Higher than the most high

In an ear­lier post, I talked about how we could deal with var­i­ants of the Heaven and Hell prob­lem—situ­a­tions where you have an in­finite num­ber of op­tions, and none of them is a max­i­mum. The solu­tion for a (de­ter­minis­tic) agent was to try and im­ple­ment the strat­egy that would reach the high­est pos­si­ble num­ber, with­out risk­ing fal­ling into an in­finite loop.

Wei Dai pointed out that in the cases where the op­tions are un­bounded in util­ity (ie you can get ar­bi­trar­ily high util­ity), then there are prob­a­bil­is­tic strate­gies that give you in­finite ex­pected util­ity. I sug­gested you could still do bet­ter than this. This started a con­ver­sa­tion about choos­ing be­tween strate­gies with in­finite ex­pec­ta­tion (would you pre­fer a strat­egy with in­finite ex­pec­ta­tion, or the same plus an ex­tra dol­lar?), which went off into some in­ter­est­ing di­rec­tions as to what needed to be done when the strate­gies can’t sen­si­bly be com­pared with each other...

In­ter­est­ing though that may be, it’s also helpful to have sim­ple cases where you don’t need all these sub­tleties. So here is one:

Omega ap­proaches you and Mrs X, ask­ing you each to name an in­te­ger to him, pri­vately. The per­son who names the high­est in­te­ger gets 1 util­ity; the other gets noth­ing. In prac­ti­cal terms, Omega will re­im­burse you all util­ity lost dur­ing the de­ci­sion pro­cess (so you can take as long as you want to de­cide). The first per­son to name a num­ber gets 1 util­ity im­me­di­ately; they may then lose that 1 de­pend­ing on the even­tual re­sponse of the other. Hence if one per­son re­sponds and the other doesn’t, they get the 1 util­ity and keep it. What should you do?

In this case, a strat­egy that gives you a num­ber with in­finite ex­pec­ta­tion isn’t enough—you have to beat Mrs X, but you also have to even­tu­ally say some­thing. Hence there is a duel of (likely prob­a­bil­is­tic) strate­gies, im­ple­mented by bounded agents, with no max­i­mum strat­egy, and each agent try­ing to com­pute the max­i­mal strat­egy they can con­struct with­out fal­ling into a loop.