Choose be­tween the fol­low­ing two op­tions:

1A. $24,000, with cer­tainty. 1B. 3334 chance of win­ning$27,000, and 134 chance of win­ning noth­ing.

Which seems more in­tu­itively ap­peal­ing? And which one would you choose in real life?

Now which of these two op­tions would you in­tu­itively pre­fer, and which would you choose in real life?

2A. 34% chance of win­ning $24,000, and 66% chance of win­ning noth­ing. 2B. 33% chance of win­ning$27,000, and 67% chance of win­ning noth­ing.

The Allais Para­dox—as Allais called it, though it’s not re­ally a para­dox—was one of the first con­flicts be­tween de­ci­sion the­ory and hu­man rea­son­ing to be ex­per­i­men­tally ex­posed, in 1953. I’ve mod­ified it slightly for ease of math, but the es­sen­tial prob­lem is the same: Most peo­ple pre­fer 1A > 1B, and most peo­ple pre­fer 2B > 2A. In­deed, in within-sub­ject com­par­i­sons, a ma­jor­ity of sub­jects ex­press both prefer­ences si­mul­ta­neously.

This is a prob­lem be­cause the 2s are equal to a one-third chance of play­ing the 1s. That is, 2A is equiv­a­lent to play­ing gam­ble 1A with 34% prob­a­bil­ity, and 2B is equiv­a­lent to play­ing 1B with 34% prob­a­bil­ity.

Among the ax­ioms used to prove that “con­sis­tent” de­ci­sion­mak­ers can be viewed as max­i­miz­ing ex­pected util­ity, is the Ax­iom of In­de­pen­dence: If X is strictly preferred to Y, then a prob­a­bil­ity P of X and (1 - P) of Z should be strictly preferred to P chance of Y and (1 - P) chance of Z.

All the ax­ioms are con­se­quences, as well as an­tecedents, of a con­sis­tent util­ity func­tion. So it must be pos­si­ble to prove that the ex­per­i­men­tal sub­jects above can’t have a con­sis­tent util­ity func­tion over out­comes. And in­deed, you can’t si­mul­ta­neously have:

• U($24,000) > 3334 U($27,000) + 134 U($0) • 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0) Th­ese two equa­tions are alge­braically in­con­sis­tent, re­gard­less of U, so the Allais Para­dox has noth­ing to do with the diminish­ing marginal util­ity of money. Mau­rice Allais ini­tially defended the re­vealed prefer­ences of the ex­per­i­men­tal sub­jects—he saw the ex­per­i­ment as ex­pos­ing a flaw in the con­ven­tional ideas of util­ity, rather than ex­pos­ing a flaw in hu­man psy­chol­ogy. This was 1953, af­ter all, and the heuris­tics-and-bi­ases move­ment wouldn’t re­ally get started for an­other two decades. Allais thought his ex­per­i­ment just showed that the Ax­iom of In­de­pen­dence clearly wasn’t a good idea in real life. (How naive, how fool­ish, how sim­plis­tic is Bayesian de­ci­sion the­ory...) Surely, the cer­tainty of hav­ing$24,000 should count for some­thing. You can feel the differ­ence, right? The solid re­as­surance?

(I’m start­ing to think of this as “naive philo­soph­i­cal re­al­ism”—sup­pos­ing that our in­tu­itions di­rectly ex­pose truths about which strate­gies are wiser, as though it was a di­rectly per­ceived fact that “1A is su­pe­rior to 1B”. In­tu­itions di­rectly ex­pose truths about hu­man cog­ni­tive func­tions, and only in­di­rectly ex­pose (af­ter we re­flect on the cog­ni­tive func­tions them­selves) truths about ra­tio­nal­ity.)

“But come now,” you say, “is it re­ally such a ter­rible thing, to de­part from Bayesian beauty?” Okay, so the sub­jects didn’t fol­low the neat lit­tle “in­de­pen­dence ax­iom” es­poused by the likes of von Neu­mann and Mor­gen­stern. Yet who says that things must be neat and tidy?

Why fret about el­e­gance, if it makes us take risks we don’t want? Ex­pected util­ity tells us that we ought to as­sign some kind of num­ber to an out­come, and then mul­ti­ply that value by the out­come’s prob­a­bil­ity, add them up, etc. Okay, but why do we have to do that? Why not make up more palat­able rules in­stead?

There is always a price for leav­ing the Bayesian Way. That’s what co­her­ence and unique­ness the­o­rems are all about.

In this case, if an agent prefers 1A > 1B, and 2B > 2A, it in­tro­duces a form of prefer­ence re­ver­sal—a dy­namic in­con­sis­tency in the agent’s plan­ning. You be­come a money pump.

Sup­pose that at 12:00PM I roll a hun­dred-sided die. If the die shows a num­ber greater than 34, the game ter­mi­nates. Other­wise, at 12:05PM I con­sult a switch with two set­tings, A and B. If the set­ting is A, I pay you $24,000. If the set­ting is B, I roll a 34-sided die and pay you$27,000 un­less the die shows “34”, in which case I pay you noth­ing.

Let’s say you pre­fer 1A over 1B, and 2B over 2A, and you would pay a sin­gle penny to in­dulge each prefer­ence. The switch starts in state A. Be­fore 12:00PM, you pay me a penny to throw the switch to B. The die comes up 12. After 12:00PM and be­fore 12:05PM, you pay me a penny to throw the switch to A.

I have taken your two cents on the sub­ject.

If you in­dulge your in­tu­itions, and dis­miss mere el­e­gance as a pointless ob­ses­sion with neat­ness, then don’t be sur­prised when your pen­nies get taken from you...

(I think the same failure to pro­por­tion­ally de­value the emo­tional im­pact of small prob­a­bil­ities is re­spon­si­ble for the lot­tery.)

Allais, M. (1953). Le com­porte­ment de l’homme ra­tionnel de­vant le risque: Cri­tique des pos­tu­lats et ax­iomes de l’école améri­caine. Econo­met­rica, 21, 503-46.

Kah­ne­man, D. and Tver­sky, A. (1979.) Prospect The­ory: An Anal­y­sis of De­ci­sion Un­der Risk. Econo­met­rica, 47, 263-92.

• For $24,000, you can have my two cents. ;) • Ac­tu­ally, that makes me think of an­other ex­pla­na­tion be­sides over­re­ac­tion to small prob­a­bil­ities: if a per­son takes 1B and loses, they know they would have won if they’d cho­sen differ­ently. If they take 2B and lose, they can tell them­selves (and oth­ers) they prob­a­bly would have lost any­way. • Ok that is ex­actly my line of think­ing and why i can’t un­der­stand the broader point of this ar­gu­ment. Yes I can see the statis­ti­cal similar­ity that makes it “the same”- but the situ­a­tion is to­tally differ­ent in that one offers “cer­tain win or risk” and the other is “risk vs risk” with a barely no­tice­able differ­ence be­tween them. So my de­ci­sion on both ques­tions goes like this 1a > 1b be­cause even if i was offered MUCH less, i’d still likely take that de­cid­ing that i’m not greedy and free money always feels good but giv­ing away free money (by try­ing to get a bit more) always feels fool­ish and greedy. 2b > 2a be­cause if the statis­tic played out over 100 times, the av­er­age per­son will think it was equal value be­tween them- un­less they logged the statis­tics to find the slight differ­ence. There­fore if it takes that much at­ten­tion to feel the differ­ence it’s easy to pre­tend they are the same risk but one is 11.12% more money- which is a lot eas­ier to no­tice with­out log­ging statis­tics. I don’t see how these de­ci­sions con­flict with each other. • I seem to agree with you, but I think how you ar­rived to 11.12% is wrong. Did you di­vide 3000/​27000? You can´t do that, since you won´t have 27000 un­less you get those 3000 dol­lar ex­tra. Shouldn´t you do 3000/​24000 = 12,5%? • Peo­ple don’t max­i­mize ex­pec­ta­tions. Ex­pec­ta­tion-max­i­miz­ing or­ganisms—if they ever ex­isted—died out long be­fore rigid spines made of ver­te­brae came on the scene. The rea­son is sim­ple, ex­pec­ta­tion max­i­miza­tion is not ro­bust (out­liers in the en­vi­ron­ment can cause large be­hav­ioral changes). This is as true now as it was be­fore evolu­tion in­vented in­tel­li­gence and in­tro­spec­tion. If peo­ple’s be­hav­ior doesn’t agree with the ax­iom sys­tem, the fault may not be with them, per­haps they know some­thing the math­e­mat­i­cian doesn’t. Fi­nally, the ‘money pump’ ar­gu­ment fails be­cause you are chang­ing the rules of the game. The origi­nal ques­tion was, I as­sume, ask­ing whether you would play the game once, whereas you would pre­sum­ably iter­ate the money pump un­til the pen­nies turn into mil­lions. The prob­lem, though, is if you asked peo­ple to make the origi­nal choices a mil­lion times, they would, cor­rectly, max­i­mize ex­pec­ta­tions. Be­cause when you are talk­ing about a mil­lion tries, ex­pec­ta­tions are the ap­pro­pri­ate frame­work. When you are talk­ing about 1 try, they are not. • I was re­ally con­fused about what point EY made that went over my head but i think I get it now. It to­tally changes the game to play it in­finite amount of times rather than 1 go to win or lose. I made my choices based on 1 game and not a hy­brid be­tween the two of them played mul­ti­ple times. If I play once, choos­ing 1a is just tak­ing money that’s already mine. If I play in­finite times, 1b earns money faster be­cause failing can be evened out. • If I knew the offer wouldn’t be re­peated, I might take 1A be­cause I’d re­ally rather not have to ex­plain to peo­ple how I lost$24,000 on a gam­ble.

• This was my thought ex­actly. If I was given the op­tion to keep the rest pri­vate if I lost, 1A would be a dis­tinctly prefer­able choice. If I had a 134 chance of hav­ing to ex­plain how I “lost” $24,000 vs an av­er­age loss of$2,200, I might well take choice 1B. (at a later time in my life, when I could af­ford to lose $2,200, and had sig­nifi­cant fi­nan­cial risk from be­ing per­ceived ask a risk-taker with money). • I think these kinds of ‘side chan­nel’ loss in­for­ma­tion are what make your in­tu­ition value 1A > 1B. In a way the im­plicit as­sump­tions in the offer are what cause the trou­ble. Naive sub­jects are naive only to pure math not to real life. • It took me 30 min­utes of sit­ting down and do­ing math be­fore I could fi­nally ac­cept that 1A+2B was an ir­ra­tional prefer­ence. I fi­nally re­al­ized that a lot of it came down to: with a 66% vs 67% chance of los­ing, I could take the riskier op­tion and not feel as bad, be­cause I could sweep it un­der the rug with “oh, I prob­a­bly would have lost any­ways.” Once I ran a sce­nario where I’d KNOW whether it was that 1% that I con­trol­led, or the 66% that I didn’t con­trol, that com­fort evap­o­rated. I learned a lot about my­self by work­ing through this ex­er­cise, so thank you very much :) • The idea is that$ amount equals your util­ity, while in re­al­ity the his­tory of how you got this amount also mat­ters (re­gret, emo­tions, etc.).

There’s no para­dox here—as your util­ity ex­pressed in $just doesn’t match util­ity of the sub­jects. As for money pump—you just have a win win situ­a­tion—you earn money, and the sub­jects earn good feel­ings. • Eliezer, I see from this ex­am­ple that the Ax­iom of In­de­pen­dence is re­lated to the no­tion of dy­namic con­sis­tency. But, the log­i­cal im­pli­ca­tion goes only one way. That is, the Ax­iom of In­de­pen­dence im­plies dy­namic con­sis­tency, but not vice versa. If we were to re­place the Ax­iom of In­de­pen­dence with some sort of Ax­iom of Dy­namic Con­sis­tency, we would no longer be able to de­rive ex­pected util­ity the­ory. (Similarly with dutch book/​money pump ar­gu­ments, there are many ways to avoid them be­sides be­ing an ex­pected util­ity max­i­mizer.) I’m afraid that the Ax­iom of In­de­pen­dence can­not re­ally be jus­tified as a ba­sic prin­ci­ple of ra­tio­nal­ity. Von Neu­mann and Mor­gen­stern prob­a­bly came up with it be­cause it was math­e­mat­i­cally nec­es­sary to de­rive Ex­pected Utility The­ory, then they and oth­ers tried to jus­tify it af­ter­ward be­cause Ex­pected Utility turned out to be such an el­e­gant and use­ful idea. Has any­one seen In­de­pen­dence pro­posed as a prin­ci­ple of ra­tio­nal­ity prior to the in­ven­tion of Ex­pected Utility The­ory? • I’m equally afraid ;). The Ax­iom of In­de­pen­dence is in­tu­itively ap­peal­ing to me, but I don’t posit it to be a ba­sic prin­ci­ple of ra­tio­nal­ity, be­cause that smells like a mind pro­jec­tion fal­lacy. I sus­pect you’re right, also, about dutch book/​money pump ar­gu­ments. I ten­ta­tively con­clude that a ra­tio­nal agent need not evince prefer­ences that can be rep­re­sented as an at­tempt to max­i­mize such a util­ity func­tion. That doesn’t mean Ex­pected Utility The­ory can’t be use­ful in many cir­cum­stances or for many agents, but this still seems like im­por­tant news, which mer­its more dis­cus­sion on Less Wrong. • which mer­its more dis­cus­sion on Less Wrong. Have you read these posts? • Yes, philoso­phers, and oth­ers, do of­ten too eas­ily ac­cept the ad­vice of strong in­tu­itions, for­get­ting that strong in­tu­itions of­ten con­flict in non-ob­vi­ous ways. • The large sums of money make a big differ­ence here. If it were for dol­lars, rather than thou­sands of dol­lars, I’d do what util­ity the­ory told me to do, and if that meant I missed out on$27 due to a very un­lucky chance then so be it. But I don’t think I could bring my­self to do the same for life-chang­ing amounts like those set out above; I would kick my­self so hard if I took the very slightly riskier bet and didn’t get the money.

