Originally Posted by
ugard
Would this distribution give an indication of whether the expected "expected return of future opportunities" was higher or lower than current expected returns? If so, how would this affect current choices? Are you referring to some sort of bizzare attempt at consumption smoothing, whereby people see returns to future bets will be higher and so become less risk averse and chase expected returns rather than percentage growth in the present? Of course, as you state, the future returns would also have to be discounted.
Maybe a simple example will illustrate:
Assume zero time-value-of-money and full-Kelly. Let's say that today we have a bet at 2.000 with and edge of 20%. This implies a win probability of 60%. Single-period Kelly stake is then 50%. The bet is settles tomorrow night. Tomorrow afternoon, after the other underlying event has already begun, there's a 10% chance there will be a betting opportunity at odds of 2.000 and edge of 90%. This implies a win probability of 100% the opportunity exists. Single-period Kelly stake on that would be 100%.
Call the quantity bet on the first event x
1 and the quantity bet on the second event x
2.
For multi-period optimization we'd have this:
Code:
maximize U =
90% * [ 60% * ln(1+x1) +
40% * ln(1-x1) ] +
10% * [ 60% * ln(1+x1+x2) +
40% * ln(1-x1+x2) ]
wrt x1, x2
subject to a budget constraint of x1+x2 ≤ 1
Solving, we see that utility is maximized at:
(x1, x2) ≈ (14.89%, 85.11%).
Obviously, that's a pretty contrived example, and could still be solved with standard single-period contemporaneous Kelly (using a 90% push probability for bet # 2), but you should get the idea.
Originally Posted by
ugard
Yes, I was careless in my definition of edge, it should be ER - 1.
I think you mean to say that you are defining edge as ER + 1, right?
Originally Posted by
ugard
Originally Posted by
Ganchrow
a Kelly player will be indifferent between betting and not betting. For any positive edge (assuming no minimum bet size) the player will strictly prefer to bet. For a bet paying out at 1:1, a Kelly player will choose to bet his edge.
This I think is the point I am misunderstanding. The relationship between Kelly (maximising expected
growth) and standard EU theory (maximising expected
value). According to Kelly, one is indifferent between a gamble at 0% edge and no gamble, but according to standard EU, the same person (who has log preferences) would refuse a bet with 0% edge (a fair bet), and hence the observed behaviour of payment of insurance premiums.
That's actually untrue. According to Kelly, one would strictly prefer no gamble to a gamble at 0% edge (0% edge => p = 1/o)
Code:
U(no gamble) = ln(1) = 0
maximize U(gamble of x at no edge) = [ln(1+(o-1)*x) + (o-1)*ln(1-x)]/o
wrt x
subject to 0 ≤ x < 1
U' = [ (o-1) / (1+(o-1)*x) - (o-1)/(1-x) ] / o = 0
implies x* = 0, U = 0 and U'' < 0 for o > 1.
Hence U(no gamble) = U(gamble of x at no edge) iff x = 0.
and U(no gamble) > U(gamble of x at no edge) iff 0 < x < 1.
Originally Posted by
ugard
Is Kelly an extension of standard EU that allows one to choose the optimal amounts to stake, rather than simply deciding whether the fixed gamble presented should be taken or not?
It's not an extension, it's just expected utility given log prefs. The stake that maximizes E(U) given log prefs is the Kelly stake.
Originally Posted by
ugard
I'm in the UK, and assume by "program" you mean "course", in which case I'm reading largely economics (a bit of politics too).
I kind of guessed the UK part based on the combination of your proper diction and usage of "behaviour" and "maximisation". I meant in what school's economics department are you studying? Although perhaps you're currently an undergrad?