# MichaelStJules comments on Teruji Thomas, ‘The Asymmetry, Uncertainty, and the Long Term’

• In­ter­est­ing!

What if we re­define ra­tio­nal­ity to be rel­a­tive to choice sets? We might not have to de­part too far from vNM-ra­tio­nal­ity this way.

The ax­ioms of vNM-ra­tio­nal­ity are jus­tified by Dutch books/​money pumps and stochas­tic dom­i­nance, but the lat­ter can be weak­ened, too, since many out­comes are in­deed ir­rele­vant, so there’s no need to com­pare to them all. For ex­am­ple, there’s no Dutch book or money pump that only in­volves chang­ing the prob­a­bil­ities for the size of the uni­verse, and there isn’t one that only in­volves chang­ing the prob­a­bil­ities for log­i­cal state­ments in stan­dard math­e­mat­ics (ZFC); it doesn’t make sense to ask me to pay you to change the prob­a­bil­ity that the uni­verse is finite. We don’t need to con­sider such lot­ter­ies. So, if we can gen­er­al­ize stochas­tic dom­i­nance to be rel­a­tive to a set of pos­si­ble choices, then we just need to make sure we never choose an op­tion which is stochas­ti­cally dom­i­nated by an­other, rel­a­tive to that choice set. That would be our new defi­ni­tion of ra­tio­nal­ity.

Here’s a first at­tempt:

Let be a set of choices or prob­a­bil­is­tic lot­ter­ies over out­comes (ran­dom vari­ables), and let be the set of all pos­si­ble out­comes which have nonzero prob­a­bil­ity in some choice from (or some­thing more gen­eral to ac­com­mo­date gen­eral prob­a­bil­ity mea­sures). Then for , we say stochas­ti­cally dom­i­nates with re­spect to if:

for all , and the in­equal­ity is strict for some . This can lift com­par­i­sons us­ing , a re­la­tion , be­tween el­e­ments of to ran­dom vari­ables over the el­e­ments of . need not even be com­plete over or tran­si­tive, but stochas­tic dom­i­nance thus defined will be tran­si­tive (per­haps at the cost of los­ing some com­par­i­sons). could also ac­tu­ally be spe­cific to , not just to .

We could play around with the defi­ni­tion of here.

When we con­sider choices to make now, we need to model the fu­ture and con­sider what new choices we will have to make, and this is how we would avoid Dutch books and money pumps. Per­haps this would be bet­ter done in terms of de­ci­sion poli­cies rather than a sin­gle de­ci­sion at a time, though.

(This ap­proach is based in part on “Ex­ceed­ing Ex­pec­ta­tions: Stochas­tic Dom­i­nance as a Gen­eral De­ci­sion The­ory” by Chris­tian Tarsney, which also helps to deal with Pas­cal’s wa­ger and Pas­cal’s mug­ging.)