One of the oldest questions in the field of investment – and sports betting of course – is how to maximise the return on a diverse portfolio of possible plays, each of which may be a very risky bet on its own.

The aim of diversification is to reduce the risk, whilst maintaining as much upside as possible. Many strategies have been developed to do this and they tend to involve the notion of ‘utility’: the idea that one may attach a value to all the possible outcomes of a position and that these values may not necessarily be equal to the monetary return. For example, would one be willing to bear a 50% risk of losing 90% of an investment in return for a 50% chance of making a 1000% return? The answer depends on your attitude to risk and this may be encapsulated in a ‘utility function’, which attaches a subjective value to these two outcomes.

One of the most famous utility functions ever studied is log-utility, or the so-called ‘Kelly criterion’, after John Kelly, Jr, who in 1956 first showed how to apply it. This criterion says that the utility of a monetary return will be equated to the logarithm of that return. It doesn’t matter what base one uses for the logarithm, as all that matters is the relative utility of the various outcomes. In essence, this criterion claims that the satisfaction of doubling your investment should be the same as the dissatisfaction associated with the risk of losing half your investment. Immediately, one can see that this is a subjective judgement. However, there are certain theoretical benefits associated with this criterion: It has been shown that maximising log-utility will always beat any other strategy ‘in the long run’, notwithstanding John Maynard Keynes’ pithy observation that “in the long run, we’re all dead!” So, if you’re willing to wait for the long run, perhaps log-utility is the function for you. In any case, it is certainly popular with sports bettors, who generally temper its volatile nature by using a so-called ‘fractional Kelly’ strategy, in which they apply the criterion to a fixed fraction (say 50%) of their bankroll, rather than all of it. This fractional Kelly strategy is known to reduce the risk considerably, whilst maintaining most of the upside.

…there is no guarantee that your bankroll will not get very close to zero in the short term – if you have a run of terrible luck.

Another useful property of log-utility is that it is risk-averse in one important sense: the utility associated with wiping out your entire bankroll is minus infinity (log x grows larger and more negative without limit as x approaches zero). This means that if you arrange your investments or bets to maximise log-utility, you should never completely wipe out your bankroll. On the other hand, there is no guarantee that your bankroll will not get very close to zero in the short term – if you have a run of terrible luck. There are, however, many other utility functions which have this desirable risk-averse property (zero risk of ruin) and which are more risk averse than the Kelly criterion. For example, the family of utility functions of the form 1–x^(-p) where p is positive parameter, are also well worth considering. The log-utility function is recovered in the limit as p tends to zero.

The problem of applying the Kelly criterion to a set of mutually exclusive outcomes, such as horses running in the same race, was solved by Kelly in his original paper. However, the problem faced by sports bettors is usually how to distribute their stakes among a diverse set of independent events, each with various possible outcomes. This is the situation if you are considering a number of Asian handicap bets in the European football leagues, where the matches are all played at the same time and you judge that there is some positive expectation in each but that this expectation also varies among those matches. This is a much harder task than the horse racing problem and it has no analytical, closed-form solution.

I addressed this problem in a paper I wrote for the Journal of the Royal Statistical Society in 2007. In this paper, I showed that if you had some model which could simulate the outcomes, then it is possible, using various numerical algorithms, to optimise not only the log utility, but also any utility function of the form given in the previous paragraph. This is useful, because usually we do have simple models which we can use to simulate outcomes even from quite complex sets of events, which needn’t necessarily be independent. As long as one can run a Monte Carlo simulation of the dependencies between events and their various outcomes, it should be possible to optimise your chosen utility function. There is a link to a free version of that paper below. In it, you can also find references and further discussion of the various points made in this short article:

Algorithms for optimal allocation of bets