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Modelling preferences under uncertainty. Utilities  q o(q   FS ?Text Box 3 7E :< David Ros david.rios@icmat.es Skype: david.rios.insua AXA-ICMAT Chair in ARA and Royal Academy of Sciences N\ 2N( 2 2CNCF\        >|  hI ?81 Marcador de pie de pgina^   m DRI. USST (    +  xT ?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i.P+D=' = @B +"g  (     rc 0e0e?Ttulo 1 `}   1So far     b 0e0e?0Marcador de contenido 2 `  ,Conceptual framework in RA IDs as problem structuring tool Probabilities for modeling uncertainty This deck Utilities for modeling preferences Xc $c ('   hn ?8Marcador de pie de pgina 3^   r DRI. USST (    /  xPo ?HMarcador de nmero de diapositiva 4^ `  v* (   3380___PPT10."   0(  +  x0U ?H5 Marcador de nmero de diapositiva^ `  r*(  ;   xX 0e0e?Rectangle 2 `}   o'Modeling preferences under uncertainty ( (( &   x? 0e0e?Rectangle 6 ` ~v___PPT9XP Build a function associating a number (utility) to each consequence, so that we compute the expected utility of each action u (a) = " u (a,) P ()  Optimal alternative: maximum expected utility alternative r " }c c %b      =   hF ?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +" @(  +  xpG ?H5 Marcador de nmero de diapositiva^ `  r*(  ;   x`[ 0e0e?Rectangle 2 `}   o'Modeling preferences under uncertainty ( (( I   x@6 0e0e?Rectangle 4 `  }'Which of these lotteries do you prefer?( (( l  DA??Picture 7}P   h\ ?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +"/ 6.P (   +  x0 ?H5 Marcador de nmero de diapositiva^ `  r*(  ;   x6 0e0e?Rectangle 2 `}   o'Modeling preferences under uncertainty ( ((    xp1 0e0e?Rectangle 4 `  Which of these lotteries do you prefer? The EMV of A is 200 The EMV of B is 25000 Yet many subjects fear the high potential loss of 50000 and end up preferring A to B  + l  DA??Picture 7}P   h7 ?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.ּ|+D=' = @B +" `$(  $+ $ x 2 ?H5 Marcador de nmero de diapositiva^ `  r*(  : $  x2 0e0e?Rectangle 2 `}   n&Modeling preferences under uncertainty' ' '  $  x9 0e0e?Rectangle 3 `b >6___PPT9 Consider this situation Throw a (balanced) coin until head appears. If this happens in the n-th draw, you get 2^n euros. How much are you ready to pay to enter in the game?? "   $ h; ?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +"| {p(c(  (+ ( xi ?H5 Marcador de nmero de diapositiva^ `  r*(  : (  xph 0e0e?Rectangle 2 `}   n&Modeling preferences under uncertainty' ' '  (  xl 0e0e?Rectangle 3 `b >6___PPT9 .Consider this situation Throw a (balanced) coin until head appears. If this happens in the n-th draw, you get 2^n euros. How much are you ready to pay to enter in the game?? The EMV of the game is 1+1+1+& . Yet most people would pay at most a few $ to enter in the game& & " `  ( h_ ?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.ּ`&T+D=' = @B +" ,(  ,+ , xf?H5 Marcador de nmero de diapositiva^ `  r*(  ; ,  xf0e0e?Rectangle 2 `}   o'Modeling preferences under uncertainty ( ( (  ,  x0g0e0e?Rectangle 4 `b >6___PPT9  Expected monetary value is not always an appropriate criterion for decision making under uncertainty Expected utility provides an appropriate solution (at least normatively)  "   , hg?