post on -EV hedging

[Ganchrow]

[ QUOTE ]
[ QUOTE ]
So you should bet less on riskier events, even with the same edge? I guess this would be to reduce variance/risk of ruin?

[/ QUOTE ]

That's correct. Here's a link about that:

http://en.wikipedia.org/wiki/Kelly_criterion

[/ QUOTE ]You can also check out my first two Kelly articles, Expected Value vs Expected Growth (Kelly criterion Part I) and Maximizing Expected Growth (Kelly criterion Part II).

I'm hoping to have Part III done shortly.

[Performify]

Great contributions to the thread, Ganchrow. Thank you...

-P

[jelly]

[ QUOTE ]

If you knew with 100% certainty that you'd later be able to hedge your position at +100, then assuming zero opportunity cost, the Kelly stake would be 50% of bankroll on the +150 bet and 50% on the +100 hedge. See my Kelly Calculator.


[/ QUOTE ]

I do not think this is optimal result from Kelly perspective. It is optimal combination to maximize EV without risking any money. It gives you 50-50 shot at 25% (of your BR) for free, which is EV of 12.5%. Your other options include 44% of +150 and 56% of +100 for perfect hedge at 12% or maxing wrong side of the freeroll (40% +150, 60% +100) for a 50-50 shot at 20% with EV of 10%.
However say you bet 60% at +150 and hedge the remaining 40% at +100. If you lose your +150 bet, you are left with 80% of BR, if you win the bet, your BR goes up to 150%. So you are betting 20% of your BR to win 50% of BR. In other words, you get +250 odds on a coinflip. Kelly says you can bet 30% of BR here, so you are still pretty safe. Taking it to the next level, you can even argue, that your BR is already at 112% figure, as this would be result of the perfect hedge (ie refusal to bet in the sense of "risking money".
I will try to follow with precise numbers later.

[trixtrix]

[ QUOTE ]
[ QUOTE ]

If you knew with 100% certainty that you'd later be able to hedge your position at +100, then assuming zero opportunity cost, the Kelly stake would be 50% of bankroll on the +150 bet and 50% on the +100 hedge. See my Kelly Calculator.


[/ QUOTE ]

I do not think this is optimal result from Kelly perspective. It is optimal combination to maximize EV without risking any money. It gives you 50-50 shot at 25% (of your BR) for free, which is EV of 12.5%. Your other options include 44% of +150 and 56% of +100 for perfect hedge at 12% or maxing wrong side of the freeroll (40% +150, 60% +100) for a 50-50 shot at 20% with EV of 10%.
However say you bet 60% at +150 and hedge the remaining 40% at +100. If you lose your +150 bet, you are left with 80% of BR, if you win the bet, your BR goes up to 150%. So you are betting 20% of your BR to win 50% of BR. In other words, you get +250 odds on a coinflip. Kelly says you can bet 30% of BR here, so you are still pretty safe. Taking it to the next level, you can even argue, that your BR is already at 112% figure, as this would be result of the perfect hedge (ie refusal to bet in the sense of "risking money".
I will try to follow with precise numbers later.

[/ QUOTE ]

actually ganchrow is correct, kelly theory is to MAXIMIZE the optimal growth of your br, it is not concerned w/ reducing volatility. (which is one of the reasons why it is imperfect, and why most people use fractional kelly; which leads to another discussion i'm not going into, as i believe sportsbetting can be better priced as a series of binary options using the black-scholes model which does take volatility into account)

in your example you can see already see which proportional br betting that will lead to the MAX ev, and that is your answer in terms of kelly

[jelly]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

If you knew with 100% certainty that you'd later be able to hedge your position at +100, then assuming zero opportunity cost, the Kelly stake would be 50% of bankroll on the +150 bet and 50% on the +100 hedge. See my Kelly Calculator.


[/ QUOTE ]

I do not think this is optimal result from Kelly perspective. It is optimal combination to maximize EV without risking any money. It gives you 50-50 shot at 25% (of your BR) for free, which is EV of 12.5%. Your other options include 44% of +150 and 56% of +100 for perfect hedge at 12% or maxing wrong side of the freeroll (40% +150, 60% +100) for a 50-50 shot at 20% with EV of 10%.
However say you bet 60% at +150 and hedge the remaining 40% at +100. If you lose your +150 bet, you are left with 80% of BR, if you win the bet, your BR goes up to 150%. So you are betting 20% of your BR to win 50% of BR. In other words, you get +250 odds on a coinflip. Kelly says you can bet 30% of BR here, so you are still pretty safe. Taking it to the next level, you can even argue, that your BR is already at 112% figure, as this would be result of the perfect hedge (ie refusal to bet in the sense of "risking money".
I will try to follow with precise numbers later.

