university in saint jannah paulo louis 2018 expected utility paulo natenzon l joint with jannah he)...
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Moderate Expected Utility
PauloNatenzon l joint with
JannahHe )
WashingtonUniversity in Saint
LouisMicroeconomicTheory Seminar
Princeton University September 13 th,
2018
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
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
( ) 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
'
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¥↳ ,
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 ( )