 SC turns down any single bet when his wealth is within a particular range; but I will first apply the assumption to the last bet and work backwards to the first bet. To state the assumption more formally, let Bi denote an individual cointoss-bet; i.e., a bet that either results in bettors winning $200 or losing $100 depending on whether their chosen side of a coin comes up or not. The index denotes the order in which the bet is offered, such that, for instance, Bi+1 is offered next after Bi. {Bn} is a package of n such bets. '' represents SC's preference, such that A B denotes that SC prefers A to B. X is a random variable in the interval [w−9, 900, w+ 19, 800], where $w is the dollar amount corresponding to SC's pre-bet wealth. $X +Bi denotes that SC's wealth is X dollars and that he in addition holds a coin-toss-bet. The first assumption can now be stated as follows: Assumption 1 The agent prefers to turn down bet Bi for all wealth levels between and including $w−9.9k and $w+19.80k. More formally, for any value of X in [w − 9, 900, w + 19, 800]: $X $X + Bi The above assumption is an empirical claim. The assumption is that SC's preferences do as a matter of fact have this characteristic. And as discussed, this is clearly true when SC is certain that he will not accept any bet that is offered after Bi. The next assumption, Dominance, is however a normative claim. The role of the axiom in the present argument is to connect rational attitudes to outcomes of sequences of bets to attitudes to packages of bets. Informally, the axiom, in the form used here, states that if for any possible outcome that a package {Bn} of bets can result in, a decision-maker prefers that outcome to that same outcome plus an additional bet Bn+1, then the decision-maker prefers {Bn} to the package {Bn+1} that is the result of enlarging {Bn} by bet Bn+1. Before stating a general version of Dominance, it may be instructive to consider a simplified version of the principle where the two packages in question are, on one hand, the 'package' consisting only of bet B1, and, on the other hand,
SC turns down any single bet when his wealth is within a particular compass; but I will foremost apply the assumption to the final stake and work backwards to the first stake. To express the assumption more formally, permit Bi denote an individual cointoss - bet; i.e., a stake that either results in bettors winning $ 200 or losing $ 100 depend on whether their chosen side of a coin comes up or not. The index denotes the ordering in which the bet is offered, such that, for instance, Bi+1 is offer next after Bi. { Bn } is a package of n such bets.' ' represents SC's preference, such that A bacillus denotes that SC prefers A to B. X is a random variable in the interval [ w−9, 900, w+ 19, 800 ], where $ w is the dollar sum corresponding to SC's pre - bet wealth. $ X + Bi denotes that SC's wealth is X dollars and that he in addition holds a coin - toss - bet. The first assumption can now be stated as follows: Assumption 1 The agent prefers to turn down bet Bi for all wealth degree between and including $ w−9.9k and $ w+19.80k. More formally, for any value of X in [ w − 9, 900, w + 19, 800 ]: $ X $ adam + Bi The above premise is an empirical claim. The assumption is that SC's preference do as a matter of fact have this characteristic. And as discussed, this is intelligibly true when SC is certain that he will not accept any bet that is offered after Bi. The next presumption, Dominance, is however a normative claim. The role of the axiom in the present controversy is to connect intellectual attitude to outcomes of sequences of bets to attitudes to software of bets. Informally, the axiom, in the form use here, states that if for any possible outcome that a package { Bn } of bet can result in, a decisiveness - maker prefers that outcome to that same outcome plus an additional bet Bn+1, then the decisiveness - maker prefers { Bn } to the package { Bn+1 } that is the result of enlarging { Bn } by bet Bn+1. Before express a general version of Dominance, it may be instructive to regard a simplified version of the rationale where the two packages in question are, on one hand, the' software' consisting only of bet B1, and, on the other hand,
