post on -EV hedging

[Ganchrow]

[ QUOTE ]
Max EV is obviously betting full BR on +150.
If you only restrict your options to the area where you do not risk any loss, fine. But I do not think it is optimal so I would like to know if Ganchrow's calculator works with this restriction too, or if it is something else.

[/ QUOTE ]I am unclear as to your question here.

What I can tell you, however, is that the calculator does indeed yield the expected growth maximizing wager allocation given any set of input data.

[crockpot]

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and think a 5% edge at +100 is the same thing as a 5% edge at +250.

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This is true crockpot. You seem to be experienced, why do u size your bets according to your edge and don't consider your odds of winning?

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i do consider them. that change was made a long time ago, and i was only ignoring it at the time because i was betting about 1/10 kelly on average.

crazily enough, i'd like to thank ben for actually contributing to the thread, douche-y as his post was. he is still wrong, by the way, although i'm impressed that his kids' college fund and his wife's kidney constitute only 2% of his bankroll.

i never said you needed a 100% probability of a hedge being available to make this a good play. if you want to be really nitty, you can assign probabilities to all the realistic hedge odds you will be offered. that changes the numbers, not the analysis as a whole.

[Ganchrow]

So anyway, here's a quick and dirty back-of-the-envelope type proof of the proposition I made above. (Specifically, In the face of a positive EV multi-way wager set spanning every possible outcome of an event, the optimal Kelly allocation will be to invest that fraction of bankroll on each outcome equal to the probability of that outcome occurring.) I'm just going to prove it for the two-outcome case. (If memory serves correctly, Kelly's original paper does prove it for the n-outcome case).

Given two events, occurring with probability p and 1-p, and paying out at decimal odds of n and o (where 1/n + 1/o < 1 -- the "arbitrage condition", meaning a risk-free arbitrage exists), with bet allocations (as percent of bankroll) of x and y.

Maximize wrt x,y:
E(U(x,y)) = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y)

Subject to:
x >= 0, y >= 0, x+y <= 1

There's no loss of generality in assuming the third constraint binds as the bettor may always place cancelling bets by wagering in n/(o+n) on event 1 o/(o+n) on event 2, yielding a guaranteed return of no/(n+o) - 1. This return will be strictly positive due the arbitrage condition, 1/n + 1/o < 1. It should also be readily apparent from the arbitrage condition that the non-negativity constraints won't bind, so for the sake of brevity we ignore them here.

So, after introducing the Lagrangean, L, on the third constraint:

Maximize wrt x,y,L:
F = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y) - L * (y + x - 1)

Yielding (the d's should actually be partial derivatives, but I can't get the special character to appear on UBB):
dF/dx = -L + ((-1 + n)*p)/(1 + (-1 + n)*x - y) - (1 - p)/(1 - x + (-1 + o)*y)
dF/dy = -L - p/(1 + (-1 + n)*x - y) + ((-1 + o)*(1 - p))/(1 - x + (-1 + o)*y)
dF/dL = y + x - 1

Setting to zero and solving gives us:
L = 1 - 1/o - 1/n
x = p
y = 1-p


QED

[rjp]

Ganchrow, great stuff again.

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If anyone's interested I'm sure I could easily throw together a proof

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Well if it can be easily done then why not? [img]/images/graemlins/wink.gif[/img]

[Ganchrow]

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and think a 5% edge at +100 is the same thing as a 5% edge at +250.

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This is true crockpot. You seem to be experienced, why do u size your bets according to your edge and don't consider your odds of winning?

[/ QUOTE ]

i do consider them. that change was made a long time ago, and i was only ignoring it at the time because i was betting about 1/10 kelly on average.

[/ QUOTE ]
At tenth-Kelly, the optimal Kelly stake for a 5% edge bet at +100 will still be over 2.5x greater than the stake on a 5% edge bet at +250 (0.5004% versus 0.1974%).

No matter how small the fraction of Kelly that you're betting, payout odds received remain a significant component of the Kelly stake.

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i never said you needed a 100% probability of a hedge being available to make this a good play. if you want to be really nitty, you can assign probabilities to all the realistic hedge odds you will be offered. that changes the numbers, not the analysis as a whole.

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I suggest you read my earlier post in this thread where I discuss this particular issue.

