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G7\:?f孓?????)?J'?q??}/⒆h?&v?? ?????%9?2* ?gcZ:Hv?Xd??3k韧???B七?栎??A?淞???eM?Gr颥 位 ?-???1/?X?教b9?N ???1??z ?+R ?hB.} ?/??????窄赞%r?K4uK~?Ij'U?鳅;??范&芪?????/U卉?tKG?簧?t???K?q?&翱?郜?蔹a?泪O? ??~钔vD??q?5^S孬???ejC眉*C????q ?Q ? )Z+ ??`G叩d2!???h圪遣?颟"b???M?麟P}????t"mm笫/?a ??n?).谁N??-茕?箐?e7?_???3U?G?? G#??? ?r?EJ???-?^?_A鲺?9KCSc绑S?A???A?%}k?k?%Y2?+ +?6?pY?J!?5)????固?Q?7呒2[{m???MfP石忙々?1蔼邬fz钓????捞Y ?诳?jm弄i几T ^? ?????龙GE天???41??恶@?诹4Nv?&eM[???}???ck晌怙s{????c??m?45x??滦ぺ???%>?盍Y??)i??h=& ?G3u???PK! ??'theme/theme/_rels/themeManager.xml.rels??M ???woo雍?&?协?? 6?$Q祉 ?.?a?i??c2?h:?q??胳?@RN??d???o7g?K(M&$R(.1?r'J???T???V?＆A然??}??|??{?P??除8?g/]?As?(⑵锑#???汉倪??PK-!檗?[Content_Types].xmlPK-!ブх6 0_rels/.relsPK-!ky???theme/theme/themeManager.xmlPK-!鳌Z]4theme/theme/theme1.xmlPK-! ??'g theme/theme/_rels/themeManager.xml.relsPK]b : ? 痃鹣?  ?   ??Z ?0e???0e????????Rectangle 2> ???? 鹇?p From risk analysis to adversarial risk analysis Part III. Modelling preferences under uncertainty. Utilities  q o(q 鸢  ? ??? 犊???Text Box 3 7E ??< David R?s david.rios@icmat.es Skype: david.rios.insua AXA-ICMAT Chair in ARA and Royal Academy of Sciences N\ 2N( 2? 2CNCF\       ? >|?  ? ???I 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (    ?  ? ???T 犊????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i??置P+D?' ????= @B +"g? 鹂 皈?  瘅   ???c ?0e???0e????????Ttulo 1 `} ????  ??So far     ??? ?0e???0e?????????Marcador de contenido 2 ` ???? ???Conceptual framework in RA IDs as problem structuring tool Probabilities for modeling uncertainty This deck Utilities for modeling preferences Xc $c ('?   ? ???n 犊????Marcador de pie de pgina 3^?   ?? DRI. USST (    ?  ? ??Po 犊????Marcador de nmero de diapositiva 4^ `?  ???* (   ??????秽33?????8?0___PPT10??虮?" ?  0痱?  ?  ? ??0U 犊????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ???X ?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( (( ?   ??? ?0e???0e????????Rectangle 6 `稷 ?????~?v___PPT9?XP??????????????? 鹪??Build a function associating a number (utility) to each consequence, so that we compute the expected utility of each action u (a) = " u (a,) P ()  Optimal alternative: maximum expected utility alternative r? " }c c %b?      =   ? ??? 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 稹@??  ?  ? ??pG 犊????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ??`[ ?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( (( ?   ??@6 ?0e???0e????????Rectangle 4 ` ???? ??'Which of these lotteries do you prefer?( (( ?  ? ??A????Picture 7}P   ? ??\ 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"/? 6?P ?   ?  ? ??? 犊????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ??? ?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( (( 鹬   ??p1 ?0e???0e????????Rectangle 4 ` ???? ?Which of these lotteries do you prefer? The EMV of A is 200 The EMV of B is 25000 Yet many subjects fear the high potential loss of 50000 and end up preferring A to B  +? ?  ? ??A????Picture 7}P   ? ???7 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?旨鸠|?+D?' ????= @B +"? 瘀`$疝?  $? $ ? ?? 2 犊????5 Marcador de nmero de diapositiva^ `?  ???*(  ? $  ??? ?0e???0e????????Rectangle 2 `} ????  ??&Modeling preferences under uncertainty' ' '  $  ??9 ?0e???0e????????Rectangle 3 `? ?????>?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??? "   $ ? ??? 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"|? ??p(??  (? ( ? ??? 犊????5 Marcador de nmero de diapositiva^ `?  ???*(  ? (  ??ph ?0e???0e????????Rectangle 2 `} ????  ??&Modeling preferences under uncertainty' ' ' ? (  ???l ?0e???0e????????Rectangle 3 `? ?????>?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& &? " `  ( ? ???_ 犊????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?旨`&T+D?' ????= @B +"? 痱?,疒?  ,? , ? ????悼????5 Marcador de nmero de diapositiva^ `?  ???*(  ? ,  ??俐f?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( ( (  ,  ??0g?0e???0e????????Rectangle 4 `? ?????>?6___PPT9???? 瘘? Expected monetary value is not always an appropriate criterion for decision making under uncertainty Expected utility provides an appropriate solution (at least normatively) ? "   , ? ??g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"U? \??0??  0? 0 ? ???g悼????5 Marcador de nmero de diapositiva^ `?  ???*(  ? 0  ???g?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( ( ( ? 0  ??p g?0e???0e????????Rectangle 3]?? ?????>?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 ? ?? !g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"z? ???4??  4? 4 ? ???g悼????5 Marcador de nmero de diapositiva^ `?  ???*(  ? 4  ???g?0e???0e????????Rectangle 2 `} ????  ??&Modeling preferences under uncertainty' ' ' ?  4  ???"g?0e???0e????????Rectangle 3"癜 ┟ PK!瘅?积[Content_Types].xml????秋??mi?]`H?? ?Xw咿t?. g????3E掇)?+浯7瞩 >?N ???1??z ?+R ?hB.} ?/??????窄赞%r?K4uK~?IV?t?冖??h,3a>?T?s? ?G}??RYa1Q?6Z?G?L$??c(5}?淹疮Y2.ы#`??ioe$?xc>溃?Gf??dFZY股??5????6___PPT9???? ??DM 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 ? 4 ? ??A????Picture 4 @ ?0 4 ? ??0.g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 鹪8鸺?  8? 8 ? ???$g悼????5 Marcador de nmero de diapositiva^ `?  ???*(  ? 8  ?? ,g?0e???0e????????Rectangle 2 `} ????  ??%Modeling preference. Single criteria & & & 痱 8  ???/g?0e???0e????????Rectangle 3/  q ???? ??VDetermine range of interest Assign utility 0 to worst value and utility 1 to best value Assign utilities to a few intermediate values (eg EP) Fit a utility function (e.g. through nonlinear least squares) Check for consistency Perform sensitivity analysis Take into account biases It. Imprecision Details http://www.jstor.org/pss/2631564 6, P+? " PVF ;     B5U 8 ? ???g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"?? ??<??  <? <  ??@0g?0e???0e????????Rectangle 2 `} ????  ??%Modelling preferences. Risk attitudes& &(& ? < ? ??`)g悼???Text Box 4P9$  疳?a Risk aversion. Concave utility Risk proneness. Convex utility Risk neutrality. Linear utility 8a 2 2CaAb ? <  ???&g?0e???0e????????Rectangle 5 ` ???? ?? Three basic risk attitudes      < ? ??0#g悼???Text Box 6? ?z ??"Consequence varying risk attitudes# 2#C"    < ? ???g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? < ? ???g悼????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 鸹@穑?  @? @  ???1g?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 @  ????A????Picture 6H\? ???? @ ? ??*g悼???Text Box 7 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5    @ ? ??g悼????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? @ ? ??@%g悼????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"?? ?D??  D? D  ??氨岜?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 D  ????A????Picture 6H\? ???? D ? ??翅笨???Text Box 7 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   疔 D ? ??`结笨???Text Box 8  ??If 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 ? ??p瘁笨????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? D ? ???羔笨????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸?唢+D?' ????= @B +"? ?H??  H? H  ??0丰?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 H  ????A????Picture 3H\? ???? H ? ?? 滇笨???Text Box 4 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   疠 H ? ??喾岜???Text Box 5  ??If 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 ? ??P漆笨????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? H ? ??欣岜????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 鸲L??  L? L  ??轻?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 L  ????A????Picture 3H\? ???? L ? ?? 泪笨???Text Box 4 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   痼 L ? ???玲笨???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 ? ???⑨笨????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? L ? ??0箩笨????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"-? ??P??  P瘙 P  ??嗦岜?0e???0e????????Ttulo 1 `} ????  ??Risk attitudes 疝 P  ????A?????Marcador de contenido 5?    ???? P ? ???冕笨????Marcador de pie de pgina 3^?   ?? DRI. USST (    ? P ? ??鹉岜????Marcador de nmero de diapositiva 4^ `?  ???*(   ??????秽33?????8?0___PPT10??虮???" ?  鸶 T? ?  T? T  ?? 轴?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes ( 痃 T ? ??兄岜???Text Box 3S 9:  ?? ( 2CA 瘙 T  ?? 酸?0e???0e????????Rectangle 4 ` ???? ??? 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 ? ???提笨????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? T ? ??0歪笨????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 瘌0X疣?  X? X  ?????0e???0e????????Rectangle 2 `} ????  ??$Modelling preferences. Risk Aversion% %(% 痃 X ? ??綮笨???Text Box 3S 9:  ?? ( 2CA ? X  ??篱毂?0e???0e????????Rectangle 4 ` ???? ??? Risk premium=CE-EMV Absolute Risk Aversion.  u  (c)/u (c) Constant Absolute Risk Aversion. Exponential utility function.   ~  k  X ? ??@潇笨????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? X ? ????笨????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 瘕@\疱?  \? \ ? ???裴笨????5 Marcador de nmero de diapositiva^ `?  ???*(  ? \  ??`柔?0e???0e????????Rectangle 2 `} ????  ??Which risk attitude??   疬 \ ? ??舍笨???Text Box 3S 9:  ??   2  \  ??郎岜?0e???0e????????Rectangle 4 ` ???? ??   ? \ ? ??A????Picture 6dZ鸫 \ ??A?????MC900323614[1]Picture 7MC900323614[1] g \ ? ??@ａ笨????1 Marcador de pie de pgina^?   ?? DRI. 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