Escolhas aleatórias com categorias

UNIVERSIDADE DE BRASÍLIA

FACE / DEPARTAMENTO DE ECONOMIA

PROGRAMA DE PÓS-GRADUAÇÃO EM ECONOMIA

MARCELO CRUZ DE SOUZA

ESCOLHAS ALEATÓRIAS COM CATEGORIAS

Dissertação apresentada como requisito

parcial para obtenção do título de Mestre em

Economia pelo Programa de Pós-Graduação

em Economia da Universidade de Brasília

Orientador: Prof. Gil Riella

Brasília

2015

MARCELO CRUZ DE SOUZA

ESCOLHAS ALEATÓRIAS COM CATEGORIAS

Dissertação apresentada como requisito

parcial para obtenção do título de Mestre em

Economia pelo Programa de Pós-Graduação

em Economia da Universidade de Brasília

Aprovado em 7 de dezembro de 2015

BANCA EXAMINADORA

Gil Riella (presidente)

Universidade de Brasília

Leandro Gonçalves do Nascimento

Universidade de Brasília

José Guilherme de Lara Resende (suplente)

Universidade de Brasília

Resumo

Caracterizamos um modelo de escolhas aleatórias em que o tomador de decisão classifica seu

universo de opções em categorias e tem preferências completas e transitivas dentro de cada

categoria. Em face de um problema de escolha, uma categoria é sorteada segundo uma função

de probabildade, e ele maximiza sua preferência dentro da categoria considerada. O

comportamento aleatório resulta do sorteio da cateogira a ser considerada. Este modelo

estende a ideia central de Furtado et al (2015), focado em escolhas determinísticas, ao

paradigma das escolhas aleatórias desenvolvido de forma pioneira por Luce (1959). É um

caso particular do modelo geral de Maximização Randômica de Utilidade criado por Block e

Marschak (1959) e Marshak (1960) e caracterizado por Falmagne (1978) e Barberá e

Pattanaik (1986), e mais geral que o modelo de Manzini e Mariotti (2014).

Abstract

We characterize a model of random choices in which the decision maker classifies her

universe in categories and has complete and transitive preferences in each category. When

faced with a decision problem, she sorts one category according to a probability function,

and maximizes her preference in the sorted category. The random behavior results from the

sorting of the category to be considered. This model extends the central idea of Furtado et al

(2015), which is focused in deterministic, to the paradigm of the random choices pioneered by

Luce (1959). It is a special case of the general model of Random Utility Maximization

introduced into Economics by Block and Marschak (1959) and Marshack (1960) and

characterized by Falmagne (1978) and Barberá and Pattanaik (1986), and is more general

than Manzini and Mariotti (2014).

À minha esposa Talita

Aos meus filhos Davi, Marcela, Isabela e Giovana

AGRADECIMENTOS

A Deus pelos dons que me concedeu, pela minha família, meus amigos, meu trabalho e pelas

oportunidades de aprendizado com que me agraciou.

À minha família, pela paciência e pelo apoio durante esse projeto.

Ao Gil, pelas aulas, pelos insights, pelos desafios e pela orientação acadêmica ao longo desses

dois anos.

Aos meus professores, dos quais, além do Gil, gostaria de mencionar, sem excluir os demais,

o Leandro e o José Guilherme, por tudo que me ensinaram.

À minha turma, especialmente Maurício, Felipe e Rafael, verdadeiros amigos e também

pacientes professores.

Ao Banco Central do Brasil pelo suporte através do Programa de Pós-Graduação, e à

Universidade de Brasília, pelo curso ofertado.

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E&'&+!# π(S) ≥ 05

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σ(A, x) = γ(x) Y

y∈A:yx

(1 − γ(y))

:& $+7*#&'+!# Px∈Xσ(X, x) = 1"&##& +!%&-!? 1(2&"%! S = 2X\∅& π(S) =

Q

x∈Sγ(x)

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09( #"#$ 6 0( ,(&#)B$/(3 4 ,+.)# *+ Px∈Aσ(A, x) =

P

R∈Ω:R∩A6=∅π(R)

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B,)/%:%.)- 'C),' ') /;A/1A) :% σ %. &1023) :% π4 D%E'. A ∈ Ω* xk∈ A%

