlinear equations. A counterfactual analysis shows that callers react to longer delay announcements by abandoning earlier, that less patient callers as characterized by their reward and cost parameters react more to delay announcements, and that congestion in the call center at the time of the call affects caller reactions to delay announcements.

A nonlinear programming formulation is introduced to solve infinite-horizon dynamic programming problems. This extends the linear approach to dynamic programming by using ideas from approximation theory to approximate value functions. Our numerical results show that this nonlinear programming is efficient and accurate, and avoids inefficient discretization.

Discrete-choice demand models are important and fundamental tools for understanding consumers' choice behavior and for analyzing firms' operations and pricing strategies. In these models, products are often described as a vector of observed characteristics. A consumer chooses the product that maximizes her utility, assumed to be a function of the observed product characteristics and the consumer's preference over these product characteristics. One central task in the demand estimation literature is to infer, based on observed data, consumers' preferences on product characteristics. We consider such an estimation problem for pure characteristics models, a class of random coefficients demand models without the idiosyncratic logit error term in a consumer's utility function. The absence of the logit error term and the use of numerical integration to approximate the integral in aggregate market shares lead to a nonsmooth formulation of approximated market share equations. As a result, solving the approximated market share equations and estimating the model by using existing methods proposed in the econometrics literature remain computationally intractable. To overcome this difficulty, we first characterize consumers' purchase decisions by a system of complementarity constraints. This new characterization leads to smooth approximated market share equations and allows us to cast the corresponding generalized method of moments (GMM) estimation problem essentially as a quadratic program with linear complementarity constraints, parameterized by an exponential, thus nonlinear, function of the structural parameter on price. We also extend this estimation framework to incorporate an endogenous pricing mechanism that captures the competitive profit maximization behavior of the producing firms. We provide existence results of a solution for the GMM estimator and present numerical results to demonstrate the computational effectiveness of our approach.

We investigate the estimation of models of dynamic discrete-choice games of incomplete information, formulating the maximum-likelihood estimation exercise as a constrained optimization problem that can be solved using state-of-the-art constrained optimization solvers. Under the assumption that only one equilibrium is played in the data, our approach avoids repeatedly solving the dynamic game or finding all equilibria for each candidate vector of the structural parameters. We conduct Monte Carlo experiments to investigate the numerical performance and finite-sample properties of the constrained optimization approach for computing the maximum-likelihood estimator, the two-step pseudo-maximum-likelihood estimator, and the nested pseudo-likelihood estimator, implemented by both the nested pseudo-likelihood algorithm and a modified nested pseudo-likelihood algorithm.

We investigate the computational aspect of estimating discrete-choice games under incomplete information. In these games, multiple equilibria can exist. Also, different values of structural parameters can result in different numbers of equilibria. Consequently, under maximum-likelihood estimation, the likelihood function is a discontinuous function of the structural parameters. We reformulate the maximum-likelihood estimation problem as a constrained optimization problem in the joint space of structural parameters and economic endogenous variables. Under this formulation, the objective function and structural equations are smooth functions. The constrained optimization approach does not require repeatedly solving the game or finding all the equilibria. We use two static-game models to demonstrate this approach, conducting Monte Carlo experiments to evaluate the finite-sample performance of the maximum-likelihood estimator, two-step estimators, and the nested pseudo-likelihood estimator.

A pure characteristics model is a class of discrete-choice random-coefficients demand models in which there is no idiosyncratic logit error term in a consumer’s utility. The absence of the logit error term and the use of numerical integration to approximate the integral in aggregate market shares lead to a nonsmooth formulation of approximated market share equations. As a result, solving the approximated market share equations for the unobserved product characteristics and estimating the model by using the nested fixed-point approach as proposed in the existing econometrics literature become computationally intractable. To overcome this difficulty, we first characterize consumers’ purchase decisions by a system of complementarity constraints. This new characterization leads to smooth approximated market share equations and allows us to cast the generalized method of moments (GMM) estimation of a pure characteristics model essentially as a quadratic program with linear complementarity constraints, parameterized by an exponential, thus nonlinear, function of the structural coefficient of price. We also extend this estimation problem to incorporate an endogenous pricing mechanism that captures the competitive profit maximization behavior of the producing firms. We provide existence results of a solution for the GMM estimator and present numerical results to demonstrate the computational effectiveness of our approach.

We model the decision-making process of callers in call centers as an optimal stopping problem. After each waiting period, a caller decides whether to abandon a call or continue to wait. The utility of a caller is modeled as a function of her waiting cost and reward for service. We use a random-coefficients model to capture the heterogeneity of the callers and estimate the cost and reward parameters of the callers using the data from individual calls made to an Israeli call center. We also conduct a series of counterfactual analyses that explore the effects of changes in service discipline on resulting waiting times and abandonment rates. Our analysis reveals that modeling endogenous caller behavior can be important when major changes (such as a change in service discipline) are implemented and that using a model with an exogenously specified abandonment distribution may be misleading. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

The widely used estimator of Berry, Levinsohn, and Pakes (1995 )p roduces estimates of consumer preferences from a discrete-choice demand model with random coefficients, market-level demand shocks, and endogenous prices. We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP’s estimator. We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates. As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops. The advantages are even more pronounced for forward-looking demand models where the Bellman equation must also be solved repeatedly. Several Monte Carlo and real-data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization. For static BLP, the constrained optimization approach can be as much as ten to forty times faster for large-dimensional problems with many markets.

Estimating structural models is often viewed as computationally difficult, an impression partly due to a focus on the nested fixed-point (NFXP) approach. We propose a new constrained optimization approach for structural estimation. We show that our approach and the NFXP algorithm solve the same estimation problem, and yield the same estimates. Computationally, our approach can have speed advantages because we do not repeatedly solve the structural equation at each guess of structural parameters. Monte Carlo experiments on the canonical Zurcher bus-repair model demonstrate that the constrained optimization approach can be significantly faster.