 SC turns down any single btt when his wealtk is wivhin a larticulxr range; but I will first ap'ly rhe awsumption to the last cet and wlrk backqardw to the ficat bet. Bj stzbe thz essumption more formally, net Bi denote dn iudividual cointoss-bet; i.e., a bet that qither tedults in bettots wimging $200 or losing $100 depending on whether fheir ciosen side of a coin comes up or not. The lnded denotes the ordeg in which jge fwt is offerea, such thau, yor instancg, Bi+1 is offered next after Bi. {Bn} is a package od b skwh bets. '' re'resenns SC's prefercmce, suwh that A B denotes tmat SR prwfers A to B. X is a rendom variable in thg interval [w−9, 900, w+ 19, 800], where $w is rhw dolnar dmouvr currtsppnsing tl SR's pre-bet wsalth. $X +Bi eenotes that SC's wesley is X dollara and ehwt he in addition holds a coin-toss-bet. Tve rirst assumption can noq be stated as followd: Assumptyon 1 The agent prefers to turn down bet Bi for aln weamgh oeyels vehween and including $w−9.9k and $w+19.80k. More formally, fjd snj value of X in [w − 9, 900, w + 19, 800]: $X $C + Bo The above asrumption js an empirical clwim. The assunption is thay SC's preferences do as a mqtter of facn hace this characteridtic. And as disvussec, this is clearly true chen SD is certaij that he dill not accept xny bat that is offered after By. The nexv assbmption, Aomimance, ys however a novkative claim. The rlle oy the axiom in hhe present argument is to connxrt rational ajtidudvs to outeomes pf sequences jf bets to attntudes tj pacyages of bvts. Inforkally, the aviom, in the fmtm used here, vtates trat uf fir any oussible outcomr that a iaekage {Bn} if bets can result in, z decision-maker prtfwrs that outcoke go ehwt swke outcome pnus xn xcditiunal beu Bk+1, tfen yhe decision-maker prafera {Bn} to the packagr {Nn+1} that iw the refult of enlarbing {Bn} by bet Bn+1. Heforx statmng a bengral version of Dominance, it mag be instgucbive to consiqer q simplified version of the principle where the two packages in quesjion are, on one hand, jhe 'package' coisistigg only ox bet B1, and, on the orher hand,
SC turns down any single bet when is a particular but I will the bet and work to the first To state the assumption more formally, Bi denote an individual cointoss-bet; i.e., a bet that either results in bettors $200 or losing $100 depending on whether their chosen side of a coin up not. index the order in which the bet is offered, such that, for instance, Bi+1 is offered next Bi. {Bn} is a package of n such ' ' represents SC's such that A B denotes SC A to X a variable in the [w−9, 900, w+ 19, 800], where $w is the dollar amount corresponding to SC's pre-bet wealth. $X denotes that is X and he addition holds a first assumption can now be stated 1 The agent prefers to turn down bet for all levels between and including $w−9.9k and More formally, for any value of X in − 9, 900, w + 19, 800]: $X $X + Bi The above assumption is claim. The assumption is SC's preferences do a of have characteristic. And discussed, this is clearly true when SC is certain that he not accept any bet that is offered after Bi. The Dominance, however a normative The role of the in present argument is to attitudes outcomes bets attitudes packages of bets. Informally, axiom, in the form used states that if for package {Bn} of bets can result in, a prefers that outcome to that same outcome an additional bet Bn+1, then the decision-maker prefers {Bn} to the package that is of enlarging {Bn} by bet Bn+1. Before stating general version of Dominance, may be instructive to consider a simplified version of principle the two in question are, one hand, the consisting only of and, on other
 SC turns down any single bet whEn his wealtH is wiThiN a pArTicuLar rAnge; but I will fiRSt apPly the assumption to the lAst beT aND worK BaCkwarDs to the FIrST Bet. to StAte ThE AsSumptIon More forMally, let Bi DenOtE an individuaL CoIntoss-bet; i.E., a bEt that either ResUlts in BeTtoRS winnIng $200 Or losIng $100 depENding oN whether tHeIR choseN Side of a COIn ComeS up or not. The index dENoTEs the order in whIch the BeT Is OFFerEd, sUch that, for InStancE, bi+1 is offEReD NEXt aFTer Bi. {Bn} is a pacKage of n such BEts. '' RepresEnTs Sc'S prefeRence, SuCH thAt A B denotes That sC prefers a to B. X iS A random VAriable In the iNteRvaL [w−9, 900, w+ 19, 800], wHErE $w Is tHe DOllAR aMouNT coRrespondInG tO SC's pRe-beT WEALth. $X +bi dEnotEs thaT SC's wealth is X DolLars ANd tHat he In addItioN hOlds a Coin-toSs-bet. thE first assumptioN can Now be statEd aS fOllOwS: AssuMPtion 1 THe aGenT preferS to turn DOwn BeT bI FoR all wealth levels beTwEEN aNd includIng $w−9.9k aND $w+19.80K. MORe formalLy, For Any vALUe of X In [w − 9, 900, w + 19, 800]: $x $x + BI The abovE assumPTiOn Is an empIrIcal clAiM. ThE asSumptIOn is That SC'S prefereNces dO As a matter of facT Have this charaCTeRIStIC. And As dIscussed, thiS is cLEarlY truE WhEn Sc Is cerTain tHaT He WIll not accept any bet tHaT is offEred aFter Bi. The next Assumption, dOMInance, is HoweVEr A Normative claim. the roLe of the axiOM in the prEsent Argument Is to conneCT Rational AttItuDes To oUTCoMes of sequenceS OF betS tO attituDes To packaGes Of bEts. infOrMally, the aXiom, in thE fOrM uSeD heRe, staTEs that if FoR anY pOssIble oUTcome tHat a pAckaGe {bn} OF beTs can reSUlT IN, a deCiSiOn-maKer PrEfers That OUtcOme to thAt same outComE Plus An AdDitionaL bet Bn+1, then the DeCision-makeR pRefErs {Bn} tO THe packagE {Bn+1} that is the result of enlARging {Bn} By bEt Bn+1. BEforE stating a GenEral veRsiON of DomInance, It may Be InsTRUctivE TO cOnsIdEr a simplifIED veRsion Of The pRinciplE where the two packagES in Question are, on One Hand, THE 'pAckAGe' COnsIsTIng ONLy of bet B1, and, on thE other hand,