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +"U \T0<(  0+ 0 x,g?H5 Marcador de nmero de diapositiva^ `  r*(  ; 0  x(g0e0e?Rectangle 2 `}   o'Modeling preferences under uncertainty ( ( ( p 0  xp g0e0e?Rectangle 3]b >6___PPT9 ^vA sketch of formalisation http://www.econ2.jhu.edu/people/karni/chapter.pdf Provides a rigorous description \ " ZU Z Z " Z Zw8   1 * @M 0 h !g?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +"z y4a(  4+ 4 x*g?H5 Marcador de nmero de diapositiva^ `  r*(  : 4  x!g0e0e?Rectangle 2 `}   n&Modeling preferences under uncertainty' ' ' "  4  x"g0e0e?Rectangle 3" ê PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! %CXdrs/shapexml.xml\n6;YǎgY$A0#NǂXD$9Yh}~$-q`wIJ3{Rpf8?Y&U7?s&Uu]pfP#Jv\Ί)>VStڢhh,3a>Ts G}(4RYa1Q6ZGL$c(5}ZʹY2.#`Dioe$Sxc>GfdFZhzY:i#@[\į2Vtkr3Odl5 G-@oDhYO6PȵW{=j]PFrYh^YYqɛ5TP>tiʺ{ w}E7+x!yw,* G~3.zoYyʞ%{ul/*z/[Lj?c٪ 2 3 +O);SB @D6PvC:O}B}ăăc7;)|nt7E.3~T6_kw˛l4PK!%drs/downrev.xmlDN0 @HCd$n,ee 5^hd[cGYzEo[qw nGr 6DL4ޑ",3̵?T &.2V Y#ߑcb1R<ܶrei8`G U_U3 kf˻ޗRW#D}?~?Y2pm p-S_PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-! %CX)drs/shapexml.xmlPK-!%drs/downrev.xmlPKb >6___PPT9 XDM expresses preferences over lotteries Preferences rational: transitive and complete Independence property Continuity property (no heaven, no hell) Preferences modeled through expected utility Utility is affine unique H( " Zn ZH Z Z l 4 DA??Picture 4 @ 0 4 h0.g?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +" 8(  8+ 8 x$g?H5 Marcador de nmero de diapositiva^ `  r*(  9 8  x ,g0e0e?Rectangle 2 `}   m%Modeling preference. Single criteria & & &  8  x/g0e0e?Rectangle 3/  q  &VDetermine range of interest Assign utility 0 to worst value and utility 1 to best value Assign utilities to a few intermediate values (eg EP) Fit a utility function (e.g. through nonlinear least squares) Check for consistency Perform sensitivity analysis Take into account biases It. Imprecision Details http://www.jstor.org/pss/2631564 6, P+ " PVF ;     B5U 8 h%g?81 Marcador de pie de pgina^   m DRI. USST (     33___PPT10i.<Q+D=' = @B +" <x(  <9 <  x@0g0e0e?Rectangle 2 `}   m%Modelling preferences. Risk attitudes& &(& W < F`)g?Text Box 4P9$  a Risk aversion. Concave utility Risk proneness. Convex utility Risk neutrality. Linear utility 8a 2 2CaAb ; <  x&g0e0e?Rectangle 5 `  o Three basic risk attitudes      < F0#g?Text Box 6 z "Consequence varying risk attitudes# 2#C"    < h#g?81 Marcador de pie de pgina^   m DRI. USST (    + < x0g?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. 1ٝ+D=' = @B +" @(  @# @  x1g0e0e?Rectangle 2 `}   WRisk attitudes  (  @  |A??Picture 6H\  @ F*g?Text Box 7 5Compare lotteries A and B. Which one do you prefer?6 26C5    @ hg?81 Marcador de pie de pgina^   m DRI. USST (    + @ x@%g?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. 1ٝ+D=' = @B +" D(  D# D  x᱁0e0e?Rectangle 2 `}   WRisk attitudes  (  D  |A??Picture 6H\  D F᱿?Text Box 7 5Compare lotteries A and B. Which one do you prefer?6 26C5    D F`᱿?Text Box 8  eIf you prefer B to A, you seem to prefer the sure prize to the lottery (risk aversion) In such case, the expected utility of B should be bigger than that of A: * u(0) + * u(100000) = < 1 * u(50000) >X 2y  2CX  z  D hp᱿?81 Marcador de pie de pgina^   m DRI. USST (    + D x᱿?