• When I made the (pre­dictable, wrong) choice, I wasn’t us­ing prob­a­bil­ity at all. I was us­ing in­tu­itive rules of thumb like: “don’t gam­ble”, “treat small differ­ences in prob­a­bil­ity as unim­por­tant”, and “if you have to gam­ble against similar odds, go for the larger win”.

How do you find time to use au­then­tic prob­a­bil­ity math for all your chance-tak­ing de­ci­sions?

• That’s ex­actly how i felt too.

“Don’t gam­ble” is the key. 1a al­lowed me to in­dulge that even if i was boxed into be­ing in the game.

So in ques­tion 2 I want to fol­low “don’t gam­ble” but both are gam­bling. Ad­di­tion­ally, both gam­bles would feel the same risk to most hu­man who didn’t record statis­tics (other than sub­con­scious and nor­mal mem­ory effected ob­ser­va­tions) so could be cheaply rounded off to say they are the same. If they are “the same” but 1 pays more money...

Oh one more point “easy come easy go”. If you can lose 2 ei­ther way you won’t feel like you ever had any­thing. How­ever even be­fore you pick 1a and they phys­i­cally hand you the money, it’s already yours (by virtue of the abil­ity to choose 1a ) un­til you choose 1b and in­tro­duce the prob­a­bil­ity that you won’t be paid. I say already yours be­cause if you are guaran­teed the choice of 1a for­ever and un­con­di­tion­ally un­less un­til you choose 1b- that’s no less “hav­ing money” than when you “have money” but it’s in your pocket or in your wallet in the other room. It might not be your money any­more if you fling your wallet out the win­dow hop­ing it will boomerang back (1b) but it was un­til you in­tro­duced that gam­ble rather than just choos­ing to clutch the wallet (1a).

I feel like i must be miss­ing the point or some­thing be­cause they seems so ob­vi­ously right...

• Nain­odelac, if you pre­fer 1A to 1B and 2A to 2B, as you should if you need ex­actly $24,000 to save your life, that is a perfectly con­sis­tent prefer­ence pat­tern. • How trust­wor­thy is the ran­dom­izer? I’d pick B in both situ­a­tions if it seemed likely that the offer were trust­wor­thy. But in many cases, I’d give some chance of foul play, and it’s FAR eas­ier for an op­po­nent to weasel out of pay­ing if there’s an ap­par­ently-ran­dom part of the wa­ger. Some­one says “I’ll pay you$24k”, it’s rea­son­ably clear. They say “I’ll pay you $27k un­less these dice roll snake eyes” and I’m go­ing to ex­pect much worse odds than 3536 that I’ll ac­tu­ally get paid. So for 1A > 1B, this may be based on ex­pec­ta­tion of cheat­ing. For 2A < 2B, both choices are roughly equally amenable to cheat­ing, so you may as well max­i­mize your ex­pec­ta­tion. It seems likely that this kind of think­ing is un­con­scious in most peo­ple, and there­fore gets ap­plied in situ­a­tions where it’s not rele­vant (like where you CAN ac­tu­ally trust the prob­a­bil­ities). But it’s not au­to­mat­i­cally ir­ra­tional. • Your de­scrip­tion is not a money pump. A money pump oc­curs when you pre­fer A > B and B > C and C > A. Then some­one can trade you in a round robin tak­ing a lit­tle out for them­selves each cy­cle. I don’t feel like typ­ing in an illus­tra­tion, so see Robyn Dawes, Ra­tional Choice in an Uncer­tain World. There is a sig­nifi­cant differ­ence be­tween sin­gle and iter­a­tive situ­a­tions. For a sin­gle play I would pre­fer 1A to 1B and 2B to 2A. If it were re­peated, es­pe­cially open-end­edly, I would pre­fer 1B to 1A for its slightly greater ex­pected pay­off. This is analo­gous, I think, to the iter­ated ver­sus one-time pris­oner’s dilemma, see Ax­elrod’s Evolu­tion of Co­op­er­a­tion for an in­ter­est­ing dis­cus­sion of how they differ. • Risk and cost of cap­i­tal in­tro­duce very strange twists on ex­pected util­ity. As­sume that liv­ing has a greater ex­pected util­ity to me than any mon­e­tary value. If I need a$20,000 op­er­a­tion within the next 3 hours to live, I have no other fund­ing, and you make me offer 1, it is com­pletely ra­tio­nal and un­bi­ased to take op­tion 1A. It is the differ­ence be­tween a 100% of liv­ing and a 97% chance of liv­ing.

If I have $1,000,000,000 in the bank and com­mand of le­gal or oth­er­wise armed forces, I may just have you kil­led—for I would not tol­er­ate such frivolous philoso­phiz­ing. • So here’s why I pre­fer 1A and 2B af­ter do­ing the math, and what that math is. 1A = 24000 1B = 26206 (rounded) 2A = 8160 2B = 8910 Now, if you take (iB-iA)/​iA, which rep­re­sents the per­cent in­crease in the ex­pected value of iB over iA, you get the same num­ber, as you stated. (iB-iA)/​iA = .0919 (rounded) This num­ber’s re­cip­ro­cal rep­re­sents the num­ber of times greater the ex­pected value of iA is than the marginal ex­pected value of iB iA/​(iB-iA) = 10.88 (not rounded) Now, take this num­ber and di­vide it by the quan­tity p(iA wins)-p(iB wins). This rep­re­sents how much you have to value the first$24000 you re­ceive over the next $3000 to pick iA over iB. Keep in mind that 243 = 8, so if$1 = 1 utilon in all cases, you should pick iA only when this quo­tient is less than 8.

1A/​(1B-1A)/​[p(1A wins)-p(1B wins)] = 369.92
2A/​(2B-2A)/​[p(2A wins)-p(2B wins)] = 1088

I have li­a­bil­ities in ex­cess of my as­sets of around $15000. That first$15000 is very im­por­tant to me in a very quan­tized, thresh­oldy way, but it is not ab­solute. I can make the money some other way, but not need­ing to—hav­ing it available to me right now be­cause of this game—rep­re­sents more util­ity than a lin­ear map­ping of dol­lars to util­ity sug­gests, by a large fac­tor.

The next thresh­old like this in my life that I can think of is “enough money to buy a house in Los An­ge­les with­out tak­ing out a mort­gage,” of which $3000 is a neg­ligible por­tion. I’d say that the util­ity I as­sign the first$24000 be­cause of this lies be­tween 370 and 1080 times the util­ity I as­sign the next $3000. This is why I take 1A and 2B given that this en­tire thing is performed only once. Once my debts are paid, all bets (on 1A) are off. If we’re deal­ing with utilons rather than dol­lars, or I have re­peated op­por­tu­nity to play (which is nec­es­sary for you to “money pump” me) iB is the ob­vi­ous choice in both cases. • 26 Jun 2015 12:24 UTC 4 points The Allais “Para­dox” and Scam Vuln­er­a­bil­ity by Karl Ham­mer is a much needed up­date for any­one who reads the OP. • I won­der how the re­sults would change if the ex­per­i­ment changes so that the out­comes of 2B are, “You have a 33% chance of re­ceiv­ing$27k, a 66% chance of not get­ting any­thing, and a 1% chance of hav­ing some­one laugh in your face for not pick­ing 2A”

• When we speak of an in­her­ent util­ity of cer­tainty, what do we mean by cer­tainty? An ac­tual prob­a­bil­ity of unity, or, more rea­son­ably, some­thing which is merely very much cer­tain, like prob­a­bil­ity .999? If the lat­ter, then there should ex­ist a func­tion ex­press­ing the “util­ity bonus for cer­tainty” as a func­tion of how cer­tain we are. It’s not im­me­di­ately ob­vi­ous to me how such a func­tion should be­have. If prob­a­bil­ity 0.9999 is very much more prefer­able to prob­a­bil­ity 0.8999 than prob­a­bil­ity 0.5 is prefer­able to prob­a­bil­ity 0.4, then is 0.5 very much more prefer­able to 0.4 than 0.2 is to 0.1?

• My in­tu­itions match the stated naive in­tu­itions, but I re­ject your as­ser­tion that the pair of prefer­ences are in­con­sis­tent with Bayesian prob­a­bil­ity the­ory.

You re­ally un­der­es­ti­mate the util­ity of cer­tainty. “Nain­odelac and Tar­leton Nick”’s ex­am­ple in these com­ments about the op­er­a­tion is a perfect counter.

With a 33% vs. 34% chance, the im­pact on your life is about the same, so you just do the straight­for­ward prob­a­bil­ity calcu­la­tion for ex­pected value and take the max­i­mum.

But when offered 100% of some pos­i­tive out­come, vs. a prob­a­bil­ity of noth­ing, it seems perfectly ra­tio­nal to pre­fer the guaran­tee. Max­i­miz­ing ex­pected dol­lar win­nings is not nec­es­sar­ily the same as max­i­miz­ing util­ity. And you’re right, the is­sue isn’t de­creas­ing re­turns. But the is­sue is the cost of risk.

Your money pump doesn’t con­vince me ei­ther. I’d be happy to pay the two cents, both times, and not re­gret the cost at the end, just as I don’t re­gret pay­ing for in­surance even if I hap­pen not to get sick.

• A bird in the hand...

Cer­tainty is a form of util­ity, too.

• That goes hand in hand with his com­ments about com­plex­ity.

The straight­for­ward ex­pected util­ity anal­y­sis doesn’t in­clude the cost of the anal­y­sis into the anal­y­sis. Nor the in­creased cost to all sub­se­quent analy­ses for the un­cer­tainty.

We have limited com­pu­ta­tional power for ex­ec­u­tive func­tions. No doubt we have util­ity built into us to con­serve those limited re­sources. Most peo­ple hate un­cer­tainty and think­ing, and they hate it much more than we do. I doubt I’m the only one here who has no­ticed that.

• For me, the choice be­tween 1A and 1B would de­pend on how badly I needed the money, which is why I dis­agree with Eliezer when the says that “marginal util­ity of the money doesn’t count”.

For ex­am­ple, let’s say I needed $20,000 in or­der to keep a roof over my head, food on my plate, and to gen­er­ally sur­vive. In this case, my penalty for failure is quite high, and IMO it would be more ra­tio­nal for me to take 1A. Sure, I could win more money if I picked 1B, but I could also die in that case. Thus, my util­ity in case of 1B would be some­thing like 3334 U($27,000, al­ive) + 134 U($0, dead) and U($any­thing, dead) is a very nega­tive num­ber.

On the other hand, if I was a billion­aire who makes $20,000 per sec­ond just by ex­ist­ing, then I would ei­ther pick 1B, or re­fuse to play the game al­to­gether, be­cause my time could be bet­ter spent on other things. • Reread the post; that’s not the para­dox. The para­dox is that, if you need the 20k to sur­vive, then you should pre­fer 2A to 2B, be­cause the ex­tra 3k 33% of the time doesn’t out­weigh an ad­di­tional 1% chance of dy­ing. If some­one prefers A in both cases, and B in both cases, they can have a con­sis­tent util­ity func­tion. When some­one prefers A in one case, and B in an­other, then they can­not have a con­sis­tent util­ity func­tion. • Reread the post; that’s not the para­dox. Right, I didn’t mean to im­ply that it was. But Eliezer seemed to be say­ing that pick­ing 1A is ir­ra­tional in gen­eral, in ad­di­tion to the para­dox, which is the no­tion that I was dis­put­ing. It’s pos­si­ble that I mis­in­ter­preted him, how­ever. • He makes it clearer in com­ments. What Cale­do­nian is dis­cussing is the cer­tainty effect- es­sen­tially, hav­ing a term in your util­ity func­tion for not hav­ing to mul­ti­ply prob­a­bil­ities to get an ex­pected value. That’s differ­ent from risk aver­sion, which is just a state­ment that the util­ity func­tion is con­cave. • I know this was posted 4 years ago, but I had a thought. If I was offered a cer­tainty of$24,000 vs a 3334 chance of $27,000, my prefer­ence would de­pend on whether this was a once-off. If this was a once-off, my pri­mary con­cern would be se­cur­ing the money and be­ing able to put food on the table tonight. Op­tion 1 will put food on the table with 100% cer­tainty, while Op­tion 2 will not. If, how­ever, the op­tion was to be offered many times, I would op­ti­mise for great­est re­turn—Op­tion 2. If I miss out this month, I’ll just scrape for food un­til next month, when chance are I’ll get the money. I think I just an­swered my own ques­tion. If my goal can be reached with$24,000, then Op­tion 1 is the best one be­cause it reaches the goal in one guaran­teed fell swoop. How­ever, if my goal is to make lots of money, then Op­tion 2 is the way to go, be­cause it makes the most over time.

