## Poll

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**37 members have voted**

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.

Quote:kubikulannThere 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?

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(V

^{2}) = 1/w * w

^{2}+ (w-1)/w * 2

^{2}= (w

^{2}+4(w-1))/w

VarV = E(V

^{2}) - (EV)

^{2}= (w

^{2}+4w-4)/w - (w-2)

^{2}/w

^{2}

= w+4- 4/w - 1 +4/w -4/w

^{2}= w + 3 - (2/w)

^{2}.

The variance ratio is

VarV / (EV)

^{2}= [w + 3- (2/w)

^{2}] / [ 1 - 4/w + (2/w)

^{2}]

= [w

^{3}+ 3w

^{2}- 4] / [ w

^{2}- 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.

One has to define precisely « long term growth » and « risk of ruin ».Quote:unJonCan’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?

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.

Ah. I see. I guess what I would do is call my utility function equal to ln(bankroll).Quote:kubikulannOne 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.

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

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.

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

geometric average of relative growth factor.

1+R = [(1+r

_{1})(1+r

_{2})…(1+r

_{n})]^(1/n)

and not

R = [(r

_{1})+(r

_{2})+…+(r

_{n})]/n

.

Quote:kubikulannThere 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.

<sigh>Quote:RSIsn'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.

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.