On substitutability and complementarity in discrete choice models

Feng, Guiyun; Li, Xiaobo; Wang, Zizhuo

doi:10.1016/j.orl.2017.11.016

Cited by 16 publications

(9 citation statements)

References 14 publications

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“…D nÀ1 is said to have the substitutability (or its corresponding choice model is substitutable) if q i ðpÞ is locally decreasing in p j for any i,j 2 [n], i 6 ¼ j. Feng et al (2018) show that a welfare-based choice model with a differentiable choice welfare function w(p) is substitutable if and only if w(p) is submodular. For the representative agent model, though it is equivalent to the welfarebased 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: Portfolio Contractmentioning

confidence: 99%

“…(2017), only a necessary condition for the substitutability of

q^{r}

is provided in Feng et al. (2018). By establishing the relationship between M‐convexity and choice welfare functions, we can give a sufficient and necessary condition as follows.…”

Section: Applicationsmentioning

confidence: 99%

“…Feng et al. (2018) observe that the choice probability function

q^{r}

in the representative agent model (22) is the same as the choice probability function

q^{w}

in the welfare‐based choice model with the choice welfare function w ( π ) given by

V^{*} false(π false)

. Based on this observation, the representative agent model with the penalty function V ( x ) is substitutable if and only if the welfare‐based choice model with the choice welfare function

w false(π false) = V^{*} false(π false)

is substitutable.…”

Section: Applicationsmentioning

confidence: 99%

“…Theorem 3.1 allows us to directly derive two results on special representative agent models in Feng et al. (2018). For

V false(x false) = x^{T} A x

with A being a positive definite matrix, they show that the corresponding representative agent model is substitutable only if

A_{italicjk} - A_{italicik} - A_{italicij} + A_{italicii} \geq 0

for all distinct i , j , k ∈ [ n ].…”

Section: Applicationsmentioning

confidence: 99%

“…To prove Theorem 4.1, first note that for a given essentially strictly convex function V(x) whose domain is contained in D nÀ1 , its conjugate function V Ã ðpÞ, which is exactly the optimal objective value function of the representative agent model ( 22), is a choice welfare function. Feng et al (2018) observe that the choice probability function q r in the representative agent model ( 22) is the same as the choice probability function q w in the welfare-based choice model with the choice welfare function w(p) given by V Ã ðpÞ. Based on this observation, the representative agent model with the penalty function V(x) is substitutable if and only if the welfare-based choice model with the choice welfare function wðpÞ ¼ V Ã ðpÞ is substitutable.…”

Section: Define a Modified One-period Cost Function Bymentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Portfolio Contractmentioning

confidence: 99%

“…(2017), only a necessary condition for the substitutability of

q^{r}

is provided in Feng et al. (2018). By establishing the relationship between M‐convexity and choice welfare functions, we can give a sufficient and necessary condition as follows.…”

Section: Applicationsmentioning

confidence: 99%

“…Feng et al. (2018) observe that the choice probability function

q^{r}

in the representative agent model (22) is the same as the choice probability function

q^{w}

in the welfare‐based choice model with the choice welfare function w ( π ) given by

V^{*} false(π false)

. Based on this observation, the representative agent model with the penalty function V ( x ) is substitutable if and only if the welfare‐based choice model with the choice welfare function