Allais Malaise

Con­tinu­a­tion of: The Allais Para­dox, Zut Allais!

Judg­ing by the com­ments on Zut Allais, I failed to em­pha­size the points that needed em­pha­sis.

The prob­lem with the Allais Para­dox is the in­co­her­ent pat­tern 1A > 1B, 2B > 2A. If you need $24,000 for a life­sav­ing op­er­a­tion and an ex­tra $3,000 won’t help that much, then you choose 1A > 1B and 2A > 2B. If you have a mil­lion dol­lars in the bank ac­count and your util­ity curve doesn’t change much with an ex­tra $25,000 or so, then you should choose 1B > 1A and 2B > 2A. Nei­ther the in­di­vi­d­ual choice 1A > 1B, nor the in­di­vi­d­ual choice 2B > 2A, are of them­selves ir­ra­tional. It’s the com­bi­na­tion that’s the prob­lem.

Ex­pected util­ity is not ex­pected dol­lars. In the case above, the util­ity-dis­tance from $24,000 to $27,000 is a tiny frac­tion of the dis­tance from $21,000 to $24,000. So, as stated, you should choose 1A > 1B and 2A > 2B, a quite co­her­ent com­bi­na­tion. The Allais Para­dox has noth­ing to do with be­liev­ing that ev­ery added dol­lar is equally use­ful. That idea has been re­jected since the dawn of de­ci­sion the­ory.

If satis­fy­ing your in­tu­itions is more im­por­tant to you than money, do what­ever the heck you want. Drop the money over Ni­a­gara falls. Blow it all on ex­pen­sive cham­pagne. Set fire to your hair. What­ever. If the largest util­ity you care about is the util­ity of feel­ing good about your de­ci­sion, then any de­ci­sion that feels good is the right one. If you say that differ­ent tra­jec­to­ries to the same out­come “mat­ter emo­tion­ally”, then you’re at­tach­ing an in­her­ent util­ity to con­form­ing to the brain’s na­tive method of op­ti­miza­tion, whether or not it ac­tu­ally op­ti­mizes. Heck, run­ning around in cir­cles from prefer­ence re­ver­sals could feel re­ally good too. But if you care enough about the stakes that win­ning is more im­por­tant than your brain’s good feel­ings about an in­tu­ition-con­form­ing strat­egy, then use de­ci­sion the­ory.

If you sup­pose the prob­lem is differ­ent from the one pre­sented - that the gam­bles are un­trust­wor­thy and that, af­ter this mis­trust is taken into ac­count, the pay­off prob­a­bil­ities are not as de­scribed - then, ob­vi­ously, you can make the an­swer any­thing you want.

Let’s say you’re dy­ing of thirst, you only have $1.00, and you have to choose be­tween a vend­ing ma­chine that dis­penses a drink with cer­tainty for $0.90, ver­sus spend­ing $0.75 on a vend­ing ma­chine that dis­penses a drink with 99% prob­a­bil­ity. Here, the 1% chance of dy­ing is worth more to you than $0.15, so you would pay the ex­tra fif­teen cents. You would also pay the ex­tra fif­teen cents if the two vend­ing ma­chines dis­pensed drinks with 75% prob­a­bil­ity and 74% prob­a­bil­ity re­spec­tively. The 1% prob­a­bil­ity is worth the same amount whether or not it’s the last in­cre­ment to­wards cer­tainty. This pat­tern of de­ci­sions is perfectly co­her­ent. Don’t con­fuse be­ing ra­tio­nal with be­ing short­sighted or greedy.

Added: A 50% prob­a­bil­ity of $30K and a 50% prob­a­bil­ity of $20K, is not the same as a 50% prob­a­bil­ity of $26K and a 50% prob­a­bil­ity of $24K. If your util­ity is log­a­r­ith­mic in money (the stan­dard as­sump­tion) then you will definitely pre­fer the lat­ter to the former: 0.5 log(30) + 0.5 log(20) < 0.5 log(26) + 0.5 log(24). You take the ex­pec­ta­tion of the util­ity of the money, not the util­ity of the ex­pec­ta­tion of the money.