 the package consisting of bets B1 and B2. Suppose that B1 will result either in outcome x if a coin comes heads up on its first toss-an event I will refer to as H1-or in outcome y if the coin comes tails up on its first toss-an event I will refer to as T1. (I assume that T1 = ¬H1.) The outcome of B2, however, depends on how the coin lands on the second toss, which is assumed to be independent of how it lands on the first toss. Table 1 represents the two alternatives on offer-i.e., to take the first bet only and to take both bets- where the state space has been partitioned according to how the coin lands on its first toss. Dominance entails that if x is preferred to x + B2 and y is preferred to y + B2, then the first bet only-i.e., package {B1}-is preferred to the 8 H. Orri Stefánsson H1 T1 {B1} x y {B2} x + B2 y + B2 Table 1 Take one bet or two? package consisting of the first and the second bet-i.e., package {B2}. And that may seem plausible, as a condition on rational attitudes to bets; and could be seen as a rationality constraint on how sequential preferences in the case under consideration relate to'static' preferences between packages. If no matter what outcome the first bet results in, the agent prefers to not add the second bet, then it might seem that she has no good reason for preferring the combination of the first and the second bet to the first bet only. (As will become apparent, I think this justification of Dominance is mistaken. But it certainly has some intuitive appeal.) We can state the axiom more formally as follows: Assumption 2 (Dominance) If for all possible outcomes X from {Bi}, X X + Bj, then: {Bi} {Bi}+ Bj = {Bi+1} As we have seen, Assumption 1 clearly holds when Bi = B100 (i.e., when Bi is the last bet on offer). Together with Dominance the assumption entails: {B99} {B100} (1) In other words, from Assumption 1 and Dominance, it follows that even if the agent has accepted a package of 99 bets, she prefers not to increase the size of the package by one additional bet. But recall that we
the package consisting of bets B1 and B2. Suppose that B1 will leave either in result x if a coin comes heads up on its inaugural pass - an event I will refer to as H1 - or in consequence y if the coin fall tails up on its inaugural toss - an event I will denote to as T1. (I assume that T1 = ¬H1 .) The outcome of B2, however, depend on how the coin lands on the second toss, which is assumed to be independent of how it lands on the inaugural toss. Table 1 represents the two option on offer - i.e., to take the inaugural bet merely and to take both bets- where the state space has been partitioned according to how the coin land on its first toss. Dominance entails that if x is preferred to x + B2 and y is prefer to y + B2, then the first bet only - i.e., package { B1}-is preferred to the 8 H. Orri Stefánsson H1 T1 { B1 } x y { B2 } x + B2 y + B2 Table 1 Take one bet or two? package consisting of the inaugural and the second bet - i.e., package { B2 }. And that may appear plausible, as a circumstance on intellectual attitudes to bets; and could be seen as a rationality constraint on how sequential preferences in the case under consideration associate to'static' preferences between packages. If no matter what outcome the first bet results in, the agent choose to not add the second bet, then it might seem that she has no effective reason for prefer the combination of the first and the second bet to the first bet only. (As will become apparent, I remember this justification of Dominance is mistaken. But it certainly has some intuitive entreaty .) We can state the axiom more formally as follows: Assumption 2 (authority) If for all possible consequence X from { Bi }, X X + Bj, then: { Bi } { Bi}+ Bj = { Bi+1 } As we have watch, Assumption 1 clearly holds when Bi = B100 (i.e., when Bi is the concluding bet on crack). Together with Dominance the assumption entails: { B99 } { B100 } (1) In early words, from Assumption 1 and Dominance, it follows that even if the agent has accepted a package of 99 bets, she prefers not to increase the size of the package by one additional bet. But recall that we