The widely-used estimator of Berry, Levinsohn and Pakes (1995) produces estimates of consumer preferences from a discrete-choice demand model with random coefficients, market-level demand shocks and endogenous prices.We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP's estimator.We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates.As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops.The advantages are even more pronounced for forward-looking demand models where Bellman's equation must also be solved repeatedly.Several Monte Carlo and real-data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization.

The two major paradigms in the theoretical agency literature are moral hazard (i.e., hidden action) and adverse selection (i.e., hidden information). Prior research typically solves these problems in isolation, as opposed to simultaneously incorporating both adverse selection and moral hazard features. We formulate two complementary generalized principal-agent models that incorporate features observed in real-world contracting environments (e.g., agents with power utility and limited liability, lognormal stock price distributions, and stock options) as mathematical programs with equilibrium constraints (MPEC). We use state-of-the-art numerical algorithms to solve the resulting models. We find that many of the standard results no longer obtain when wealth effects are present. We also develop a new measure of incentives calculated as the change in the agent's certainty equivalent under the optimal contract for a change in action evaluated at the optimal action. This measure facilitates interpretation of the resulting contracts and allows us to compare contracts across different contracting environments.

Estimation of games with multiple equilibria has received much attention in the recent econometrics literature. Unlike other estimation problems such as single-agent dynamic decision models or demand estimation, in which there is a unique solution in the underlying structural models, games usually admit multiple equilibria and the number of equilibria in a game can vary for different structural parameters. This fact makes the estimation of games far more challenging because the likelihood function or other criterion function defined in the space of structural parameters can be discontinuous or non-differentiable. Two-step estimators by Bajari et al. (2007) and Pesendorfer and Schmidt-Dengler (2008) and Nested Pusedo Likelihood (NPL) estimators by Aguirregabiria and Mira (2007) are proposed to address this problem. We recast the estimation problem as a constrained optimization problem with the Bayesian-Nash equilibrium condition being the constraints. The advantage of our formulation is that the likelihood function, now defined in the equilibrium probability space, is continuous and smooth. This allows researchers to use state-of-the-art optimization software to solve the estimation problem. In a Monte Carlo study, we compare the performance of a two-step estimator, NLP estimator, and our constrained optimization estimator.

Berry, Levinsohn and Pakes (1995), or BLP, introduce a widely-used estimator that handles market-level demand shocks and price endogeneity in structural discrete choice demand models. This estimator is necessary to work with demand data from differentiated products industries. The estimator is computationally intensive and difficult to program, largely because a system of market share equations must be repeatedly numerically inverted. This paper reviews typical computational implementations of the estimator, and shows that some common implementations lead to incorrect parameter estimates. We present suggestions on how to avoid these errors. We also introduce a new computational formulation of the estimator that is much simpler to program and much quicker to compute because it avoids repeatedly numerical inverting the market share equations. Our alternative formulation avoids the problems of false parameter estimates.

We establish the existence results for the Allaz–Vila [B. Allaz, J.-L. Vila, Cournot competition, forward markets and efficiency, J. Econ. Theory 59 (1993) 1–16] forward market equilibrium model when the M producers have different linear cost functions. We also consider an example with three asymmetric producers. The computational results supplement the conclusion in that the forward trading would increase market efficiency.

Although stock options are commonly observed in chief executive officer (CEO) compensation contracts, there is theoretical controversy about whether stock options are part of the optimal contract. Using a sample of Fortune 500 companies, we solve an agency model calibrated to the company-specific data and we find that stock options are almost always part of the optimal contract. This result is robust to alternative assumptions about the level of CEO risk-aversion and the disutility associated with their effort. In a supplementary analysis, we solve for the optimal contract when there are no restrictions on the contract space. We find that the optimal contract (which is characterized as a state-contingent payoff to the CEO) typically has option-like features over the most probable range of outcomes. Paper published as: "Endogenous Selection and Moral Hazard in Compensation Contracts" in Operations Research, Linthicum 58 (July/August 2010): 1090-1106.

Maximum likelihood estimation of structural models is regarded as computationally difficult by many who want to apply the Nested Fixed-Point approach. We present a direct optimization approach to the problem and show that it is significantly faster than the NFXP approach when applied to the canonical Zurcher bus repair model. The NFXP approach is inappropriate for estimating games since it requires finding all Nash equilibria of a game for each parameter vector considered, a generally intractable computational problem. We reformulate the problem of maximum likelihood estimation of games as an optimization problem qualitatively no more difficult to solve than standard maximum likelihood estimation problems. The direct optimization approach is also applicable to other structural estimation problems such as auctions and RBC models, and also to other estimation strategies, such as the methods of moments. It is also easily implemented on standard software implementing state-of-the-art nonlinear programming algorithms

Abstract.,Maximum,likelihood estimation of structural models,is regarded as computationally,difficult. This impression,is due to a focus on the Nested Fixed-Point approach. We present a direct optimization,approach,to the problem,and show that it is significantly faster than the NFXP approach,when applied to the canonical Zurcher bus repair model. The NFXP approach,is inappropriate,for estimating,games,since it requires finding all Nash equilibria of a game for each parameter vector considered, a generally intractable computational,problem. We reformulate,the problem,of maximum likelihood estimation,of games,so into an optimization,problem,that is qualitatively no more,difficult to solve than standard,problems.,The direct optimization,approach is also applicable to other structural estimation,problems,such as auctions and RBC models, and also to other estimation strategies, such as the methods of moments. It is also easily implemented,on standard,software.