[lotus guardian]

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Kelly is about maximizing your bankroll growth in the long run, not about maximizing your EV. If you bet your whole bankroll on +150 you will go broke eventually and that's not a very good bankroll growth rate.

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While perfectly true, your statement is about 5 levels below the current level of the discussion.

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That was a discussion? I thought the discussion was over the moment Ganchrow gave the right answer and I was just pointing how you were wrong because your examples didn't maximize bankroll growth.

Go suck at football. [img]/images/graemlins/grin.gif[/img]

[jelly]

Ganchrow,
Actually, math is a democracy because everyone can set his own goals, that is why I challenged your solution until you showed the details. Thank you, it was helpful. Obviously I made a wording mistake, make it:
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I do not think this is optimal result from sportsbetting perspective.


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Anyway my counterexample was wrong for the very reason I noted. The BR size really needs to be viewed as 112% so now we are actually risking 32% to win 38%, which is no longer attractive.

[Ganchrow]

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Obviously I made a wording mistake, make it:
[ QUOTE ]

I do not think this is optimal result from sportsbetting perspective.


[/ QUOTE ]

I think what you really mean to say is that it is not optimal from your own personal perspective.

Kelly serves to maximize the expected growth of bankroll. The fact that this isn't your goal (implying, that like most people, your preferences are not logarithmic) is completely acceptable -- economics is very nonjudgmental that way. But that doesn't change the objective fact that for an investor looking to maximize bankroll growth, there is only one solution to the allocation problem above, and that's the one given by Kelly.


[ QUOTE ]
Anyway my counterexample was wrong for the very reason I noted. The BR size really needs to be viewed as 112% so now we are actually risking 32% to win 38%, which is no longer attractive.

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One might argue that your alternate solution was actually wrong (at least from the perspective of bankroll growth maximization) because it was based on personal preference rather than on the underlying mathematics. [img]/images/graemlins/wink.gif[/img]

[trixtrix]

[ QUOTE ]
So anyway, here's a quick and dirty back-of-the-envelope type proof of the proposition I made above. (Specifically, In the face of a positive EV multi-way wager set spanning every possible outcome of an event, the optimal Kelly allocation will be to invest that fraction of bankroll on each outcome equal to the probability of that outcome occurring.) I'm just going to prove it for the two-outcome case. (If memory serves correctly, Kelly's original paper does prove it for the n-outcome case).

Given two events, occurring with probability p and 1-p, and paying out at decimal odds of n and o (where 1/n + 1/o < 1 -- the "arbitrage condition", meaning a risk-free arbitrage exists), with bet allocations (as percent of bankroll) of x and y.

Maximize wrt x,y:
E(U(x,y)) = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y)

Subject to:
x >= 0, y >= 0, x+y <= 1

There's no loss of generality in assuming the third constraint binds as the bettor may always place cancelling bets by wagering in n/(o+n) on event 1 o/(o+n) on event 2, yielding a guaranteed return of no/(n+o) - 1. This return will be strictly positive due the arbitrage condition, 1/n + 1/o < 1. It should also be readily apparent from the arbitrage condition that the non-negativity constraints won't bind, so for the sake of brevity we ignore them here.

So, after introducing the Lagrangean, L, on the third constraint:

Maximize wrt x,y,L:
F = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y) - L * (y + x - 1)

Yielding (the d's should actually be partial derivatives, but I can't get the special character to appear on UBB):
dF/dx = -L + ((-1 + n)*p)/(1 + (-1 + n)*x - y) - (1 - p)/(1 - x + (-1 + o)*y)
dF/dy = -L - p/(1 + (-1 + n)*x - y) + ((-1 + o)*(1 - p))/(1 - x + (-1 + o)*y)
dF/dL = y + x - 1

Setting to zero and solving gives us:
L = 1 - 1/o - 1/n
x = p
y = 1-p


QED

[/ QUOTE ]

years of using fincad and matlab instead of paper+ pen has deterioted my already meager math skills

mega reps to ganchrow for providing the proof and reminding me what i had already forgotten..

i realize now that the goal of kelly is to maximize the log-normal return of your br so that your overall br size will grow at an exponential rate.

the fact that appropriate bet-size of each stake is = to the probability of occurrence is merely a by-product of the solution, not the intended goal...

vnh sir