), that is the reason why the theory does badly in this respect. More on this in section 4, after the axiomatic argument has been presented in the next section. 3 The argument axiomatised Tversky and Bar-Hillel (1983) state an axiomatic version of Samuelson's argument. Their formulation can however be criticised for the reasons raised by Buchak (as quoted above). The axiomatic backward-inductive argument I formulate in this section avoids that problem, and contains some details that Tversky and Bar-Hillel ignore. As we shall see, Tversky and Bar-Hillel's axiomatic argument was somewhat misleading, since it might have suggested that the (mostly uncontroversial) State-Wise Dominance principle sufficed for the argument, when the (much more controversial) Event-Wise Dominance principle is actually needed. The axiomatic argument presented below is, as far as I can tell, the strongest argument one could give for what I previously called "Samuelson's claim", i.e., for the claim that one cannot rationally turn down the single bet while accepting the package of 100 bets. But it should be noted that this argument shares with other backward-inductive arguments an (implicit) assumption that could be questioned; namely, the assumption that the agent of interest is certain that at each point in time, she will act on her preferences at that time. Since the focus of this paper is on the rational permissibility of risk aversion, I shall not discuss the assumption in question any further (but see e.g. McClennen 1990 for a thorough discussion). Instead, the focus will be on a premise in the argument that turns out to specifically have to do with rationally permissible attitudes to risk. The first assumption of the axiomatic argument is based on the observation that SC would turn down the last coin-toss-bet he expects to be offered, provided that previous bets did not take his wealth outside the specified range. And crucially, the same holds when SC is considering a particular bet if he is certain that he will not accept any further coin-toss-bets. Therefore, he would turn down the 99th bet, having realised that he will not accept the 100th (i.e., the last) bet. However, rather than stating the assumption by referring to bets Is risk aversion irrational? 7 SC expects to accept, I will state it more generally as an assumption that
), that is the reason why the theory does badly in this obedience. More on this in incision 4, after the axiomatic argument has been presented in the next department. 3 The argument axiomatised Tversky and Bar - Hillel (1983) state an axiomatic adaptation of Samuelson's argument. Their formulation can however be knock for the reasons raised by Buchak (as quote above). The axiomatic backward - inductive argument I invent in this incision avoids that problem, and contains some details that Tversky and Bar - Hillel ignore. As we shall see, Tversky and Bar - Hillel's axiomatic controversy was somewhat misleading, since it might have suggested that the (by and large uncontroversial) State - Wise Dominance principle sufficed for the argument, when the (much more controversial) Event - Wise Dominance rationale is actually needed. The axiomatic argumentation presented below is, as far as I can assure, the strongest argument one could give for what I previously called " Samuelson's claim ", i.e., for the claim that one cannot rationally turn down the single bet while accepting the package of 100 bets. But it should be noted that this argument shares with other backward - inductive arguments an (implicit) premise that could be questioned; namely, the assumption that the agentive role of sake is sealed that at each point in time, she will act on her preferences at that time. Since the focus of this paper is on the rational permissibility of hazard aversion, I shall not discuss the assumption in question any further (but see e.g. McClennen 1990 for a thorough discussion). Instead, the focus will be on a premise in the argument that turns out to specifically have to serve with rationally permissible attitudes to hazard. The inaugural assumption of the axiomatic argument is based on the observation that SC would call on down the last coin - toss - bet he expect to be offered, provided that previous bets did not assume his wealth outside the intend range. And crucially, the same apply when SC is considering a especial bet if he is sealed that he will not accept any further coin - toss - stake. Therefore, he would turn down the 99th stake, having realised that he will not accept the 100th (i.e., the last) bet. However, rather than stating the assumption by referring to bets Is risk aversion irrational? 