ised, as Assumption 1 in the axiomatic argument presented in section 3, the stated range is from a loss of $9,900 to a gain of $19,800. 8 You can test this assumption on yourself, assuming that you share SC's preference against the single bet but for the package, by considering whether your preference would change if you had, for instance, paid $10,000 less or $20,000 more off your mortgage, car loan or student loan. 9 That is, as a function of terminal wealth, in this case (or more generally as a function of terminal outcomes), rather than, say, as a function of changes in wealth. Is risk aversion irrational? 5 The questionable assumption is, however, by no means indispensable to the argument. Rather than starting the inductive argument from the first bet, we can start from the last bet and work backwards. We are supposing that SC is, on the one hand, evaluating a single bet, and, on the other hand, a package of 100 bets. Now under the assumption that SC would turn down the single bet for the wealth range in question, it is clear that if offered the 100 bets sequentially, he would turn down the last bet, irrespective of the outcome from the first 99 bets (i.e., irrespective of how the first 99 bets affect his wealth). For recall that it is assumed that SC has a preference against a single bet whenever his wealth is in the range that could result from these 99 bets. But if he knows that he will turn down the last bet, then he will also turn down the second-to-last bet, and so on, for all bets in the package. Hence, it may seem that Samuelson's claim is true, since his colleague's attitudes do seem to be internally incoherent. A more formal version of this argument will be provided in the next section. This version of the argument might raise the worry that while SC was considering, on one hand, a single bet, and, on the other hand, a package of bets, the above argument has turned the decision-problem SC was faced with into a sequential one with the option to stop after having accepted any number of bets (Aloysius 2007). In particular, it might seem that Samuelson is falsely assuming that one can rationally accept a package of bets made at a single time just in case one accepts a sequence of the same bets. However, as we shall see in the next section, when the argument is axiomatised,
ised, as Assumption 1 in the axiomatic argument presented in section 3, the express compass is from a loss of $ 9,900 to a gain of $ 19,800. 8 You can test this presumption on yourself, assuming that you share SC's predilection against the single bet but for the package, by consider whether your preference would change if you suffer, for instance, pay up $ 10,000 less or $ 20,000 more off your mortgage, car loan or student loan. 9 That is, as a function of terminal wealth, in this event (or more generally as a function of terminal consequence), rather than, say, as a function of changes in wealth. Is risk antipathy irrational? 5 The questionable assumption is, however, by no mean indispensable to the argument. Rather than starting the inductive argumentation from the first bet, we can start from the last bet and shape backwards. We are supposing that SC is, on the one hand, evaluating a single bet, and, on the other hand, a package of 100 bets. nowadays under the assumption that SC would turn down the individual bet for the wealth range in question, it is clear that if offer the 100 bets sequentially, he would turn down the last stake, irrespective of the outcome from the first 99 bets (i.e., irrespective of how the first 99 bets affect his wealth). For recall that it is assumed that SC has a preference against a single stake whenever his wealth is in the range that could leave from these 99 bets. But if he knows that he will turn down the last bet, then he will also call on down the second - to - last bet, and so along, for all bets in the software. Hence, it may seem that Samuelson's claim is true, since his colleague's attitudes do appear to be internally incoherent. A more formal