The Allais Paradox

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.