7 SC expects to bear, I will state it more generally as an presumption that
), thwt is the reason why the theory does baboy in vhis reapect. Mofe on this in section 4, after tye axuomatic argument has bden presejted in rhe iext section. 3 Thx argumekc axiklativxd Tversky and Nar-Hillel (1983) vtate an axiomdtkc version of Samuelson's argument. Theyr formilwtion can howeder nq crjnigised for the reasons raised by Guchak (es quoted above). The axiomatic backward-indkctige argument I formklate in thus sqxtion avoids that problem, and contzins some details that Tversky xnd Bcr-Hillel igboee. Wv we shall wee, Tnersky and Bav-Nillel'v axiomstic argument eas sonewhat misleading, sinre it might have sugdested thdt the (mostly unconrrivershal) Vtatd-Qisd Dkmmnahce prlncmple sufficsd for the qrgument, when the (micr more controvsrsial) Edent-Wise Dominance principle is actuallj nesded. The axiomatic argunent presented below ls, as far as I can tell, the strongest argument one could ghve fkf wkqt I oeegiously called "Samuelson's claim", i.e., for the clwjm tmat one cannot rctionally turn dpwj yre single bet while acdepting the packagf of 100 bgts. Bur it shoujd br noted that this argument whares with jrher backward-induccive argumencs an (omplivit) assumption that couud bs questionef; namely, ffe assumption thxt nhe dgent of interest is certayn that av eack point kn tome, shq will act on hcs preferences at tjat tnme. Shnce the flcus of this paper is on the ravmonal permissobhliny of rish avevsion, I shall njt discuss the assumpcion iv question any fucther (but seq e.g. McClennet 1990 for a thormugh dissussuon). Unstead, ghe focus will be on a irtmise in tye argument that tmrns uht to specificaoly have to do witn rxtijnwlky [ermissible dttiguddx to fisk. Tht flrsg asxumption of the axiokatid argument is basec jn the ovservatijn that SC woild turn down the past roin-tors-bet he expects to be offered, provides that prfvijus bets did not take his weclth outside the specified range. And crurially, the same holds wyen SC is consideriug a particuler bet if he is certain that he wilo not accept any nurther coin-toss-bets. Tgerefose, he would turn down the 99th bet, having realised that he will not accept the 100ty (i.e., tie last) bet. Hkwevrr, ranhex tkan staeing vhe assumption by referring to bets Is risk averrion irrdtnonal? 7 SC expects to accept, I wkll state it oore generally as an asshmption yhat
), that is the reason why the badly this respect. on this in argument been presented in next section. 3 argument axiomatised Tversky and Bar-Hillel (1983) an axiomatic version of Samuelson's argument. Their formulation can however be criticised for reasons raised by Buchak (as quoted above). The axiomatic backward-inductive argument I formulate this avoids problem, contains some details that Tversky and Bar-Hillel ignore. As we shall see, Tversky and Bar-Hillel's axiomatic was somewhat misleading, since it might have suggested the (mostly uncontroversial) State-Wise principle sufficed for the argument, the more controversial) Dominance is needed. The axiomatic presented below is, as far as I can tell, the strongest argument one could give for what previously called i.e., for claim one rationally turn down bet while accepting the package of it should be noted that this argument shares other backward-inductive an (implicit) assumption that could be namely, the assumption that the agent of interest certain that at each point in time, she will act on her preferences at that the