Each Way Kelly Criterion

I have developed a chrome extension that generates the win overrounds and place overrounds on top of the oddschecker website.  You can get it from chrome store here.

This can help highlight markets where the place book is overbroke and may help to unlock some each way value.

Now I have created a simple javascript calculator to help guide recommendations on staking based on Kelly but taking account of the nuance of each way betting.

Warning.  This calculator is next to useless.  It assumes only one horse is value in a bad each way race, and we know that often several of the rags can be value each time.  If you are making multiple bets on the same race you have to consider the "Real" Kelly Criterion as discussed here. This is going to be a difficult calculation because not only do you need to know the probability for each horses win/place, you also need to know the probability of all the possible combos.  

Warning.  It also assumes that you cannot hedge the bet.  Which in reality you most definitely can (probably).  If you can hedge the bet by laying the win part, the place part or both on the exchange then you can often increase your stakes and lower your cost of variance (read here) and ultimately grow your bank faster than Kelly staking and with less risk.  

Warning.  I also have some misgivings on Kelly in general, are we maximising the correct parameter.  Do most bettors want to maximise their growth?  Or do most bettors want to survive first and then maximise the chance of replacing their income with a betting income?  I think I am in the 2nd camp, Kelly is a high risk strategy that does maximise wealth growth but at the expense of increased risk of large drawdowns/ruin.  This issue is suggested here - "The problem with using Kelly is that no matter what you calculate your expected return to be, your variance will be ridiculously, and to my eye, “uninvestably” high."

However, I will continue.

The Calculator

* enter odds in decimal
* the place odds input will default based on the each way terms
* for true win/place inputs either enter the prob or the odds, no need to enter both

Bank: €
Each Way terms: 1 /
Win odds:
Place odds:
True Win:Prob % Odds
True Place:Prob % Odds



Stake

This is the each way stake that the Kelly Criterion recommends to maximise your expected growth.  For an each way bet you have to place the same stake on the win part of the bet (which may be negative EV) and the place part of the bet.  It's effectively two different bets but you can't take the place bet at the each way terms offered without taking the win bet.

Total Stake

This is the total stake, the stake on the win part plus the stake on the place part of the bet.

EV

This is the Expected Value of the bet defined as (decimal odds - 1) * probability for the win part and the place part.  If you could make the same bet at the same stake over and over again this is how much you would win on average over the long term.

Certainty Equivalent

Kelly seeks to optimise the log of the expected growth (referred to as expected geometric growth rate on the wikipedia page).  So the exponential of this value that we have optimised is the expected geometric growth rate or referred to as the Certainty Equivalent in some of the articles on this subject.  The Certainty Equivalent is what this bet is really worth to you - its how much you expect to grow your bank to.  The bettor would be indifferent between having the listed set of bets and having the Certainty Equivalent.

I am still trying to get my head around why the Certainty Equivalent differs so much from the Expected Value.  The concept that drives it home for me though is the idea of taking a neutral EV bet over and over.  You're long term expected value is break even.  However there is a chance that you will go broke over a period of bad variance.  So if you could take this neutral EV bet forever, with a non-infinite bankroll, then the long term prospect is either you break even or you go broke.  So its not fair to say that it is neutral in terms of cost.  There is a cost of variance and the Certainty Equivalent takes this into account, where as the Expected Value does not.

Optimise

This is the value that Kelly seeks to optimise. It is the sum of the logs of your final positions weighted by the probability.  The reason to optimise the log is because it gives easier equations, and log is an always ascending function.  If you maximise the log of expected growth it is equivalent to maximising expected growth itself.

Kelly Criteria and Each Way

First a brief review of the basic Kelly formula.

The Kelly Criterion seeks to maximise the geometric expected growth (after a number of bets you will have increased your bank by (1+fb) p times and reduced by (1-f) q times):

r = (1 + fb)^p . (1 - f)^ q

where 
f = fraction of your bank roll
p = probability win
q = probability lose (1-p)
b = payout for win (which is decimal odds - 1)


Or more specifically you maximise the log of this:

log(r) = plog(1 + fb) + qlog(1 - f)

You can do this by differentiating and setting to zero and solving the equation.

pb/(1 + fb) - q/(1-f) = 0

As you can see this just requires simply rearranging the equation to write it out in terms of f.  This is nice and straight forward for a simple bet.

