Conditional Choice Probabilities Research Articles

Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

Abstract We propose a new sequential Efficient Pseudo-Likelihood (k-EPL) estimator for dynamic discrete choice games of incomplete information. k-EPL considers the joint behaviour of multiple players simultaneously, as opposed to individual responses to other agents’ equilibrium play. This, in addition to reframing the problem from conditional choice probability (CCP) space to value function space, yields a computationally tractable, stable, and efficient estimator. We show that each iteration in the k-EPL sequence is consistent and asymptotically efficient, so the first-order asymptotic properties do not vary across iterations. Furthermore, we show the sequence achieves higher-order equivalence to the finite-sample maximum-likelihood estimator with iteration and that the sequence of estimators converges almost surely to the maximum-likelihood estimator at a nearly superlinear rate when the data are generated by any regular Markov perfect equilibrium, including equilibria that lead to inconsistency of other sequential estimators. When utility is linear in parameters, k-EPL iterations are computationally simple, only requiring that the researcher solve linear systems of equations to generate pseudo-regressors which are used in a static logit/probit regression. Monte Carlo simulations demonstrate the theoretical results and show k-EPL’s good performance in finite samples in both small- and large-scale games, even when the game admits spurious equilibria in addition to one that generated the data. We apply the estimator to analyse competition in the U.S. wholesale club industry.

The Biases in Applying Static Demand Models Under Dynamic Demand

This article investigates the mechanisms that underlie the biases in price elasticities of demand in applying static demand models under dynamic demand, which have been pointed out by previous empirical studies. It studies three sources of biases: disregard of state variables (affecting short-run elasticity); inconsistent utility parameter estimates; and changing expectations of consumers (affecting long-run elasticity). Disregard of state variables, such as durable product holdings, which is not negligible but not paid much attention to in the literature, leads to an overestimate of short-run own elasticities. Inconsistent utility parameter estimates arises due to the failure to account for consumers’ future expectations and unobserved state variables. Changing expectations of consumers are not explicitly specified in the static model, and this also leads to biased results when applying static models. Regarding the magnitude of the biases, the first and the third sources of biases might induce large biases in price elasticities, especially when the focus is on the large conditional choice probability products. Possible remedies for the use of static demand models are also discussed.

Binary choice with misclassification and social interactions, with an application to peer effects in attitude

The interaction of economic agents is one of the most important elements in economic analyses. Social interactions on subjective outcomes, behavior, or decisions, are inherently difficult to identify and estimate because these variables are prone to misclassification errors. This paper puts forth a binary choice model with misclassification and social interactions to rectify the misclassification problems in social interactions studies. We achieve the identification of the conditional choice probability of the latent dependent variable by the technique of repeated measurements and a monotonicity condition. We construct the complete likelihood function from the two repeated measurements and propose a nested pseudo likelihood algorithm for estimation. Consistency and asymptotic normality results are shown for the proposed estimation method. We illustrate the finite sample performance of the model and the estimation method by three Monte Carlo experiments and an application to the study of peer effects among students in their attitudes towards learning.

Modeling activity-travel behavior under a dynamic discrete choice framework with unobserved heterogeneity

The major challenges in dynamic activity-based models include predicting activity-related choices and understanding inherent heterogeneous preferences. The dynamic discrete choice model (DDCM) has been used for daily activity-travel planning. However, ignoring unobservable heterogeneity can bias the estimation and prediction results. To solve this problem, we propose a DDCM that accounts for unobserved heterogeneity to capture useful hidden information on travelers’ characteristics. The conditional choice probability estimator and expectation–maximization (EM) algorithm are used in conjunction to estimate the dynamic model. The algorithm iteration depends mainly on the mapping relationship between posterior distributions and conditional choice probabilities. Meanwhile, a less complex log-likelihood function is proposed in the maximization step to estimate two types of parameters simultaneously. The proposed techniques are verified using household travel survey data from Chongqing (China). Two unobserved types of travelers, time and cost sensitivity, are identified based on the Bayesian information criterion (BIC) value. Time- and segment-varying sensitivity analyses are conducted to present choice probability differences of mode and activity scheduling under a one-unit increase in the number of schoolchildren and cars. Different impacts on activity-travel patterns, such as trip frequency, mode share, and activity schedule, generated by changes in auto ownership, are analyzed. Finally, the adjusted rho-squared value, BIC values compared with other single-effect models, and aggregate validation results demonstrate the exceptional performance of the proposed model.

