I don’t think I understand the point of the temporal average. I think I follow how to calculate it, but I don’t see any justification here for why we should care about the value we calculate that way, or why it’s given that name. (Maybe I just missed these? Maybe they’re answered in the paper?)
I’ve written about this myself, though not recently enough to remember that post in depth. My answer for why to bet Kelly is “over a long enough time, you’ll almost certainly get more money than someone else who was offered the same bets as you and started with the same amount of money but regularly bet different amounts on them”.
I happen to know that in this type of game, maximizing temporal average is the way to get that property, which is neat. That’s the justification I’d give for doing that calculation in this type of game. But it’s not clear to me what justification you’d give.
The temporal average is pretty much just the average exponential growth rate. The reason that it works to use this here is that it’s an ergodic quantity in this problem, so the statistic that it gives you in the one-step problem matches the statistic that it would give you for a complete time-sequence. That means if you maximize it in the one-step problem, you’ll end up maximizing it in the time-series too (not true of expected value here).
I think your justification for Kelly is pragmatically sufficient, but theoretically leaves me a bit cold. I’m interested in knowing why Kelly is the right choice here, and Ole Peters’ paper blew my mind when I read it the first time because it finally gave an answer to this.
and I don’t think this converges as the number of steps n grows. If I’m not getting myself mixed up, then what does converge is 1nlogT(Un), but… I can do the same with “expected money following the bet-everything strategy”.
I think your justification for Kelly is pragmatically sufficient, but theoretically leaves me a bit cold. I’m interested in knowing why Kelly is the right choice here
So I feel like any justification eventually has to boil down in either pragmatics (“here’s something we care about”) or pretend-pragmatics (“here’s something we’re pretending to care about for the purposes of this hypothetical; presumably we think there’s some correspondence to the real world but we may not specify exactly what we think it is”). If we don’t have something like that, why pick one theoretical justification over another?
And I don’t feel like my justification is lacking in theory. It’s not that I’ve done a bunch of experiments and said “this seems to satisfy my pragmatic desires but I don’t know why”. I have a theoretical argument for why it satisfies my pragmatic desires.
I don’t think I understand the point of the temporal average. I think I follow how to calculate it, but I don’t see any justification here for why we should care about the value we calculate that way, or why it’s given that name. (Maybe I just missed these? Maybe they’re answered in the paper?)
I’ve written about this myself, though not recently enough to remember that post in depth. My answer for why to bet Kelly is “over a long enough time, you’ll almost certainly get more money than someone else who was offered the same bets as you and started with the same amount of money but regularly bet different amounts on them”.
I happen to know that in this type of game, maximizing temporal average is the way to get that property, which is neat. That’s the justification I’d give for doing that calculation in this type of game. But it’s not clear to me what justification you’d give.
The temporal average is pretty much just the average exponential growth rate. The reason that it works to use this here is that it’s an ergodic quantity in this problem, so the statistic that it gives you in the one-step problem matches the statistic that it would give you for a complete time-sequence. That means if you maximize it in the one-step problem, you’ll end up maximizing it in the time-series too (not true of expected value here).
I think your justification for Kelly is pragmatically sufficient, but theoretically leaves me a bit cold. I’m interested in knowing why Kelly is the right choice here, and Ole Peters’ paper blew my mind when I read it the first time because it finally gave an answer to this.
I don’t follow, sorry.
What statistic is this? If I calculate the time-average for one step, using the Kelly strategy, I get roughly 1.02:
T(U1)=1.20.6⋅0.80.4≈1.02
If I calculate it for two steps, if I’ve done it right, I get roughly 1.04:
T(U2)=(1.2⋅1.2)0.6⋅0.6⋅(1.2⋅0.8)0.6⋅0.4⋅(0.8⋅1.2)0.4⋅0.6⋅(0.8⋅0.8)0.4⋅0.4≈1.04
and I don’t think this converges as the number of steps n grows. If I’m not getting myself mixed up, then what does converge is 1nlogT(Un), but… I can do the same with “expected money following the bet-everything strategy”.
E(U1)=2⋅0.6+0⋅0.4=1.2E(U2)=(2⋅2)⋅(0.6⋅0.6)+(2⋅0)⋅(0.6⋅0.4)+...=1.44log(E(Un))=nlog(1.2)
So I feel like any justification eventually has to boil down in either pragmatics (“here’s something we care about”) or pretend-pragmatics (“here’s something we’re pretending to care about for the purposes of this hypothetical; presumably we think there’s some correspondence to the real world but we may not specify exactly what we think it is”). If we don’t have something like that, why pick one theoretical justification over another?
And I don’t feel like my justification is lacking in theory. It’s not that I’ve done a bunch of experiments and said “this seems to satisfy my pragmatic desires but I don’t know why”. I have a theoretical argument for why it satisfies my pragmatic desires.