Now lets tackle the Each Way derivation in a very similar way.

For each way we have another term, the probability and payout if the horse places but does not win (in that case you win b2 from the place part but you lose your win stake, therefore the payout is b2 - 1).

So now 

r = (1 + f*(b1 + b2))^p1 . (1 + f*(b2 - 1)) ^ p2 . (1 - 2f) ^ q

f = fraction of your bank roll
p1 = probability win
p2 = probability place only (probability place - probability win)
q = probability lose (1- p1 - p2)
b1 = payout for win part (which is decimal odds - 1)
b2 = payout for place part (which is decimal odds - 1)

The other terms are the same but take account of the fact that you lose twice your stake when you lose and you win both win and place bet when you win.  [Note for later in the special case where b2 = 1 the middle term simplifies to just 1 so disappears.]

And now

log (r) = p1log(1 + f*(b1 + b2)) + p2log(1 + f*(b2 - 1) + qlog(1 - 2f)

And we have an extra term when we differentiate which makes the algebra a little more difficult.

p1(b1+b2) / (1 + f*(b1 + b2)) + p2(b2-1) / (1 + f*(b2 - 1)) - 2q/(1-2f) = 0

It gets complicated from here (for me!) but you can see already that the common denominator here will introduce terms like f squared in the numerator.  Ultimately we will have a quadratic equation to solve using [-b-sqrt(b^2-4ac)]/2a.  You'll recognise this formula from your Leaving Cert.

Going back to the special case where b2 = 1.  When you have the equation in the form of a quadratic you notice that where b2 = 1 that the coefficient of f squared "a" necessarily has to be equal to 0.  But you are dividing by 2a to solve the quadratic roots, you can't divide by zero.  So in the special case  you have to use a different equation.  But luckily you notice that the simplified formula where that term disappears for b2=1 can be solved quite easily.

Dig into the "view page source" for the equations (you can see the formula and copy them into your spreadsheets for more convenience).

Here is a snippet.
    
    if (b2 == 1)
    {
      f = (p1 * (b1 + b2) - 2*q) / (2 * (b1 + b2) * (p1 + q)) ;
    }
    else
    {
      a = -2 * (b1 + b2) * (b2 - 1) ;
      b = -2 * q * (2 * b2 + b1 - 1) + p1 * (b1 + b2) * (b2 - 3) + p2 * (b2 - 1) * (b1 + b2 -2) ;
      c = -2 * q + p1 * (b1 + b2) + p2 * (b2 - 1) ;
      
      f = (-b - Math.sqrt(b*b - 4*a*c)) / ( 2*a) ;
    }

Compared

How does this compare to the Basic Kelly stakes.   Suppose you find a horse that the books are pricing to 11.0 (1/5 means you are getting 3.0 on the place). You make 10.0 true to win and 2.0 true to place.  The calculator above indicates to bet €5.24 each way off a bank of €100.  

Taking both bets separately Kelly would recommend €1 on the win and €25 on the place.  If you bet €1 each way then you are way understaking the place.  If you bet €25 each way then you are overstaking the win.  The average could be a fair compromise at €13 x 2.  Our calculator comes in well under this.  It is weighted more towards the Basic Kelly win stake recommendation, this is the riskier side of the bet.  It is also taking into consideration the variance and how the bets are related, so builds in a little more caution.



Further reading (linked above as well)

https://www.pinnacle.com/en/betting-articles/Betting-Strategy/the-real-kelly-criterion/HZKJTFCB3KNYN9CJ

https://www.pinnacle.com/en/betting-articles/Betting-Strategy/placing-a-bet-the-cost-of-variance/46M23L4G43DB8LPG

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

I also came across this on an old forum from "slapdash".  I don't know if slapdash has a typo in what they posted, or it's different just because they are basing calculations off 0.5 stake on win and place, so 1 unit total.  I can't get his formula to marry up with what I have derived.

https://forum.punterslounge.com/topic/58094-kelly-criterion-for-ew-bets/

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