Kelly bets / Pool size and infinit wealth

[aerokid240]

My question has to the with the pool size and kelly sized bets and bettors' current bankroll

I read the following
"...Even if a bettor possesses infinite wealth, there is a maximum bet producing the greatest expected profit, and any amount above lowers the expected profit.

The maximum bet can be calculated by writing the equation for expected profit as a function of betsize, and solving for the bet size which maximized expected profit.

Example:

calculated probability = 6%
dividend (Decimal odds) = 20
edge = 0.20


total pool size = 100,000
maximum er bet = $416
expected profit = 39.60

.. in the above example the maximum expected profit is a bet of 416. If one made a bet of 2/3 of the maximum, i.e. $277, the expected profit will be $35.5 or 90% of the maximum."

I don't know how they derived the maximum er bet, i.e: $416 and the expected profit of 39.60. Can anyone explain how they arrive at their values ?

[FourLengthsClear]

http://forum.sbrforum.com/handicapper-think-tank/29009-expected-value-vs-expected-growth-kelly-criterion-part-i.html#post250260

http://forum.sbrforum.com/handicapper-think-tank/29841-maximizing-expected-growth-kelly-criterion-part-ii.html

[aerokid240]

When i work out the kelly stakes i am getting 1.0526% as the % of bank roll to wager and 0.2105% as the expected profit

if one has infinite wealth how can get arrive at a maximum bet ? I am just not seeing it.

[FourLengthsClear]

When i work out the kelly stakes i am getting 1.0526% as the % of bank roll to wager and 0.2105% as the expected profit

if one has infinite wealth how can get arrive at a maximum bet ? I am just not seeing it.

There is no such thing as infinite wealth therefore it is moot.

[aerokid240]

is there some way we can factor in the pool size when making bets ?
what is the recommendation when your bankroll is larger that pool size ?

example:
calculated probability = 6%
dividend (Decimal odds) = 20
edge = 0.20


total pool size = 100,000

you bankroll: 100,000,000

[FourLengthsClear]

is there some way we can factor in the pool size when making bets ?
what is the recommendation when your bankroll is larger that pool size ?

example:
calculated probability = 6%
dividend (Decimal odds) = 20
edge = 0.20


total pool size = 100,000

you bankroll: 100,000,000

What sort of pool are you talking about? Is it a Pari-Matuel whereby any bet you make will change the odds?

If so you would need to calculate how much your bet is going to change the odds/dividend and base the 'edge' on that.

[aerokid240]

..yes, for a pari-mutuel pool.
thank you

[mathdotcom]

If you have infinite wealth why would you ever gamble

[aerokid240]

The question was in relation to situations when you bankroll is greater than pool size.

In that given situation you cannot simply bet the recommended kelly bet because of the impact on the odds and thus rate of return.

[mathdotcom]

The question was in relation to situations when you bankroll is greater than pool size.

In that given situation you cannot simply bet the recommended kelly bet because of the impact on the odds and thus rate of return.

Then you make a max bet

Done

[FourLengthsClear]

Then you make a max bet

Done

Not into a pari-mutuel pool you don't!

[mathdotcom]

Not into a pari-mutuel pool you don't!

Thought by pool he meant limit.

I know you'll forgive me for quickly skimming his post

[RickySteve]

Incalculable unless you're the last money in.

[FourLengthsClear]

Incalculable unless you're the last money in.

True.

[MonkeyF0cker]

Incalculable unless you're the last money in.

Even then, you're at the mercy of the track's pool update interval. It's not all too uncommon for odds/pool totals to update around the first turn.

[buby74]

I think the reference in the original post to “infinite wealth” probably means that your bank roll is so great that you are betting far less than Kelly and so the question is how much profit to extract from a positive EV bet in a pari-mutuel pool as if you bet too much you will destroy the EV but if you bet too little you are leaving money on the table.

Leaving aside Kelly momentarily to maximise your profit from a pari-mutual or tote pool this is my formula. It assumes you are the last person to bet and ignores the fact that your bet will increase the size of the pool slightly so it is slightly conservative but adding in the increase in the pool makes the formula very messy.

Let T =the price in decimal odds available when you bet
Let F=The fair price in decimal odds
Let M= T/F (this is equal to the edge +1)
Let P= the amount of cash in the pool with the track take removed.
The fraction of the Pool you should bet is SQRT(M)-1 divided by T
I call this the Murphy fraction in homage to Kelly

The final price after your bet will be T/SQRT(M)

You should never bet more than the Murphy fraction of the Pool to do so reduces your profit.
The example In the original post
T=20
M=1.2
F=1/6%=16.66
P=100,000

Murphy is (SQRT(1.2)-1)/20*100,000=$477

My answer is greater than the answer in the original post (416) which could be due to the 100,000 representing the amount bet before the track take is removed if the track take is 12.5% so that P=87,500 I get 417.


Coming back to Kelly

Using this notation the Kelly fraction of your bankroll to bet is (M-1)/(T-1) so the two formulas look similar but remember Murphy is the percentage of the pool not your bankroll

To work out which amount to use I suggest the following approach

Work out the cash value of Murphy and of Kelly using sqrt(M) and use the smaller of the two. If Kelly is a lot smaller than the optimum amount lies between Kelly calculated based on M and Kelly based on SQRT(M) but I haven’t worked this out.

Apologies if this is already published somewhere I worked this out about 10 years ago but never used it (couldn’t calculate F well enough!).

I calculated the Murphy fraction by maximising profit which is= x*(M-1) where X is the amount staked by you but remembering that the final M will be lower because M=T/F and while the initial T is P/S where S is the amount already staked on the horse, the final T is P/(S+X) where X is the amount staked by you. But the calculus is reasonably straightforward.

[aerokid240]

buby74: I think you got it. when i worked it out using another formula i got 447 as well, I guess originally i didn't take into account the track take.

Thank you