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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?  ?   ??P驶?0e???0e????????Rectangle 2> ???? 鹇?p From risk analysis to adversarial risk analysis Part III. Modelling preferences under uncertainty. Utilities  q o(q 鸢  ? ??鹬劭???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\       ? {|?  ? ??乍蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ?  ? ??P骚蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i??置P+D?' ????= @B +"g? 鹂 皈?  瘅   ??罢м?0e???0e????????Ttulo 1 `} ????  ??So far     ??嗯м?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 $c ('?   ? ???惜蘅????Marcador de pie de pgina 3^?   ?? DRI. USST (    ?  ? ???学蘅????Marcador de nmero de diapositiva 4^ `?  ???* (   ??????秽33?????8?0___PPT10??虮?"? 瘊0疴?  ?  ? ???颧蘅????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ??`支?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( ((    ??鹨м?0e???0e????????Rectangle 6 `稷 ?????~?v___PPT9?XP??????????????? 鹉??Build a function associating a number (utility) to each consequence, so that we compute the expected utility of each action u (a) = " u (a,) P ()  Optimal alternative: maximum expected alternative r? " }c c b?      5   ? ??P鸳蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 稹@??  ?  ? ??@千蘅????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ??忻м?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( (( ?   ??失?0e???0e????????Rectangle 4 ` ???? ??'Which of these lotteries do you prefer?( (( ?  ? ??A????Picture 7}P   ? ???魔蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 瘀P 疝?   ?  ? ??p阀蘅????5 Marcador de nmero de diapositiva^ `?  ???*(  ?   ????0e???0e????????Rectangle 2 `} ????  ??&Modeling preferences under uncertainty' ' '    ??癌м?0e???0e????????Rectangle 3 `? ?????>?6___PPT9???? ?Consider this situation Throw a (balanced) coin until head appears. If this happens in the n-th draw, you get 2^n euros. How much are you ready to pay to enter in the game??? "    ? ??穰м????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 瘃`$鹳?  $? $ ? ??咖蘅????5 Marcador de nmero de diapositiva^ `?  ???*(  ? $  ?? ??0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( ( (   $  ??揣?0e???0e????????Rectangle 4 `? ?????>?6___PPT9???? 瘥? Expected monetary value is not always an appropriate criteria for decision making under uncertainty Expected utility provides an appropriate solution (at least normatively) ? "   $ ? ??0?м????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"U? \?p(??  (? ( ? ?? 抚蘅????5 Marcador de nmero de diapositiva^ `?  ???*(  ? (  ??鸺м?0e???0e????????Rectangle 2 `} ????  ??'Modeling preferences under uncertainty ( ( ( ? (  ??蓝м?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 ( ? ???м????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"z? ???,??  ,? , ? ????м????5 Marcador de nmero de diapositiva^ `?  ???*(  ? ,  ??按м?0e???0e????????Rectangle 2 `} ????  ??&Modeling preferences under uncertainty' ' ' ?  ,  ??P?м?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 ? , ? ??A????Picture 4 @ ?0 , ? ??协м????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"? 鹪?0鸺?  0? 0 ? ???м????5 Marcador de nmero de diapositiva^ `?  ???*(  ? 0  ??┃?0e???0e????????Rectangle 2 `} ????  ??%Modeling preference. Single criteria & & & 痱 0  ??`?м?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 ;     5U 0 ? ???м????1 Marcador de pie de pgina^?   ?? DRI. USST (     ??????秽33????????___PPT10?i?沸<Q+D?' ????= @B +"?? ???4??  4? 4  ???涧?0e???0e????????Rectangle 2 `} ????  ??%Modelling preferences. Risk attitudes& &(& ? 4 ? ???ェ蘅???Text Box 4P9$  疳?a Risk aversion. Concave utility Risk proneness. Convex utility Risk neutrality. Linear utility 8a 2 2CaAb ? 4  ???м?0e???0e????????Rectangle 5 ` ???? ?? Three basic risk attitudes      4 ? ??喁м???Text Box 6? ?z ??"Consequence varying risk attitudes# 2#C"    4 ? ??P厂蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? 4 ? ??孝м????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 鸹8穑?  8? 8  ??p搂?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 8  ????A????Picture 6H\? ???? 8 ? ??`掸蘅???Text Box 7 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5    8 ? ??P晶蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? 8 ? ??@Η蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? ? ??<??  <? <  ???'?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 <  ????A????Picture 6H\? ???? < ? ??阔м???Text Box 7 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   鹉 < ? ?? 莽蘅???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) Graphically& >X 2?  2CX  ?  < ? ????蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? < ? ???恭蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸?唢+D?' ????= @B +"? ? ??@??  @? @  ???钵?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 @  ????A????Picture 3H\? ???? @ ? ??@?м???Text Box 4 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   鹑 @ ? ??p¨蘅???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) Graphically& >Z 2?  2CZ  ?  @ ? ??翱м????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? @ ? ??p?蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +" ? 鸱D??  D? D  ???м?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes  ( 鹑 D  ????A????Picture 3H\? ???? D ? ??0ぇ蘅???Text Box 4 皓?5Compare lotteries A and B. Which one do you prefer?6 26C5   痿 D ? ???м???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) Graphically& >x 2}  2Cx  ~  D ? ??0?蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? D ? ??锭蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"-? ??H??  H瘙 H  ??0脓?0e???0e????????Ttulo 1 `} ????  ??Risk attitudes 疝 H  ????A?????Marcador de contenido 5?    ???? H ? ???僻蘅????Marcador de pie de pgina 3^?   ?? DRI. USST (    ? H ? ??笆м????Marcador de nmero de diapositiva 4^ `?  ???*(   ??????秽33?????8?0___PPT10??虮???" ?  鸶 L? ?  L? L  ???哀?0e???0e????????Rectangle 2 `} ????  ??Risk attitudes ( 痃 L ? ??懈м???Text Box 3S 9:  ?? ( 2CA 瘙 L  ??0骇?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    L ? ????м????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? L ? ??嗪м????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 瘌P疣?  P? P  ?? ⅶ?0e???0e????????Rectangle 2 `} ????  ??$Modelling preferences. Risk Aversion% %(% 痃 P ? ??啶м???Text Box 3S 9:  ?? ( 2CA ? P  ??`困?0e???0e????????Rectangle 4 ` ???? ??? Risk premium=CE-EMV Absolute Risk Aversion.  u  (c)/u (c) Constant Absolute Risk Aversion. Exponential utility function.   ~  k  P ? ???Ё蘅????1 Marcador de pie de pgina^?   ?? DRI. USST (    ? P ? ???户蘅????2 Marcador de nmero de diapositiva^ `?  ???*(   ??????秽33????????___PPT10?i? 沸??+D?' ????= @B +"? 瘕 T疱?  T? 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