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. +D=' = @B +" H(  H# H  x0᱁0e0e?Rectangle 2 `}   WRisk attitudes  (  H  |A??Picture 3H\  H F ᱿?Text Box 4 5Compare lotteries A and B. Which one do you prefer?6 26C5    H F᱿?Text Box 5  gIf you prefer A to B, you seem to prefer the lottery to the sure prize (risk proneness) In such case, the expected utility of A should be bigger than that of B: * u(0) + * u(100000) = > 1 * u(50000) >Z 2y  2CZ  z  H hP᱿?81 Marcador de pie de pgina^   m DRI. USST (    + H x᱿?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. 1ٝ+D=' = @B +" L(  L# L  x᱁0e0e?Rectangle 2 `}   WRisk attitudes  (  L  |A??Picture 3H\  L F ᱿?Text Box 4 5Compare lotteries A and B. Which one do you prefer?6 26C5    L F᱿?Text Box 5  }If you are indifferent between A and B, you seem indifferent between the lottery and the sure prize (risk neutrality) In such case, the expected utilities of A and B should be equal: * u(0) + * u(100000) = = 1 * u(50000) >x 2q  2Cx  r  L h᱿?81 Marcador de pie de pgina^   m DRI. USST (    + L x0᱿?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. 1ٝ+D=' = @B +"- Pm(  P P  r᱁0e0e?Ttulo 1 `}   8Risk attitudes  P  A??0Marcador de contenido 5     P h᱿?8Marcador de pie de pgina 3^   m DRI. USST (    + P x᱿?HMarcador de nmero de diapositiva 4^ `  r*(   3380___PPT10.ʎ"    T (  T" T  x ᱁0e0e?Rectangle 2 `}   VRisk attitudes (  T F᱿?Text Box 3S 9:  q ( 2CA  T  x ᱁0e0e?Rectangle 4 `  2 Formally, for a lottery (p1,x1;p2,x2;& ;pn,xn) consider its monetary expected value MEV p1*x1+p2*x2+& +pn*xn and its certainty equivalent CE u^{-1}(p1*u(x1)+p2*u(x2)+& +pn*u(xn)) If MEV>CE, risk aversion If MEV<CE, risk proneness If MEV=CE, risk neutrality PN P      +            N    T h᱿?81 Marcador de pie de pgina^   m DRI. USST (    + T x0᱿?H2 Marcador de nmero de diapositiva^ `  r*(   33___PPT10i. 1ٝ+D=' = @B +" 0X(  X8 X  x챁0e0e?Rectangle 2 `}   l$Modelling preferences. Risk Aversion% %(%  X F챿?Text Box 3S 9:  q ( 2CA ( X  x챁0e0e?Rectangle 4 `  \ Risk premium=CE-EMV Absolute Risk Aversion.  u  (c)/u (c) Constant Absolute Risk Aversion. Exponential utility function.   ~  k  X h@챿?81 Marcador de pie de pgina^   m DRI. 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Risk AversionWhich risk attitude??Presentacin de PowerPoint-Multicriteria preferences under uncertainty -Multicriteria preferences under uncertainty Presentacin de PowerPoint In summary Wrapping up Fuentes usadasTemaServidores OLE incrustadosTtulos de diapositivaL 8@ _PID_HLINKSAH2http://www.econ2.jhu.edu/people/karni/chapter.pdf!http://www.jstor.org/pss/2631564mailto:david.rios@icmat.esmailto:david.rios@icmat.es2http://www.econ2.jhu.edu/people/karni/chapter.pdf!http://www.jstor.org/pss/2631564mailto:david.rios@icmat.es,https://www.youtube.com/watch?v=gWUi3gHRPVA,https://www.youtube.com/watch?v=gWUi3gHRPVAmailto:david.rios@icmat.es,https://www.youtube.com/watch?v=gWUi3gHRPVA,https://www.youtube.com/watch?v=gWUi3gHRPVA_<DavidDavid  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Pictures3Current UserSummaryInformation( PowerPoint Document(<DocumentSummaryInformation8