That make sense to any­one?

• It ab­solutely can make sense to pre­fer op­tion 1A over op­tion 1B (which I think is what you mean). What does not make sense is to pre­fer op­tion 1A over 1B, AND pre­fer 2B over 2A. It’s worth read­ing the two fol­lowup ar­ti­cles be­fore you get into this fur­ther: Zut Allais and Allaise Malaise. Wel­come to Less Wrong!

• Would I pay $24k to play a game where I had a 3334 prob­a­bil­ity of win­ning an ex­tra$3k? Let’s con­sult our good friend the Kelly Cri­te­rion.

We have a bet that pays 1/​8:1 with a 3334 prob­a­bil­ity of win­ning, so Kelly sug­gests stak­ing ~73.5% of my bankroll on the bet. This means I’d have to have an ex­tra ~$8.7k I’m will­ing to gam­ble with in or­der to choose 1b. If I’m risk-averse and pre­fer a frac­tional Kelly scheme, I’d need to start with ~$20k for a three-fourths Kelly bet and ~$41k for a one-half Kelly bet. Since I don’t have that kind of money ly­ing around, I choose 1a. In case 2, we come across the in­ter­est­ing ques­tion of how to an­a­lyze the costs and benefits of trad­ing 2a for 2b. In other words, if I had a voucher to play 2a, when would I be will­ing to trade it for a voucher to play 2b? Un­for­tu­nately, I’m not ex­pe­rienced with such analy­ses. Qual­i­ta­tively, it ap­pears that if money is tight then one would pre­fer 2a for the greater chance of win­ning, while some­one with a big­ger bankroll would want the bet­ter re­turns on 2b. So, there’s some amount of wealth where you be­gin to pre­fer 2b over 2a. I don’t find it ob­vi­ous that this should be the same as the bound­ary be­tween 1a and 1b. This is a prob­lem be­cause the 2s are equal to a one-third chance of play­ing the 1s. That is, 2A is equiv­a­lent to play­ing gam­ble 1A with 34% prob­a­bil­ity, and 2B is equiv­a­lent to play­ing 1B with 34% prob­a­bil­ity. Equiv­alence is tricky busi­ness. If we look at the win­nings dis­tri­bu­tion over sev­eral tri­als, the 1s look very differ­ent from the 2s and it’s not just a mat­ter of scale. The dis­tri­bu­tions cor­re­spond­ing to the 2s are much more diffuse. Surely, the cer­tainty of hav­ing$24,000 should count for some­thing. You can feel the differ­ence, right? The solid re­as­surance?

A cer­tain bet has zero volatility. Since much of the the­ory of gam­bling has to do with man­ag­ing volatility, I’d say cer­tainty counts for a lot.

• I don’t like many of the stan­dard ar­gu­ments against cap­i­tal pun­ish­ment. In par­tic­u­lar, I’m tired of the ar­gu­ment “if you just put an in­no­cent per­son in jail, they might be ex­on­er­ated later. If you ex­e­cute an in­no­cent per­son, and they are ex­on­er­ated later, it’s too late.”

Of course, I then point out that peo­ple can be ex­on­er­ated in the time be­tween be­ing con­victed and be­ing ex­e­cuted (which can be quite long some­times), and the re­sponse is gen­er­ally that in the life sen­tence there’s always some chance of be­ing freed due to ex­on­er­a­tion while in the cap­i­tal pun­ish­ment case, there’s a seg­ment of time where there’s no chance of be­ing freed.

My re­sponse is that a chance X of be­ing freed due to ex­on­er­a­tion when sen­tenced to life in prison is, for some Y, equiv­a­lent to hav­ing a chance Y of be­ing freed due to ex­on­er­a­tion be­fore your ex­e­cu­tion and zero chance of be­ing freed af­ter be­ing ex­e­cuted. Since there are val­ues of X that are con­sid­ered ac­cept­able, there are val­ues of Y that must be ac­cept­able too and there­fore this ar­gu­ment can­not be used as a ba­sis for an ab­solutist anti-cap­i­tal-pun­ish­ment stance.

I have yet to have any­one un­der­stand my re­sponse (the few times I’ve tried it, any­way). But it seems to me that I’ve stum­bled onto some­thing equiv­a­lent to the Allais prob­lem. Peo­ple don’t think of “chance X of be­ing freed” and “chance Y of be­ing freed be­fore ex­e­cu­tion and no chance of be­ing freed af­ter ex­e­cu­tion” as state­ments that can ever be equiv­a­lent, be­cause they re­ally don’t like the cer­tain failure in the last ex­am­ple, even though the two may be math­e­mat­i­cally equiv­a­lent.

• Since there are val­ues of X that are con­sid­ered ac­cept­able, there are val­ues of Y that must be ac­cept­able too and there­fore this ar­gu­ment can­not be used as a ba­sis for an ab­solutist anti-cap­i­tal-pun­ish­ment stance.

I agree.

Have you con­sid­ered that life in prison has more value than be­ing dead? Also, why com­pare cap­i­tal pun­ish­ment to life sen­tences? What if there were no life sen­tences? Of course you can still die in prison for what­ever that’s worth, but the chance is sig­nifi­cantly smaller.

• Have you con­sid­ered that life in prison has more value than be­ing dead?

I didn’t post that be­cause it was about cap­i­tal pun­ish­ment, I posted it be­cause I thought this par­tic­u­lar anti-cap­i­tal pun­ish­ment ar­gu­ment was rele­vant to the Allais prob­lem. I don’t see how life in prison be­ing more valuable than be­ing dead is rele­vant to the Allais prob­lem.

What if there were no life sen­tences? Of course you can still die in prison for what­ever that’s worth, but the chance is sig­nifi­cantly smaller.

In­so­far as that’s rele­vant, it just changes the val­ues of X and Y; the ab­solutist “we can’t do it be­cause an in­no­cent may be ex­on­er­ated only af­ter he is kil­led” po­si­tion still has the same flaw.

• Ok, good to know you weren’t try­ing to sneak in poli­tics. I agree it’s not rele­vant.

In­so­far as that’s rele­vant, it just changes the val­ues of X and Y; the ab­solutist “we can’t do it be­cause an in­no­cent may be ex­on­er­ated only af­ter he is kil­led” po­si­tion still has the same flaw.

Yes, if we’re strictly log­i­cal this is true.

• Please cor­rect me if any of my as­sump­tions are in­nacu­rate, and I apol­o­gize if this com­ment comes off as com­pletely tau­tolog­i­cal.

Ex­pected util­ity is ex­plic­ity defined as the statistic

$\sum_{{x}\in{X}}{p(x\$U(x)})

where X is the set of all pos­si­ble out­comes as­so­ci­ated with a par­tic­u­lar gam­ble, p(x) is the pro­por­tion of times that out­come x oc­curs within the gam­ble, and U(x) is the util­ity of out­come x, a func­tion that must be strictly in­creas­ing with re­spect to the mon­e­tary value of out­come x.

To re­duce am­bi­guity:

• 1A, 1B, 2A, and 2B are in­stances of gam­bles.

• For 1B, the pos­si­ble out­comes are $27000 and$0.

• For 1B, the ex­pected util­ity is p($27000) * U($27000) + p($0) * U($0) = 3334 * U($27000) + 134 * U($0).

If you choose 1A over 1B and 2B over 2A, what can we con­clude?

• that you are not us­ing the rule “max­i­mize ex­pected util­ity” to make your de­ci­sions. Thus you do not fit the defi­ni­tion, as given by the Ax­iom of In­de­pen­dence, of con­sis­tent de­ci­sion mak­ing.

If you choose 1A over 1B and 2B over 2A, what can we not con­clude?

• that your de­ci­sion rule changes ar­bi­trar­ily. You could, for ex­am­ple, always fol­low the rule, “Max­i­mize min­i­mum net util­ity. In the case of a tie, max­i­mize ex­pected util­ity.” In this case, you would choose 1A and 2B.

• that you would be wrong or stupid for us­ing a differ­ent de­ci­sion rule when you only get to play one time, than the rule you would use when you get to play 100 times.

• That all seems pretty un­con­tro­ver­sial.

• How do I alle­vi­ate feel­ing pleased at my­self for hav­ing read the state­ment of the para­dox—that peo­ple preferred 1A>1B but 2B>2A—and im­me­di­ately go­ing “WHAT?” and bog­gling at the screen and pul­ling con­fused faces for about thirty sec­onds, so flab­ber­gasted I had to reread that this choice pat­tern was com­mon?

(Per­son­ally I’m re­ally strongly bi­ased these days to­ward a bird in the hand and would have cho­sen 1A and 2A ev­ery time. I oc­ca­sion­ally do bits of sysad­min for dodgy dot-coms that friends are work­ing for. There are peo­ple who offer equity; I take an hourly fee. “No, no, that’s fine, I am but hum­ble roadie.” This may not always be the best life strat­egy, but it seems to work for me at pre­sent.)

• There are peo­ple who offer equity; I take an hourly fee.

Pe­nal­ise ex­pected value of equity be­cause prob­a­bil­ity is lower than I have been led to be­lieve—an in­cred­ibly use­ful heuris­tic.

How do I alle­vi­ate feel­ing pleased at my­self

In 33/​34ths of the wor­lds where you make choice A in 1, you are mer­cilessly teased and mocked by your in­fe­ri­ors, a la this, thirty sec­onds in, for not pick­ing B. As­sum­ing coun­ter­fac­tual out­comes are re­vealed.