Comparison of Fully Probabilistic and Partially Probabilistic Choice Set Models for Mode Choice

Contemporary models consider choice sets to be either fully deterministic or fully probabilistic. Deterministic choice set models do not account for stochasticity in the choice set formation, whereas probabilistic choice set models fail to recognize that exclusion and inclusion can be deterministic for some alternatives and individuals and yet random for others. A more general scenario is, therefore, where some alternatives are deterministically included or excluded and others probabilistically included. This paper proposes a richer framework that combines the features of both deterministic and probabilistic choice set models and explicitly allows an alternative to be deterministically included, deterministically excluded, or probabilistically considered in the choice set. This framework is better than the conventional models in four aspects: (a) the factors influencing consideration type are explicitly and parametrically analyzed instead of assumption as 0 or 1; (b) the specification can disentangle factors that affect the inclusion outcome from the type of consideration; and (c) the specification also permits differential sensitivity to factors in conditional choice probability among those who consider an alternative deterministically versus probabilistically. The partially probabilistic choice set model, a special case of the proposed generalized framework, developed using empirical data collected from working commuters in Chennai city, is benchmarked against the fully probabilistic choice set models. The results show that the former had improved goodness-of-fit, realistic consideration probability estimates, and better predictability of mode shares than the latter. Relevant policies have been evaluated by identifying the appropriate target segments at both the consideration and choice stages using the proposed model.

A discrete choice model for partially ordered alternatives

In this paper, we analyze a discrete choice model for partially ordered alternatives. The alternatives are differentiated along two dimensions: the first an unordered “horizontal” dimension, and the second an ordered “vertical” dimension. The model can be used in circumstances in which individuals choose among products of different brands, wherein each brand offers an ordered choice menu, for example, by offering products of varying quality. The unordered–ordered nature of the discrete choice problem is used to characterize the identified set of model parameters. Following an initial nonparametric analysis that relies on shape restrictions inherent in the ordered dimension of the problem, we then provide a specialized analysis for parametric specifications that generalize common ordered choice models. We characterize conditional choice probabilities as a function of model primitives with particular analysis focusing on cases in which unobservable taste for quality of each brand offering is multivariate normally distributed. We provide explicit formulae used for estimation and inference via maximum likelihood, and we consider inference based on Wald and quasi‐likelihood ratio statistics, the latter of which can be robust to a possible lack of point identification. An empirical illustration is conducted using data on razor blade purchases in which each brand has product offerings vertically differentiated by quality.

The whole day path planning problem incorporating mode chains modeling in the era of mobility as a service

The growing popularity of Mobility as a Service (MaaS) has led researchers to realize that it can be used for the optimal management of the transportation system of a city. This study is to solve the whole day multimodal path planning problem considering user-specific modal preference, which can be used on the recommended system of the MaaS platform. The mathematical formulations of the whole day multimodal shortest hyperpath problem are proposed while accounting for individual constraints, such as the time window, multicriteria, and park-and-ride demands, to develop customized travel schemes. The primary obstacle involves consideration of the travelers’ modal preferences to develop an optimal whole day journey. The dynamic discrete choice model (DDCM) that accounts for unobservable heterogeneity is proposed to characterize dynamic mode choice behavior and generate a set of feasible mode chains. The conditional choice probability estimator and expectation–maximization (EM) algorithm are used in conjunction to estimate the dynamic model. In this way, the posterior probability can be adapted to the conditional choice probability estimation. The label correcting concept is used to develop a user-constrained shortest hyperpath algorithm that can solve the multimodal shortest hyperpath problem and calculate the Pareto set of the whole day travel schemes according to the feasible mode chains and different constraints. The proposed techniques are verified by utilizing household travel survey data and multimodal network data in Nanjing (China). Two unobserved states of travelers, namely time- and cost-sensitive, are identified based on the BIC value. The adjusted rho-squared value, accuracies of predicted mode chains, along with the aggregate validation results, confirm the model’s effectiveness. The top two resulting hyperpaths are generated for a randomly chosen traveler on the basis of his or her preferences. This study is an important step toward promoting MaaS and improving the sales of MaaS bundles.