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P

y0_∈Aσ(A, y0) −

σ(A ∪ {x}, x) = 1 −P

y0_∈Axσ(A ∪ {x}, y0) = σ(A ∪ {x}, null)+ ! "#$ #.3$#

70%# σ(A, y) 6= σ(A ∪ {x}, y) "0$0 07F.6 y ∈ A, !'3*# "#$ "=>'%!"!'%?'@(0 σ(A, null) = σ(A ∪ {x}, null).26 1.071.!$ @0-#, σ(A, null) ≥ σ(A ∪ {x}, null)+ <#$30'3#, "0$0 3#%# A, B ∈ Ω @#6 A ⊆ B, .6 'G6!$# &'(3# %! "0--#- 6#-3$0 1.! σ(A, null) ≥ σ(B, null)+ HF#$0, -!I0 x ∈ A ⊆ B+ <!70 H&$60)*# J, K0-30 "$#40$ 1.! σ(A, x) = 0 ⇒ σ(B, x) = 0+ ! σ(A, x) = 0, %! 6#%# -(6(70$ 0# @0-# null 0@(60, %!4! !5(-3($ .6 A0_{6('(607 @#6 x ∈ A}0_{⊆ A307 1.!} σ(A0_{, x) = 0+ 0K!6#- 1.! {x} 6= A}0_{, "#$30'3# "#%!6#- !-@#7/!$ z ∈ A}0_{\ {x}} ! 3!6#- σ(A0_{\ {z}, x) = P} x6= σ(A0, x)! "#$30'3#, "#$ "=>'%!"!'%?'@(0, @#6# z ∈ A, σ(B, x) = 0+ !"!" $&(7&1+)%)+* )+ .! J+ L;$0'-(3(4(%0%!M x . y . z ⇒ x . z+ <#$ N#'#3#'(@(%0%!, 3!6#- σ({x, y, z}, y) = σ({x, y, z}, z) = 0, 7#F# σ({x, y, z}, x)+ σ({x, y, z}, null) = 1+ 2'3*#, σ({x, z}, z) = 1−[σ({x, z}, x)+σ({x, z}, null)] ≤ 1 − [σ({x, y, z}, x) + σ({x, y, z}, null)] = 0+ O('076!'3!, @#6# σ({z}, z) = pz6= 0! σ({x, z}, z) = 0, "#$ %!&'()*# x . z+ P+ x . y ⇒ px= py+ PQ

!" #$%&'(#(&')&*+,- σ({x, y}, y) = 0 ( σ({x, y}, null) = σ({y}, null) = 1−py. !"/,&/!- σ({x, y}, x) = py. 0 ,+&',- #(1, 23"4,56! 7- σ({x, y}, x) 6=

0 ⇒ σ({x, y}, x) = px.

8. x . y, z ⇒ y . z !9 z . y.

!" 7- :,;(4!: <9( px = py = pz. =>,4( p (::( ?,1!". @,;(4!: /,4$

;A4 #!" #$%&'(#(&')&*+, <9( σ({x, y, z}, y) = σ({x, y, z}, z) = 0 ( ',B σ({x, y, z}, x) = p- ( <9( σ({y, z}, null) = σ({x, y, z}, null) = 1 − p. !"/,&/!- σ({y, z}, y) + σ({y, z}, z) = p. C+&,14(&/(- #(1, 23"4,56! 7-σ({y, z}, y) = 0!9 σ({y, z}, z) = 0.

D. x, y . z ⇒ x . y !9 y . x.

E!?,4(&/( /(4!: px= py= pz. !" #$%&'(#(&')&*+,- σ({x, y, z}, z) = 0

( σ({x, y, z}, null) = σ({x, z}, null) = 1 − p. 0&/6! /(4!: <9( /(" (F,$ /,4(&/( 94 '!: ?,1!"(: σ({x, y, z}, x) ( σ({x, y, z}, y) +G9,1 , p- ! !9$ /"! :(&'! &(*(::,"+,4(&/( H("!. %::! +4#1+*, <9( σ({x, y}, x) = 0 !9 σ({x, y}, y) = 0- *,:! *!&/"I"+! /("B,4!: 94, *!&/",'+56! *!4 #$%&'(#(&')&*+,. =!&*19+4!: <9( x . y !9 y . x.

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