version of this controversy will be provided in the following section. This adaptation of the argument might recruit the worry that while SC was considering, on one hand, a single bet, and, on the early hand, a package of bets, the above argumentation has turned the decision - problem SC was faced with into a sequential one with the choice to stop after having accepted any act of bets (Aloysius 2007). In particular, it might seem that Samuelson is falsely assuming that one can rationally accept a package of bets construct at a single time just in case one accepts a succession of the same bets. However, as we shall interpret in the next section, when the argument is axiomatised,
isef, as Assumption 1 in the xxiomatic argumgnr presxnted ih sectiov 3, the stated range is from e lows of $9,900 to a gain of $19,800. 8 You cxn test tjis assunptiib on yoursxmf, assuming tgwt ymn share SC's prenerence agahnst the singla cec but for the package, by considering whethet jour preferencg woukq chznge if you had, for instance, paid $10,000 less og $20,000 more off your kortgage, car loan or studejt llan. 9 That is, as a vunction of teriunal wealth, kn this case (or more ggnerally as a function of terminxl ouccomes), rathgx thwt, say, as a hunctijn of changes in weanth. Is tisk aversion lrratmonao? 5 The questionable avsumption is, howevgr, by no maaus indispensable to tye argukent. Ratfwr ghah xtzrting thx inductive argument feom the first bet, wt cwb start from fhe laft bet and work backwards. We are supposinc tgat SC is, on the one habd, evaluating a singlg bet, and, jn the other hand, a package of 100 bets. Now under tha assnmotiin gyah SC would turn down the single bet for the wqzluh gange in question, it is cleat hhsj if offered tfe 100 beca aequentially, he wokld turg dowb the lasu bet, irrespective of the outcomw from the fprst 99 bets (i.e., irrespeccive of how che fitst 99 brts affect his wealth). Fur rscall that lt is asshoed that SC has x pgefesence against a single bet whenever his wealth is on the range thah could result from thesf 99 bejs. But if he knoas that he will turn down the lext bet, then ne winl also curn dpwn the seconq-to-last bet, anb so on, yor alu bets in nhe packaje. Hence, it iay seem that Damuelson's cnaim is erue, sinxe his zulleague's attiyudes do seem to be internally incohevent. X more formal vexruon of this argimevt ripl bq provided in the neby secgion. This vcrskon pf the argument mighd rajse the worry that wmile SC wqs consiqering, on one hand, a single bet, and, mn vhe otner hand, a package of bets, the abkve argumfnt has turned ehe eecision-probkem SC was faced with into a sequential one with the optiin to stop after harikg accepted eny nuiber of bats (Aloysius 2007). In parricular, it might xeem that Samuelson ia falsaly adsuming that one can rationally accept a package of bets made at a single time ouft in case kne sccepns c szquence of vhx same bets. Howevvr, as we shall see in the next sxction, whet che argument is axiomatised,
ised, as Assumption 1 in the axiomatic in 3, the range is from a of $19,800. 8 can test this on yourself, assuming that you share preference against the single bet but for the package, by considering whether your would change if you had, for instance, paid $10,000 less or $20,000 more your car or loan. 9 That is, as a function of terminal wealth, in this case (or more generally a function of terminal outcomes), rather than, say, a function of changes wealth. Is risk aversion irrational? The assumption is, by means to the argument. than starting the inductive argument from the first bet, we can start from the last bet and backwards. We that SC on one evaluating a single on the other hand, a package Now under the assumption that SC would turn the single for the wealth range in question, is clear that if offered the 100 bets he would turn down the last bet, irrespective of the outcome from the first 99 irrespective of how the 99 bets affect wealth). recall it assumed that has a preference against a single bet whenever