De Econometrist neemt een statistische kijk op de wereld.
With over 1.6 billion people gambling at some point during a given year and over 4.2 billion people having gambled at some point in their lives, it is safe to say that gambling is booming. The risk versus reward element gives players a small rush of excitement and this is also the reason why gambling is so addictive for a lot of people. Lots of social events also include gambling, for example a poker evening. Long story short: there is a big chance that you will be gambling at some point in your life. But if we gamble, we also want to win, right? So why not use our mathematical and probability skills to minimize the probability of losing? In this article, I will present to you the so-called martingale betting technique, which is commonly referred to as the ‘never-lose’ or ‘infinite wealth’ betting strategy. How does this technique work and is it really that safe? Let’s find out!
Martingale betting is focussed on 50/50 betting, that is, there is a 0.5 probability of winning and a 0.5 probability of losing. Thus, examples of events where we can apply martingale betting are coin flips or betting on either red or black coming up on a two colored die. The technique was invented in the 18th century in France. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.The best way to show how the technique works is by giving a mathematical example. Suppose we’re betting ‘M’ dollars (M > 0) on a coin landing heads, where, if it lands head we get our bet doubled back. Obviously, a coin lands either heads or tails with a 50/50 probability, so we can apply the martingale technique. Suppose we lose the first bet, which implies that we lost ‘M’ dollars. We now double the amount of our previous bet, thus we bet ‘2M’ dollars. Suppose we lose again, which implies that we lost ‘2M + M = 3M’ dollars. We double the amount of our bet again, so we bet ‘2M*2=4M’ dollars. Since the probability of losing three times in a row is quite small (0.5*0.5*0.5 = 0.125) let us assume we won the bet which yields us ‘8M’ dollars. In total, we’ve now spent ‘M + 2M + 4M = 7M’ dollars and we won ‘8M’ dollars. Clearly, 8M > 7M for M > 0 and we made a profit of M dollars. The martingale technique was successful!
Notice that how long we can keep on doubling our bet depends on how much we want to spend (and how big our savings account is). The martingale technique always works if we have infinite wealth, since our probability of losing ‘n’ times, where n is approaching infinity, is equal to . But no one has infinite wealth, so we conclude that the martingale betting technique is not a rock-solid never-lose betting strategy. The martingale trap is the phenomenon where people assume that the probability that for example a coin lands heads six times in a row is impossible. Intuitively, we have to agree that it is not very likely. But it will happen eventually and people will eventually lose their money. Let’s look at the following fun R experiment: let the number ‘0’ represent a coin landing heads and let the number ‘1’ represent a coin landing tails. Assume that the coin flips are identically and independently distributed. Clearly, the coin flips follow a Bernoulli distribution with p = 0.5 (recall that for p = 0.5, the Bernoulli PMF is given by . We apply the martingale betting technique on the coin landing heads. Let us first simulate ten draws from a Bernoulli distribution with p = 0.5 by running ‘rbern(10, 0.5)’. R gives us: 0 0 1 1 0 0 1 0 0 0. Let us look at this sequence of numbers. We are betting on the coin landing heads, which is represented by a ‘0’. Clearly, we made a profit using the martingale betting technique in this case and we only had to double our money once (the longest series of 1’s has length 2). But now, suppose we simulate 50 coin flips. We run ‘rbern(50, 0.5)’ and obtain 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0. As you can see, this time we were quite lucky and made a big profit if we would apply the martingale betting strategy (all 0’s are wins). But suppose we were betting on the coin landing tails. With only 50 coin flips, there was already a case where we had seven 0’s, which means seven losses in a row. If your starting bet was pretty big and considering the fact that your bets increase exponentially we can assume quite some people would be out by now. Then, the martingale technique would not have worked, since we do not have infinite wealth. I’m sure that you agree with me that if we simulate say 1000 draws we would obtain even crazier losing streaks.
Let me show you that the probability that you will lose all your money is even bigger if you apply the martingale betting technique on a bet where the probability of losing is even a tiny bit larger than the probability of winning. Let q be the probability of losing (q > 0.5), let B be the amount of the initial bet (B > 0) and let n be a finite number of bets the gambler can afford to lose. Clearly, the probability of losing all n bets is (i.i.d., multiplication rule). When all bets are lost, we have a total loss of . The probability that the gambler does not lose all n bets is (1-). Then, the expected profit per round is . Since, clearly, B > 0, we get a negative expected profit if q > 0.5. The larger n becomes, the lower our expected profit is. So say the probability of losing is 0.6 and we do 9 bets where we start with 1 USD, our expected profit is 1(1 – ) = -4.159. So we can conclude you should most definitely not apply the martingale betting technique on bets where the odds are against you.
The martingale betting technique, intuitively, seems like a technique in which you’ll barely ever lose. But taking a closer look at it, we see that it is not as solid as you might think. The event of a long losing streak is way more likely than most of us think, in particular if the total amount of bets we place increases. As I have also shown, the probability of winning using the martingale betting technique becomes even smaller if we bet on something where the odds are against you. If you do not want to lose any money, the best thing to do is to simply not gamble. But, if you can keep yourself under control, it’s never wrong to try your luck!
Dit artikel is geschreven door Lars Beute