Counterfactual Mugging

Re­lated to: Can Coun­ter­fac­tu­als Be True?, New­comb’s Prob­lem and Re­gret of Ra­tion­al­ity.

Imag­ine that one day, Omega comes to you and says that it has just tossed a fair coin, and given that the coin came up tails, it de­cided to ask you to give it $100. What­ever you do in this situ­a­tion, noth­ing else will hap­pen differ­ently in re­al­ity as a re­sult. Nat­u­rally you don’t want to give up your $100. But see, Omega tells you that if the coin came up heads in­stead of tails, it’d give you $10000, but only if you’d agree to give it $100 if the coin came up tails.

Omega can pre­dict your de­ci­sion in case it asked you to give it $100, even if that hasn’t ac­tu­ally hap­pened, it can com­pute the coun­ter­fac­tual truth. Omega is also known to be ab­solutely hon­est and trust­wor­thy, no word-twist­ing, so the facts are re­ally as it says, it re­ally tossed a coin and re­ally would’ve given you $10000.

From your cur­rent po­si­tion, it seems ab­surd to give up your $100. Noth­ing good hap­pens if you do that, the coin has already landed tails up, you’ll never see the coun­ter­fac­tual $10000. But look at this situ­a­tion from your point of view be­fore Omega tossed the coin. There, you have two pos­si­ble branches ahead of you, of equal prob­a­bil­ity. On one branch, you are asked to part with $100, and on the other branch, you are con­di­tion­ally given $10000. If you de­cide to keep $100, the ex­pected gain from this de­ci­sion is $0: there is no ex­change of money, you don’t give Omega any­thing on the first branch, and as a re­sult Omega doesn’t give you any­thing on the sec­ond branch. If you de­cide to give $100 on the first branch, then Omega gives you $10000 on the sec­ond branch, so the ex­pected gain from this de­ci­sion is

-$100 * 0.5 + $10000 * 0.5 = $4950

So, this straight­for­ward calcu­la­tion tells that you ought to give up your $100. It looks like a good idea be­fore the coin toss, but it starts to look like a bad idea af­ter the coin came up tails. Had you known about the deal in ad­vance, one pos­si­ble course of ac­tion would be to set up a pre­com­mit­ment. You con­tract a third party, agree­ing that you’ll lose $1000 if you don’t give $100 to Omega, in case it asks for that. In this case, you leave your­self no other choice.

But in this game, ex­plicit pre­com­mit­ment is not an op­tion: you didn’t know about Omega’s lit­tle game un­til the coin was already tossed and the out­come of the toss was given to you. The only thing that stands be­tween Omega and your 100$ is your rit­ual of cog­ni­tion. And so I ask you all: is the de­ci­sion to give up $100 when you have no real benefit from it, only coun­ter­fac­tual benefit, an ex­am­ple of win­ning?

P.S. Let’s as­sume that the coin is de­ter­minis­tic, that in the over­whelming mea­sure of the MWI wor­lds it gives the same out­come. You don’t care about a frac­tion that sees a differ­ent re­sult, in all re­al­ity the re­sult is that Omega won’t even con­sider giv­ing you $10000, it only asks for your $100. Also, the deal is unique, you won’t see Omega ever again.