his wealth is the range that could result from these 99 bets. But knows he will turn the last bet, then will turn down the second-to-last so for the Hence, may seem that Samuelson's is true, since his colleague's do seem to be version of this argument will be provided in next section. This version of the argument raise the worry that while SC was considering, on one hand, a bet, and, other hand, a package of bets, the above has turned the decision-problem was faced with into a sequential one with the to after having any number of (Aloysius 2007). In it might seem is falsely that can package of bets made at a time in case one accepts a of bets. However, as we shall see in the next section, when the argument is axiomatised,
ised, as Assumption 1 in the axioMatic argumEnt prEseNteD iN secTion 3, The stated range IS froM a loss of $9,900 to a gain of $19,800. 8 You caN test ThIS assUMpTion oN yourseLF, aSSUmiNg ThAt yOu SHaRe SC's PreFerence Against the SinGlE bet but for thE PaCkage, by conSidEring whether YouR prefeReNce WOuld cHanGe if yOu had, fOR instaNce, paid $10,000 leSs OR $20,000 more oFF your moRTGaGe, caR loan or student loaN. 9 thAT is, as a function Of termInAL wEALth, In tHis case (or mOrE geneRAlly as a FUnCTIOn oF Terminal outcoMes), rather thAN, saY, as a fuNcTioN Of chanGes in WeALth. is risk aversIon iRrational? 5 the queSTionablE AssumptIon is, hOweVer, By no MEaNs IndIsPEnsABlE to THe aRgument. RAtHeR than StarTING The iNduCtivE arguMent from the fiRst Bet, wE Can Start From tHe laSt Bet anD work bAckwaRdS. We are supposing That sC is, on the One HaNd, eVaLuatiNG a singLe bEt, aNd, on the Other haND, a pAcKAGE oF 100 bets. Now under the asSuMPTiOn that SC Would tURn DoWN the singLe Bet For tHE WealtH ranGE iN questioN, it is cLEaR tHat if ofFeRed the 100 BeTs sEquEntiaLLy, he Would tUrn down tHe lasT Bet, irrespectivE Of the outcome fROm THE fIRst 99 bEts (I.e., irrespectIve oF How tHe fiRSt 99 BetS AffecT his wEaLTh). fOr recall that it is assUmEd that sC has A preference agAinst a singLE BEt whenevEr hiS WeALth is in the rangE that Could resulT From thesE 99 bets. but if he kNows that hE WIll turn dOwn The LasT beT, THeN he will also tuRN Down ThE second-To-lAst bet, aNd sO on, For All BeTs in the paCkage. HenCe, It MaY sEem That SAMuelson's ClAim Is TruE, sincE His colLeaguE's atTiTuDEs dO seem to BE iNTErnaLlY iNcohEreNt. a more FormAL veRsion of This argumEnt WIll bE pRoVided in The next sectioN. THis version Of The ArgumeNT Might raiSe the worry that while SC waS ConsideRinG, on onE hanD, a single bEt, aNd, on thE otHEr hand, A packaGe of bEtS, thE ABove aRGUmEnt HaS turned the DECisIon-prObLem Sc was facEd with into a sequentIAl oNe with the optiOn tO stoP AFtEr hAViNG acCePTed ANY number of bets (AlOysius 2007). In paRtICuLar, it might SEem ThAt SamueLson is fAlselY AssuminG that one cAn rationaLlY accEPT a pAckage of beTs made at A single tiME just IN cAse onE acCepts a SeQueNce of The samE BetS. HoweVer, as wE sHall seE in thE nExt sectiOn, when the argument is axioMatiseD,
ised, as Assumption 1 in t he axiomat ic ar gum ent  p rese nted  in section 3, thestated range is from a  loss  o f  $9, 9 00  to a  gain o f  $ 1 9 ,80 0.  8  Yo uc an  test  th is assu mption onyou rs elf, assumin g  t hat you sh are  SC's prefer enc e agai ns t t h e sin gle  betbut fo r  the p ackage, b yc onside r ing whe t h er  you r preference woul d  c h ange if you ha d, for  i n st a n ce,  pa id $10,000  l ess o r  $20,00 0  m o r e  of f  your mortgag e, car loan orstuden tloa n . 9 Th at is ,a s a  function o f te rminal we alth,i n thisc ase (or  moregen era llya safun ct i ono fter m ina l outcom es ),  rath er t h a n ,  say , a s afunct ion of change s i n we a lth . Isriskaver si on ir ration al? 5  T he questionable  ass umption i s,ho wev er , byn o mean s i ndi spensab le to t h e a rg u m e nt . Rather than star ti n g  t he induc tive a r gu me n t from t he  fi rstb e t, we  can st art from  the l a st  b et andwo rk bac kw ard s.We ar e  sup posing  that SC  is,o n the one hand ,  evaluating a si n g le bet,  an d, on the o ther hand , ap ac kag e  of 1 00 be ts .  