Regularised orthogonal machine learning for nonlinear semiparametric models

Summary This paper proposes a Lasso-type estimator for a high-dimensional sparse parameter identified by a single index conditional moment restriction (CMR). In addition to this parameter, the moment function can also depend on a nuisance function, such as the propensity score or the conditional choice probability, which we estimate by modern machine learning tools. We first adjust the moment function so that the gradient of the future loss function is insensitive (formally, Neyman orthogonal) with respect to the first-stage regularisation bias, preserving the single index property. We then take the loss function to be an indefinite integral of the adjusted moment function with respect to the single index. The proposed Lasso estimator converges at the oracle rate, where the oracle knows the nuisance function and solves only the parametric problem. We demonstrate our method by estimating the short-term heterogeneous impact of Connecticut’s Jobs First welfare reform experiment on women’s welfare participation decision.

Identification of dynamic games with unobserved heterogeneity and multiple equilibria

This paper provides sufficient conditions for nonparametrically identifying dynamic games with incomplete information, allowing for multiple equilibria and payoff-relevant unobservables. Our identification involves two steps. We first identify the equilibrium conditional choice probabilities and state transitions using the Markov property and four-period data. The first step of our identification relies on eigenvalue-eigenvector decomposition, and thus incurs the same issue of identification up-to-label-swapping as the existing literature. This makes it difficult to identify payoff primitives in the second step, which requires consistent matching of unobserved types across different values of the observed variables. Instead of imposing assumptions such as monotonicity, we address this type-matching problem by exploiting the Markov property and longitudinal variations of observables in the intermediate periods to link different decompositions.

Inverse Reinforcement Learning with Explicit Policy Estimates

Various methods for solving the inverse reinforcement learning (IRL) problem have been developed independently in machine learning and economics. In particular, the method of Maximum Causal Entropy IRL is based on the perspective of entropy maximization, while related advances in the field of economics instead assume the existence of unobserved action shocks to explain expert behavior (Nested Fixed Point Algorithm, Conditional Choice Probability method, Nested Pseudo-Likelihood Algorithm). In this work, we make previously unknown connections between these related methods from both fields. We achieve this by showing that they all belong to a class of optimization problems, characterized by a common form of the objective, the associated policy and the objective gradient. We demonstrate key computational and algorithmic differences which arise between the methods due to an approximation of the optimal soft value function, and describe how this leads to more efficient algorithms. Using insights which emerge from our study of this class of optimization problems, we identify various problem scenarios and investigate each method's suitability for these problems.

Some remarks on CCP-based estimators of dynamic models

This note provides several remarks relating to the conditional choice probability (CCP) based estimation approaches for dynamic discrete-choice models. Specifically, the Arcidiacono and Miller (2011) estimation procedure relies on the ”inverse-CCP” mapping ψp from CCPs to choice-specific value functions. Exploiting the convex-analytic structure of discrete choice models, we discuss two approaches for computing this mapping, using either linear or convex programming, for models where the utility shocks can follow arbitrary parametric distributions. Furthermore, the ψ function is generally distinct from the ”selection adjustment” term (i.e. the expectation of the utility shock for the chosen alternative), so that computational approaches for computing the latter may not be appropriate for computing ψ.