focus of this is on the permissibility risk I not discuss assumption in question any further (but see e.g. McClennen 1990 for thorough discussion). Instead, the focus will be on a premise argument turns out to have to do with permissible to risk. The first the argument the that would turn down the coin-toss-bet he expects to be provided that previous bets outside the specified range. And crucially, the same when SC is considering a particular bet he is certain that he will not accept any further coin-toss-bets. Therefore, would turn 99th bet, having realised that he will not the 100th (i.e., the bet. However, rather than stating the assumption by referring bets risk aversion 7 SC expects accept, I will it more generally assumption that
), that is the reason why the theoRy does badlY in thIs rEspEcT. MorE on tHis in section 4, afTEr thE axiomatic argument has bEen prEsENted IN tHe nexT sectioN. 3 thE ARguMeNt AxiOmATiSed TvErsKy and BaR-Hillel (1983) staTe aN aXiomatic versIOn Of SamuelsoN's aRgument. Their ForMulatiOn Can HOweveR be CritiCised fOR the reAsons raisEd BY BuchaK (As quoteD ABoVe). ThE axiomatic backwarD-InDUctive argument i formuLaTE iN THis SecTion avoids ThAt proBLem, and cONtAINS soME details that TVersky and BaR-hilLel ignOrE. As WE shall See, TvErSKy aNd Bar-Hillel'S axiOmatic argUment wAS somewhAT misleaDing, siNce It mIght HAvE sUggEsTEd tHAt The (MOstLy uncontRoVeRsial) statE-wISE DomInaNce pRinciPle sufficed foR thE argUMenT, when The (muCh moRe ContrOversiAl) EveNt-wise Dominance prInciPle is actuAllY nEedEd. the axIOmatic ArgUmeNt preseNted belOW is, As FAR As i can tell, the strongeSt ARGuMent one cOuld giVE fOr WHat I prevIoUslY calLED "SamuElsoN'S cLaim", i.e., foR the clAIm ThAt one caNnOt ratiOnAllY tuRn dowN The sIngle bEt while aCceptINg the package of 100 BEts. But it shoulD Be NOTeD That ThiS argument shAres WIth oTher BAcKwaRD-induCtive ArGUmENts an (implicit) assumpTiOn that Could Be questioned; nAmely, the asSUMPtion thaT the AGeNT of interest is cErtaiN that at eacH Point in tIme, shE will act On her prefEREnces at tHat TimE. SiNce THE fOcus of this papER Is on ThE rationAl pErmissiBilIty Of rIsk AvErsion, I shAll not diScUsS tHe AssUmptiON in questIoN anY fUrtHer (buT See e.g. MCClenNen 1990 fOr A tHOroUgh discUSsION). InsTeAd, The fOcuS wIll be On a pREmiSe in the Argument tHat TUrns OuT tO specifIcally have to dO wIth rationaLlY peRmissiBLE attitudEs to risk. The first assumptIOn of the AxiOmatiC argUment is baSed On the oBseRVation That SC Would TuRn dOWN the lAST cOin-ToSs-bet he expECTs tO be ofFeRed, pRovided That previous bets diD Not Take his wealth OutSide THE sPecIFiED raNgE. and CRUcially, the same hOlds when SC Is COnSidering a pARtiCuLar bet iF he is ceRtain THat he wiLl not accePt any furtHeR coiN-TOss-Bets. TherefOre, he wouLd turn dowN The 99th BEt, HavinG reAlised ThAt hE will Not accEPt tHe 100th (i.E., the laSt) Bet. HowEver, rAtHer than sTating the assumption by reFerrinG to beTs IS risk averSioN IrrAtional? 7 SC ExpeCts to accepT, I wIll State It mORe genEralLY aS an ASsumpTion THat
), that is the reason whythe theory  does  ba dly  i n th is r espect. More o n  thi s in section 4, afterthe a xi o mati c  a rgume nt hasb ee n pre se nt edin th e nex t s ection.  3 The arg ume nt  axiomatised Tv ersky andBar -Hillel (198 3)statean  ax i omati c v ersio n of S a muelso n's argum en t . Thei r  formul a t io n ca n however be crit i ci s ed for the rea sons r ai s ed b y B uch ak (as quo te d abo v e). The ax i o m ati c  backward-ind uctive argu m ent  I for mu lat e  in th is se ct i onavoids that  pro blem, and  conta i ns some details  thatTve rsk y an d  B ar -Hi ll e l i g no re. Aswe shall  s ee , Tve rsky a n d  Bar -Hi llel 's ax iomatic argum ent  was som ewhat  misl eadi ng , sin ce itmight  h ave suggested t hatthe (most lyun con tr overs i al) St ate -Wi se Domi nance p r inc ip l e su fficed for the arg um e n t,  when th e (muc h  m or e  controv er sia l) E v e nt-Wi se D o mi nance pr incipl e  i sactuall yneeded .The  ax iomat i c ar gument  present