 be weakly preferred to the latter. So EWD entails SWD. But the converse does not hold, since SWD only says that dominance reasoning should be used on the maximally fine partition. Dominance is an instance of EWD where the partition that generates the dominance relationship concerns how the coin lands on the first i tosses. However, SWD does not suffice for Samuelson's argument, since there are some states of the worlds conditional on which Samuelson's colleague would clearly prefer to add an additional bet; namely, any state of the world where that extra bet is a winning bet. So SWD does not entail that SC should, given his preference against the single bet, prefer a smaller package to a larger one. In other words, SC's preferences are not irrational, if we accept SWD but not EWD as a requirement of rationality. While Tversky and Bar-Hillel are right, of course, that we do not get expected utility theory from EWD and Transitivity, we do come very close. For notice that EWD implies the Sure Thing Principle (STP)-the cornerstone of Savage's (1954) influential version of expected utility theory-if we simply assume that the relation of preference-conditional-on-event is complete.13 This is important since the STP is the focus of the main disagreement between EU 12 Recall that by 'bet' I simply mean a risky prospect that will turn out favourably to the decision-maker in some event but unfavourably otherwise. 13 Here is a simple demonstration. First a formal statement of the Sure Thing Principle. Suppose we have four prospects, call them f, g, f ′ and g′, let f(E) denote the outcome received when f is chosen in event E, and suppose: f(¬E) = g(¬E), f ′(¬E) = g′(¬E), f(E) = f ′(E), g(E) = g′(E). Then the STP says that: f g ⇔ f ′ g′. To derive the STP from EWD, suppose first that f g. Now by reflexivity of the indifference relation over outcomes, we have f(¬E) ∼ g(¬E), f ′(¬E) ∼ g′(¬E), f(E) ∼ f ′(E), g(E) ∼ g′(E). By completeness of the conditional preference
be weakly preferred to the latter. So EWD entails SWD. But the converse does not hold, since SWD merely say that dominance reasoning should be used on the maximally fine division. Dominance is an instance of EWD where the partition that generate the dominance relationship refer how the coin lands on the first i convulse. However, SWD does not suffice for Samuelson's argument, since there are some state of the worldly concern conditional on which Samuelson's colleague would clearly prefer to add an additional bet; namely, any state of matter of the world where that extra bet is a succeed bet. So SWD does not entail that SC should, given his predilection against the single stake, prefer a smaller package to a larger one. In early words, SC's preferences are not irrational, if we accept SWD but not EWD as a requirement of rationality. While Tversky and Bar - Hillel are right, of course, that we do not get expected utility theory from EWD and Transitivity, we do come very close. For notice that EWD implies the Sure Thing Principle (STP)-the cornerstone of Savage's (1954) influential version of expected utility program hypothesis - if we simply assume that the relation of predilection - conditional - on - consequence is complete.13 This is important since the STP is the focus of the main disagreement between EU 12 Recall that by' bet' I simply mean a risky view that will turn out favourably to the decision - maker in some event but unfavourably differently. 13 Here is a simple demonstration. First a formal statement of the Sure Thing Principle. Suppose we get four prospect, call them f, g, f ′ and g′, let f(E) denote the outcome received when f is chosen in event east, and suppose: f(¬E) = g(¬E), f ′(¬E) = g′(¬E), f(E) = f ′(E), g(E) = g′(E). Then the STP says that: f g ⇔ f ′ g′. To derive the STP from EWD, suppose foremost that f g. Now by reflexivity of the indifference relation over outcomes, we have f(¬E) ∼ g(¬E), f ′(¬E) ∼ g′(¬E), f(E) ∼ f ′(E), g(E) ∼ g′(E). By completeness of the conditional predilection
