I am rewriting a computational modelling project on modelling the stock market, so I am doing a bit of background reading. Fortune’s Formula by William Poundstone is a good general-interest description of some work from the 1950s onwards, on developing models for both the stockmarket, and gambling in casinos. In terms of mathematical modelling, gambling (aka investing) in the stock market, and gambling in casinos are almost the same — the aim is the same in both: maximise the money acquired while minimise the risk.
As it always easiest to understand a concrete example, let’s consider how a simple model can help us balance risk and reward, in the context of betting on a roulette wheel. In a conventional European roulette wheel, if you place a £1 bet on red your probability of winning is 18/37=0.485, and if you win you get a total of £2, your £1 bet plus winnings of £1. Now, 0.485 < 0.5, so on average you lose money, as you gain £1 a fraction 0.485 of the time but lose £1 0.515 of the time. It is then not worth betting.
However, what if you know a roulette wheel is biased, so that red comes up not 0.485 of the time, but say 0.55 of the time. Then on average you win 0.55 and lose only 0.45 of the time. Thus you should be able to make money betting on this roulette wheel. But how?
Naively, you might think that you should just go in, put all your money down, and start gambling. But this is risky, say you start with £1. If you place all of your money down, then you have a 0.45 chance of losing this, and then you are bankrupt, and will go home broke. Indeed, even if you bet 50 p a time, then with probability 0.45² = 0.2025 you will lose the first two bets and again go home broke. However, if you only bet tiny amounts then although it will then be very unlikely that you will go broke, you won’t win much.
Like most problems involving making money, people have been motivated to think about this problem of making as much money as possible while minimising the probability of going bankrupt — strictly speaking the problem is that of maximising the probability that you make money while minimising the probability. In particular, shortly after Claude Shannon laid the foundations for modern IT, his colleague at Bell labs, John Kelly, realised that Shannon’s ideas also applied to gambling. He came up with the Kelly criterion, now used both in gambling, and in the world of finance. To see how it works, take a look at the plot above.
The plot shows what are called cumulative distribution functions, which here means that each point indicates the probability that will have at least much money, in this case after gambling 200 times, in each case investing a fraction f of the amount of money you have at time. The blue circles are for where you gamble a fraction f = 0.2 of your funds, at each round of roulette. The vertical dashed line shows £1 — your original stake. The blue circles cross this line at about a probability of 0.45, so after 200 rounds of betting you will only have won more than you have lost 45% of the time.
The orange crosses shows the results of gambling a fraction f = 0.1 of your current funds, in each round. This is the amount the Kelly criterion says is optimal here. Note that the probability you are ahead is now almost 80%. So, using the Kelly criterion has increased the probability that you come out ahead, but note that the blue and orange curves cross, with the blue curve higher (and higher is good, higher probability of at least that amount) above about £10. So, by betting larger fractions, 0.2 not 0.1, of your money each time, you are less likely to come out ahead. However, the small probability that you win a large sum is somewhat less small.
This is a nice example of a risk/reward trade-off, if you bet larger fractions you will probably lose money, but you could win big. There is no one perfect strategy to gamble, no perfect value of the fraction f. The best strategy for you depends on your appetite for risk, if you want to minimise the risk of losing money go for small f, but if you want to maximise (the always small) probability of leaving with £100, go for a large f.
Chance & Necessity 