Martingale property

So I was reviewing some notes and came across Wilmott’s explanation of a Martingale property

The coin-tossing experiment possesses another property that can be important in finance. You know how much money you have won after the fifth toss. Your expected winnings after the sixth toss, and indeed after any number of tosses if we keep playing, is just the amount you already hold. That is, the conditional expectation of your winnings at any time in the future is just the amount you already hold: E[Si|Sj, j < i] = Sj. This is called the martingale property. Isn’t this in violation of basic rules of probability? specifically, the property of large numbers? If I won the first five coin tosses, shouldn’t the odds of me winning go down in the 6th, 7th,…coin toss? Either this is wrong or I’m missing something really simple here. Pls help

i think they mean since the probability of a coin toss is 50/50, you are likely to lose in the future just as much as you are to win. so regardless of the pattern of wins/losses, the amount you can win is what you already have

this is a simplistic view, and you are taking it a step farther than what they considered. for instance any one toss will have equal odds, but consider tossing a coin 10 times - anyone would hyposthesize 5 heads and 5 tails obviously. but suppose you get 4 tails in the first few tosses, within this scope the odds of getting heads should be higher

^ Wrong, coins have no memory. If you toss a coin and heads come up the first 10 times, the probability is still 50/50.

However, this is assuming a fair coin that has not been tampered with. You can start questioning THAT hypothesis. For the above example however, it’s an assumption so you don’t question it.

This is called the Gambler’s Fallacy.

good thing i dont gamble

The Martingale System is why casinos have maximums on their table games.

Two points

  1. I should’ve used the term “expectation of my net earnings” as opposed to saying that odds of me winning go down with 6th or 7th coin toss. Apologies for this.

  2. So how am I supposed to decide when to stop gamblng on this coin toss bet? i.e. if the expectation of me winning Nth time vs N-1 are effectively the same?

Yes coins have no memory but if the law of large numbers is true (which it is) then wouldn’t the N more tosses which you choose to make, eventually even out the first 5 wins you’ve had i.e. marginally reduce your winnings by 5/N each time you toss a coin? Albeit theoretically?

I see what you’re saying and I think this is due to the initial 5 flips becoming a small percentage of the total number of flips, essentially making them insignificant. This is why the longer you are at a casino, the worse your chances of winning money (considering the odds are always in favor of the casino).

Good point and I did actually consider this, but for law of large numbers to work, N doesn’t have to be infinity, it just says “large enough”. Forget that I won first 5 tosses, say if I won first N/2 tosses, where N = 100k - then what? In that case, I don’t think 50k wins are too small?

+1

i have the craziest stories with that and martingale lol

Martingales are “fair games”-- you’re expected to lose nothing and you’re expected to gain nothing, in the long run. A random walk is an example of a martingale. Remember, the “best” we can do at predicting tomorrow is to look at today (think back to quant).

If you won in the past that has no utility in predicting what’s to come with a martingale. The expectation of “winnings” is still zero (today/most recent outcome is the best predictor of the future).

The key is to find roulette tables with a good ratio of minimum/maximum bet size…that’s what determines how many times you can martingale. From what I’ve seen, 10x is about as good as you’ll see.

…martingale betting stories are more fun than martingale process stories.

A martingale is a special form of a random walk. It basically is a Brownian motion that can terminate mid-course (gambler’s ruin problem). The expected value for a random walk/Brownian is zero, if it has no drift. So if you have won 5 bucks already and the expected value of the next toss is 0, your future expected wealth must be 5.

Hi, OP. You misunderstand the meaning of that equation. Think of Si as the cumulative winnings, or some other variable with a value that changes in every period. The Martingale property means that the expectation for Si+1 is Si. That is, there is zero drift, and all previous values before period i do not matter. So, if your winnings at time i are $50k, the Martingale property means that your expected winnings at time Sj are $50k. In other words, the expected value of the coin toss is zero.

The Martingale property has important implications in defining stochastic processes and establishing elementary rules in stochastic calculus. If you are interested in this topic, there is a more focused but accessible book, and Willmott actually recommends it. I can find the name when I get a chance.

Got it - Naren said the same thing too, but I like your explanation. I get it now. thanks.

@Book - are you referring to “Wilmott introduces Quant Finance”? bchad recommended it to me on another thread.

Yep, Ohai nailed it. I would just say that in a fair coin toss indefinitely into the future and no transaction cost (i.e no house edge), your expected “future” earnings are 0. The key element being “future” rather than “total”.

That means if you have already won $10 after 5 or 10 tosses (however much you bet), at every new toss, the best guess is that your future wealth won’t change any more. If you’ve made money in the past, lucky you, but it doesn’t affect your estimate of increased future earnings (unless you have reason to believe that the coin isn’t fair).

For stocks, this means that if prices are a random walk, the expected future price is basically the current price, grown by whatever expected rate of return you think it has. This is why stocks are often modeled as a random walk “with drift,” the drift being the required or expected return. Note that if you believe in some notion of value or justifiable multiples, stocks simply cannot be a true random walk, because that would imply that prices can get arbitrarily far away from fundamentals. Still, it is not a bad model for more short-term trends, and also can work as a way to model the fact that much of the news flow in the future is unpredictable and can go either way in favor of or against the stock.

One thing about a martingale (like coin toss counts) is that given an infinite number of tosses, you will eventually cover the entire universe of possible end values. So if you can tolerate infinite losses while wating, and can wait an arbitrarily long length of time, you can walk away with an arbitrarily large amount of money. There’s then some value in the option to stop at a time of your choosing. In practice, house bet limits affect the application of the martingale strategy to reduce risks of losses (which is different than the thing I described in the paragraph above, but related). However, even if there were no house limits, real people do not have infinite funds with which to gamble, and the exponential nature of bet sizes would eventually wipe those out too. Added: I think the book that Wilmott really liked was “Beat the Dealer” by Ed Thorpe. Though Wilmott also was big on the Kelly Criterion, and I think that book was Fortune’s Formula.

^ I read “Beat the Dealer” many, many years ago. Although I’ve forgotten almost all of it, I do recall it being an excellent read.

This made me think. Is this value in a monetary sense? Is there an optimum “walk away now” strategy? I am guessing No for both.

I don’t know enough off the top of my head to compute it, but it is clear that you should be willing to pay more for a game that plays 1000 rounds and lets you stop whenever you want than you would to play a game that forces you to play exactly 1000 rounds and take whatever the final value is.

If you have a game with drift, however, you may want to play to 1000 rounds if you think that the effect of the drift is greater than the effect of the random part.

I’m sure this calculation has been done before, and it would be interesting to think about what you would have to do to compute it. The option guys here presumably know.

I played roulette Martingale style and lost like $800 at atlantic city. problem was they had a limit on bets with not enough hands being dealt per se