• I’ll just have to cry my­self to sleep on a big bed made of $24,000! • Agree with De­nis. It seems rather ob­jec­tion­able to de­scrible such be­havi­our as ir­ra­tional. Hu­mans may well not trust the ex­per­i­menter to pre­sent the facts of the situ­a­tion to them ac­cu­rately. If the ex­per­i­menter’s dice are loaded, choos­ing 1A and 2B could well be perfectly ra­tio­nal. • I hate to dis­cuss this again, but... Is Michael Vas­sar’s var­i­ant Pas­cal’s Mug­ging (with the pigs), by­pass­ing as it does Robin’s ob­jec­tion, the re­duc­tio of ex­pected value? If you don’t care about pigs, sub­sti­tute some­thing else re­ally re­ally bad that doesn’t re­quire cre­at­ing 3^^^3 hu­mans. • Is Pas­cal’s Mug­ging the re­duc­tio ad ab­sur­dum of ex­pected value? No. I thought it might be! But Robin gave an ex­cel­lent rea­son of why we should gen­uinely pe­nal­ize the prob­a­bil­ity by a pro­por­tional amount, drag­ging the ex­pected value back down to neg­ligi­bil­ity. (This may be the first time that I have pre­sented an FAI ques­tion that stumped me, and it was solved by an economist. Which is ac­tu­ally a very en­courag­ing sign.) • This may be re­lated to the phe­nomenon of over­con­fi­dent prob­a­bil­ity es­ti­mates. I would not be sur­prised to find that peo­ple who claim a 97% cer­tainty have a real 90% prob­a­bil­ity of be­ing right. Maybe some­one who hears there’s 1 chance in 34 of win­ning noth­ing in­ter­prets that as com­ing from an over­con­fi­dent es­ti­ma­tor whereas the 34% and 33% prob­a­bil­ities are taken at face value. On the other hand, the over­con­fi­dence de­tec­tor seems to stop work­ing when faced with as­serted cer­tainty. • Upon reread­ing the thread and all of its com­ments, I sus­pect the per­son I origi­nally quoted meant some­thing along the lines of “prefer­ring 1A to 1B but 2B to 2A is ir­ra­tional”, which seems more defen­si­ble. There is noth­ing ir­ra­tional about prefer­ring 1A and 2B by them­selves, it’s choos­ing the first op­tion in the first sce­nario and the sec­ond in the sec­ond that’s dodgy. • You can define a util­ity func­tion on gam­bles that is not the ex­pected value of a util­ity func­tion on amounts of money, but then that func­tion is not ex­pected util­ity, and you’re out­side of nor­mal mod­els of risk aver­sion, and you’re vi­o­lat­ing ra­tio­nal­ity ax­ioms like the one Eliezer gave in the OP. Hav­ing a util­ity func­tion de­ter­mined by any­thing other than amounts of money is ir­ra­tional? WTF? • Choos­ing 1A and 2B is ir­ra­tional re­gard­less of your level of risk aver­sion. No, only if the util­ity of avoid­ing risk is worth less than the money at risk. Duh. • “Nain­odelac and Tar­leton Nick”, why are you us­ing my (re­versed) name? steven: not if you’re non­lin­early risk averse. As many have sug­gested, what if you take a large one-time util­ity hit for tak­ing any risk, but you’re not averse be­yond that? • tcp­kac: no one is as­sum­ing away risk aver­sion. Choos­ing 1A and 2B is ir­ra­tional re­gard­less of your level of risk aver­sion. • I con­fess, the money pump thing some­times strikes me as … well… con­trived. Yes, in the­ory, if one’s prefer­ences vi­o­late var­i­ous rules of ra­tio­nal­ity (acyclic­ity be­ing the eas­iest), one could con­ceiv­ably be money-pumped. But, uh, it never ac­tu­ally hap­pens in the real world. Our prefer­ences, once they vi­o­late ideal­ized ax­ioms, lead to messes in highly un­re­al­is­tic situ­a­tions. Big deal. • James D. Miller has a pro­posal for Lot­tery Tick­ets that Usu­al­lly Pay Off. Robin, were you think­ing of a cer­tain col­league of yours when you men­tioned ac­cept­ing in­tu­ition too read­ily? • Let’s say you pre­fer 1A over 1B, and 2B over 2A, and you would pay a sin­gle penny to in­dulge each prefer­ence. The switch starts in state A. Be­fore 12:00PM, you pay me a penny to throw the switch to B. I don’t un­der­stand why I would pay you a penny to throw the switch gefore 12:00? • Oh, here I come again, I’ve already com­mented in similar fash­ion el­se­where, and sev­eral peo­ple said the same here: noth­ing vs. non-noth­ing as a bi­nary switch may work bet­ter if the situ­a­tion is not re­peated to “add up to nor­mal­ity” but only played once. One can ar­gue that re­peats may seem as be­ing played once each time, but, be­ing crea­tures gifted with mem­ory, we can no­tice a catch of en­coun­ter­ing such situ­a­tions of­ten and mod­ify be­havi­our. • As­sum­ing this is a one off and not a re­peated iter­a­tion; I’d take 1A be­cause I’d be *re­ally* up­set if I lost out on$27k due to be­ing greedy and not tak­ing the sure $24k. That 134 is a small risk but to me it isn’t worth tak­ing—the$24k is too im­por­tant for me to lose out on.

I’d take 2B in­stead of 2A be­cause the differ­ence in odds is ba­si­cally neg­ligible so why not go for the ex­tra $3k? I have ~2/​3rds chance to walk away with noth­ing ei­ther way. I don’t re­ally see the para­dox there. The point is to win, yes? If I play game 1 and pick B and hit that 134 chance of loss and walk away with noth­ing I’ll be feel­ing pretty stupid. Let’s say you pre­fer 1A over 1B, and 2B over 2A, and you would pay a sin­gle penny to in­dulge each prefer­ence. The switch starts in state A. Be­fore 12:00PM, you pay me a penny to throw the switch to B. The die comes up 12. After 12:00PM and be­fore 12:05PM, you pay me a penny to throw the switch to A. But why would I pay to switch it back to A when I’ve already won given the con­di­tions of B? And as Doug_S. men­tions, you can take my pen­nies if I’m get­ting paid out tens of thou­sands of dol­lars. I do see the point in it be­ing difficult to pro­gram this type of de­ci­sion mak­ing, though. • For­give me if I’m mi­s­un­der­stand­ing some­thing, but the way I see it, if I choose 1A, it means that I am will­ing to forgo (i.e. pay) 3000$ for an ad­di­tional 134 ~ 3% chance of get­ting money. Then if I choose 2B, if means I am un­will­ing to forgo an ad­di­tional 3000$in ex­change for an ad­di­tional 1% chance of get­ting money. So what I learn from this is that the value I as­sign an ex­tra per­centage chance of get­ting money is some­where be­tween 1000$ and 3000$. • This is an old post, but I guess one re­s­olu­tion is that: U($24,000) > 3334 U($27,000) + 134 U($0 & Re­gret that I didn’t take the $24000) Which is con­sis­tent with: 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0) It’s an in­ter­est­ing psy­cholog­i­cal fact that the re­gret is trig­gered in one case, but not the other. • While Elez­ier’s ar­gu­ment is still cor­rect (that you should mul­ti­ply to make de­ci­sions based on prob­a­bil­is­tic knowl­edge), I see a perfectly ra­tio­nal and util­i­tar­ian ex­pla­na­tion for choos­ing 1A and 2B in the stated prob­lem. The clue lies in Colin Reid’s com­ment: “peo­ple do not as­cribe a low pos­i­tive util­ity to win­ning noth­ing or close to noth­ing—they ac­tively fear it”. This fear is ex­plained by Kin­greaper: “in sce­nario 1B if you lose you know it’s your fault you got noth­ing”. That makes the two cases, stated as they are, differ­ent. In game 1 the util­ity of U1($0) has nega­tive value: a sense of guilt (or shame) over hav­ing made the bad choice, which doesn’t seem pos­si­ble in game 2 (be­cause game 2 is stated in terms of ab­stract prob­a­bil­ities, see be­low).

This makes the in­equa­tions com­pat­i­ble:

U($24,000) > 33/​34 U($27,000) + 1/​34 U1($0)  e.g. 24 > 3334 · 27 + 134 · −1000 0.34 U($24,000) + 0.66 U2($0) < 0.33 U($27,000) + 0.67 U2($0)  e.g. 0.34 · 24 + 0.66 · 0 < 0.33 · 27 + 0.67 · 0 Note that stat­ing the game with the “switch” rule turns game 2 into one (let’s call it 3) in which the guilt/​shame reap­pears, mak­ing U3=U1 -- so a ra­tio­nal player with the de­scribed nega­tive U1 would choose A in game 3 and there would be no money pump. This solu­tion to the para­dox is less valid if it is made clear that the sub­ject will be al­lowed to play the game many times. Another in­ter­est­ing way to re­move this as a pos­si­ble solu­tion would be to restate case 2 in more con­crete terms, to make it clear that you won’t get away not know­ing that “it was your fault” if you loose: 4A. If a 100-face dice falls on <=34, win$24,000, oth­er­wise win noth­ing.

4B. If a 100-face dice falls on <=33, win $27,000, oth­er­wise win noth­ing.  Just to pre­vent the sub­ject be­ing pat­tern-match­ing and not think­ing, we should add the phrase “note that if the dice falls on a 34 and you’ve cho­sen A, you win 24k, but if you’ve cho­sen B, you get noth­ing”. I be­lieve game 4 is pretty equiv­a­lent to game 3 (the one with the switch). I’ve checked Allais’ doc­u­ment and it suffers the same flaw: it’s not an ac­tual ex­per­i­ment in which peo­ple are asked to choose A or B and ac­tu­ally al­lowed to play the game, but a ques­tion­naire ask­ing sub­jects what they would choose. This is not the same, among other rea­sons be­cause it doesn’t force the ex­per­i­menter or sub­ject to de­tail the me­chan­ics of the game (and hence it is not stated whether the sub­ject will be given that sense of shame or even al­lowed to “chase the rab­bit”). It would be in­ter­est­ing to know the re­sult of an ac­tual ex­per­i­ment with this de­sign, pos­si­bly with smaller figures to re­duce the non-lin­ear­ity of the util­ity func­tions—since that’s not what’s be­ing dis­cussed here --, and with sub­jects filtered against in­nu­mer­acy (since those are out of hope any­way). • That makes the two cases, stated as they are, differ­ent. In game 1 the util­ity of U1($0) has nega­tive value: a sense of guilt (or shame) over hav­ing made the bad choice, which doesn’t seem pos­si­ble in game 2 (be­cause game 2 is stated in terms of ab­stract prob­a­bil­ities, see be­low).

If you could choose whether or not to have this guilt, would you choose to have it? Does it make you bet­ter off?

• Ummm, no. The money pump fails be­cause of the REASON for the prefer­ence differ­ence.

The rea­son is, as some have already stated, that in sce­nario 1B if you lose you know it’s your fault you got noth­ing. In sce­nario 2B if you lose, you can ra­tio­nal­ise it eas­ily as “Would have lost any­way”

In your money pump sce­nario, we have a 1/​3rd chance of play­ing 1. If we get to play 1, we know we’re play­ing 1. So your money pump fails, be­cause a stan­dard player would pre­fer that the switch be on A at all times.

• I would love to know if the re­sults are differ­ent if you re­peat­edly ex­pose peo­ple to the situ­a­tion rather than com­mu­ni­cate it in a for­mal way. They are likely to ob­serve the out­comes of their strat­egy and adapt. Per­haps what is be­ing mea­sured is sim­ply the nu­mer­acy of the sub­jects and not their prac­ti­cal in­abil­ity to de­ter­mine op­ti­mal strate­gies.

The lot­tery is an­other in­ter­est­ing ex­am­ple, what is be­ing bought is the prob­a­bil­ity of a big win, not a statis­ti­cally op­ti­mal in­vest­ment. Play­ing the lot­tery gen­uinely in­creases the chance of you sud­denly gain­ing a life chang­ing amount of money. This is a perfectly ra­tio­nal choice.

• This is a perfectly ra­tio­nal choice.

What about the Allais para­dox? Imag­ine some­one who is happy to play the lot­tery but would re­fuse to play an al­ter­na­tive ver­sion where the ticket merely con­fers a slight in­crease on a sig­nifi­cant pre-ex­ist­ing prob­a­bil­ity of win­ning ‘life chang­ing money’. (As I un­der­stand it, most/​all lot­tery play­ers would in fact re­fuse the ‘al­ter­na­tive’ gam­ble.) Do you want to say that such a per­son is ‘perfectly ra­tio­nal’? Would you call them perfectly ra­tio­nal if they ac­cepted both gam­bles (de­spite both of them hav­ing nega­tive EV)?

To be fair, It is pos­si­ble to tell a con­sis­tent story about a per­son for whom ei­ther gam­ble would be ra­tio­nal: Per­haps the Earth is go­ing to be de­stroyed soon and the cost of en­try into the new self-sus­tain­ing Mars colony equals the lot­tery jack­pot.

But need­less to say, most peo­ple aren’t in situ­a­tions re­motely re­sem­bling this one.

• Imag­ine some­one who is happy to play the lot­tery but would re­fuse to play an al­ter­na­tive ver­sion where the ticket merely con­fers a slight in­crease on a sig­nifi­cant pre-ex­ist­ing prob­a­bil­ity of win­ning ‘life chang­ing money’. (As I un­der­stand it, most/​all lot­tery play­ers would in fact re­fuse the ‘al­ter­na­tive’ gam­ble.)

This is likely be­cause play­ing the lot­tery gives you “hope” of a life-chang­ing event. It means that you KNOW there is a pos­si­ble life-chang­ing event available.

If you already have that knowl­edge, then pay­ing for the lot­tery be­comes just about the money; which isn’t worth­while. If you don’t, pay­ing for the lot­tery is buy­ing that knowl­edge, and the knowl­edge has value to you.

• Thank you for your com­ments.

I think the Allais para­dox is fas­ci­nat­ing, how­ever, al­though it is very re­veal­ing about our likely mo­tives for play­ing the lot­tery it doesn’t change the po­ten­tial ra­tio­nal­ity of ac­tual play­ing it. I.e. that money and value don’t nec­es­sar­ily have a lin­ear re­la­tion­ship, and so op­ti­mis­ing for EV is not ra­tio­nal.

Although, I feel that the likely an­swer is that the brain is op­ti­mised for rapid re­sponses to sur­vival prob­lems and these solu­tions may well be an op­ti­mal re­sponse given con­straints on both pro­cess­ing and ex­pected out­come.

Another per­spec­tive is that in gen­eral speci­fi­ca­tions are not ac­cu­rate but in­stead a com­mu­ni­ca­tion of ex­pe­rience. If the prob­lem speci­fi­ca­tion is viewed in­stead as a mea­sure­ment of a sys­tem where the plac­ing of bets is an in­put and the out­put is not ran­dom but the out­come of an un­known set of in­ter­ac­tions. Sys­tems en­coun­tered in the past will form a prob­a­bil­ity dis­tri­bu­tion over their be­havi­our, the fre­quency of ob­served con­se­quences then act as a mea­sure­ment of the like­li­hood that the sys­tem in ques­tion is equiv­a­lent to one of these types. This would ex­plain the feel­ing of switch­ing be­tween the two ex­am­ples (they con­sti­tute the likely out­comes of two types of sys­tem) and thus rep­re­sent situ­a­tions where dis­tinct be­havi­ours were ap­pro­pri­ate.

