A note on the invariance of the distribution of the maximum

Mogens Fosgerau; Per Olov Lindberg; Lars Göran Mattsson; Jörgen Weibull

doi:10.1016/j.jmateco.2017.10.005

A note on the invariance of the distribution of the maximum

Mogens Fosgerau, Per Olov Lindberg, Lars Göran Mattsson, Jörgen Weibull

2 Citations (Scopus)

Abstract

Many models in economics involve discrete choices where a decision-maker selects the best alternative from a finite set. Viewing the array of values of the alternatives as a random vector, the decision-maker draws a realization and chooses the alternative with the highest value. The analyst is then interested in the choice probabilities and in the value of the best alternative. The random vector has the invariance property if the distribution of the value of a specific alternative, conditional on that alternative being chosen, is the same, regardless of which alternative is considered. This note shows that the invariance property holds if and only if the marginal distributions of the random components are positive powers of each other, even when allowing for quite general statistical dependence among the random components. We illustrate the analytical power of the invariance property by way of examples.

Original language	English
Journal	Journal of Mathematical Economics
Volume	74
Pages (from-to)	56-61
ISSN	0304-4068
DOIs	https://doi.org/10.1016/j.jmateco.2017.10.005
Publication status	Published - Jan 2018

Keywords

Discrete choice
Extreme value
Invariance
Leader-maximum
Random utility

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title = "A note on the invariance of the distribution of the maximum",

abstract = "Many models in economics involve discrete choices where a decision-maker selects the best alternative from a finite set. Viewing the array of values of the alternatives as a random vector, the decision-maker draws a realization and chooses the alternative with the highest value. The analyst is then interested in the choice probabilities and in the value of the best alternative. The random vector has the invariance property if the distribution of the value of a specific alternative, conditional on that alternative being chosen, is the same, regardless of which alternative is considered. This note shows that the invariance property holds if and only if the marginal distributions of the random components are positive powers of each other, even when allowing for quite general statistical dependence among the random components. We illustrate the analytical power of the invariance property by way of examples.",

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T1 - A note on the invariance of the distribution of the maximum

AU - Fosgerau, Mogens

AU - Lindberg, Per Olov

AU - Mattsson, Lars Göran

AU - Weibull, Jörgen

PY - 2018/1

Y1 - 2018/1

N2 - Many models in economics involve discrete choices where a decision-maker selects the best alternative from a finite set. Viewing the array of values of the alternatives as a random vector, the decision-maker draws a realization and chooses the alternative with the highest value. The analyst is then interested in the choice probabilities and in the value of the best alternative. The random vector has the invariance property if the distribution of the value of a specific alternative, conditional on that alternative being chosen, is the same, regardless of which alternative is considered. This note shows that the invariance property holds if and only if the marginal distributions of the random components are positive powers of each other, even when allowing for quite general statistical dependence among the random components. We illustrate the analytical power of the invariance property by way of examples.

AB - Many models in economics involve discrete choices where a decision-maker selects the best alternative from a finite set. Viewing the array of values of the alternatives as a random vector, the decision-maker draws a realization and chooses the alternative with the highest value. The analyst is then interested in the choice probabilities and in the value of the best alternative. The random vector has the invariance property if the distribution of the value of a specific alternative, conditional on that alternative being chosen, is the same, regardless of which alternative is considered. This note shows that the invariance property holds if and only if the marginal distributions of the random components are positive powers of each other, even when allowing for quite general statistical dependence among the random components. We illustrate the analytical power of the invariance property by way of examples.

KW - Discrete choice

KW - Extreme value

KW - Invariance

KW - Leader-maximum

KW - Random utility

U2 - 10.1016/j.jmateco.2017.10.005

DO - 10.1016/j.jmateco.2017.10.005

M3 - Journal article

SN - 0304-4068

VL - 74

SP - 56

EP - 61

JO - Journal of Mathematical Economics

JF - Journal of Mathematical Economics

ER -

A note on the invariance of the distribution of the maximum

Abstract

Keywords

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