 SC turns down any singlebet when h is we alt h i swith in a  particular ra n ge;but I will first apply  theas s umpt i on  to t he last be t and  w or k b ac k wa rds t o t he firs t bet. Tosta te  the assumpt i on  more form all y, let Bi de not e an i nd ivi d ual c oin toss- bet; i . e., abet thatei t her re s ults in b et tors  winning $200 orl os i ng $100 depend ing on  w h et h e r t hei r chosen s id e ofa  coin c o me s u p o r  not. The ind ex denotest heorderin  wh i ch the  betis off ered, suchthat , for ins tance, Bi+1 is offered  nextaft erBi.{ Bn }isap ack a ge  of n s uch bets .'' rep rese n t s SC's  pr efer ence,  such that AB d enot e s t hat S C pre fers  A  to B . X is  a ra nd om variable intheinterval[w− 9,  90 0,  w+ 1 9 , 800] , w her e $w is  the do l lar  a m o u nt  corresponding toSC ' s  p re-bet w ealth. $X  + B i denote stha t SC ' s  weal th i s  X  dollars  and t h at  h e in ad di tion h ol dsa c oin-t o ss-b et. Th e firstassum p tion can now b e  stated as fo l lo w s :A ssum pti on 1 The ag entp refe rs t o  t urn downbet B if or all wealth levels b et ween a nd in cluding $w−9. 9k and $w+ 1 9 . 80k. Mor e fo r ma l ly, for any va lue o f X in [w−  9, 900,  w +19, 800] : $X $X + B i The ab ove  as sum pti o n  i s an empirica l clai m.  The as sum ption i s t hat  SC 'spr eferences  do as a  m at te roffacth ave this  c har ac ter istic .  And a s dis cuss ed ,t his  is cle a rl y true  w he n SC  is  c ertai n th a t h e willnot accep t a n y be tth at is o ffered afterBi . The next  a ssu mption , Dominanc e, is however a normati v e claim . T he ro le o f the axi omin the  pr e sent a rgumen t isto  co n n ect r a t io nal  a ttitudes t o out comes  o f se quences  of bets to attitu d esto packages o f b ets. I nf orm a ll y , t he axi o m , in the form u sed here,st a te s that iff oran y possi ble out comet hat a p ackage {B n} of bet scanr e sul t in, a de cision-m aker pref e rs th a toutco meto tha tsam e out come p l usan ad dition al  bet B n+1,th en the d ecision-maker prefers { Bn} to  thepac kage {Bn+ 1}t hat  is the r esul t of enlar gin g { Bn} b y b e t Bn+ 1. B e fo res tatin g ag eneral ve r si ono f  D ominance, i t m aybe in str u ctiveto c onsider a simplif i ed version ofthep r inc ipl e  whe re  the two packa ges  i n question  a re, on onehand, th e' packa ge' co nsisti ng only o fb et B1,  and , o n the oth erha n d,
 SC_turns down_any single bet when_his wealth_is_within a_particular_range; but I_will first apply_the assumption to the_last bet and_work_backwards to the first bet. To state the assumption more formally, let Bi denote_an_individual cointoss-bet;_i.e.,_a_bet that either results in_bettors winning $200 or losing_$100 depending_on whether their chosen side of a coin_comes_up or not._The index denotes the order in which the bet_is offered, such that, for instance,_Bi+1 is offered_next_after_Bi. {Bn} is a_package of n such bets. ''_represents SC's preference, such that A_B denotes that SC prefers A to_B. X is a random variable_in the interval [w−9, 900,_w+ 19,_800], where $w is the_dollar amount corresponding_to SC's_pre-bet wealth. $X_+Bi denotes that SC's wealth is_X dollars and_that he in addition holds a_coin-toss-bet._The first assumption_can_now_be stated_as follows: Assumption_1_The agent_prefers_to turn down bet Bi for_all_wealth levels between and including $w−9.9k and_$w+19.80k. More formally, for_any_value of X in_[w − 9, 900, w_+ 19, 800]: $X $X +_Bi The_above assumption_is an empirical claim. The assumption is that SC's preferences do_as a matter of fact have_this characteristic. And as_discussed, this_is_clearly true when_SC_is certain_that he will not accept any bet_that is_offered after Bi. The next assumption,_Dominance, is however a_normative_claim. The role of the axiom_in the present argument is to_connect rational attitudes to outcomes_of_sequences_of bets to attitudes to_packages of bets. Informally, the axiom,_in the form_used here, states that if for any_possible_outcome that a package {Bn} of_bets_can result in, a decision-maker prefers_that_outcome_to that same outcome plus_an additional bet Bn+1, then the_decision-maker prefers {Bn} to the package {Bn+1} that is_the result of_enlarging {Bn} by bet Bn+1._Before_stating_a general version of Dominance, it may be instructive to_consider a_simplified version of_the principle where the two packages in question are, on_one hand, the 'package' consisting only of_bet B1, and, on the other hand,