 thf package consisting of nets B1 and B2. Supkowe thav B1 wilm result either in outcome x if a comn cimes yeads up on its first goss-an evvnt I wilo reher to as H1-or in outcome y if fme conn comes tails uk on its firvt toss-an evend K cill refer to as T1. (I assume that T1 = ¬R1.) The oitfome of B2, howeder, cqpenss on how the coin lands on the sedond tovs, which is axsumed to be independent ov hoa it lands on the virst toss. Jzblq 1 represents the two alternatives kn offer-i.e., to take the first beg onlv and to taje bohv bets- wherx the ftate space mss beet partiyioned accordikg to hoq the coin lands on ivs first toss. Dominagce entains that if x is predeered jo x + B2 ave y is pcefsrred ho b + B2, then tge first ber only-i.e., package {B1}-ix [grferred to tge 8 H. Jrwi Stefánsson H1 T1 {B1} x y {B2} x + B2 y + B2 Tablt 1 Taie one bet or two? packate consisting of the virst and the second bet-i.e., package {B2}. And that may seem plagsiblx, xs c condkriln on rational attitudes to bets; and could be aetn ss a rationaliby constraint on hpw srzuential prefgrences in the case under cojsiderajion rwlate to'suatic' preferences between packagws. If no matner qhat outcome the fnrst bet resblts im, the agent prefers to not abd the second bet, then it jkght seem that sfe mas no good reason for preferwing the rombiuation ow thg first and the sfcond bet to the first bft onpy. (As will bfcome apparent, I think this jusvmfication of Cokindnce is iistalen. But it cewtainly has soke intuntive xppeal.) We ban state the axiom iore formally ws follows: Avsumptiog 2 (Diminqnce) If wor all possibke outcomvs X feom {Bi}, X X + Bj, thek: {Bi} {Cj}+ Bj = {Bi+1} As we kcvt ween, Assumptiom 1 zlewrpy hjnds when Bi = B100 (i.d., wfrn Bi is the lasb bdt om offer). Together witv Dojinance the assumpyijn entaios: {B99} {B100} (1) Yn other wordx, from Assumption 1 and Vominaice, it fojlows that even if the agent hzs acceptfd w package of 99 bebs, sre prefers not to increase the size of the packagx by one additional bej. But recall that we
the package consisting of bets B1 and that will result in outcome x up its first toss-an I will refer as H1-or in outcome y if coin comes tails up on its first toss-an event I will refer to T1. (I assume that T1 = ¬H1.) The outcome of B2, however, depends how coin on second toss, which is assumed to be independent of how it lands on the first toss. 1 represents the two alternatives on offer-i.e., to the first bet only to take both bets- where state has been according how coin lands on first toss. Dominance entails that if x is preferred to x + B2 and y is preferred y + the first only-i.e., {B1}-is to the 8 Stefánsson H1 T1 {B1} x y B2 y + B2 Table 1 Take one or two? consisting of the first and the bet-i.e., package {B2}. And that may seem plausible, a condition on rational attitudes to bets; and could be seen as a rationality constraint sequential preferences in the under consideration relate preferences packages. no what outcome first bet results in, the agent prefers to not add the bet, then it might seem that she has no good preferring combination of the and the second bet the bet only. (As will I this is But certainly has some intuitive We can state the axiom formally as follows: Assumption possible outcomes X from {Bi}, X X + then: {Bi} {Bi}+ Bj = {Bi+1} As have seen, Assumption 1 clearly holds when Bi = B100 (i.e., when is the on offer). Together with Dominance the assumption entails: {B100} (1) In other from Assumption 1 and Dominance, it follows that even the has accepted package of 99 she prefers not increase the size package by additional But
 the package consisting of betS B1 and B2. SuppOse thAt B1 WilL rEsulT eitHer in outcome x iF A coiN comes heads up on its firsT toss-An EVent i WiLl refEr to as H1-OR iN OUtcOmE y If tHe COiN comeS taIls up on Its first toSs-aN eVent I will refER tO as T1. (I assumE thAt T1 = ¬H1.) The outcoMe oF B2, howeVeR, dePEnds oN hoW the cOin lanDS on the Second tosS, wHIch is aSSumed to BE InDepeNdent of how it lands ON tHE first toss. TablE 1 repreSeNTs THE twO alTernatives On Offer-I.E., to take THe FIRSt bET only and to takE both bets- whERe tHe statE sPacE Has beeN partItIOneD according tO how The coin laNds on iTS first tOSs. DominAnce enTaiLs tHat iF X iS pRefErREd tO X + B2 And Y Is pReferred To Y + B2, Then tHe fiRST BEt onLy-i.E., pacKage {B1}-Is preferred to The 8 h. OrrI steFánssOn H1 T1 {B1} X y {B2} x + b2 y + b2 TablE 1 Take oNe bet Or Two? package consiStinG of the firSt aNd The SeCond bET-i.e., pacKagE {B2}. ANd that mAy seem pLAusIbLE, AS a Condition on rationaL aTTItUdes to beTs; and cOUlD bE Seen as a rAtIonAlitY COnstrAint ON hOw sequenTial prEFeReNces in tHe Case unDeR coNsiDeratIOn reLate to'Static' prEfereNCes between packAGes. If no matter WHaT OUtCOme tHe fIrst bet resuLts iN, The aGent PReFerS To not Add thE sECoND bet, then it might seem ThAt she hAs no gOod reason for pReferring tHE COmbinatiOn of THe FIrst and the secoNd bet To the first BEt only. (As Will bEcome appArent, I thiNK This justIfiCatIon Of DOMInAnce is mistakeN. bUt it CeRtainly Has Some intUitIve AppEal.) we Can state tHe axiom mOrE fOrMaLly As folLOws: AssumPtIon 2 (doMinAnce) IF For all PossiBle oUtCoMEs X From {Bi}, X x + bj, THEn: {Bi} {bi}+ bj = {bi+1} As We hAvE seen, assuMPtiOn 1 clearLy holds whEn BI = b100 (i.e., wHeN BI is the lAst bet on offer). toGether with doMinAnce thE ASsumptioN entails: {B99} {B100} (1) In other words, fROm AssumPtiOn 1 and domiNance, it foLloWs that EveN If the aGent haS accePtEd a PACkage OF 99 BeTs, sHe Prefers not TO IncRease ThE sizE of the pAckage by one additioNAl bEt. But recall thAt wE
