How advanced mathematics and gaming theory can help you as a trader

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I have shown that for a given percentage win you gain is greater if you increase your position size when you are having a string of wins than if you keep your position size the same. That's why all the blue dots are on the right hand side of the red blobs exept for win/loss ratios close to 50%

The strings are highly relevant nt because the bets are correlated but because they occur all the time in random sequences. You can calculate what size strings to expect for a given win/loss rate and they are much longer than most people expect.

Your point about risk of ruin is mathmaticaly impossible. If your normal risk is $500 and you increase by $250 per win when you are betting $5000 you have just made profits of $47250 because you must have had 17 wins in a row to be betting $5000

Last edited by Nickemp; April 12th, 2011 at 05:08 AM.

What you have created is a bet size modulator. You let the betsize grow and then reduce it back to the starting value. This just increases the average bet size. If you have a higher win rate, it will give you a higher average bet size than for a smaller win rate.

The outcome of your bets are not influenced by or correlated to the bet size. If you have doubled your bet size this does not change the win rate of the following bet, so it just doubles your expectancy.

The bet size modulator is a funny machine, which modulates the bet size all the time for no reason. Very entertaining, but not useful for non-correlated bets. Instead of doing this you can calculate the probability for each winning string and once adjust the bet size. Let us look at an example and take a win rate of 60% and look at the probabilities for strings of wins

loss 40% -> bet size 1.0
1 win (actually a string loss/win) 24% -> bet size 1.5
2 wins (loss/win/win) 14.4% -> bet size 2.0
3 wins (loss/win/win/win) 8.64% -> bet size 2.5
4 wins (loss/win/win/win/win) 5.18% -> bet size 3.0
5 wins (loss/win/win/win/win/win) 3.11% -> bet size 3.5
6 wins (loss/win/win/win/win/win/win) 1.87% -> bet size 4.0

I have added the prior loss to make the events mutually exclusive, so that the probabilities will add up to 100%.
Your betting modulator will then vary the bet size according to the probabilities of strings. The average bet size will be.

This will probably converge to a value close to the double of the original bet size. So instead of using the betsize modulator, you can also simply double up your bet size for all bets and will get a similar result.

Where this approach makes sense

In trading the events are non-random, and there are correlations. Otherwise technical analysis would be useless junk and markets could not be traded by technical strategies.

The approach that you presented is part of the Anti-Martingale betting systems. There are two advantages linked to this:

(1) Typically the win rate of any strategy is not constant over time, but will vary. Let us assume that the win rate of a breakout strategy with defined target and stop loss increases from 55% to 70%. In this case the anti-Martingale betting system will increase the bets automatically. Then decrease the bets again, when after a few months the system returns lower win rates again. The anti-Martingale strategy therefore lets you trade size, when the win rate (in the recent past) has been higher, but reduces size, when the win rate gets lower. This is the opposite of doubling, when in a losing position.

(2) In case that you have positively correlated bets, your expectancy after a win will be higher than after a loss, so you should bet more after a win and less after a loss.

However, if bets are negatively correlated, the anti-Martingale system may wreak havoc, as it bets high, when the expectancy is low.

Think about it. It is like a puzzle.

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This is where the rubber meets the road I think. In a thought experiment you can establish the win rate by definition; in the real world the win rate can be guessed at but is subject to change without notice.

What is the implication for real-world trading? Should bet size be varied based on prior outcomes? This might be a good idea if you theorize, for instance, that the market goes through cycles when your trading method is favored and those when it is not.

Then again, we've also seen that this "string" approach can be affirmatively harmful if the win rate is right around 50%, as you get chopped up by all the W-L-W-L sequences. So if you regard the current win rate as an unknown then this approach might be a bad idea.

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Just to be clear: The issue is not the win rate, but the expectancy. You can have a high positive expectancy with a win rate of 30%, such as is typically the case for trend following strategies. If you refer to a win rate of 50% with equal outcomes (which was the original assumption), then you should not bet anyhow, as there is no edge. In this case, the Anti-Martingale approach is harmful, because it simply amplifies the outcome of "no edge".

Where it is very harmful, are bets that are negatively correlated. Imagine that you play Black Jack and you increase your bet size after a string of wins. That is the moment, when you should stop playing immediately and certainly not increase your bet size. So the approach is particularly harmful, if consecutive bets are negatively correlated, such as is the case with Black Jack.

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Question No.1 : Are price moves at a given resolution (1 sec, 1 min , 30 min, etc.) positively or negatively correlated? Or otherwise put, is the market trending or range bound?

Question No. 2: Are your trades positively correlated, negatively correlated or not correlated?

If you can answer these questions, you are a certainly a genius!

However, what could be done, is to evaluate your trade history and check for auto-correlation. But you would need a rather large sample, as correlations are typically changing a lot.

So you should assume that your trades are not correlated, or that the current correlation cannot be established. In this case the concept of progressive betting does not make sense.

This is the whole point: Progressive betting is the key to success in playing Black Jack, but if you apply it to trading, well, you should better read the book by Ralph Vince to focus on money management and forget about progressive betting.

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To push the thought experiment a bit further, let's assume you have a system with an overall positive expectancy of $50 per trade and an overall success rate of 60%.

If you break the data down further, however, let's say you can classify this as two different systems. Sometimes the system is actually a loser, with an expectancy of ($100) per trade and a success rate of 50%. During successful periods, the system has an expectancy of $200 per trade and a success rate of 70%. The difference is whether market conditions favor the system's approach on a given day.

The key question for the trader is whether, right now on any given day, you're trading the losing system or the winning system. One thing that is clear is that you would expect there to be more (and longer) strings of winners under the winning system than under the losing system. In fact, a sizeable cluster of wins over any given period would be the primary evidence you'd use to determine which system had been in place over that period. If wins are occurring more frequently (or perhaps more accurately, if expectancy is higher), this is what tells you that conditions were favorable for your trading approach over that period.

Might it therefore make sense to increase bet size when you've noticed more winners, or higher expectancy, across the most recent sample of trades?

I don't expect a definitive answer, just raising this as a thought to mull over. I think the answer really depends on your view of how frequently market conditions change. If conditions can be expected to change constantly, so it's basically random whether or not the system will be favored on a given day, then that would be one thing. If conditions can be expected to linger for weeks or months at a time, that would be another. These are empirical questions that you can't really answer with a thought experiment, but that may be worth thinking about.

Hi guys, I'm new here and apologize if this has already been posted as I didn't read this entire thread, but Yale has an entire semester on game theory at academic earth. Lot's of math but good stuff.

I'd post the link but this is my first post.

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Unless I've missed something, I think the issue isn't that the formula is flawed per se, but rather that the results are not optimal.

This is because the profits posted in those charts would be higher if you used a larger bet on all trades than if you used a larger bet only during strings of wins. To see this clearly you'd need another chart showing the results of a fixed position size using a larger bet.

Increasing bet size after a win would make sense if the odds of a win increased with each win, but the assumption in these examples is that the odds of a win are fixed. In that case what you want to do is find the optimal bet size and use it on each trade since each trade is equally likely to give you a profit.

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