Does Kelly have the answer?

In my last two blogs, I have asked the question what is the right amount to bet on any given bet, whilst all the time making sure I bet sensibly and gamble responsibly.

In the first blog in the series I posed the question and discussed in more detail exactly what I meant by the title, “Do you lose too often?”. 

I came to the conclusion that it was only possible to answer this question if the average odds of winning bets were known. 

If these were sufficient to cover losing bets and then to make a profit in the long run, then a bettor is making value bets and not betting at bad odds.

Staking Level

Once we know that this is the case, then we need to find a staking level which will mean that the long term winning strategy of value betting is not undone by a short term spell of bad luck. 

As part of the first blog I took two examples of bettors who both make value bets, but are both very different.

1. Andy strikes bets at even money and wins 60% of the time
2. Ben strikes bets at 2/5 and wins 80% of the time

I suggested that if Andy has a betting bank of £200, then he might stake £33.33 per bet. This would mean there is a less than 0.5% chance of losing all of his bank without winning a bet. 

The comparable bet for Ben would be £66.66; this is more than I suggested for Andy because Ben is only half as likely to lose a single bet as Andy. 

Obviously the way I worked this out could be used to find a staking level that gave a different percentage chance of being wiped out without winning a bet. 

If Andy is happy to accept a 5% of wiping his bank out then he can alter his staking level to £50 a bet, this is the maximum he could bet and give himself less than a 5% chance of wiping himself out without a win achieve this. 

Conversely if 0.5% is too much of a chance for Ben, then he can change his staking level to £33.33 per bet, meaning that there is then only a 0.032% of wiping out his bank without getting a winning bet. 

The important thing about this method is that the amount bet on the second bet would remain the same, whether the first one is won or lost and so would the amount bet on the third bet.

Looking at the two examples of bettors above, they were useful to illustrate the argument for level stakes betting when bets have similar levels of risk and return, but in practice they are not realistic examples; very few bettors stake bets with the same level of risk every time. 

Andy's bets in the example are quite simple to calculate, but if he starts striking some bets at 2/1 some at 1/3, then it becomes much more difficult. So how can we know what is the right stake to bet in real life when the assumptions that simplified my example don't hold. 

Well one suggestion was developed in the 1950s by a physicist called Dr. John Kelly. 

Kelly was an American working for a company called Bell Laboratories when he wrote a paper in which he published a mathematical formula known as the Kelly Criterion. 

It claimed to calculate (f) the optimum proportion of a betting bank to bet in any bet where 3 things were known:

1. the net odds received on the bet (b)
2. the probability of winning (p)
3. the probability of losing (q)

Kelly said that in any situation the right proportion of the bank to bet would be given by ((b*p)-q)/b. 

So for the example above, Andy's Kelly bet would be ((1*0.6)-0.4)/1 = 0.2 or £40, and Ben's Kelly bet would be ((0.25*0.8)-0.2)/0.25 = 0.3 or £60. 

This gets around the problem I had with my method of people taking different bets with different odds and different percentages of winning. 

The same calculation can be carried out for each bet individually, and the answer will always be the optimum proportion of the remaining betting bank to use. It is important to note that the Kelly criterion won't lead to the same bet being made each time. 

If Adam bets £40 the first time and wins, then his bank will then be £240, and his Kelly bet the next time would be £48.

Similarly if Adam lost his first bet, then his betting bank would have gone down to £160 and his Kelly bet the second time would be £32. 

The Kelly Criterion is clearly more complicated than deciding on a set amount to stake for each bet and sticking to it. 

If we take Adam for example and look at two of the many scenarios that can happen over six bets, either losing six bets in a row or losing four bets in a row. The tables below compare the staking plan I suggested earlier £33.33 per bet, meaning Adam has at least six bets before he runs out of money, and the Kelly system.

Adam Bets	Kelly Betting	Level Stakes
Adam Wins Six Bets in a Row	£597.20	£399.98
Adam Loses Six Bets in a Row	£52.43	0.02

It can be seen in both of these examples, that the Kelly bet works better than the straight £33.33 a time strategy whether Adam loses all of his bets or wins them all. 

When he loses, the £33.33 a time strategy leaves his bank at zero (0.02 because of rounding). Whilst of course the Kelly bet has also lost him money, he would at least have £52 left after the six bets with that strategy. 

When Adam wins all of his bets, he almost doubles his money with the £33.33 a time strategy (not quite because of rounding), but if he followed the Kelly strategy then he would be £14.72 better off. 

Of course this is a limited illustration of the Kelly system versus the consistent stake strategy. In these examples we are looking at just two of the 64 outcomes that it is possible can happen. Losing six bets in a row would happen 0.41% of the time, and winning six in a row would happen 4.67% of the time. This begs the obvious question of what about the other 95% of the time. 

To show the details of all 64 permutations would be too much here. 

But as a summary, although the Kelly system showed the best results for the situation where all bets were won and when all bets were lost, these two situations were two of only 14 where the Kelly system did best. In 50 of the situations, it was the consistent staking strategy that did best, and the combined probability of these 50 outcomes was over 72%. 

Or another way, if there are two people making the same bets as described for Adam, both with a bank of £200 at the start, and one follows the Kelly system and one bet 33.33 each time, then there is a 72% chance that the bettor with the steady stakes will end up on top after six bets.

Kelly Criterion Conclusion

As well as this objection another argument against the Kelly bet is that it is too complicated for many people. 

In practice, I think the biggest danger with any staking plan is the danger of not sticking to it and therefore having no staking plan at all. 

With my suggestion, a punter who tends to bet on outcomes with similar odds, can calculate the amount they can stake each time to give them a certain percentage chance of losing enough bets in a row to wipe out their bank. 

Once this calculation has been made, it is then simple to stick to that stake for every bet, regardless of the results of previous bets. 

The main advantage of this is that it does not require the punter to do another calculation every time they want to strike a bet; 

I think the main downfall of the Kelly bettor might be that after a while he gets bored of doing the calculation and reverts to gut feeling, or worse, he loses some bets and then starts to chase his losses as described in my last blog.

This is the third in a series of blogs about staking.

The first blog can be found here and the second here.