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kubikulann
kubikulann
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August 18th, 2019 at 7:15:02 AM permalink
There is no ‘‘solution’’. Because there is no exact definition of optimal.

What is optimal depends on the individual preferences. In gambling or investing, that includes risk aversion level.
IOW, what do you mean by ‘‘too high’’? There is no objective answer. Each his own. « Risk neutrality » means just that: no risk is deemed too high.

Kelly answered one specific goal: optimizing safe long term growth. In a specific environment (repetition, indépendance, proportionality, risk neutrality,…).
This game, or the St Petersburg one, do not apply to Kelly reasoning, because wager sizes affect the form of the bet.

Long term reasoning and targeting only Expectation leads to infinity paradoxes. A Kelly bet is never the whole bankroll. In these two examples, using it leads to wagering the whole lot. Paradox.
Last edited by: kubikulann on Aug 18, 2019
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unJon
unJon
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August 18th, 2019 at 8:10:30 AM permalink
Quote: kubikulann

There is no ‘‘solution’’. Because there is no exact definition of optimal.

What is optimal depends on the individual preferences. In gambling or investing, that includes risk aversion level.
IOW, what do you mean by ‘‘too high’’? There is no objective answer. Each his own. « Risk neutrality » means just that: no risk is deemed too high.

Kelly answered one specific goal: optimizing safe long term growth. In a specific environment (repetition, indépendance, proportionality, risk neutrality,…).
This game, or the St Petersburg one, do not apply to Kelly reasoning, because wager sizes affect the form of the bet.

Long term reasoning and targeting only Expectation leads to infinity paradoxes. A Kelly bet is never the whole bankroll. In these two examples, using it leads to wagering the whole lot. Paradox.



Can’t we still intelligibly ask the question about this game of what bet sizing strategy maximizes long term growth while holding the risk of ruin to zero?
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
kubikulann
kubikulann
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August 18th, 2019 at 8:16:37 AM permalink
INTRODUCING VARIANCE
It was shown that
EV = 1/w * (-w) + (w-1)/w * 2 = (w-2)/w = 1 - 2/w .
(Please note this is negative for w<2)

What is the variance?
E(V2) = 1/w * w2 + (w-1)/w * 22 = (w2+4(w-1))/w
VarV = E(V2) - (EV)2 = (w2+4w-4)/w - (w-2)2/w2
= w+4- 4/w - 1 +4/w -4/w2 = w + 3 - (2/w)2 .

The variance ratio is
VarV / (EV)2 = [w + 3- (2/w)2] / [ 1 - 4/w + (2/w)2 ]
= [w3 + 3w2 - 4] / [ w2 - 4w + 4]
= (w-2) + 9 + 24/(w-2) + 20/(w-2)2
It increases with w (w>2).

This is the usual case: to increase return you have to increase variance. Except that here, your return rises marginally to 1 while your variance rises proportionally with w. Going to the max is definitely not a good idea.

Maximizing some sort of « utility function » F= aEV - bSD would lead to a much more reasonable
a/b is the individual’s preference.
Last edited by: kubikulann on Aug 18, 2019
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kubikulann
kubikulann
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August 18th, 2019 at 8:27:35 AM permalink
Quote: unJon

Can’t we still intelligibly ask the question about this game of what bet sizing strategy maximizes long term growth while holding the risk of ruin to zero?

One has to define precisely « long term growth » and « risk of ruin ».
With Kelly’s definition of long term growth (i.e. average relative rate of growth of the capital), there is incompatibility with the second condition.
Why is that? Well, grossly, you can win 2$ max. Amassing those 2+2+2+…, your bankroll increases. Yet the gain does not, so it represents an ever lower share of your current bankroll. Your relative rate of growth is diminishing. So the Kelly criterion decides to favor the case with the maximal rate, i.e. the soonest.

On the other hand, technically whenever you keep 0.01$ of your capital untouched, you are not ruined. So I guess there should be some more explicit condition on how much you want to safeguard... Suppose you possess a capital K and you want to protect part p from loss. You then simply establish that your gambling (or investment) bankroll is B=(1-p)K. This bankroll then can meet ruin.