 the package consisting of  bets B1 a nd B2 . S upp os e th at B 1 will resulte ithe r in outcome x if a co in co me s  hea d sup on  its fi r st t oss -a neve nt Iwillref er to a s H1-or in  ou tc ome y if the co in comes t ail s up on itsfir st tos s- ane ventI w ill r efer t o  as T1 . (I assu me that T 1  = ¬H1. ) Th e ou tcome of B2, howe v er ,  depends on ho w theco i nl a nds  on  the secon dtoss, which i s  a s s u med to be indepen dent of how itlandson  th e  first  toss .T abl e 1 represe ntsthe two a lterna t ives on offer-i .e., t o t ake  the fi rs t b et onl y  a ndt o t ake both  b et s- wh eret h e stat e s pace  hasbeen partitio ned  acc o rdi ng to  howtheco in la nds on  itsfi rst toss. Domin ance  entailstha tifxis pr e ferred  to  x+ B2 an d y isp ref er r e d  t o y + B2, then the  f i r st  bet onl y-i.e. ,  p ac k age {B1} -i s p refe r r ed to  the 8H. OrriStefán s so nH1 T1 { B1 } x y{B 2}x +  B2 y + B2  Table  1 Takeone b e t or two? pack a ge consisting of t he firs t a nd the seco nd b e t-i. e.,p ac kag e  {B2} . And  t h at may seem plausible,  a s a co nditi on on rationa l attitude s t o bets;andc ou l d be seen as a  rati onality co n strainton ho w sequen tial pref e r ences in  th e c ase  un d e rconsideration r elat eto'stat ic'  prefer enc esbet wee npackages.  If no m at te rwh atoutco m e the fi rs t b et  re sults in, th e age nt p re fe r s t o not a d dt h e se co nd  bet , t he n itmigh t  se em that  she hasnog oodre as on forpreferring th ecombinatio nofthe fi r s t and th e second bet to the fir s t bet o nly . (As  wil l becomeapp arent,  It hink t his ju stifi ca tio n of Do m i na nce  i s mistaken . But  it c er tain ly hassome intuitive app e al. ) We can stat e t he a x i om  mo r ef orm al l y a s follows: Assump tion 2 (Do mi n an ce) If for all  p ossible  outcom es Xf rom {Bi }, X X +Bj, then:  { Bi}{ B i}+  Bj = {Bi+ 1} As we  have see n , Ass u mp tion1 c learly  h old s whe n Bi = B10 0 (i. e., wh en  Bi is  thela st bet o n offer). Together with  Domin ancethe  assumpti one nta ils: {B99 } {B 100} (1) I n o the r wor ds, fromAssu m pt ion 1 and  Dom i nance, it fo llo w s  t hat even if t h e a genthas accept ed a  package of 99 be t s, she prefers  not t o i ncr e aseth e size of thepac ka g e  by onead ditional be t. But r ec a ll th at we
 the_package consisting_of bets B1 and_B2. Suppose_that_B1 will_result_either in outcome_x if a_coin comes heads up_on its first_toss-an_event I will refer to as H1-or in outcome y if the coin comes_tails_up on_its_first_toss-an event I will refer_to as T1. (I assume_that T1_= ¬H1.) The outcome of B2, however, depends_on_how the coin_lands on the second toss, which is assumed to_be independent of how it lands_on the first_toss._Table_1 represents the two_alternatives on offer-i.e., to take the_first bet only and to take_both bets- where the state space has_been partitioned according to how the_coin lands on its first_toss. Dominance_entails that if x is_preferred to x_+ B2_and y is_preferred to y + B2, then_the first bet_only-i.e., package {B1}-is preferred to the_8_H. Orri Stefánsson_H1_T1_{B1} x_y {B2} x_+_B2 y_+_B2 Table 1 Take one bet_or_two? package consisting of the first and_the second bet-i.e., package_{B2}._And that may seem_plausible, as a condition on_rational attitudes to bets; and could_be seen_as a_rationality constraint on how sequential preferences in the case under consideration_relate to'static' preferences between packages. If_no matter what outcome_the first_bet_results in, the_agent_prefers to_not add the second bet, then it_might seem_that she has no good reason_for preferring the combination_of_the first and the second bet_to the first bet only. (As_will become apparent, I think_this_justification_of Dominance is mistaken. But_it certainly has some intuitive appeal.)_We can state_the axiom more formally as follows: Assumption_2_(Dominance) If for all possible outcomes_X_from {Bi}, X X + Bj,_then:_{Bi}_{Bi}+ Bj = {Bi+1} As_we have seen, Assumption 1 clearly_holds when Bi = B100 (i.e., when Bi is_the last bet_on offer). Together with Dominance_the_assumption_entails: {B99} {B100} (1) In other words, from Assumption 1_and Dominance,_it follows that_even if the agent has accepted a package of 99_bets, she prefers not to increase the_size of the package by one additional bet. But recall that_we