Moderate Expected Utility

PauloNatenzon l joint with

JannahHe )

WashingtonUniversity in Saint

LouisMicroeconomicTheory Seminar

Princeton University September 13 th,

2018

Main Results : two representation Theorems

Motivation : Transitivity

-

Single notion of transitivity in deterministic models

- Different strengths of transitivity in random choice

IfPla,

b) 3 1/2 and Pcb,

c) 31/2,

then

Pla,

c) 3 112 ( WST ) -

Pla,

c ) 3 min I Pla ,b ),

Pcb,

c ) I ( MST )

Pla,

c ) 7 max I Pla ,b ),

Pcb,

c ) I ( SST )

Data example I : Lea and Ryan120151Female

TangaraFrogs,

Mating Decisions

am

84% a WST V

MSTUI. 63%

SST X

69%

^

Data example II : Soltani,

De Martine, Camerer Cede )

Male Caltech students

Choice over money lotteries

If

• by . F' ÷÷ :go.iq#

Data example II : Soltani,

De Martine, Camerer Code )

Male Caltech students

Choice over money lotteries

÷i÷÷÷÷÷:

Data example # : Tversky and Russo,

1969

Prisoners in Detroit

Perceptual

task

90% so % • WST ✓

77%MST -SSTX

Failures of SST and degree of comparability

L.J.Savage 's example :

Trip To Rome vs. Trip To Paris :

plR

,P ) = 42

Trip To Rome €5 vs . Trip To Rome :

plR

't

,R ) =/

Trip To Rome €5 vs . Trip To Paris : pl Rt,Pl= ?

SSTrequires pl R

"

,P ) 3 I unreasonable

MST only requires pl Rt,

P ) 3 pl Rip )

Objectives

:

accommodate

common failures of SST;

retain more empirical bite then WST.

Ex : 2- = La ,

b, c )

plaits) >e ( b. c) >pla,

c ) s' 12

p( b. c) >plaits) >pla,

c ) s' 12 WSTplaits) > pla,

c ) >p

( b,

c ) s' 12::::::c:::::::::Plaid

>

plb.cl>

pla .by , yz ) SST

4/6

1761720

213296

to!

Z O set of choice options

p: -22 →

0,1,

play) t

ply, x ) = I

.

Def : p is a moderate utility model I MUM ) ifZu: Z → IR utility

3- d. : -22 → Rt distance metric on z f "Nif(m*di)such that for all w x and

y z

plw.tl ply,-21 ulwl ulxl

ulyluld .

dlw,

x ) dly,z )

Easy implication : plx , y) 342 iffnlx ) July )

.

Ex : Thurstone 's binary Probit is a MUM

IV.lxez ~ N (M ,

E ),

I full rank

play)

PlU .

>

Uy)

IP IUx-Uy- IMx-MH> -Cmx -MHStd ( U.

-%)Stock(Mx-Mystalk.% ) )

InYi

'

Ix y

Ex 2 : Tversky 's

Elimination

- by - Aspects is a MUM

Every a mapped to a set of "

aspects"

A; m measure

ula) = m ( A )

d ( a. b) = ml AIB ) t m I BIA )

F I t ) = It It

⇒plain= It Im%÷mYh

= m ( AIB )m I AIB ) t m ( BIA )

StrengthenMST to MSTT :

[ MST ]

plaits)

ells,

c) 31/2

⇒

plainplaits)

plb.cl( MSTT )[ Mstt ] plaits)plb.cl> 112plainplaits)plb.cl⇒ 47am

, ca

,n=elb.cle e

Steps of the proof :

( 7) Easy .

X I Construct n,

d

Show d is a metric

Show u,

d represent p

( ) Steph I Halff,

1976 ) : Every

MUMsatisfies MST

Proof : Suppose

MSTfails

µI x

, y ) A

ply,

z ) 31/2

elx,

z ) <

play) A

ply,

z )

ulxl- Az) a dlx,z ) (UH-HHAulYf )

)dlx, y)

y z

sLdlxiyltdly,#(MH-MHAmlYf '

)dlx, y)

y z

s

dlx.gl/uH-mlH)+dly,z)fuH-mCz

)

)dlx, y

) dly , -2 )

=

ulx) -

mlz) I I

1) Step 2 : Every MUM satisfies MST.

Letplxiy)

ply,z) 31/2

.

By Step 1, plx ,z)3p( x.y ) ply,z ) .

Suppose plx ,-2 ) plxiy) ply,z ) plxiy ) . sen

,ulx) - Hy) tidy) -ulz) =dlx.zlf.mx/-uly) )dlxiy)

s [

dlxiyltdly.zbf.mx) -rely ) )dlxiy )

=

ulx) .

rely)

tdlyitfuxdi.ly) )

Y Z'

(

MUMSMSTT ) Construct u and di

MSTT ⇒ WST ⇒ In : 2- → I I,

. . . ,k ) onto such that

play ) 342 iff ulx ) July )

Order pairs x -4 y by lplx , y) - 1121 :

elx'

, y' l > play

' ) > pls ;D > . - - - > elxm , yn ) > Ik

i I 2 3 - - . m

m - 2

Di O I In - D ( n . 1)

C'

,

= ( n . 1)Chin - Dlztl ]