So, in the end, the so-called Kelly answer to RS’s question seems to be:
Establish the amount that you are comfortable with losing, and bet it entirely.
Last edited by: kubikulann on Aug 18, 2019
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unJon
unJon
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August 18th, 2019 at 8:51:38 AM permalink
Quote: kubikulann

One has to define precisely « long term growth » and « risk of ruin ».
With Kelly’s definition of long term growth (i.e. average relative rate of growth of the capital), there is incompatibility with the second condition.
Why is that? Well, grossly, you can win 2$ max. Amassing those 2+2+2+…, your bankroll increases. Yet the gain does not, so it represents an ever lower share of your current bankroll. Your relative rate of growth is diminishing. So the Kelly criterion decides to favor the case with the maximal rate, i.e. the soonest.

On the other hand, technically whenever you keep 0.01$ of your capital untouched, you are not ruined. So I guess there should be some more explicit condition on how much you want to safeguard... Suppose you possess a capital K and you want to protect part p from loss. You then simply establish that your gambling (or investment) bankroll is B=(1-p)K. This bankroll then can meet ruin.

Ah. I see. I guess what I would do is call my utility function equal to ln(bankroll).
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
kubikulann
kubikulann
Joined: Jun 28, 2011
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August 18th, 2019 at 8:57:08 AM permalink
Quote: unJon

I guess what I would do is call my utility function equal to ln(bankroll).

Many economists do that. But it has been shown to be a less than accurate estimate of real people utility functions — admitting there are such things, which other research has proven false.
Actually, there has to be a utility for negative values (debts and the such) and U should be concave on some domain to account for lottery and casino gambling behaviour.
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kubikulann
kubikulann
Joined: Jun 28, 2011
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August 18th, 2019 at 9:24:40 AM permalink
Quote: kubikulann

With Kelly’s definition of long term growth (i.e. average relative rate of growth of the capital)

CORRECTION
geometric average of relative growth factor.

1+R = [(1+r1)(1+r2)…(1+rn)]^(1/n)

and not

R = [(r1)+(r2)+…+(rn)]/n
.
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RS
RS
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August 18th, 2019 at 11:30:56 AM permalink
Quote: kubikulann

There is no ‘‘solution’’. Because there is no exact definition of optimal.

What is optimal depends on the individual preferences. In gambling or investing, that includes risk aversion level.
IOW, what do you mean by ‘‘too high’’? There is no objective answer. Each his own. « Risk neutrality » means just that: no risk is deemed too high.

Kelly answered one specific goal: optimizing safe long term growth. In a specific environment (repetition, indépendance, proportionality, risk neutrality,…).
This game, or the St Petersburg one, do not apply to Kelly reasoning, because wager sizes affect the form of the bet.

Long term reasoning and targeting only Expectation leads to infinity paradoxes. A Kelly bet is never the whole bankroll. In these two examples, using it leads to wagering the whole lot. Paradox.


Isn't the kelly criterion optimal betting for maximized bankroll growth? Certainly there is an optimal bet or at least there exists a bet which is too high. If you bet minimum ($2), then you make no EV. If you bet 100% of BR then you have 100% ROR.
01000101 01110000 01110011 01110100 01100101 01101001 01101110 00100000 01100100 01101001 01100100 01101110 00100111 01110100 00100000 01101011 01101001 01101100 01101100 00100000 01101000 01101001 01101101 01110011 01100101 01101100 01100110 00101110
kubikulann
kubikulann
Joined: Jun 28, 2011
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August 18th, 2019 at 12:12:36 PM permalink
Quote: RS

Isn't the kelly criterion optimal betting for maximized bankroll growth? Certainly there is an optimal bet or at least there exists a bet which is too high. If you bet minimum ($2), then you make no EV. If you bet 100% of BR then you have 100% ROR.

<sigh>
Do you read what I write?

Kelly optimizes one specific objective, under a series of sepcific conditions. Those conditions are not met by your game.

« Optimal » and « too high » are subjective concepts. What is better in the eye of one player is worse for another.

EV and Risk of ruin are two different measures. One is an amount, the other is a probability.
« If I spend nothing, I get no butter. If I spend all, I have no money anymore. There *must* be an optimal level of spending. »
Yes... depends on your personal preference between butter and money.
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