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The Timeform Knowledge: Staking and the Kelly Criterion

How you stake can mark you out from the crowd
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Simon Rowlands examines a popular staking plan in the latest edition of the Timeform Knowledge.

"“A New Interpretation of Information Rate, by J. L. Kelly, Jr” came out in 1956 to very little acclaim – not least because the maths behind it is beyond most mere mortals..."

Now then: you have a bet, possibly arrived at through some of the processes described in earlier modules of The Timeform Knowledge; you know what are its odds; you know what you think should be its odds; and you know how much money in total you have at your disposal. What should you do with that information?

In terms of general principles, it would be difficult to improve greatly upon the guidance issued in tongue-in-cheek manner by Timeform's founder and lifelong atheist, Phil Bull, in his "Ten Commandments", published in 1970:

"Let thy stake be related to the depth of thy pocket and to what thou regardest as the true chance of the horse; that which hath the greater chance deserveth the greater stake."

Or, as it is more technically known, the Kelly Criterion.

The eponymous John Kelly was a remarkable character: a gun-toting Texan daredevil, physicist, early computer scientist and bona-fide genius who was reportedly inspired by corrupt wagering on the pre-recorded TV show "The $64,000-Dollar Question" to look into how a bettor, believing they have an edge, should stake optimally.

Optimally, in this context, meant the bettor never risked losing their entire bank. Kelly's findings had a more immediate and intentional application at the time in what was the nascent field of information theory.

Anecdotally, Kelly initially presented his idea to his bosses at Bells Laboratories as a system for betting on hypothetically fixed horse races (this was the US: that sort of thing never happens in Britain, obviously) but was persuaded to make his content less controversial.

"A New Interpretation of Information Rate, by J. L. Kelly, Jr" came out in 1956 to very little acclaim - not least because the maths behind it is beyond most mere mortals - but gained popularity among a small number of investors and professional card players in the 1960s and 1970s and is now considered by many to be the final word on optimal staking, be it on the Stock Exchange or in betting markets. At least in theory.

The "at least in theory" bit is because there are a few real-world issues which mean that many bettors - including some who turn over vast sums on horseracing markets - use an adapted version of the Kelly Criterion rather than its purest form.

Fortunately, the abstruse maths have been simplified by some over the years, so that mere mortals can use a modified version of it also, including by Nick Mordin in his 2002 book "Winning Without Thinking".

Mordin recommends the following: 1) Work out the chance, in % terms, that you think the horse has of winning; 2) Multiply this by the horse's actual odds; 3) Subtract the probability, again in % terms, you believe the horse has of losing; and 4) Divide the result of the above by the horse's actual odds.

The result is the % of your betting capital you should risk.

This can also be expressed as: (((P*O) - Q )/O) = % of bank risked, where P is the % chance of success, Q is the % chance of failure (equal to 100 minus P), and O is the horse's actual fractional odds (or decimal odds minus one).

In an example in which you think a horse should be even money (50% chance of winning) and its odds are 2/1 (or 3.0 decimally), the equation gives you (((50*2) - 50)/2) = 25% of your bank.

The real-world issues referred to a bit earlier include: that the bet you strike could affect the odds you secure (especially in pool betting); the unrealistic assumption that you will always get your bet accepted or matched to its full stake (or even at all); the difficulty in applying Kelly to coinciding and contingent events; and, perhaps most importantly, that - however good our judgement and/or algorithms - our assessment of probabilities may be flawed.

Anecdotally, some betting syndicates and big bettors use "Half-Kelly", "Quarter-Kelly" or "Discounted Kelly", whereby the suggested stake is factored down by an amount linked to the confidence in the assessment of the probabilities in the first place.

It is mathematically sub-optimal but seemingly a sensible ploy in reality.

Nonetheless, the basic principle of the Kelly Criterion - that you should stake according to the edge you have and the likelihood of the event occurring - is sound and should inform a sensible bettor's approach, even if that bettor does not use the Kelly Criterion itself.

Discipline is highly important in staking as well as in selecting the bets you bet upon, with a rigid adoption of a version of the Kelly Criterion perhaps being the most disciplined approach to staking imaginable.

As with many other things, you either have discipline or you don't, but it is possible to acquire it if you put your mind to it (and it is definitely possible to lose it if you don't).

Remember, whatever else you do, to avoid staking which varies alarmingly for no good reason, and always be aware that no staking "system" ever devised can make up for fundamental deficiencies in selecting your bets.

***

Kelly is clear about what you "should" do if striking a conventional bet - in which you allow the bet to run its course - but not about what to do in the at-times far more complex world of modern betting. In particular, the Kelly Criterion does not specifically allow for the possibility of closing your bet, for better or worse, before the event in question has reached its conclusion.

For instance, you might think a horse is rotten value but still back it because you are even more convinced that you will be able to trade it to secure a guaranteed profit. A horse you priced up at 8.0 (7/1 in fractional odds) is available at 5.0 (4/1), which is bad value, but you think you will be able to trade out at 4.0 (3/1) and lock in a profit regardless.

Strictly according to Kelly, you should not strike the first bet, as it has a negative expectation in the event itself. However, a bet of, say, £100 at 5.0 (yielding £400 profit) gives you the opportunity of laying between £100 and £133.33 at 4.0 and guaranteeing a neutral or profitable outcome, come what may.

Even if you replace your assessment of the odds (8.0), with your assessment of the odds at which you anticipate being able to lay back your initial bet (4.0), you may face a Kelly dilemma. For if the intention is always to trade - sometimes advantageously, sometimes disadvantageously - then you will never lose all of your stake and may in reality lose only a small proportion of it on those occasions on which you lose at all.

Far from "Discounted Kelly", or one of its cousins, a trading strategy might be seen to encourage more aggressive staking, for the down side (and the up side!) will not be as great as if striking a typical win/lose bet, in which you scoop the pot or lose your stake.

The principles of value still apply, and the suggested strategy for the above is to attempt to end up with an outcome which approximates your original assessment of the horse's "true" odds.

By laying back £132 at 4.0, you would be left with a profit of £4 if the horse wins (profit of £400 less profit of £132*3, or £396) and £32 (the £132 stake you have taken less the £100 stake you first laid out) if it loses.

A much simpler and valid, if sometimes less optimal, alternative is to use Betfair's "Cash Out" facility to lock in profit or minimise loss, of course!

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