Some Remarks on CCP-based Estimators of Dynamic Models

This note provides several remarks relating to the conditional choice probability (CCP) based estimation approaches for dynamic discrete-choice models. Specifically, the Arcidiacono and Miller [2011] estimation procedure relies on the “inverse-CCP” mapping from CCP’s to choice-specific value functions. Exploiting the convex-analytic structure of discrete choice models, we discuss two approaches for computing this, using either linear or convex programming, for models where the utility shocks can follow arbitrary parametric distributions. Furthermore, the inverse-CCP mapping is generally distinct from the “selection adjustment” term (i.e. the expectation of the utility shock for the chosen alternative), so that computational approaches for computing the latter may not be appropriate for computing inverse-CCP mapping.

Dynamic Games in Empirical Industrial Organization

This survey is organized around three main topics: models, econometrics, and empirical applications. Section 2 presents the theoretical framework, introduces the concept of Markov Perfect Nash Equilibrium, discusses existence and multiplicity, and describes the representation of this equilibrium in terms of conditional choice probabilities. We also discuss extensions of the basic framework, including models in continuous time, the concepts of oblivious equilibrium and experience-based equilibrium, and dynamic games where firms have non-equilibrium beliefs. In section 3, we first provide an overview of the types of data used in this literature, before turning to a discussion of identification issues and results, and estimation methods. We review different methods to deal with multiple equilibria and large state spaces. We also describe recent developments for estimating games in continuous time and incorporating serially correlated unobservables, and discuss the use of machine learning methods to solving and estimating dynamic games. Section 4 discusses empirical applications of dynamic games in IO. We start describing the first empirical applications in this literature during the early 2000s. Then, we review recent applications dealing with innovation, antitrust and mergers, dynamic pricing, regulation, product repositioning, advertising, uncertainty and investment, airline network competition, dynamic matching, and natural resources. We conclude with our view of the progress made in this literature and the remaining challenges.

Dynamic Games in Empirical Industrial Organization

This survey is organized around three main topics: models, econometrics, and empirical applications. Section 2 presents the theoretical framework, introduces the concept of Markov Perfect Nash Equilibrium, discusses existence and multiplicity, and describes the representation of this equilibrium in terms of conditional choice probabilities. We also discuss extensions of the basic framework, including models in continuous time, the concepts of oblivious equilibrium and experience-based equilibrium, and dynamic games where firms have non-equilibrium beliefs. In section 3, we first provide an overview of the types of data used in this literature, before turning to a discussion of identification issues and results, and estimation methods. We review different methods to deal with multiple equilibria and large state spaces. We also describe recent developments for estimating games in continuous time and incorporating serially correlated unobservables, and discuss the use of machine learning methods to solving and estimating dynamic games. Section 4 discusses empirical applications of dynamic games in IO. We start describing the first empirical applications in this literature during the early 2000s. Then, we review recent applications dealing with innovation, antitrust and mergers, dynamic pricing, regulation, product repositioning, advertising, uncertainty and investment, airline network competition, dynamic matching, and natural resources. We conclude with our view of the progress made in this literature and the remaining challenges.

Diamonds in the Rough: Leveraging Click Data to Spotlight Underrated Products

Problem Definition. Inspired by a real data set from the Chinese retailer JD.com, we study the click and purchase behavior of customers in an online retail setting by employing a structural estimation approach. In particular, we aim to understand what characteristics of products drive the click versus purchase decisions of customers.With the boom in e-commerce, which has been further fueled due to the current COVID-19 pandemic, an ever increasing number of customers are shopping online. Such a rapid growth in online shopping provides retailers with abundance of data to study customer shopping behavior. For example, online retailers can benefit from a better understanding of the customer search (click) and purchase behavior to improve their operational decisions such as assortment planning. From an academic standpoint, even though there is an extensive body of work on consumer search models, most prior work do not incorporate search sequences and purchase decisions of customers jointly in their analysis. We, however, combine the click and order data to estimate both the click and purchase behavior of customers.Methodology/Results. Using a large data set from JD.com, we propose a structural model to estimate the click and purchase behavior of customers according to a dynamic discrete choice model. In particular, we assume that the customer's utility from each product has an observed and (pre-click) unobserved part (in addition to a random shock). The observed part of the utility is known to the customer prior to the click; however, the unobserved part of the utility can only be learned after the customer clicks on the product. We assume that the customer solves a dynamic program in order to find the optimal search strategy. Due to the curse of dimensionality, we propose a novel value function approximation scheme inspired by the Conditional Choice Probability approach. This, in turn, reduces the estimation into a computationally tractable two-stage process. By combining the click and order data, our proposed structural framework allows us to disentangle and estimate the observed and unobserved parts of product utilities. Our estimation results show that the value of click for customers can be quite significant. This is evidenced by the fact that the unobserved utilities of products vary significantly across products. Most importantly, we are able to identify underrated products which we call diamonds in the rough: these are products with low “face” values (i.e., low observed utilities), but high total utilities due to their high unobserved utilities. Thus, even though such products have a low chance of being clicked (due to their low observed utilities), they have a high chance of being purchased, if clicked.Managerial Implications. Our structural framework provides an online retailer with new tools and insights to better manage the product assortment based on customer click and purchase behavior. In particular, our structural model allows the retailer to disentangle the observed and unobserved parts of product utilities and identify underrated diamond-in-the-rough products. The online retailer can increase her revenue by bringing such products into the spotlight by promoting them on the search page or using tags such as “spotlight product” (similar to “Amazon's Choice” tags on Amazon.com) to entice customers to click on them.

