Technical Note—On the Relation Between Several Discrete Choice Models

Feng, Guiyun; Li, Xiaobo; Wang, Zizhuo

doi:10.1287/opre.2017.1602

Cited by 35 publications

(21 citation statements)

References 16 publications

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“…Interestingly, Theorem 2 below tells use that the two general frameworks are actually equivalent. The proof makes use of Theorem 1 and some results in [10]. The equivalence provides a new way to view regularized MDPs through the lens of stochastic MDPs.…”

Section: Propositionmentioning

confidence: 99%

“…where w s is a vector of size |A| with entries w sa = r(a|s) + γ s q(s |s, a)V (s ). Using Theorem 1 in [10], we have that for any strictly convex function φ s (•), there is always a distribution set Ξ s such that T RE, φ φ φ [V ](s) = T DS, Ξ [V ](s) for any w s ∈ R |A| , and vise-versa. As a result, for any set of convex functions φ φ φ = { φ s , s ∈ S} there always a set Ξ = { Ξ s , s ∈ S} such that T RE,φ φ φ [V ] = T DS,Ξ [V ] for any V ∈ R |S| , and vice-versa.…”

Section: A3 Proof Of Theoremmentioning

confidence: 99%

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A Relation Analysis of Markov Decision Process Frameworks

Mai,

Jaillet

2020

Preprint

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We study the relation between different Markov Decision Process (MDP) frameworks in the machine learning and econometrics literatures, including the standard MDP, the entropy and general regularized MDP, and stochastic MDP, where the latter is based on the assumption that the reward function is stochastic and follows a given distribution. We show that the entropy-regularized MDP is equivalent to a stochastic MDP model, and is strictly subsumed by the general regularized MDP. Moreover, we propose a distributional stochastic MDP framework by assuming that the distribution of the reward function is ambiguous. We further show that the distributional stochastic MDP is equivalent to the regularized MDP, in the sense that they always yield the same optimal policies. We also provide a connection between stochastic/regularized MDP and constrained MDP. Our work gives a unified view on several important MDP frameworks, which would lead new ways to interpret the (entropy/general) regularized MDP frameworks through the lens of stochastic rewards and vice-versa. Given the recent popularity of regularized MDP in (deep) reinforcement learning, our work brings new understandings of how such algorithmic schemes work and suggest ideas to develop new ones.

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Section: Propositionmentioning

confidence: 99%

Section: A3 Proof Of Theoremmentioning

confidence: 99%

A Relation Analysis of Markov Decision Process Frameworks

Mai,

Jaillet

2020

Preprint

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“…For the representative agent model, though it is equivalent to the welfare‐based choice model as shown by Feng et al. (2017), only a necessary condition for the substitutability of

q^{r}

is provided in Feng et al. (2018).…”

Section: Applicationsmentioning

confidence: 99%

Discrete Convex Analysis and Its Applications in Operations: A Survey

Chen

2021

Production and Operations Management

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D iscrete convexity, in particular, L \ -convexity and M \ -convexity, provides a critical opening to attack several classical problems in inventory theory, as well as many other operations problems that arise from more recent practices, for instance, appointment scheduling and bike sharing. As a powerful framework, discrete convex analysis is becoming increasingly popular in the literature. This review will survey the landscape of the approach. We start by introducing several key concepts, namely, L \ -convexity and M \ -convexity and their variants, followed by a discussion of some fundamental properties that are most useful for studying operations models. We then illustrate various applications of these concepts and properties. Examples include network flow problem, stochastic inventory control, appointment scheduling, game theory, portfolio contract, discrete choice model, and bike sharing. We focus our discussion on demonstrating how discrete convex analysis can shed new insights on existing problems, and/or bring about much more simpler analyses and algorithm developments than previous methods in the literature. We also present several results and analyses that are new to the literature.

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“…Representative agent model: Our MICONVP formulation is based on a characterization of logit probabilities as solutions of a concave maximization problem where the decision variables correspond to the choice probabilities and the objective function is the entropy-regularized expected utility. This concave maximization problem is an example of a representative agent model, and has been studied in a number of papers in the economics and operations management literatures (Anderson et al 1988, Hofbauer and Sandholm 2002, Feng et al 2017. The goal of our paper is not to contribute directly to this literature, but rather to leverage one such result so as to obtain an exact and computationally tractable reformulation of the logit-based SOCPD problem.…”

Section: Literature Reviewmentioning

confidence: 99%

Exact Logit-Based Product Design

Akçakuş¹,

Mišić²

2021

Preprint

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The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product design problem. At first glance, this problem appears hopeless: one must optimize an objective function that is neither convex nor concave, over an exponentially-sized discrete set of attribute combinations. In this paper, we develop an exact methodology for solving this problem based on modern integer, convex and conic optimization. At the heart of our methodology is the surprising result that the logit-based SOCPD problem can be exactly reformulated as a mixed-integer convex program.Using both synthetic problem instances and instances derived from real conjoint data sets, we show that our methodology can solve large instances to provable optimality or near optimality in operationally feasible time frames.

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Technical Note—On the Relation Between Several Discrete Choice Models

Cited by 35 publications

References 16 publications

A Relation Analysis of Markov Decision Process Frameworks

A Relation Analysis of Markov Decision Process Frameworks

Discrete Convex Analysis and Its Applications in Operations: A Survey

Exact Logit-Based Product Design

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