I.e. as one starts to un­der­stand an ex­ist­ing sys­tem one gets diminish­ing re­turns for op­ti­mis­ing in­ter­ac­tion with it (a good ex­am­ple is AI pro­gram­ming it­self), how­ever sys­tems may be un­known to the user. Th­ese un­known sys­tems may demon­strate rare, but highly benefi­cial or un­ex­pected events, like notic­ing an anomaly in a physics ex­per­i­ment. In this case it is ra­tio­nal to play/​in­ter­act as do­ing so pro­vides more in­for­ma­tion which may be used to iden­tify the sys­tem and thus lead to un­der­stand­ing and thus an ex­pected benefit in the fu­ture.

• I think the Allais para­dox is fas­ci­nat­ing, how­ever, al­though it is very re­veal­ing about our likely mo­tives for play­ing the lot­tery it doesn’t change the po­ten­tial ra­tio­nal­ity of ac­tual play­ing it. I.e. that money and value don’t nec­es­sar­ily have a lin­ear re­la­tion­ship, and so op­ti­mis­ing for EV is not ra­tio­nal.

Of course, that just means you max­imise ex­pected util­ity rather than ex­pected money. (I was al­most go­ing to write “ex­pected value” in­stead of “ex­pected util­ity” as you used the word “value”, but ob­vi­ously that would be con­fus­ing in this con­text...)

• Yes, ab­solutely, apolo­gies for my un­fa­mil­iar­ity with the terms.

The point I’m try­ing to make is that lot­tery play­ing op­ti­mises util­ity (as­sum­ing util­ity means what is con­sid­ered valuable to the per­son). Say­ing that lot­tery play­ing is ir­ra­tional is mak­ing a state­ment about what is valuable more that it does about what is rea­son­able.

• I would have cho­sen 1A and 2B, for the fol­low­ing rea­sons: Any sum of the or­der of $20,000 would rev­olu­tionize my per­sonal cir­cum­stances. The likely pay­off is enor­mous. There­fore, I’d pick 1A be­cause I’d get such a sum guaran­teed, rather than run the 3% risk (1B) of get­ting noth­ing at all. Whereas choice 2 is a gam­ble ei­ther way, so I am led to treat both op­tions as qual­i­ta­tively the same. But that’s a mis­take: if the value of get­ting ei­ther nonzero pay­off at all is so great, then I should have fa­vored the 34% chance of win­ning some­thing over the 33% chance, just as I fa­vored the 100% chance over the ~97% chance in choice 1. In­ter­est­ing. • Not sure if any­one pointed this out, but in a situ­a­tion where you don’t trust the or­ga­nizer, the proper ex­e­cu­tion of 1A is a lot eas­ier to ver­ify than the proper ex­e­cu­tion of 1B, 2A and 2B. 1A min­i­mizes your risk of be­ing fooled by some hid­den clev­er­ness or vi­o­la­tion of the con­tract. In 1B, 2A and 2B, if you lose, you have to ver­ify that the ran­dom num­ber gen­er­a­tor is truly ran­dom. This can be ex­tremely costly. In op­tion 1A, ver­ifi­ca­tion con­sists of check­ing your bank ac­count and see­ing that you gained$24,000. Straight­for­ward and sim­ple. Hardly any risk of be­ing de­ceived.

• This dis­cus­sion re­minded me of the Tor­ture vs. Dust Specks dis­cus­sion; i.e. in that dis­cus­sion, many com­ments, per­haps a ma­jor­ity, amounted to “I feel like choos­ing Dust Specks, so that’s what I choose, and I don’t care about any­thing else.” In the same way, there is a perfectly con­sis­tent util­ity func­tion that can pre­fer A1 to B1 and B2 to B1, namely one that sets util­ity on “feel­ing that I have made the right choice”, and which does not set util­ity on money or any­thing else. Both in this case and in the case of the Tor­ture and Dust Specks, many com­ments in­di­cate a util­ity func­tion which places value on the feel­ing of hav­ing made a right choice, with­out re­gard for any­thing else, es­pe­cially for whether or not the choice was ac­tu­ally right, or for the con­se­quences of the choice.

• “Nain­odelac and Tar­leton Nick”: This is not about risk aver­sion. I agree that if it is vi­tal to gain at least $20,000, 1A is a su­pe­rior choice to 1B. How­ever, in that case, 2A is also a su­pe­rior choice to 2B. The er­ror is not in prefer­ring 1A, but in si­mul­ta­neously prefer­ring 1A and 2B. • Is Pas­cal’s Mug­ging the re­duc­tio ad ab­sur­dum of ex­pected value? • Do you re­ally want to say that it can be ra­tio­nal to ac­cept a 13 chance of par­ti­ci­pat­ing in a lot­tery, already know­ing that if you got to par­ti­ci­pate you would change your mind? Risk aver­sion is (or at least, can be) a mat­ter of taste, this is just a mat­ter of not be­ing stupid. • Cale­do­nian, Nick T: “Risk aver­sion” in the stan­dard mean­ing is when an agent max­i­mizes the ex­pec­ta­tion value of util­ity, and util­ity is a func­tion of money that in­creases slower than lin­early. When an agent doesn’t max­i­mize ex­pected util­ity at all, that’s some­thing differ­ent. • My ex­pe­rience of watch­ing game shows such as ‘Deal or No Deal’ sug­gests that peo­ple do not as­cribe a low pos­i­tive util­ity to win­ning noth­ing or close to noth­ing—they ac­tively fear it, as if it would make their life worse than be­fore they were se­lected to ap­pear on the show. It seems this fear is in some sense in­versely pro­por­tional to the ‘so­cially ex­pected’ prob­a­bil­ity of the bad event—so if the player is aware that very few play­ers win less than £1 on the show, they start get­ting very un­com­fortable if there is a high chance of this hap­pen­ing to them, be­cause win­ning less than £1 is some­how em­bar­rass­ing, and win­ning 1p is some­how sig­nifi­cantly worse than win­ning say 50p. In con­trast, on game shows where there’s a ‘dou­ble or noth­ing’ op­tion at the end, it is so­cially ac­cepted that there’s a high chance of win­ning noth­ing, so play­ers seem to be much more san­guine about the gam­ble. I think the psy­chol­ogy of ‘face’ has a lot to an­swer for when it comes to such de­ci­sions. • I am in­tu­itively cer­tain that I’m be­ing money-pumped all the time. And I’m very, very cer­tain that trans­ac­tion costs of many forms money-pump peo­ple left and right. • It’s ra­tio­nal to take the cer­tain out­come if gam­bling causes psy­cholog­i­cal stress. Notwith­stand­ing that stress is in­trin­si­cally un­pleas­ant, it in­creases your risk of pep­tic ul­cers and stroke, which could eas­ily can­cel out the ex­pected gain. • But such psy­cholog­i­cal stress arises from your per­cep­tion of re­al­ity. If it is caused by an er­ro­neous per­cep­tion of re­al­ity, then the ra­tio­nal thing to do is cor­rect your per­cep­tion, not take the er­ror for granted. If you are cer­tain that you made the right de­ci­sion, then you shouldn’t feel stressed when you “lose”. • Since I know my­self, I know what I will do af­ter mid­night (pay to switch it to A), and so I re­sign my­self to do­ing it im­me­di­ately (i.e., leav­ing the switch at A) so as to save ei­ther one cent or two, de­pend­ing on what hap­pens. I will do this even if I share Don’s in­tu­ition about cer­tainty. Why pay be­fore mid­night to switch it to B if I know that af­ter mid­night I will pay to switch it back to A? *[if the first die comes up 1 to 34] • I think defenses of the sub­ject’s choices by re­course to non­mon­e­tary val­ues is miss­ing the point. Any­thing can be ra­tio­nal with a suffi­ciently weird util­ity func­tion. The ques­tion is, if sub­jects un­der­stood the de­ci­sion the­ory be­hind the prob­lem, would they still make the same choice? After see­ing a valid ar­gu­ment that your prefer­ences make you a money pump, you cer­tainly could per­sist in your origi­nal judg­ment, by in­sist­ing that your feel­ings make your first judg­ment the right one. But se­ri­ously?---why? • 16 Mar 2015 17:22 UTC 0 points I don´t re­ally see how me chos­ing 1A > 1b and 2b >2A is a flaw of mine. First of all, my util­ity func­tion, which i have in­her­ited from mil­lions of years of evolu­tion, tells me to SOMETIMES take risks IF I CAN AFFORD IT, es­pe­cially when the in­creas­ing stake out­weighs the in­creas­ing risk. This is how I see it: If it was my life at stake, I would of course try to raise the odds. But this is ex­tra money. I don´t even starve if i don´t get the money. If I am not cer­tain I can get the money in case 2, I think that low­er­ing my win-chance with 1100 is worth to raise the stake with 3000 dol­lars, which is 3000/​24000 = 18 of the origi­nal stake. When I lower my odds with 1 % I raise the stake with 12,5 %. Since the out­come is ran­dom any­how, AND not in my fa­vor, and the risk in­crease is only 1100, I take my chances. • My re­s­olu­tion to this, with­out chang­ing my in­tu­itions to pick things that I cur­rently per­ceive as ‘sim­ply wrong’, would be that I value cer­tainty. A 910 chance of win­ning x dol­lars is worth much less to me than a 1010 chance of win­ning 9x/​10 dol­lars. How­ever, a 210 chance of win­ning x dol­lars is worth only barely less than a 410 chance of win­ning x/​2 dol­lars, be­cause as far as I can tell the added util­ity of the lack of wor­ry­ing in­creases mas­sively as the more cer­tain op­tion ap­proaches 100%. Now, this be­comes less pow­er­ful the closer the odds, are, but slower than the dol­lar differ­ence be­tween the two change. So a 99% chance of x is barely effected by this com­pared to a 100% chance of .99x, but still by a greater value than .01x, and the more likely op­tion still dom­i­nates. I might take a 99% chance of x over a 100% chance of .9x, how­ever, and I would definitely pre­fer a 99% chance of x over a 100% chance of 0.8x. EDIT: Upon fur­ther con­sid­er­a­tion, this is wrong. If pre­sented with the ac­tual choice, I would still pre­fer 1A to 1B, but to main­tain con­sis­tency I will now choose 2A > 2B. • 1A.$24,000, with cer­tainty.

1B. 3334 chance of win­ning $27,000, and 134 chance of win­ning noth­ing. 2A. 34% chance of win­ning$24,000, and 66% chance of win­ning noth­ing.

2B. 33% chance of win­ning $27,000, and 67% chance of win­ning noth­ing. I would choose 1A over 1B, and 2B over 2A, de­spite the 9.2% bet­ter ex­pected pay­out of 1B and the small in­creased risk in 2B. If the op­tion was re­peat­able sev­eral times, I’d choose 1B over 1A as well (but switch back to 1A if I lost too many times). This does not make me sus­cep­ti­ble to a money pump or a Dutch book (you’re wel­come to try, but note that I don’t ac­cept trades with nega­tive ex­pected util­ity). I sim­ply think that my util­ity func­tion at this time is such that Utility($24,000)>Utility(97% chance $27,000 + 3% chance$0), yet also Utility(34% chance $24,000 + 66% chance$0)<Utility(33% chance $27,000 + 67% chance$0)

I ac­knowl­edge that in one case, I trade ex­pected pay­out for cer­tainty, and in the other, I trade in­creased risk (not cer­tainty) for ex­pected pay­out. I’m not sure I see any­thing wrong with this, un­less you’re offended that I am will­ing to pay for cer­tainty. Cer­tainty is valuable in this world of over­con­fi­dent peo­ple, ac­ci­dents, and cheaters.

• This does not make me sus­cep­ti­ble to a money pump or a Dutch book (you’re wel­come to try, but note that I don’t ac­cept trades with nega­tive ex­pected util­ity). I sim­ply think that my util­ity func­tion at this time is such that Utility($24,000)>Utility(97% chance$27,000 + 3% chance $0), yet also Utility(34% chance$24,000 + 66% chance $0)<Utility(33% chance$27,000 + 67% chance $0) This… means you’re vuln­er­a­ble to the Dutch Book de­scribed in the post. Why do you think oth­er­wise? I’m not sure I see any­thing wrong with this, un­less you’re offended that I am will­ing to pay for cer­tainty. Ba­si­cally, this. The point of util­ity is that it’s lin­ear in prob­a­bil­ity, which dis­al­lows a pre­mium for cer­tainty. If I know your util­ity for$27,000, and your util­ity for $24,000, and$0, then I can calcu­late your prefer­ences over any gam­ble con­tain­ing those three out­comes. If your de­ci­sion pro­ce­dure is not equiv­a­lent to a util­ity func­tion, then there are cases where you can be made worse off even though it looks to you like you’re be­ing made bet­ter off.