Hierarchical Bayes Conjoint Choice Models - Model Framework, Bayesian Inference, Model Selection, and Interpretation of Estimation Results

Choice-based conjoint (CBC) is nowadays the most widely used variant of conjoint analysis, a class of methods for measuring consumer preferences. The primary reason for the increasing dominance of the CBC approach over the last 35 years is that it closely mimics real choice behavior of consumers by asking respondents repeatedly to choose their preferred alternative from a set of several offered alternatives (choice sets). Within the framework of CBC analysis, the multinomial logit (MNL) model is the most frequently used discrete choice model due to the existence of closed form solutions for conditional choice probabilities. The popularity of CBC and the MNL model has grown even more since the introduction of hierarchical Bayesian (HB) estimation techniques that accommodate individual consumer heterogeneity in choice data, and which have now become state-of-the-art in marketing theory and practice. Still, researchers and practitioners have to make further decisions under this framework (CBC, MNL, HB estimation), such as how to represent preference heterogeneity. Here, using a normal distribution (and therefore a unimodal distribution) has become the standard approach in the marketing literature. However, the thin tails of the normal distribution suggest that the standard HB-MNL model should not be the “go-to” approach to approximate multimodal preference distributions, because individual preference patterns lying at the tails of the normal distribution (i.e., that do not fit well with the assumption of a unimodal distribution) tend to be shrunk to the population mean. This shrinkage, especially in multimodal data settings, could mask important information (e.g., new or different structures in the data). A mixture of normal distributions avoids this limited flexibility of the most simple continuous approach of assuming a unimodal prior heterogeneity distribution. There are currently two prominent HB-CBC modeling approaches embedding the mixture-of-normals (MoN) approach: the more widespread MoN-HB-MNL model, and the Dirichlet process mixture (DPM)-HB-MNL model. In this article, we review the prominent HB-MNL model (with its normal prior), the MoN-HB-MNL model, and the DPM-HB-MNL model and apply them to an empirical multi-country CBC data set. We compare the statistical performance of the three models in terms of goodness-of-fit and predictive accuracy, show how to include consumer background characteristics in the upper level of these models, and illustrate how to interpret the estimation results (with a special focus on cross-county heterogeneity). In sum, our article serves as a kind of user guide to the estimation and interpretation of Hierarchical Bayes Conjoint Choice Models. For our data, we observed that all three choice models (both with and without consumer background characteristics) resulted in a one-component solution. The DPM-HB-MNL model nevertheless yielded a higher cross-validated hit rate compared to the MoN-HB-MNL and the HB-MNL models due to its even more flexible prior assumptions. The two latter models tended to slightly overfit our empirical data, which was reflected by higher goodness-of-fit statistics but a lower predictive accuracy compared to the DPM-HB-MNL model. We showed that this result could be attributed to the weaker extent of Bayesian shrinkage of these two models. The DPM-HB-MNL model showed a stronger shrinkage effect and seems therefore somewhat more robust against overfitting. Including consumer background characteristics in terms of country of origin information for the respondents did not improve the statistical model performance (especially not the predictive performance). Still, using the country of origin information for respondents in a post-hoc segmentation analysis helped us to explain some differences in brand preferences between the five countries.