ed be l ow is, as fara s I can tell, th e st r onge stargument on e co u ld g ivef or  wh a t I p revio us l yc alled "Samuelson'scl aim",i.e.,  for the clai m that one c a nnot rat iona l ly turn down thesingl e bet whil e  accepti ng th e packag e of 100b e ts. Butitsho uld  be n ot ed that thisa r gume nt  shares  wi th othe r b ack war d-i nd uctive ar gumentsan  ( im pl ici t) as s umptionth atco uld  be q u estion ed; n amel y,  t h e a ssumpti o nt h at t he  a gent  of  i ntere st i s  ce rtain t hat at ea chp oint  i ntime, s he will act o nher prefer en ces  at th a t  time. S ince the focus of thisp aper is  on  therati onal perm iss ibilit y o f  riskaversi on, I  s hal l not d i s cu ssth e assumpti o n  in  ques ti on a ny furt her (but see e.g.M cCl ennen 1990 fo r a  tho r o ug h d i sc u ssi on ) . I n s tead, the focus  will be o na  p remise int hear gumentthat tu rns o u t to sp ecificall y have to  d o wi t h  ra tionally p ermissib le attitu d es to ri sk. T hefirstas sum ption  of th e  ax iomat ic arg um ent is  base don the o bservation that SC woul d turn  down  th e last co in- t oss -bet he e xpec ts to be o ffe red , pro vid e d tha t pr e vi ous betsdidn ot take h i swea l t houtside the s p eci fiedran g e. And  cru cially, the sameh olds when SC i s co n s ide rin g  a p ar ticular bet if  he  i s certainth at he willnot acce pt any f urther  coin- toss-be t s .T herefo re,hewould tur n d ow n  the 99 th  b e t, hav ingre alised  thath e wi l l  not accept the100th ( i.e., the  last )bet. Ho w ever , rather t han stating  the a ssum ption  by ref er ring t o b et s Is riska version i rrati onal? 7  S C ex pec ts toacce p t , I w illst ate  it moreg e ne r al ly asan a ssump ti on t hat
), that_is the_reason why the theory_does badly_in_this respect._More_on this in_section 4, after_the axiomatic argument has_been presented in_the_next section. 3 The argument axiomatised Tversky and Bar-Hillel (1983) state an axiomatic version_of_Samuelson's argument._Their_formulation_can however be criticised for_the reasons raised by Buchak_(as quoted_above). The axiomatic backward-inductive argument I formulate in_this_section avoids that_problem, and contains some details that Tversky and Bar-Hillel_ignore. As we shall see, Tversky_and Bar-Hillel's axiomatic_argument_was_somewhat misleading, since it_might have suggested that the (mostly_uncontroversial) State-Wise Dominance principle sufficed for_the argument, when the (much more controversial)_Event-Wise Dominance principle is actually needed._The axiomatic argument presented below_is, as_far as I can tell,_the strongest argument_one could_give for what_I previously called "Samuelson's claim", i.e.,_for the claim_that one cannot rationally turn down_the_single bet while_accepting_the_package of_100 bets. But_it_should be_noted_that this argument shares with other_backward-inductive_arguments an (implicit) assumption that could be_questioned; namely, the assumption_that_the agent of interest_is certain that at each_point in time, she will act_on her_preferences at_that time. Since the focus of this paper is on the_rational permissibility of risk aversion, I_shall not discuss the_assumption in_question_any further (but_see_e.g. McClennen_1990 for a thorough discussion). Instead, the_focus will_be on a premise in the_argument that turns out_to_specifically have to do with rationally_permissible attitudes to risk. The first_assumption of the axiomatic argument_is_based_on the observation that SC_would turn down the last coin-toss-bet_he expects to_be offered, provided that previous bets did_not_take his wealth outside the specified_range._And crucially, the same holds when_SC_is_considering a particular bet if_he is certain that he will_not accept any further coin-toss-bets. Therefore, he would turn_down the 99th_bet, having realised that he_will_not_accept the 100th (i.e., the last) bet. However, rather than_stating the_assumption by referring_to bets Is risk aversion irrational? 7 SC expects to_accept, I will state it more generally_as an assumption that