Cer­tainty is valuable in this world of over­con­fi­dent peo­ple, ac­ci­dents, and cheaters.

Isn’t cer­tainty im­pos­si­ble in a world of over­con­fi­dent peo­ple, ac­ci­dents, and cheaters?

• This… means you’re vuln­er­a­ble to the Dutch Book de­scribed in the post. Why do you think oth­er­wise?

I’m re­ally not. You mean, “This means that ac­cord­ing to my the­ory you’re vuln­er­a­ble to the Dutch Book de­scribed in the post” Like I said though, I’m not ac­cept­ing trades with nega­tive util­ity, and be­ing money pumped and Dutch Booked both have nega­tive util­ity.

As for the “money pump” de­scribed in the post, I gain $23,999.98 if it hap­pens as de­scribed. Also, there would have been no need to pay the first penny as the state of the switch was not rele­vant at that time. Also the game was switched from “34% for 24,000 and 33% for 27,000″ to “34% chance to play game 1, at which time you may choose” Ba­si­cally, this. The point of util­ity is that it’s lin­ear in prob­a­bil­ity, which dis­al­lows a pre­mium for cer­tainty. If I know your util­ity for$27,000, and your util­ity for $24,000, and$0, then I can calcu­late your prefer­ences over any gam­ble con­tain­ing those three out­comes. If your de­ci­sion pro­ce­dure is not equiv­a­lent to a util­ity func­tion, then there are cases where you can be made worse off even though it looks to you like you’re be­ing made bet­ter off.

I agree that if you take the prob­a­bil­ity out of my util­ity func­tion, then I am di­rectly al­ter­ing my prefer­ence in the ex­act same situ­a­tion. Even so, there is in re­al­ity at least one differ­ence: if some­one is cheat­ing or made a mis­calcu­la­tion, op­tion 1A is cheat-proof and er­ror-proof but none of the other op­tions are. And I’ve definitely at­tached util­ity to that. This as­pect would dis­ap­pear if prob­a­bil­ities were re­moved from my util­ity func­tion.

• Like I said though, I’m not ac­cept­ing trades with nega­tive util­ity, and be­ing money pumped and Dutch Booked both have nega­tive util­ity.

You’ve ex­pressed that 1A>1B, and 2B>2A. The first deal is “In­stead of 2A, I’ll give you 2B for a penny.” By your stated prefer­ence, you agree. The sec­ond deal is “In­stead of 1B, I’ll give you 1A.” By your stated prefer­ence, you agree. You are now two pen­nies poorer. So ei­ther you do not ac­tu­ally hold those stated prefer­ences, or you are vuln­er­a­ble to Dutch book­ing. (What does it mean to ac­tu­ally pre­fer one gam­ble to an­other? That you’re will­ing to pay to trade gam­bles. Sup­pose you hate sel­l­ing things; then your prefer­ences de­pend on the or­der you re­ceived things, which makes you vuln­er­a­ble to the or­der in which other peo­ple pre­sent you op­tions!)

Also the game was switched from “34% for 24,000 and 33% for 27,000” to “34% chance to play game 1, at which time you may choose”

What is the differ­ence be­tween those two games? The out­come prob­a­bil­ities are the same (mul­ti­ply them out and check!). Or are you will­ing to pay hun­dreds of dol­lars (in ex­pec­ta­tion) to have him roll two dice in­stead of one?

Even so, there is in re­al­ity at least one differ­ence: if some­one is cheat­ing or made a mis­calcu­la­tion, op­tion 1A is cheat-proof and er­ror-proof but none of the other op­tions are.

But, don’t you have some nu­mer­i­cal prefer­ence for this? If it were a cer­tain 24,000 against a 33/​34ths chance of 27 mil­lion, I hope you’d pick the lat­ter, even if there’s some chance of the die be­ing loaded in the sec­ond op­tion. What this sug­gests, then, is that you need to ad­just your prob­a­bil­ities- but if the prob­a­bil­ities are pre­sented to you as your es­ti­mate af­ter cheat­ing is taken into ac­count, then it doesn’t make sense to dou­ble-count the risk of cheat­ing!

(One use­ful heuris­tic that peo­ple of­ten have when eval­u­at­ing gam­bles is imag­in­ing the per­son on the other side of the gam­ble. If some­thing looks re­ally good on your end and re­ally bad on their end, then this is sus­pi­cious- why would they offer you some­thing so bad for them? Keep in mind, though, that gam­bles are done both against other peo­ple and against the en­vi­ron­ment. If there’s gold sit­ting in the ground un­der­neath you, and you have a 97% chance of suc­cess­fully ex­tract­ing it and be­com­ing a mil­lion­aire, you shouldn’t say “hmm, what’s in it for the ground? Why would it offer me this deal?”)

• You’ve ex­pressed that 1A>1B, and 2B>2A. The first deal is “In­stead of 2A, I’ll give you 2B for a penny.” By your stated prefer­ence, you agree. The sec­ond deal is “In­stead of 1B, I’ll give you 1A.” By your stated prefer­ence, you agree.

Note that it be­comes a differ­ent prob­lem this way than my stated prefer­ences (and note again that my stated choices (not prefer­ences) were con­text-de­pen­dent) -- there is the ad­di­tional in­for­ma­tion that the deal­maker had a good chance to cheat and didn’t take it. This in­for­ma­tion will re­duce my di­su­til­ity calcu­la­tion for the un­cer­tainty in the offer, as it in­creases my odds of win­ning 1B from [33/​34 - good chance of cheat­ing] to [33/​34 - small chance of cheat­ing]

You are now two pen­nies poorer.

Or 23,999.98 dol­lars richer.

So ei­ther you do not ac­tu­ally hold those stated prefer­ences, or you are vuln­er­a­ble to Dutch booking

If I did hold those prefer­ences, I would not be vuln­er­a­ble to Dutch book­ing, nor money pump­ing. Money pump­ing is in­finite, whereas by giv­ing me two pairs of differ­ent choices you can make me choose twice (and it’s not a prefer­ence re­ver­sal, though it would be ex­actly a prefer­ence re­ver­sal if you mul­ti­ply the first choice’s odds by 0.34 and pre­tend that changes noth­ing).

For me to be vuln­er­a­ble to Dutch book­ing, you’d have to some­how get money out of me as well. But how? I can’t buy game 1 for less than 24,000 minus the cost of var­i­ous wit­nesses if I in­tend to choose 1A, and you can’t sell game 1 for less than 26,200. You’d have an even worse time con­vinc­ing me to buy game 2. You can’t con­vince me to bid against ei­ther of the the­o­ret­i­cally su­pe­rior choices 1B and 2B. If you change my situ­a­tion I might change my choice, as I already stated sev­eral con­di­tions that would cause me to aban­don 1A.

What is the differ­ence be­tween those two games?

Op­tion 1A has a 0% chance of un­de­tected cheat­ing. Op­tions 1B, 2A, and 2B all have a 100% chance of un­de­tected cheat­ing. In Game 3, you can pay to change your de­fault choice twice, and the deal­maker shows a will­ing­ness to elimi­nate his abil­ity to cheat be­fore your sec­ond choice.

But, don’t you have some nu­mer­i­cal prefer­ence for this?

Not cur­rently. There would be a lot of fac­tors de­ter­min­ing how likely I think a mis­calcu­la­tion or cheat­ing might be, and there is no way to de­ter­mine this in the ab­stract.

• Ques­tion: do the rest of you ac­tu­ally find the choice of 1A clearly in­tu­itive?

I think my in­tu­ition for ex­am­ples like this has been safely kil­led off, so my re­place­ment in­tu­ition in­stead says: “hm, clearly 34*(27-24) > 27, so 1B!” (with­out ac­tu­ally eval­u­at­ing 27-24, just not­ing it’s ≥1). Which mainly sug­gests that I’ve grown ac­cus­tomed to calcu­lat­ing ex­pec­ta­tions out ex­plic­itly where they’re ob­vi­ous, not that I’m nec­es­sar­ily good at avoid­ing real life analogues of the prob­lem.

• do the rest of you ac­tu­ally find the choice of 1A clearly in­tu­itive?

I chose 1B. I seem to be an out­lier in that I chose 1B and 2B and did no ar­ith­metic.

• If flip­ping the switch be­fore 12:00 pm has no effect on the amount of money one ac­quires why would one pay any­thing to do it? why not just flip the switch only once af­ter 12:00 pm and be­fore 12:05PM?

• 29 Aug 2012 22:52 UTC
0 points

I won­der if this bias is some­how try­ing to com­pen­sate for some other bias. Sup­pose you think the ex­per­i­menter is over­con­fi­dent, i.e., their log-odds are twice as much as they should; so, when they say 100% they do mean 100%, but when they say 97.1% they ac­tu­ally mean 85.2% (and when they say 34% they mean 41.8%, and when they say 33% they mean 41.2%). Now, Op­tion 1B sud­denly looks much uglier, doesn’t it? (I’m not claiming this hap­pens con­sciously.)

• I ini­tially chose 1A and 2B, but af­ter read­ing the anal­y­sis of those de­ci­sions, I agree that they are in­con­sis­tent in a way that im­plies that one choice was ir­ra­tional (in the con­text of this silly lit­tle game). So I did some in­tro­spec­tion to figure out where I went wrong. Here’s what I found:

1) I may have mis­judged how small 134 is, and this only be­came ap­par­ent when the ques­tion was phased as it is in ex­am­ple 2.

2) I think I as­sumed an im­plicit costs in these gam­bles. The first cost is a de­lay in learn­ing the out­come of these gam­bles; the sec­ond is the im­plicit need to work to earn this money. I think that these as­sump­tions are rea­son­able be­cause there is es­sen­tially no re­al­is­tic con­di­tion in which I would in­stantly see the re­sults of a de­ci­sion that might earn me $27,000; there would prob­a­bly be a de­lay of sev­eral months (if work­ing) or years (if in­vest­ing) be­tween mak­ing the de­ci­sion and learn­ing whether I got the money or not. This pro­longed un­cer­tainty has a nega­tive util­ity, since I am un­able to make firm plans for the money dur­ing that in­ter­val. This nega­tive util­ity would ap­ply to all op­tions ex­cept 1A. Fur­ther­more, earn­ing$24,000 would re­al­is­ti­cally re­quire sev­eral months of work on my part. How­ever, a pro­ject that had a 13 chance of pay­ing out $24,000 might only take a month. The im­plicit differ­ence in op­por­tu­nity cost be­tween sce­nario 1 and sce­nario 2 has im­pli­ca­tions for the marginal util­ity of money in each sce­nario (mak­ing me more risk-averse in sce­nario 1, which im­plic­itly has a higher op­por­tu­nity cost). Th­ese im­plicit costs are not speci­fied in this game, so it is tech­ni­cally “ir­ra­tional” to in­cor­po­rate them into my de­ci­sion-mak­ing. How­ever, in any re­al­is­tic sce­nario, such costs will ex­ist (re­gard­less of what the sales­man says), so it is good that I/​we in­tu­itively in­clude them in my/​our de­ci­sion-mak­ing. • If you’d ask any per­son ca­pa­ble of do­ing the math whether they would want to play 1A or 1B a thou­sand times you’d prob­a­bly get a differ­ent an­swer, but not an an­swer that’s more cor­rect. Also the util­ity value of money is not di­rectly rel­a­tive to the amount of money. Imag­ine that you would need a 1000$ dol­lars of money to save your dy­ing rel­a­tive with cer­tainty by pay­ing for his/​her treat­ment. Good enough for ex­plain­ing 1A > 1B, but doesn’t re­solve the con­tra­dic­tion with 2B > 2A.

But even a more re­veal­ing edit is based ex­actly onto the cer­tainty. If you would be pre­sented with these two ques­tions, in such a fash­ion that you would get the money and get to know the re­sult in 1 month af­ter be­ing pre­sented with it. By se­lect­ing 1A you would have 0% chance that the plans you make would fail, and with 1B you would have a 134 chance that they would fail. Mean­while re­gard­less of whether you se­lect 2A or 2B you will have to face un­cer­tainty. So you would be frus­trated while try­ing to make plans that are con­di­tion­ally de­pen­dent with you get­ting the money.

As these con­di­tions are not pre­sent in the pre­sen­ta­tion it’s pos­si­ble to rule these kind of in­stinc­tive judg­ments as flawed, but as it turns out, they’re not fool­ish, on a gen­eral level. You could even make a claim that it’s costly to perform the calcu­la­tion that tells you whether the as­surance is worth it—but of course in­stead of say­ing that you should just figure out how much value this as­surance has in each given situ­a­tion.

• You’re right that cer­tainty helps out with plan­ning, and so cer­tainty can be valuable some­times. It’s still a bias to un­con­sciously add in a value for cer­tainty if you don’t need it in this case, even if it some­times pays off, and so it’s worth think­ing through the ‘para­dox.’

