Oct 18, 2009

Roulette - how (not) to make money on the Martingale Strategy


I was taught a sure way to win money when playing roulette. You bet on color, if you lose you double your bet, if you lose again, you double your bet again. You keep on doubling bets until you win, wining back all your money plus profits equal to the original stake. You have a chance of winning in the first round close to 50%, and you are sure to recover your money later on. Sounds good, doesn’t it?
Even the gamblers admit there is a problem with it. You might not have enough money to keep on gambling.

They are partly right. In reality the strategy has a negative expected value, no matter how much money you have, you could be Rothschild, Rockefeller, Gates or owner of the entire Universe you are poised to lose money when playing the Martingale Strategy.

The probability of winning in the first round is 18/37. The probability of losing your entire fortune is (19/37)^n where “n” is the number of the round in which you run out of money. The probability of winning your initial bet (recovering all money + initial bet) is (1-(19/37)^n-18/37). It turns out that the expected value of the bet is negative for n=1 (when you can cover only round 1) and is monotonically decreasing in “n” (the more rounds you can / are determined to cover the more negative the expected value of your bet gets)… So running out of cash when playing does not hamper your strategy, it saves your ass!

Obviously the expected utility theory can’t explain why anyone would want to play this strategy. You should be super afraid of big losses. This strategy tells you to take huge bets when things go sour. NO GOOD

Prospect theory can’t explain it either. You should be underweighting probabilities close to 50% and overweighting low-risk high loss events (that’s why you insure). You should never play this strategy. NO GOOD

Biases in the evaluation of disjunctive events (Put simply, breaking a chain of same events). But these should be underestimated! But the gambler apparently overestimates the probability that the chain will be broken and so he wins his money back. NO GOOD

The last line of defense is Regression to the Mean. People are stupid and think that the outcome of round X predicts the outcome in round (X+1). (Red in this round predicts black in the next). I actually think this is it…GOOD

The problem is that “Regression to the mean” is just not rich enough, it doesn’t feel right that people should so easily disregard the huge bets they are potentially taking. People probably think that all previous outcomes predict outcome in this round (the more reds have fallen previously the more confident you feel about black falling this round), on the other hand if many reds have fallen in the previous rounds maybe people will expect more reds to fall… I NEED A PSYCHOLOGIST TO EXPLAIN THIS!!!

If you think this is fascinating, you need to read Judgement Under Uncertainty:Heuristics and Biases, A.Tversky, D.Kahneman. It's one of the most fascinating econ papers ever! (Published in Science, authors are psychologists)

All the computations that show why Martingale Strategy Doesn't work are here. (p.s. It's 1 in the morning, I'll need to recheck the math tmw, but it should be OK).

3 comments:

LM said...

I think that the "regression to the mean" or "law of small numbers" really works, i.e. people often behave as if they were drawing balls from a box and not returning them back. Just remember how many times people say something like "I must get lucky on the third try", or I have heard "there was a major airplane crash last week, it is much less probable there will be one again soon".

The second thing you suggest (that people think there will be another red after a chain of red) is a mirror case: when people do not know that the results are just random and try to search for a pattern. E.g. if you believe in the efficient market hypothesis, you might say that people suffer from this illusion if they think that a fund with a good history of performance for last few periods will bring you above average return in the future.

TomasHozik said...

Yeap, exactly! That's what I was trying to say!

One can think that a line of black numbers predicts red number (regression to the mean).

Or, one can think that a line of black numbers predict more black numbers. (finding a pattern where no exists.)- lucky streak in poker and other nonsense.

I don't see any reason for people to prefer A to B. Why should they think that in rulette regression to the mean works but in poker divergence from the mean works? That's not consistent!!! And then you have no explanation why even irrational players play the Martingale Strategy!

Micki said...

I've tried this a few times, starting with a very small bet and have lost more than I've won. The Fibonacci strategy, a similar negative progression, is marginally more successfull.

I've just started blogging about these systems and I would love you to drop by.