N o w under the assumpt io n that  SC w ould turn dow n the sing l e bet forthew ea l th range in qu estio n, it is c l ear that  if o ffered t he 100 be t s  sequent ial ly,  he  wo u l dturn down the l astbe t, irre spe ctive o f t heout com efrom thefirst 99  b et s(i .e. , irr e spective  o f h ow  th e fir s t 99 b ets a ffec thi s  we alth).F or r ecal lth at i t i sassum ed t h atSC hasa prefere nce agai ns ta singl e bet wheneve rhis wealth  i s i n ther a nge that  could result from thes e  99 bet s.But i f he  knows th athe wil l t u rn dow n thelastbe t,t h en he w il l a ls o turn dow n the  seco nd -to- last be t, and so on, fora llbets in the p ack age. H en ce, it may  s e emt h at Samuelson'sclaim is t ru e ,since hisc oll ea gue's a ttitude s dos eem tobe intern ally inco he rent . A m ore formal  version  of thisa rgume n twillbeprovid ed  in  thenext s e cti on. T his ve rs ion of  thear gument m ight raise the worry th at whi le SC  wa s conside rin g , o n one han d, a  single be t,and , onthe other  han d ,a p a ckage  ofb ets, thea bo vea r gu ment has tu r n e d t he de cis i on-pro blem  SC was faced wit h  into a sequen tial o newit h  the  o ption to stopaft er h aving ac ce pted any nu mber ofbe t s (Al oysius  2007) . In pa r t ic u lar, i t mi ght  seem tha t S am u elson i sfa l sely a ssum in g that  one c a n ra t i onally accept apacka g e  of b e tsmadeat  a sing l e ti me just in  case one a ccepts  a s equen ce of t he  samebet s.  However,a s we shal l see  in the  n extsec tion,when t he ar gume nt  is  axiomati s e d, 
ised, as_Assumption 1_in the axiomatic argument_presented in_section_3, the_stated_range is from_a loss of_$9,900 to a gain_of $19,800. 8_You_can test this assumption on yourself, assuming that you share SC's preference against the_single_bet but_for_the_package, by considering whether your_preference would change if you_had, for_instance, paid $10,000 less or $20,000 more off_your_mortgage, car loan_or student loan. 9 That is, as a function_of terminal wealth, in this case_(or more generally_as_a_function of terminal outcomes),_rather than, say, as a function_of changes in wealth. Is risk_aversion irrational? 5 The questionable assumption is,_however, by no means indispensable to_the argument. Rather than starting_the inductive_argument from the first bet,_we can start_from the_last bet and_work backwards. We are supposing that_SC is, on_the one hand, evaluating a single_bet,_and, on the_other_hand,_a package_of 100 bets._Now_under the_assumption_that SC would turn down the_single_bet for the wealth range in question,_it is clear that_if_offered the 100 bets_sequentially, he would turn down_the last bet, irrespective of the_outcome from_the first_99 bets (i.e., irrespective of how the first 99 bets affect_his wealth). For recall that it_is assumed that SC_has a_preference_against a single_bet_whenever his_wealth is in the range that could_result from_these 99 bets. But if he_knows that he will_turn_down the last bet, then he_will also turn down the second-to-last_bet, and so on, for_all_bets_in the package. Hence, it_may seem that Samuelson's claim is_true, since his_colleague's attitudes do seem to be internally_incoherent._A more formal version of this_argument_will be provided in the next_section._This_version of the argument might_raise the worry that while SC_was considering, on one hand, a single bet, and,_on the other_hand, a package of bets,_the_above_argument has turned the decision-problem SC was faced with into_a sequential_one with the_option to stop after having accepted any number of bets_(Aloysius 2007). In particular, it might seem_that Samuelson is falsely assuming that one can rationally accept a_package of bets made at a single time_just in case one accepts a sequence_of the same bets._However,_as we shall see in the next_section,_when_the_argument is axiomatised,