• I wanted to point out that this flaw is not a fool­ish flaw. That’s how we cre­ate plans, we pro­ject and cre­ate ex­pec­ta­tions, and the an­ti­ci­pated feel­ing of loss is frus­trat­ing to plan for. In a the­o­ret­i­cal ex­am­ple you might make a bad de­ci­sion, but isn’t it also that this flaw causes you to make good de­ci­sions in ac­tual real-world situ­a­tions? Since they don’t tend to oc­cur in such the­o­ret­i­cal forms where you have all the re­quired in­for­ma­tion available and which lack con­text.

If you’d ac­tu­ally en­counter this prob­lem in a real-world situ­a­tion, you might end up mak­ing a bad de­ci­sion be­cause of han­dling it with a too the­o­ret­i­cal ap­proach—what if I told you get to play both games and ac­tu­ally get to choose be­tween both, when you come to visit me? But you didn’t have money to pay for the ticket to fly over? What if you took a loan? And with­out the cer­tainty of A1 you might end up in a bad situ­a­tion where you’ll lack the means to pay back your loan—in other words a de­ci­sion mak­ing agent with this flaw han­dles the situ­a­tion well. But of course you can take all that into ac­count. And as it’s a prob­lem deal­ing with ra­tio­nal­ity, I think it’s pretty im­por­tant to note these things.

Any­way I agree with you, Vaniver =)

• The prob­lem as stated is hy­po­thet­i­cal: there is next to no con­text, and it is as­sumed that the util­ity scales with the mon­e­tary re­ward. Once you con­front real peo­ple with this offer, the con­text ex­pands, and the anal­y­sis of the hy­po­thet­i­cal situ­a­tion falls short of be­ing an ad­e­quate rep­re­sen­ta­tion of re­al­ity, not nec­es­sar­ily be­cause of a fault of the real peo­ple.

Many real peo­ple use a strat­egy of “don’t gam­ble with money you can­not af­ford to lose”; this is over­all a pretty suc­cess­ful strat­egy (and if I was look­ing to make some money, my mark would be the per­son who likes to take risks—just make him sub­se­quently bet­ter offers un­til he even­tu­ally loses, and if he doesn’t, hit him over the head, take the now sub­stan­tial amount of money and run). To aban­don this strat­egy just be­cause in this one case it looks as if it is some­what less prof­itable might not be effec­tive in the long run. (In other cir­cum­stances, peo­ple on this site talk about self-mod­ifi­ca­tion to counter some ex­pected situ­a­tions as one-box­ing vs. dual-box­ing; can we con­sider this strat­egy such a self-mod­ifi­ca­tion?)

Another use­ful real-life strat­egy is, “stay away from stuff you don’t un­der­stand” - $24,000 free and clear is eas­ier to grasp than the other offer, so that strat­egy fa­vors 1A as well, and doesn’t ap­ply to 2A vs. 2B be­cause they’re equally hard to un­der­stand. The fram­ing of offer two also sug­gests that the two offers might be com­pared by mul­ti­ply­ing per­centage and val­ues, while offer 1 has no such sug­ges­tion in branch 1A. We’re look­ing at a hy­po­thet­i­cal situ­a­tion, analysed for an ideal agent with no past and no fu­ture—I’m not sur­prised the real world is more com­plex than that. • it is as­sumed that the util­ity scales with the mon­e­tary re­ward. Not nec­es­sar­ily. It is as­sumed that re­ceiv­ing$24000 is equally good in ei­ther situ­a­tion. Your util­ity func­tion can ig­nore money en­tirely (in which case 1A2A is ir­ra­tional be­cause you should be in­differ­ent in both cases). You can use the util­ity func­tion which prefers not to re­ceive mon­e­tary re­wards di­visi­ble by 9: in this case, 1A>1B and 2A>2B is your best bet, giv­ing you 100% and 34% chances to avoid 9s, rather than 0% chances. In gen­eral, your util­ity func­tion can have ar­bi­trary prefer­ences on A and B sep­a­rately; but no mat­ter what, it will pre­fer 1A to 1B if and only if it prefers 2A to 2B.

As for the rest of your re­ply—yes, it is true that real peo­ple use strate­gies (“heuris­tic” is the word used in the origi­nal post) that lead them to choose 1A and 2B. That’s sort of why it’s a para­dox, af­ter all. How­ever, these strate­gies, which work well in most cases, aren’t nec­es­sar­ily the best in all cases. The math shows that. What the math doesn’t tell us is which case is wrong.

My own judg­ment, for this par­tic­u­lar sum of money (which is high rel­a­tive to my cur­rent in­come), is that choice 1A is cor­rectly bet­ter than choice 2A, in or­der to avoid risk. How­ever, choice 1B is also bet­ter than choice 2B, upon re­flec­tion, even though my in­tu­itions tell me to go with 2B. This is be­cause my in­tu­itions aren’t dis­t­in­guish­ing 33% and 34% cor­rectly.

In re­al­ity, faced with the op­por­tu­nity to earn amounts on the or­der of $20K, I should max­i­mize my chances to walk away with some­thing. In the first case, I can max­i­mize them fully, to 100%, which trig­gers my “suc­cess!” in­stinct or what­ever: I know I’ve done ev­ery­thing I can be­cause I’m cer­tain to get lots of money. In the sec­ond case, I don’t get any satis­fac­tion from the cor­rect de­ci­sion, be­cause all I’ve done is im­prove my chances by 1%. In gen­eral, the heuris­tic that 1% chances are nearly worth­less is cor­rect, no mat­ter what’s at stake: I can usu­ally do bet­ter by work­ing on some­thing that will give me a 10% or 25% chance. In this case, this heuris­tic should be ig­nored, be­cause there is no effort spent mak­ing the im­prove­ment, and fur­ther­more, there isn’t re­ally any­thing else I can do. On the other hand, sup­pose that the amount of money at stake is$2.40 or $2.70. Sud­denly, our risk-aver­sion heuris­tic is no longer be­ing trig­gered at all (un­less you’re re­ally strapped for cash), and we have no prob­lem do­ing the util­ity calcu­la­tion. Here, 1A<1B and 2A<2B is the cor­rect choice. • The util­ity func­tion has as its in­put only the mon­e­tary re­ward in this par­tic­u­lar in­stance. Your idea that risk-avoidance can have util­ity (or that 1% chances are use­less) can­not be mod­el­led with the set of equa­tions given to analyse the situ­a­tion (the per­centage is no in­put to the U() func­tion) - the model falls short be­cause the util­ity at­taches only to the money and noth­ing else. (Another ex­am­ple of a group of in­di­vi­d­u­als for whom the risk might out-uti­lize the re­ward are gam­bling ad­dicts.) Se­cu­rity is, all other things be­ing equal, preferred over in­se­cu­rity, and we could prob­a­bly de­vise some ex­per­i­men­tal setup to trans­late this into a util­ity money equiv­a­lent (i.e. how much is the test sub­ject pre­pared to pay for se­cu­rity and pre­dictabil­ity? that is the mar­gin of in­surance com­pa­nies, btw). :-P I wanted to sug­gest that a real-life util­ity func­tion ought to con­sider even more: not just to the sin­gle case, but the strate­gies used in this case—do these strate­gies or heuris­tics have bet­ter util­ity in my life than try­ing to figure out the best pos­si­ble ac­tion for each prob­lem? In that case, an op­ti­mal strat­egy may well be sub­op­ti­mal in some cases, but work well re: a re­al­is­tic life­time filled with prob­a­ble events, even if you don’t con­trive a$24000 life-or-death op­er­a­tion. (Should I spend two years of my life study­ing more statis­tics, or work on my father’s farm? The farm might profit me more in the long run, even if I would miss out if some­body made me the 1A/​1B offer, which is very un­likely, mak­ing that strat­egy the ra­tio­nal one in the larger con­text, though it ap­pears ir­ra­tional in the smaller one.)

• Risk-avoidance is cap­tured in the as­sign­ment of U($X). If the risk of not get­ting any money wor­ries you dis­pro­por­tionately, that means that the differ­ence U($24K) - U($0) is higher than 8 times the differ­ence U($27K) - U($24K). • That’s a neat trick, how­ever, I am not sure I un­der­stand you cor­rectly. You seem to be say­ing that risk-avoidance does not ex­plain the 1A/​2B prefer­ence, be­cause you say your as­sign­ment cap­tures risk-avoidance, and it doesn’t lead to that. (It does lead to your take of the term though—your prefer­ence isn’t 1A/​2B, though). Your as­sign­ment looks like “diminish­ing util­ity”, i.e. a util­ity func­tion where the util­ity scales up sub­pro­pro­tion­ally with money (e.g. twice the money must have less than twice the util­ity). Do you think diminish­ing util­ity is equiv­a­lent to risk-avoidance? And if yes, can you ex­plain why? • I think so, but your ques­tion forces me to think about it harder. When I thought about it ini­tially, I did come to that con­clu­sion—for my­self, at least. [I re­al­ized that the math I wrote here was wrong. I’m go­ing to try to re­vise it. In the mean­time, an­other ques­tion. Do you think that risk avoidance can be mod­eled by as­sign­ing an ad­di­tional util­ity to cer­tainty, and if so, what would that util­ity de­pend on?] Also, think­ing about the para­dox more, I’ve re­al­ized that my in­tu­ition about prob­a­bil­ities re­lies sig­nifi­cantly on my ex­pe­rience play­ing the board game Set­tlers of Catan. Are you fa­mil­iar with it? • One way to do it to get to the de­sired out­come is to re­place U(x) with U(x,p) (with x be­ing the money re­ward and p the prob­a­bil­ity to get it), and define U(x,p)=2x if p=1 and U(x,p)=x, oth­er­wise. I doubt that this is a use­ful model of re­al­ity, but math­e­mat­i­cally, it would do the trick. My stated opinion is that this spe­cial case should be looked at in the light of more gen­eral starte­gies/​heuris­tics ap­plied over a va­ri­ety of situ­a­tions, and this ap­proach would still fall short of that. I know Set­tlers of Catan, and own it. It’s been awhile since I last played it, though. Your point about games made me aware of a cru­cial differ­ence be­tween real life and games, or other ab­stract prob­lems of chance: in the lat­ter, chances are always known with­out er­ror, be­cause we set the game (or prob­lem) up to have cer­tain chances. In real life, we pre­dict events ei­ther via causal­ity (100% chance, no guess­work in­volved, un­less things come into play we for­got to con­sider), or via ex­pe­rience /​ statis­tics, and that in­volves guess­work and mar­gins of er­ror. If there’s a pre­dic­tion with a 100% chance, there is usu­ally a causal re­la­tion­ship at the bot­tom of it; with a chance less than 100%, there is no such causal chain; there must be some fac­tor that can thwart the fa­vor­able out­come; and there is a chance that this fac­tor has been as­sessed wrong, and that there may be other fac­tors that were over­looked. Worst case, a 3334 chance might ac­tu­ally only be 3034 or less, and then I’d be worse off tak­ing the chance. Com­par­ing a .33 with a .34 chance makes me think that there’s gotta be a lot of guess­work in­volved, and that, with er­ror mar­gins and con­fi­dence in­ter­vals and such, there’s usu­ally a size­able chance that the un­der­ly­ing prob­a­bil­ities might be equal or re­versed, so go­ing for the higher re­ward makes sense. [rewrit­ten] Imag­ine you are a math­e­mat­i­cal ad­vi­sor to a king who asks you to ad­vise him of a course of ac­tion and to pre­dict the out­come. In situ­a­tion, you can pretty much ad­vise what­ever, be­cause you’ll pre­dict a failure; the out­come ei­ther con­firms your pre­dic­tion, or is a lucky wind­fall, so the king will be con­tent with your ad­vice in hind­sight. In situ­a­tion 2, you’ll pre­dict a gain; if you ad­vised A, your pre­dic­tion will be con­firmed, but if you ad­vised B, there’s a chance it won’t be, with the king an­gry at you be­cause he didn’t make the money you pre­dicted he would. Your ca­reer is over. -- Now imag­ine a col­lec­tion of au­tonomous agents, or a bun­dle of heuris­tics fight­ing for Dar­winist sur­vival, and you’ll see what strat­egy sur­vives. [If you like stereo­types, imag­ine the “king” as “math­e­mat­i­cian’s non-math­e­mat­i­cal spouse”. ;-)] • One way to do it to get to the de­sired out­come is to re­place U(x) with U(x,p) (with x be­ing the money re­ward and p the prob­a­bil­ity to get it), and define U(x,p)=2x if p=1 and U(x,p)=x, oth­er­wise. The prob­lem with this is that deal­ing with p=1 is iffy. Ideally, our cer­tainty re­sponse would be trig­gered, if not as strongly, when deal­ing with 99.99% cer­tainty—for one thing, be­cause we can only ever be, say, 99.99% cer­tain that we read p=1 cor­rectly and it wasn’t ac­tu­ally p=.1 or some­thing! Ideally, we’d have a de­cay­ing fac­tor of some sort that de­pends on the prob­a­bil­ities be­ing close to 1 or 0. The rea­son I asked is that it’s very pos­si­ble that a cor­rect model of “at­tach­ing a util­ity to cer­tainty” would be equiv­a­lent to a model with diminish­ing util­ity of money. If that were the case, we would be ar­gu­ing over noth­ing. If not, we’d at least stand a chance of for­mu­lat­ing gam­bles clar­ify­ing our in­tu­itions if we knew what the al­ter­na­tives are. Com­par­ing a .33 with a .34 chance makes me think that there’s gotta be a lot of guess­work in­volved, and that, with er­ror mar­gins and con­fi­dence in­ter­vals and such, there’s usu­ally a size­able chance that the un­der­ly­ing prob­a­bil­ities might be equal or re­versed, so go­ing for the higher re­ward makes sense. If the 33% and 34% chances are in the mid­dle of their er­ror mar­gins, which they should be, our un­cer­tainty about the chances can­cels out and the ex­pected util­ity is still the same. Go­ing for the higher ex­pected value makes sense. I brought up Set­tlers of Catan be­cause, if I imag­ine a tile on the board with$24K and 34 dots un­der it, and an­other tile with $27K and 33 dots, sud­denly I feel a lot bet­ter about com­par­ing the prob­a­bil­ities. :) Does this help you, or am I atyp­i­cal in this way? Imag­ine you are a math­e­mat­i­cal ad­vi­sor to a king who asks you to ad­vise him of a course of ac­tion and to pre­dict the out­come. Ob­vi­ously with the ad­vi­sor situ­a­tion, you have to take your ad­visee’s bi­ases into ac­count. The one most rele­vant to risk avoidance is, I think, the sta­tus quo bias: rather than tak­ing into ac­count the util­ity of the out­comes in gen­eral, the king might be an­gry at you if the util­ity be­comes worse, and not as picky if the util­ity be­comes bet­ter (than it is now). You have to take your own util­ity into ac­count, which de­pends not on the out­come but on your king’s satis­fac­tion with it. • The prob­lem is not with the hy­po­thet­i­cal. It is with the in­tu­ition. In­tu­itions which re­ally do prompt bad de­ci­sions in the real life cir­cum­stances along these lines. • You seem to have ex­am­ples in mind? • The lot­tery comes im­me­di­ately to mind. You can’t be ab­solutely sure that you’ll lose. • “That is, the Ax­iom of In­de­pen­dence im­plies dy­namic con­sis­tency, but not vice versa.” Really? A hy­per­bolic dis­counter can con­form to the Ax­iom of In­de­pen­dence at any par­tic­u­lar time and be dy­nam­i­cally in­con­sis­tent. • I agree with Dagon. This ex­per­i­ment as­sumes that the sub­jec­tive prob­a­bil­ities of par­ti­ci­pants were iden­ti­cal to the stated prob­a­bil­ities. In re­al­ity, I feel like peo­ple are prob­a­bly wary of stated prob­a­bil­ities due to ex­pe­riences with or fears of shys­ters and con­men. That, is if asked to choose be­tween 1A and 1B, 1B offers the pos­si­bil­ity that the ran­domis­ing mechanism’ that the ex­per­i­menter is offer­ing is in fact rigged. Even if the ex­per­i­menter is com­pletely hon­est in their state­ment of their own sub­jec­tive prob­a­bil­ities, they may sim­ply dis­agree with that of the par­ti­ci­pants. What­ever ran­domis­ing mechanism’ is sug­gested is, of course, al­most cer­tainly com­pletely pre­dictable given suffi­cient in­for­ma­tion—a die roll, or similar, pre­dictable us­ing New­to­nian me­chan­ics. That, is the ex­per­i­menter’s stated prob­a­bil­ity is purely a re­flec­tion of their own in­for­ma­tion con­cern­ing that mechanism, which may be com­pletely at odds with the par­ti­ci­pant’s knowl­edge. • Surely the an­swer is de­pend­ed­nat on goal crite­rion. If the goal is to get ‘some’ money then the 100% op­tion and the 34% op­tions are bet­ter. If your goal is get ‘the most’ money then the 97% and the 33% op­tions are bet­ter. How­ever the goal might be so­cially con­stru­ictued. This re­minded me of John Nash whom offered one of his sec­traries$15 dol­lars if she shared it equally with a co-worker but $10 if she kept it for her-self. She took the$15 and split it with her co-worker. She chose an op­tion that max­imised her so­cial cap­i­tal but was a weaker one eco­nom­i­cally.