Linear IV regression estimators for structural dynamic discrete choice models

In structural dynamic discrete choice models, unobserved or mis-measured state variables may lead to biased parameter estimates and misleading inference. In this paper, we show that instrumental variables can address such measurement problems when they relate to state variables that evolve exogenously from the perspective of individual agents (i.e., market-level states). We define a class of linear instrumental variables estimators that rely on Euler equations expressed in terms of conditional choice probabilities (ECCP estimators). These estimators do not require observing or modeling the agent’s entire information set, nor solving or simulating a dynamic program. As such, they are simple to implement and computationally light. We provide constructive arguments for the identification of model primitives, and establish the estimator’s consistency and asymptotic normality. Four applied examples serve to illustrate the ECCP approach’s implementation, advantages, and limitations: dynamic demand for durable goods, agricultural land use change, technology adoption, and dynamic labor supply. We illustrate the estimator’s good finite-sample performance in a Monte Carlo study, and we estimate a labor supply model empirically for taxi drivers in New York City.

Identification and Estimation of a Dynamic Discrete Choice Model with an Endogenous Time-Varying Unobservable State Using Proxies

This paper studies a dynamic discrete choice model including an unobservable state, which evolves by a hidden action of agents. First, this paper shows new nonparametric identification results for observable and unobservable conditional choice probabilities (CCPs) and an initial type probability in both stationary and nonstationary model. The new identification results show two periods of data (T = 2) is sufficient and invertibility assumption on model dynamics is not necessary when there are proxies for the unobservable state variable which satisfies certain conditions. The result contrasts with Hu and Shum (2012) which required at least four periods of data (T >= 4) and required the invertibility assumption. The subjective measures of attitudes and expectations can be examples of such proxy variables. If proxy variables are missing due to rotating module design, which is the common cause of missing attitude and expectation variables in Panel data, the identification can be obtained under stronger assumptions on the model dynamics. Second, this paper proposes a two-step estimator using proxy variables, which generalizes Arcidiacono and Miller (2011) to endogenize the dynamics of the unobservable state variable. The proposed estimator is root-n consistent and asymptotically normal, conditional on that the first stage EM algorithm is initialized not too far from the global optimum of the objective function. The Monte Carlo experiment shows that the estimator performs well even when proxy variables are missing due to rotating module design.

Merger, Product Repositioning and Firm Entry: The Retail Craft Beer Market in California

We study the effects of merger on firm entry, product variety and prices in the retail craft beer market in California. We develop a new method to estimate multiple-discrete choice models in order to recover fixed costs. The method is based on bounds of conditional choice probabilities and does not require solving a game. Using the estimated model, we simulate a counterfactual merger where a large brewery acquires multiple craft breweries. In most markets, we find that new firms enter, non-merging incumbents add products, and merging firms drop products. However, the net effects of product variety from new firm entry and incumbent product portfolio adjustment differ considerably across markets. Larger markets are more likely to see an increase in product variety, which moderates the loss of consumer surplus from the merger's price effects. In a majority of smaller markets, product variety decreases, exacerbating the welfare loss from the price effects.

Unnesting the Fixed Point in Dynamic Program Estimation

A conditional choice probability (CCP) estimator of a dynamic empirical model solves both a dynamic programming problem and a maximum likelihood problem. The estimator can dispatch the former problem before tackling the latter when the utility function is linearly parameterized; otherwise it must nest the former within the latter. This nested fixed point bogs down the estimator, requiring it to solve hundreds or thousands of difficult value function equations. I develop a method to disentangle the two problems under any utility function (thus circumventing the onerous value function calculations). My estimator is asymptotically efficient and has a closed-form characterization when the utility function is linearly parameterized.