So in the triple-your-even-odds-bet situation, the normal setup is to take the expectation of f={(1,1,1,...): inf, otherwise: 0}, and EV(f)=0. But you’re saying we should change that game from f:Ω->[0,inf] to g:Ω,?R?->[0,inf] where ?R? is a domain I don’t really understand, a “source of randomness”, and then we can try many times, averaging, and take the limit?
I’m suspicious that I don’t understand how the “source of randomness” actually operates with infinities and limits, and it seems like it’s important to make it formal to make sure nothing’s being swept under the rug. Do you have a link to something that shows how “source of randomness” is generally formalized, or if not, how you’re thinking it works more explicitly?
But you’re saying we should change that game from f:Ω->[0,inf] to g:Ω,?R?->[0,inf]
No. The key change I’m making is from assigning every strategy an expected value (normally real, but including infinity as you do should be possible) to having the essential math thing be a comparison between two strategies. With your version, all we can say is that all-in has EV 0, don’t bet has EV 1, and everything else has EV infinity—but by doing the comparison inside the limits, we get some more differentiation there.
R isn’t distinct from Ω. The EV function “binds” the randomness inside what its applied to, so when I roll it out I need to have it occur explicitly inside the limit. I think its fine to say the Rs are normal random variables. Lets say that each r(i) is uniformly distributed in [0;1) iid.game() then uses that for its randomness. For the game at hand, we could say that the binary digit expansion becomes the sequence of heads and tails thrown.
As you might have noticed I wrote the post in a bit of a hurry, so sorry if not everything is hammered out.
So in the triple-your-even-odds-bet situation, the normal setup is to take the expectation of f={(1,1,1,...): inf, otherwise: 0}, and EV(f)=0. But you’re saying we should change that game from f:Ω->[0,inf] to g:Ω,?R?->[0,inf] where ?R? is a domain I don’t really understand, a “source of randomness”, and then we can try many times, averaging, and take the limit?
I’m suspicious that I don’t understand how the “source of randomness” actually operates with infinities and limits, and it seems like it’s important to make it formal to make sure nothing’s being swept under the rug. Do you have a link to something that shows how “source of randomness” is generally formalized, or if not, how you’re thinking it works more explicitly?
No. The key change I’m making is from assigning every strategy an expected value (normally real, but including infinity as you do should be possible) to having the essential math thing be a comparison between two strategies. With your version, all we can say is that all-in has EV 0, don’t bet has EV 1, and everything else has EV infinity—but by doing the comparison inside the limits, we get some more differentiation there.
R isn’t distinct from Ω. The EV function “binds” the randomness inside what its applied to, so when I roll it out I need to have it occur explicitly inside the limit. I think its fine to say the Rs are normal random variables. Lets say that each r(i) is uniformly distributed in [0;1) iid.game() then uses that for its randomness. For the game at hand, we could say that the binary digit expansion becomes the sequence of heads and tails thrown.
As you might have noticed I wrote the post in a bit of a hurry, so sorry if not everything is hammered out.