Is Kelly Dead?

[Mathcapper]

In a discussion on the topic of optimal wagering strategy last week on Derek Simon's TwinSpires podcast, Dave Schwartz (producer of HorseStreet Pars) proclaimed,

"The Kelly Criterion for all practical purposes today is dead."

His reasoning? He believes we no longer have the ability to predict final odds – one of the two main components of the Kelly formula – due to the large shifts that now seem to regularly occur in last-minute betting and after the race has begun. [If you've seen my related posts on this subject you know where I'm headed with this ;-)]

For those who may not be familiar with it, the Kelly Criterion is a formula developed by John Kelly of Bell Labs in 1956 that has been adopted by horseplayers (and gamblers in general) to determine the optimal percentage of one's bankroll to wager in order to maximize one's returns over the long run.

Under the Kelly formula for two-outcome events, known as Bernoulli distributions,

f = (pb – q )/b

where:

f = fraction of your current bankroll to wager
b = final odds on the tote board
p = probability of winning, as determined by your fair odds line
q = 1 – p = probability of losing


Your Expected Value (also known as your edge, or your expected ROI) is defined as follows:

EV = (probability of winning)x(final odds) + (probability of losing)x(-1)

So your edge = pb – q, which is the same as the numerator in the Kelly equation.

And since b in the denominator of the Kelly equation is the final odds,

f = (pb – q)/b = edge/odds

So the fraction of your bankroll you should wager on each bet is simply your edge divided by the final odds.


Example:

Fair Odds = 4-1
Final Odds = 5-1
p = 1/(Fair Odds+1) = 1/(4+1) = 0.2
b = 5
q = 1 – p = 0.8
edge = pb – q = (5 x 0.2) – 0.8 = 0.2 = +20%
f = (pb – q)/b = edge/odds = 20%/5 = 4%

So with a 20% edge getting 5-1 odds, one should wager 4% of his current bankroll.


For those of us who like to think in terms of odds rather than probabilities, your edge, along with the corresponding Kelly bet, can be more easily calculated using this alternative form of the equation:

edge = (final odds+1)/(fair odds+1) – 1     [edited]

f = [(final odds+1)/(fair odds+1) – 1]/final odds     [edited]

So in the example above,

edge = (5+1)/(4+1) – 1 = 0.2 = +20%
and f = [(5+1)/(4+1) – 1]/5 = .04 = 4%


The drawback to Kelly's formula is that it's very sensitive to the inputs. Even a small overestimate in one's perceived edge can end up decimating one's bankroll over time rather than optimizing it. To account for the sensitivity and volatility, most Kelly practitioners use what is known as "fractional Kelly," where they bet only a fraction, say 1/2 or 1/3, of what full Kelly suggests.

This is why it is so important to be able to estimate final odds (as well as fair odds) accurately. After all, if you don't have a good idea of what the final odds are going to be, how can you possibly figure out how big your edge really is, and how much you should wager accordingly?

If however, one has the means to accurately predict final odds, such as can be accomplished by calculating the equivalent win parlays from the DD Will Pays, then this is no longer a problem, and the Kelly Criterion can be used just as effectively as it ever was.

So contrary to Dave's belief, even though John Kelly himself has been long since gone from this world (he died of a stroke on a Manhattan sidewalk at the young age of 41), his theory, as well as his legacy, is alive and well in the world of horse racing.


Rocky

[TGJB]

Rocky, that one is a beauty.

[richiebee]

How many graduate level credits did I earn for reading and trying to understand
Rocky\'s post? I hope there won\'t be a pop quiz after the handicapping seminar.

[Lost Cause]

richiebee Wrote:
-------------------------------------------------------
> How many graduate level credits did I earn for
> reading and trying to understand
> Rocky\'s post? I hope there won\'t be a pop quiz
> after the handicapping seminar.


Good..i\'m not the only one...After reading that all I could say to myself is why am I betting against this guy...

Great post though Rocky...I think...

[FrankD.]

Richie,

You will have a quiz tomorrow morning at breakfast, Mike\'s @ 700; don\'t bother calling TGJB he will just be getting in!

Awesome post Rocky for sure.
Good luck,

Frank D.

[rezlegal]

Posters- including Rocky- are encouraged to Google Kelly criterion and go to  the Wikipedia link.In fact,Rocky gave us the Kelly Criterion for Dummies version. It is hysterical.

[Mathcapper]

Rez - I wasn\'t trying to give a dissertation on the subject. All us horseplayers (and gamblers in general) need to know is to wager according to our edge/odds.

The Wikipedia entry isn\'t very practical - it\'s mostly proofs and esoteric formulas that are unusable by anyone except the computer guys. For a better background on the subject (although the math is still pretty deep), check out Kelly\'s original paper:

A New Interpretation of Information Rate

[TGJB]

He wasn\'t saying you we\'re a dummy. Trust me.

[P-Dub]

I still feel like a dummy.

[Mathcapper]

LOL - maybe I am a dummy for misinterpreting his comment - sorry Rez :-)

[paniolo]

FORTUNE\'S FORMULA (The Untold Story of the Scientific Betting System That Bet the Casinos and Wall Street) by William Poundstone is an excellent and enjoyable read on Kelly Criterion. Based on the positive reaction to Mathcappers excellent post,FWIW I recommend board members check it out. It is all text no math degree required.

[alydar61]

Honestly, my eyes were already glazing over at \"In a discussion\".

But I must say, I was able to glean an important piece of information from this post while I was zoning. I am wagering too much of my bankroll in each race, especially at Saratoga this summer.

Thank you, Rocky. Sometimes the easy stuff gets overlooked.

[Mathcapper]

Thanks for the suggestion paniolo - this looks great, can\'t wait to check it out.

Rocky