[/ QUOTE ]

actually ganchrow is correct, kelly theory is to MAXIMIZE the optimal growth of your br, it is not concerned w/ reducing volatility. (which is one of the reasons why it is imperfect, and why most people use fractional kelly; which leads to another discussion i'm not going into, as i believe sportsbetting can be better priced as a series of binary options using the black-scholes model which does take volatility into account)


[/ QUOTE ]

I do not see how this applies here, my example is more volatile than Ganchrow's solution.

[ QUOTE ]

in your example you can see already see which proportional br betting that will lead to the MAX ev, and that is your answer in terms of kelly

[/ 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.

[lotus guardian]

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.

[trixtrix]

i was referring to your earlier pt about the "perfect hedge" to reduce volatility

i agree w/ your later pts actually, it does have a higher ev at risk of adding volatility, that is actually a better representative of kelly.

[jelly]

[ QUOTE ]
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.

[/ QUOTE ]

While perfectly true, your statement is about 5 levels below the current level of the discussion. You might want to reread to understand why I used the example.

OK, enough of being ugly on people just because of their Argy football icon just after our U20 team lost the WC finals [img]/images/graemlins/smile.gif[/img]

[jelly]

[ QUOTE ]
i was referring to your earlier pt about the "perfect hedge" to reduce volatility

i agree w/ your later pts actually, it does have a higher ev at risk of adding volatility, that is actually a better representative of kelly.

[/ QUOTE ]

I see, I figured that it was clear from context that I am using "perfect" in a different meaning than "optimal".

[Ganchrow]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

If you knew with 100% certainty that you'd later be able to hedge your position at +100, then assuming zero opportunity cost, the Kelly stake would be 50% of bankroll on the +150 bet and 50% on the +100 hedge. See my Kelly Calculator.


[/ QUOTE ]



I do not think this is optimal result from Kelly perspective. It is optimal combination to maximize EV without risking any money. It gives you 50-50 shot at 25% (of your BR) for free, which is EV of 12.5%. Your other options include 44% of +150 and 56% of +100 for perfect hedge at 12% or maxing wrong side of the freeroll (40% +150, 60% +100) for a 50-50 shot at 20% with EV of 10%.
However say you bet 60% at +150 and hedge the remaining 40% at +100. If you lose your +150 bet, you are left with 80% of BR, if you win the bet, your BR goes up to 150%. So you are betting 20% of your BR to win 50% of BR. In other words, you get +250 odds on a coinflip. Kelly says you can bet 30% of BR here, so you are still pretty safe. Taking it to the next level, you can even argue, that your BR is already at 112% figure, as this would be result of the perfect hedge (ie refusal to bet in the sense of "risking money".
I will try to follow with precise numbers later.

[/ QUOTE ]

[/ QUOTE ]
With all due respect, jelly, mathematics is not a democracy and its direct logical conclusions are not subject to your personal opinion. [img]/images/graemlins/wink.gif[/img]

In the optimal solution, betting 50% of bankroll on each outcome, the full Kelly bettor would realize expected utility of 50%*ln(1 + 50%*1.5 - 50%) + 50%*ln(1 - 50% + 50%) = 11.1572%. In the suboptimal solution, betting 60% at +150 and 40% at +100, the full Kelly bettor would realize expected utility of 50%*ln(1 + 60%*1.5 - 40%) + 50%*ln(1 - 60% + 40%) = 9.1161%.

The objective of Kelly is not to maximize bankroll EV (a linear optimization model), nor is it (as is true with Markowitz portfolio optimization -- a quadratic optimization model) to maximize EV - k * sigma^2. What (full) Kelly serves to accomplish is to maximize the expected logarithm of bankroll, which in turn is mathematically equivalent to maximizing the geometric mean of outcomes, exponentially weighting for outcome probability. (One rather interesting aside here is that you can show that Markowitz optimization in fact relates to the first two terms of the Taylor expansion of Kelly (recall that ln(1+x) = x - (x^2)/2 + (x^3)/3 - ... + (-1)^(n+1) * (x^n)/n + ... .) This means that Markowitz is a an acceptable approximation of Kelly for sufficiently small deviations from the initial portfolio. I don't know ... at least thought I that was neat.)

So what this mathematically implies is this rather surprising result: 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 probability of that outcome occurring. So in other words, faced with a pure arbitrage opportunity, the portion of bankroll invested in each outcome is not a function of the odds offered on that outcome. (If anyone's interested I'm sure I could easily throw together a proof).

So relating this to the example given above, whether the odds on heads and tails are, respectively, +100/+150, -120/+150, -900/+1000, or -10,000/+15,000, the Kelly optimal allocation, in each and every one of these four examples, will be 50% of bankroll on heads and 50% of bankroll on tails. Similarly, in the case of a biased coin that lands heads 60% of the time, the optimal allocation assuming a risk-free arbitrage exists, will always be exactly 60% on heads and 40% tails. Again, please do check out my Kelly Calculator.