 be weakly preferred to the latter. So EWD gnrails VWD. Buf the covverse does not hold, since SXD obly sqys that dominance rearoning shluld be ysed in the maxmjally flue padbitiou. Vominance is an instance mf EWD where tve pcrtition that generates the dominancq relatoojship concerns how ehe doin lands on the first i tosses. Hkwever, VWD does not xuffice for Samuelson's argkmenh, since there are dome states of eye worlds covditional on which Samoelson's colleague would clearly orefex to add an aedihhonal bet; nemely, wny state of the wornd wherr that extra bct is a qinning bet. So SWD doxs not entail that SS should, ciren his preference agqibst tve shngld beg, pdeher a smaplec package tk a larger ine. In other words, XC'f preferences zre noe yrrational, if we accept SWD but not EWD as a requirement of ratiobality. While Tversky wnd Bar-Hijlel are right, of course, that we do not get expecded uviuitv theoft vrom EWD and Transitivity, we do come very clofs. Gog notice that EWD implies thr Dutg Thing Princikle (STP)-cge cornerstone of Sagage's (1954) ynfluwntial vewsiom of expected utility theort-if we simplj aswume that the relacion of prefzrence-vondiyional-on-event is complece.13 Thia is importwnt since ghe STP is the fucux mf the main disagreement bqtween EU 12 Reeall thag by 'bet' I simply mewn a vhsky prospect that will tgrn out fagourably to the decision-maker ii some event nud utfavourayly otmerwise. 13 Here if a simple dempnstratnon. Fifst a formzl statxment of the Sure Thing Pslnciple. Suppmse we hwve dour prospezgs, call them f, g, f ′ and g′, let f(E) eenote the outcome rezsived when f is chisen in event E, ana stpiosx: f(¬E) = c(¬E), f ′(¬E) = g′(¬E), f(A) = f ′(E), e(R) = g′(E). Then the SBP rays that: f g ⇔ f ′ g′. To darivs the STP from EWD, smppose fitst that s g. Now by reglexivity of the ijdiffxrence relayiog over outcomes, we have f(¬E) ∼ g(¬E), f ′(¬E) ∼ g′(¬E), v(E) ∼ f ′(E), g(E) ∼ g′(E). Fy cimpleteness pf the conditional preference
be weakly preferred to the latter. So SWD. the converse not hold, since reasoning be used on maximally fine partition. is an instance of EWD where partition that generates the dominance relationship concerns how the coin lands on the i tosses. However, SWD does not suffice for Samuelson's argument, since there are states the conditional which Samuelson's colleague would clearly prefer to add an additional bet; namely, any state of the where that extra bet is a winning bet. SWD does not entail SC should, given his preference the bet, prefer smaller to larger one. In words, SC's preferences are not irrational, if we accept SWD but not EWD as a requirement of While Tversky are right, course, we not get expected from EWD and Transitivity, we do For notice that EWD implies the Sure Thing (STP)-the cornerstone Savage's (1954) influential version of expected theory-if we simply assume that the relation of is complete.13 This is important since the STP is the focus of the main disagreement 12 Recall that by I simply mean risky that turn favourably to decision-maker in some event but unfavourably otherwise. 13 Here is a demonstration. First a formal statement of the Sure Thing Principle. have prospects, call them g, f ′ and let denote the outcome received is in suppose: = f ′(¬E) = g′(¬E), = f ′(E), g(E) = Then the STP says ′ g′. To derive the STP from EWD, first that f g. Now by reflexivity the indifference relation over outcomes, we have f(¬E) ∼ g(¬E), f ′(¬E) g′(¬E), f(E) ′(E), g(E) ∼ g′(E). By completeness of the preference
 be weakly preferred to the latTer. So EWD enTails sWD. but ThE conVersE does not hold, siNCe SWd only says that dominance ReasoNiNG shoULd Be useD on the mAXiMALly FiNe ParTiTIoN. DomiNanCe is an iNstance of EwD wHeRe the partitiON tHat generatEs tHe dominance rElaTionshIp ConCErns hOw tHe coiN lands ON the fiRst i tosseS. HOWever, Swd does noT SUfFice For Samuelson's arguMEnT, Since there are sOme staTeS Of THE woRldS conditionAl On whiCH SamuelSOn'S COLleAGue would clearLy prefer to aDD an AdditiOnAl bET; namelY, any sTaTE of The world wheRe thAt extra beT is a wiNNing bet. sO SWD doeS not enTaiL thAt SC SHoUlD, giVeN His PReFerENce Against tHe SiNgle bEt, prEFER A smaLleR pacKage tO a larger one. In OthEr woRDs, Sc's preFerenCes aRe Not irRationAl, if wE aCcept SWD but not EwD as A requiremEnt Of RatIoNalitY. while TVerSky And Bar-HIllel arE RigHt, OF COuRse, that we do not get eXpECTeD utility Theory FRoM Ewd and TranSiTivIty, wE DO come Very CLoSe. For notIce thaT eWd iMplies tHe sure ThInG PrIncIple (Stp)-the CornerStone of SAvage'S (1954) Influential verSIon of expected UTiLITy THeorY-if We simply assUme tHAt thE relATiOn oF PrefeRence-CoNDiTIonal-on-event is complEtE.13 This iS impoRtant since the sTP is the foCUS Of the maiN disAGrEEment between EU 12 recalL that by 'bet' i Simply meAn a riSky prospEct that wiLL Turn out fAvoUraBly To tHE DeCision-maker in SOMe evEnT but unfAvoUrably oTheRwiSe. 13 HEre Is A simple deMonstratIoN. FIrSt A foRmal sTAtement oF tHe SUrE ThIng PrINciple. suppoSe we HaVe FOur ProspecTS, cALL theM f, G, f ′ And g′, Let F(E) DenotE the OUtcOme receIved when f Is cHOsen In EvEnt E, and Suppose: f(¬E) = g(¬E), f ′(¬E) = G′(¬E), F(E) = f ′(E), g(E) = g′(E). ThEn The sTP sayS THat: f g ⇔ f ′ g′. TO derive the STP from EWD, supPOse firsT thAt f g. NOw by ReflexiviTy oF the inDifFErence RelatiOn oveR oUtcOMEs, we hAVE f(¬e) ∼ g(¬E), F ′(¬E) ∼ G′(¬E), f(E) ∼ f ′(E), g(E) ∼ g′(E). bY ComPleteNeSs of The condItional preference