Let dlxixl :=, .ec#p..Yz

( Ci 12 t Di ) lulx ) - ily )l, play ) . plxiiyi ) >

' k

( MUMS MSTT ) Shaw dlx.zlsdlx.gl tally ,z )

di " D=

I &, % , . ,

c :*. "" " " " " "

( Ci 12 t Di ) lulx ) -

rely )l, play ) . plxiiyi ) > ' k

Case 5 of 10 : ul x ) > rely ) > u ( z )

dlx, y ) tdly ,

z ) - d ( x,

z ) = ( a "lztD ;) lulxl - idyll

t ( 4iztdjlluly-nlz.tl

- ( 42T De ) IMG ) truly ) - rely ) - NHI

MST- t ⇒

= ( Di - De ) Iulx ) - rely ) I t ( Dj- De )

Intl) -

htt) I

Di =D ; = De ✓

Des Din Dj ✓

3 ( Div Dj- De ) ' I t ( Did Dj - De ) ( n - 2) Din Die DesDiv Dj

3 In . 1)l "

- ( n . 1)l . Z

t [ O - ( n - De-

2) ( n . 2) = O

( MUMS MSTT ) Shaw in d represent e

di " D=

I &, % , ,

a "" "" " " " " "

( Ci ht Dillulx ) -

ily )l, play ) . plxiiyi ) > ' k

Case I : elw ,x ) 3 ply , z ) > 1/2

⇒ ulw ) > ulx ), rely ) > ulz )

,

dlw.de/4ztDi)(ulw7-ulxDdly,zl=l4ztDj)luly)-ulzl)

,is j

⇐ ulw ) - ulx )=

dlwix ) 42! Di3

qz! Dj=

UH - uh )

dlyiz )

Theorem

Let Z =L I,

2,

. . .

,n ) be finite

1)Halff suggests

"

MST iff Pla ,b) = F I

ulaldfa?f,b)

)

Wejust

showedcounterexamples: Pla ;b ) s Pcb , c) =P la

, c) 71k

Pcb,

c) 7 Pla,

b) =P la, c) 7112

1 : MST 't holdsifand only if Fails,

ctd

Pla,

b) 3 plc, of) ⇒Hal-Mlb) u ( c ) -Adn )

dla,

b )>died )

Identification for u,

d ?

↳ We need a

"

sufficiently rich"

set of options

Rich

setting: lotteries (

e.g .

GP 2006 )

= L I,

2,

. . .

,n ) finite set of prizes

IDIZ) lotteries over Z

e:

DI→

CoD,

pix,y) t

ply,x) =/

Def : p is a moderate expected utility model IMEM ) if

I U :$ > LO,

I ] linear,

onto

I norm11.11.

,

.

such thatplxiy) 3

plwit) ⇐

Uk) -

Uylz

Uw) -

UHHx - yll Hw -2-11

Necessary conditions for MEM :

MSTT,

continuous on D 's DiagonalDef : p is

linear

plx.yt-plaxtltalz.xyth.dz)°¥convex play) -

- ' 12,ecx,

z )ply,2-131/2£1impliespl'sxt'sy,z ) 3playta - a) x

,z )symmetricplay

) -

- ' 12,

pcx,

2) 3

ply,

2) 342,

plz,z

' I >

plz,z

"

) Vz "impliesplxiz' ) 3

ply,z

' )

Il I

Steps of the proof :'

, , ,

( 7) Easy . •

y

I ( )e

has uniquelinear extension to hyperplane H

3$v NM ⇒ U

, parallel I stochastic indifference hyperplanes I

B ( x ,y , p ) : = I znx : plz , y ) 3 p } is symmetric ,

convex,

compact ,non - empty interior ⇒ unit ball for 11/1,3

Let Hz - yll : = qllz-xlttsfuczj.nl/D

'

Uniqueness of U,

Hill in MEM

Where do we fit

WST ⇒

pla, b) 342 ha ) ? Hb )

This paper

mstt ⇐

elaisl-kfnadiav.fi'

)SSTT

" " " k " sela,b7=F( vial ,vlb ) )SST

,plain@bio ) 7'k ⇒plant> maxQuadruple:Debreu 1958pleb) ? plc,

d) ⇐play) >ecb,d )⇐fableFlvlal - rib ) )

Luce 1959Product rule : .

fab) .

elb.de/c,a)=ela.clelablelb,a

) ⇐ Logit

plaids¥↳ ,

Halff ( 1976 )

X

Relation to Random Utility Model ( RUM )

binary RUM

MUM

E!I

Ep 2

Correlated

Probit•

Relation to Random Utility Model I RUM )

Def :

p: Z' → L 0,1 ) is a binary RUM if

3Mprobability on the set of strict orderings on Z

such that

plxiy) = MI Is : x > y } )

Results : I MUM ¢ RUM

z RUM ¢ MUM

Conjecture : 3

p is MUM ⇒p

=foe '

for some

p'

RUM , f strictly increasing

I MUM ¢ RUM. sample : Z =L 1,2 , 3,4

, 5,63 ,Oc

Echoessays,1M¥⇒ Mfl > : 3>4 and 64231=0

M( I > : 652 and 5>131--0

M( I s : 3>4 and S > I } )=0

Should be plot ) tels , 1) tel 6,2151

but

Its t 's - E t 's - c = I - f- e > I ( )

z RUM ¢ MUM.

Ex : Condorcet