SUPPLEMENT TO STOCHASTIC CHOICE AND CONSIDERATION SETS : SUPPLEMENTAL APPENDICES (Econometrica, Vol. 82, No. 3, May 2014, ) = 1.

Size: px

Start display at page:

Download "SUPPLEMENT TO STOCHASTIC CHOICE AND CONSIDERATION SETS : SUPPLEMENTAL APPENDICES (Econometrica, Vol. 82, No. 3, May 2014, ) = 1."

Garey Roberts
5 years ago
Views:

1 Econometrica Supplementary Material SUPPLEMENT TO STOCHASTIC CHOICE AND CONSIDERATION SETS : SUPPLEMENTAL APPENDICES Econometrica, Vol 82, No 3, May 2014, BY PAOLA MANZINI AND MARCO MARIOTTI This supplement contains Appendixes C and D KEYWORDS: Discrete choice, random utility, logit model, Luce model, consideration sets, bounded rationality, revealed preferences I-ASYMMETRY: APPENDIX C: NECESSITY OF THE AXIOMS 1 pba\{a} Let p γ be a random consideration set rule For all A D,alla b A with b a: γa p γ a A \{b} c A\{b}:c a p γ a A γa if b a 1 γb 1 if a b so that thus, pba\{a} c A:c a 1 a b Therefore, if 1, we have b a and, I-INDEPENDENCE: pab\{b} pab and pa A\{b} pa A pa B\{b} pa B Let p γ be a random consideration set rule Then, for all AB D and for all a b A B, a b: γa p γ a A \{b} c A\{b}:c a p γ a A γa c A:c a if b a 1 γb 1 if a b γa c B\{b}:c a γa c B:c a p γa B \{b} p γ a B 2014 The Econometric Society DOI: /ECTA10575

2 2 P MANZINI AND M MARIOTTI as desired Similarly, we have p γ a A\{b} p γ a A c A\{b} c A c B\{b} 1 1 γb p γa B \{b} p γ a B c B In Manzini and Mariotti 2014, we reported that the random consideration set rule satisfies i-neutrality; below, we show that i-neutrality is indeed necessary: I-NEUTRALITY: a A,andb c A paa\{c} pba\{c} > 1 paa\{c} pba\{c} for all A D, Let p γ be a random consideration set rule For all A D, a A,and b c A with c a: γa 1 γd p γ a A \{c} d A\{c}:d a p γ a A γa if c a 1 γc 1 γd 1 if a c so that paa\{c} a a, then d A:d a > 1 paa\{c} 1 1 γc for all a A such that a a,whileif a Ab p γa A\{c} p γ a A Therefore, paa\{c} > 1and pba\{c} d A\{c}:d a d A:d a 1 γd 1 γd 1 1 γc > 1imply paa\{c} 1 1 γc pba\{c} APPENDIX D: INDEPENDENCE OF THE AXIOMS First, we recall the axioms; next, we provide examples of random choice rules that fail to satisfy only one of the axioms, and show that they are not random consideration set rules

3 STOCHASTIC CHOICE AND CONSIDERATION SETS 3 I-ASYMMETRY: 1 pba\{a} I-INDEPENDENCE: pab\{b} pab and pa A\{b} pa A pa B\{b} pa B FAILS ONLY I-ASYMMETRY: LetX {a b}, and assume choice probabilities are as in Table DI, wherex y > 0, x + y<1, and α 1 x+y Wehavethat 1 x y x+y < 1, ensuring that y y pa {b} 1 αy > 0, while the upper bound on α also ensures that pa {a}<1 This random choice rule fails i-asymmetry, since pas 1 \{b} x+1 αy < 1 since α> 1 and pbs 1\{a} α>1 i-independence pas 1 x1 αy 1 x pbs 1 holds trivially for a and b For the default alternative: pa S 1 \{a} pa S 1 pa S 1 \{b} pa S 1 pa {b} pa {a b} 1 x y pa pa {a} pa S 2 \{a} pa S 2 pa {a} pa {a b} 1 x y 1 x y 1 pa pa {b} pa S 3 \{b} pa S 3 so that i-independence holds Finally, we show directly that there is no p γ that returns the above probabilities First of all, by the arguments in the proof of Theorem 1, it must be that γa x+1 αy and γb αyifa b, then 1 αy p γ b {a b} γb 1 γa 1 x yαy y p b {a b} TABLE DI ARANDOM CHOICE RULE THAT FAILS I-ASYMMETRY pa S i pb S i pa S i S 1 {a b} x y 1 x y x+1 αy S 2 {a} 1 x+1 αy 1 αy 1 αy S 3 {b} αy 1 1 x y 1 αy

4 4 P MANZINI AND M MARIOTTI where the inequality follows from our assumption that α> 1 Next,ifb a, 1 x then x + 1 αy p γ a {a b} γa 1 γb x + 1 αy x p a {a b} where the inequality follows from α> 1 1 x > 1andy>0 FAILS ONLY I-INDEPENDENCE: Let X {a b}, and assume choice probabilities are as in Table DII, wherex y > 0, x + y<1, and α 1 1 y with α 1 Whilei-Asymmetry holds since pas 1\{b} 1and pbs 1\{a} α 1, 1 x pas 1 pbs 1 i-independence does not, since pa S 1 \{a} pa S 1 pa {b} pa {a b} 1 x y 1 1 x pa pa {a} pa S 2 \{a} pa S 2 where the inequality follows from our assumption α 1 1 x To see that there is no p γ that returns the above probabilities, observe that, by the usual arguments, it must be that γa x and γb αy Nowifa b, we have p γ b {a b} γb 1 γa αy1 x y p b {a b} where the inequality follows from α 1 If instead b a, then 1 x p γ a {a b} γa 1 γb x x p a {a b} where the inequality follows from α y > 0 TABLE DII A RANDOM CHOICE RULETHAT FAILS I-INDEPENDENCE pa S i pb S i pa S i S 1 {a b} x y 1 x y S 2 {a} x 1 x S 3 {b} αy 1

5 STOCHASTIC CHOICE AND CONSIDERATION SETS 5 REFERENCE MANZINI, P, AND MMARIOTTI 2014: Stochastic Choice and Considerations Sets, Econometrica, 82, [2] School of Economics and Finance, University of St Andrews, Castlecliffe, The Scores, St Andrews KY16 9AL, Scotland, UK, and IZA Institute for Labor Economics, Bonn, Germany; paolamanzini@st-andrewsacuk and School of Economics and Finance, University of St Andrews, Castlecliffe, The Scores, St Andrews KY16 9AL, Scotland, UK; marcomariotti@st-andrews acuk Manuscript received February, 2012; final revision received September, 2013

Stochastic Choice and Consideration Sets

DISCUSSION PAPER SERIES IZA DP No. 6905 Stochastic Choice and Consideration Sets Paola Manzini Marco Mariotti October 2012 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Stochastic