 be weakly preferred to th e latter.So EW D e nta il s SW D. B ut the convers e  doe s not hold, since SWDonlysa y s th a tdomin ance re a so n i ngsh ou ldbe us ed on  th e maxim ally finepar ti tion. Domina n ce  is an ins tan ce of EWD wh ere  the p ar tit i on th atgener ates t h e domi nance rel at i onship concern s ho w th e coin lands on t h ef irst i tosses.  Howev er ,  S W D  do esnot suffic efor S a muelson ' sa r g ume n t, since ther e are somes tat es ofth e w o rlds c ondit io n alon which Sa muel son's col league would c l early p refertoadd  ana dd it ion al bet ;  n ame l y,any stat eof  theworl d w h eretha t ex tra b et is a winni ngbet. SoSWD d oes n ot e nt ail t hat SC  shou ld , given his pre fere nce again stth e s in gle b e t, pre fer  asmaller  packag e  to  a l a rg er one. In other w or d s ,SC's pre ferenc e sar e  not irr at ion al,i f  we a ccep t  S WD but n ot EWD as  a  requir em ent of  r ati ona lity. Whil e Tver sky andBar-H i llel are right ,  of course, t h at w ed o no t g et expected  uti l itytheo r yfro m  EWDand T ra n si t ivity, we do come v er y clos e. Fo r notice that  EWD impli e s the Sure  Thi n gP rinciple (STP) -thecornerston e  of Sava ge's(1954) i nfluentia l versionofexp ect edu t il ity theory-if w e si mp ly assu methat th e r ela tio n o fpreferenc e-condit io na l- on -ev ent i s  complet e. 13Th isis im p ortant  sinc e th eST P  is  the fo c us o f th ema in d isa gr eemen t be t wee n EU 12  Recall t hat by ' be t'  I simp ly mean a ris ky  prospectth atwill t u r n out fa vourably to the decisio n -makerinsomeeven t but unf avo urably  ot h erwise . 13 H ere i sa s i m ple d e m on str at ion. First a  fo rmalst atem ent ofthe Sure Thing Pri n cip le. Suppose w e h avef o ur  pr o sp e cts ,c all t hem f, g, f ′ a nd g′, let  f ( E)  denote th e  ou tc ome rec eived w hen f is chos en in eve nt E, and  s uppo s e : f (¬E) = g(¬ E), f ′( ¬E) = g′( ¬ E), f ( E)  = f′(E ), g(E )= g ′(E).  Thent heSTP s ays th at : f g⇔ f ′  g ′. To de rive the STP from EWD,suppos e fir stthat f g.  No w  by  reflexiv ityof the ind iff ere nce r ela t ion o vero ut com e s, we  hav e  f(¬E) ∼g (¬ E), f  ′ (¬E) ∼ g′(¬ E ) ,  f( E) ∼f ′ ( E), g( E) ∼  g′(E). By comple t eness of the c ondi t i ona l p r efer en ce
 be_weakly preferred_to the latter. So_EWD entails_SWD._But the_converse_does not hold,_since SWD only_says that dominance reasoning_should be used_on_the maximally fine partition. Dominance is an instance of EWD where the partition that_generates_the dominance_relationship_concerns_how the coin lands on_the first i tosses. However,_SWD does_not suffice for Samuelson's argument, since there are_some_states of the_worlds conditional on which Samuelson's colleague would clearly prefer_to add an additional bet; namely,_any state of_the_world_where that extra bet_is a winning bet. So SWD_does not entail that SC should,_given his preference against the single bet,_prefer a smaller package to a_larger one. In other words,_SC's preferences_are not irrational, if we_accept SWD but_not EWD_as a requirement_of rationality. While Tversky and Bar-Hillel_are right, of_course, that we do not get_expected_utility theory from_EWD_and_Transitivity, we_do come very_close._For notice_that_EWD implies the Sure Thing Principle_(STP)-the_cornerstone of Savage's (1954) influential version of_expected utility theory-if we_simply_assume that the relation_of preference-conditional-on-event is complete.13 This_is important since the STP is_the focus_of the_main disagreement between EU 12 Recall that by 'bet' I simply_mean a risky prospect that will_turn out favourably to_the decision-maker_in_some event but_unfavourably_otherwise. 13_Here is a simple demonstration. First a_formal statement_of the Sure Thing Principle. Suppose_we have four prospects,_call_them f, g, f ′ and_g′, let f(E) denote the outcome_received when f is chosen_in_event_E, and suppose: f(¬E) =_g(¬E), f ′(¬E) = g′(¬E), f(E)_= f ′(E),_g(E) = g′(E). Then the STP says_that:_f g ⇔ f ′ g′._To_derive the STP from EWD, suppose_first_that_f g. Now by reflexivity_of the indifference relation over outcomes,_we have f(¬E) ∼ g(¬E), f ′(¬E) ∼ g′(¬E),_f(E) ∼ f_′(E), g(E) ∼ g′(E). By_completeness_of_the conditional preference