• Nick,

“Is Michael Vas­sar’s var­i­ant Pas­cal’s Mug­ging (with the pigs), by­pass­ing as it does Robin’s ob­jec­tion, the re­duc­tio of ex­pected value? If you don’t care about pigs, sub­sti­tute some­thing else re­ally re­ally bad that doesn’t re­quire cre­at­ing 3^^^3 hu­mans.”
The Porcine Mug­ging doesn’t by­pass the ob­jec­tion. Your es­ti­mates of the fre­quency of simu­lated peo­ple and pigs should be com­men­su­rably vast, and it is vastly un­likely that your simu­la­tion (out of many with in­tel­li­gent be­ings) will be se­lected for an ac­tual Porcine Mug­ging that will con­sume vast re­sources (enough to simu­late vast num­bers of hu­mans). Th­ese things offset to get you work­able calcu­la­tions.

• The Allais Para­dox is in­deed quite puz­zling. Here are my thoughts:

0. Some com­menters sim­ply dis­miss Bayesian rea­son­ing. This doesn’t solve the prob­lem, it just strips us of any math­e­mat­i­cal way to an­a­lyze the prob­lem. On the other hand, the fact that the in­con­sis­tent choice seems ok does mean that the Bayesian way is miss­ing some­thing. Sim­ply dis­miss­ing the in­con­sis­tent choice doesn’t solve the prob­lem ei­ther.

1. If I un­der­stand cor­rectly, you ar­gue that situ­a­tion 1 can be turned into situ­a­tion 2 by ran­dom­iza­tion. In other words, if you sell me situ­a­tion 1, I can sell some­body else (named X) situ­a­tion 2 by throw­ing some dies and us­ing your offer. More speci­fi­cally, I throw a 100-sided die. If it’s > 34, X looses. Other­wise, I play X’s op­tion with you. How­ever, this can’t be re­versed. Given only situ­a­tion 2, I can’t sell situ­a­tion 1, as­sum­ing I have only $0 ini­tial cap­i­tal. Hence, it seems that as­sum­ing in­vert­ibil­ity of situ­a­tions (I can both buy and sell them) and un­limited money buffers for that pur­pose are im­por­tant for the de­manded con­sis­tency. • I’m afraid I don’t fol­low the maths in­volved, but I’d like to know whether the equa­tions work out differ­ently if you take this premise: - Since 1A offers a cer­tainty of$24,000, it is deemed to be im­me­di­ately in your pos­ses­sion. 1B then be­comes a 3334 chance of win­ning $3,000 and 134 chance of los­ing$24,000.

Can some­one tell me how this works out math­e­mat­i­cally, and how it then com­pares to 2B?

• Well, if Pas­cal’s Mug­ging doesn’t do it, how about the St. Peters­burg para­dox? ;)

Oh wait… in­finite set athe­ist… never mind.

• It’s sim­ple to show that no ra­tio­nal per­son would ac­tu­ally give money to a Pas­cal mug­ger, as the next mug­ger might threaten 4^^^4 peo­ple. I’m not sure whether this solves the prob­lem or just sweeps it un­der the rug, though.

• Nick is right to ob­ject, but re­mov­ing the phrase “on amounts of money” makes the state­ment un­ob­jec­tion­able—and rele­vant and true.

• you’re vi­o­lat­ing ra­tio­nal­ity ax­ioms like the one Eliezer gave in the OP

No. Those ax­ioms are “if ⇒ then” state­ments. I’m vi­o­lat­ing the “if” part.

• Cale­do­nian, if util­ity is any func­tion defined on amounts of money, then if you are max­i­miz­ing ex­pected util­ity, you can­not fall prey to the Allais para­dox. You can define a util­ity func­tion on gam­bles that is not the ex­pected value of a util­ity func­tion on amounts of money, but then that func­tion is not ex­pected util­ity, and you’re out­side of nor­mal mod­els of risk aver­sion, and you’re vi­o­lat­ing ra­tio­nal­ity ax­ioms like the one Eliezer gave in the OP.

• Cale­do­nian, Nick T: “Risk aver­sion” in the stan­dard mean­ing is when an agent max­i­mizes the ex­pec­ta­tion value of utility

Oh, I agree.

I just mea­sure util­ity differ­ently than you do.

• Dawes gives a very similar 2-gam­ble ex­am­ple of a money pump on pg 105 of Ra­tional Choice.

• It seems to me that your ar­gu­ment re­lies on the util­ity of hav­ing a prob­a­bil­ity p of gain­ing x be­ing equal to p times the util­ity of gain­ing x. It’s not clear to me that this should be true.

The trou­ble with the “money pump” ar­gu­ment is that the choice one makes may well de­pend on how one got into the situ­a­tion of hav­ing the choice in the first place. For ex­am­ple, let’s as­sume some­one pre­fer 2B over 2A. It could be that if he were offered choice 1 “out of the blue” he would pre­fer 1A over 1B, yet if it were an­nounced in ad­vance that he would have a 23 chance of get­ting noth­ing and a 13 chance of be­ing offered choice 1, he would de­cide be­fore­hand that B is the bet­ter choice, and he would stick with that choice even if al­lowed to switch. This may seem odd, but I don’t see why it’s log­i­cally in­con­sis­tent.

• Con­stant’s re­sponse im­plies that if some­one prefers 1A to 1B and 2B to 2A, when con­fronted with the money pump situ­a­tion, the per­son will de­cide that af­ter all, 1A is prefer­able to 1B and 2A is prefer­able to 2B. This is very strange but at least con­sis­tent.

• As long as it was only one oc­ca­sion, I wouldn’t make the effort to cross the room for two pen­nies. If I’m play­ing the game just once, and I feel a one-off pay­ment of 2p tends to zero, I’ll play with you, sure. £1 for a lot­tery ticket crosses the thresh­old of pal­pa­bil­ity, even play­ing once. I can get a news­pa­per for a pound. Is this ir­ra­tional? I hope not.

• Risk aver­sion, and the de­gree to which it is felt, is a per­son­al­ity trait with high var­i­ance be­tween in­di­vi­d­u­als and over the lifes­pan. To ig­nore it in a util­ity calcu­la­tion would be ab­surd. Mau­rice Allais should have listened to his homonym Alphonse Allais (no ap­par­ent re­la­tion), hu­morist and the­o­ret­i­cian of the ab­surd, who fa­mously re­marked “La logique mène à tout à con­di­tion d’en sor­tir”. Logic leads to ev­ery­thing, on con­di­tion it don’t box you in.

• This re­minds me of the fool­ish de­ci­sions on “deal or no deal”. Peo­ple would fail to fol­low their own an­nounced util­ity.

• I think I missed some­thing on the alge­braic in­con­sis­tency part...

If there is some ra­tio­nal in­de­pen­dent util­ity to cer­tainty, the alge­braic claims should be more like this:

• U($24,000) + U(Cer­tainty) > 3334 U($27,000) + 134 U($0) • 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0) This seems con­sis­tent so long as U(Cer­tainty) > 134 U($27,000).

I’m not com­mit­ted to the no­tion there is a ra­tio­nal in­de­pen­dent value to cer­tainty, I’m just not see­ing how it can be dis­missed with quick alge­bra. Maybe that wasn’t your goal. For­give me if this is my over­sight.

• Since peo­ple only make a finite num­ber of de­ci­sions in their life­time, couldn’t their util­ity func­tion spec­ify ev­ery de­ci­sion in­de­pen­dently? (You could have a util­ity func­tion that is nor­mal ex­cept that it says that ev­ery­thing you hear be­ing called 1A is prefer­able to 1B, and any­thing you hear be­ing called 2B is prefer­able to 2A. If this con­tra­dicts your nor­mal util­ity func­tion, this rule is always more im­por­tant. Even if 2B leads to death, you still choose 2B.)

The util­ity func­tion would be im­pos­si­ble to come up with in ad­vance, but it ex­ists.

• No, only if the util­ity of avoid­ing risk is worth less than the money at risk. Duh.

Some­one did not read the OP care­fully enough.

Hint: re-read the defi­ni­tion of the Ax­iom of In­de­pen­dence.

• Some­one isn’t think­ing care­fully enough.

Hint: I did not as­sert that X is strictly preferred to Y.

• If you crunch the num­bers differ­ently, you can come to differ­ent con­clu­sions. For ex­am­ple, if I choose 1B over 1A, I have a 1 in 34 chance of get­ting burned. If I choose 2B over 2A, my chance of get­ting burned is only 1 in 100.