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

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How could you identify when these strings would be most probable to occur given that there is a fixed 50% in each flip? Seems like simple math so I'll be blown away if you have an edge.

Six years ago I was on pace to become well off from online poker but my game couldn't adapt to make it worth it anymore. It was hard to throw in the towel.

ESPN's WSOP program got people interested because they gave away player's hands and only showed exciting pots. I rode a 2 year poker bubble of easy money at pretty high stakes driven by loose credit, covenant access and lack of government regulation. Now that the pots are 1/3 the size and people are betting small ball - forget about it.

No. Again, you're looking at the wrong resolution. I'm not making ANY predictions about singular events. I'm talking about STRINGS.

For strategies that have shown/observed a greater than 50% win rate, it's mathematically impossible to have an equal number or depth of strings for winners and losers. Thus, you know that going forward, IF you are able to observe a greater than 50% win rate, then there will be more positive strings and they will generally be of greater length. The higher the win rate, the more pronounced the phenomenon becomes.

Your argument is exactly why the progressive strategy works. I do not have a considerable edge in determining the next coil flip (50/50), but I do know that if I progressively increase my bets 50% on each winner, over the course of strings, I can take advantage winning strings and minimize losing strings.

it's like this.....

What if I bet you, 10 to 1 odds, (My $1000 against your $100) that if you flipped a coin 10 times, that the distribution would NOT be even (i.e. that you'd observe H-T-H-T-H-T-H-T-H-T exactly or vice versa with T starting)...would you take it? You'd be a fool if you did. The odds of that happening are .5^9 or just under 2%. That would mean that I was giving you 10:1 on a 50:1 probability, a good bet for me.

Heck, even a shorter string, it becomes apparent. what are the odds of tossing 4 times, and having the outcome be EXACTLY H-T-H-T or T-H-T-H?

.5^3 or .125 or 12.5% or 1 out of 8 times roughly.

That is to say, that 1 out of 8 times, the prob will be exactly even and evenly distributed along the sequence.

With a coin flip, over the course of infinite sample series, the even distrubution negates the string contributions from the progressive strategy (i.e. for every time there's a string that helps a 50% add to winner strategy, there's another string of oscillating outcomes that has a negative effect for the same strategy).

However, as I said, with outcomes that are NOT 50/50, this type of strategy starts to have merit.

No, the probability of any individual outcome is still 50%. So you cannot predict anything going forward. However, you know that the likelihood of an uneven distribution (for a finite sample set) is MORE than the likelihood for an even one....so you know that if you stick to a string progressive strategy, it works out.

The problem here is that the longer you go, the more an oscillating outcome will hurt you (for every string of 3, a H-T-H-T sequence hurts you)...in cards, as discussed previously, there are more inherent strings becaust it's not mutually exclusive, a winning hand in BJ indicates an increased probability in losing hand (which is why/how people tend to count cards).

However, as I stated earlier, for outcomes that are NOT 50%, the strings HAVE to be unevenly distributed, regardless of the sample size (if you truly observe win ratio above 50%).

I agree with this. I've walked through the logic and convinced myself that if you have say a 60% success rate and therefore can expect more winning strings than losing strings over a large sample, this progressive betting method could make good sense.

The hard part is determining in real time, amid changing market conditions, what your system's success rate is right now. You can develop pretty good idea of the system's success rate in the past, but you're up against a constantly moving target. A drawdown could be an unlucky streak, or alternatively the first sign that the market has changed in a way that removes whatever edge you would have expected based on the testing process.

That's just my own psychological bugaboo though and not a commentary on this progressive betting system in general.

Dangerous conclusion. A 60% success rate does in no way justify on its own to make use of progressive betting.

Take a dice and have a look at the two events

-> 1 or 2 or 3 or 4 -> you win (probability 2/3)
-> 5 or 6 -> you lose (probability 1/3)

Now your winning probability is about 66.67%, but there is no way you can make use of progressive betting, as the next event (throwing the dice) is not contingent on past results.

The following 2 users say Thank You to Fat Tails for this post:

Again, no one is disputing that. What I'm simply saying is that IF you develop a strategy based on experience, backtesting, whatever, and it enjoys a greater than 50% win rate in the past, and you have a reasonable expectation that it will in the future...then strings become useful.

What you're essentially trying to argue is that there's no way of predicting the outcome of a single event, and that's true....but I'm not talking about singular events...I'm talking about sequences of multiple samplings and the observations you gain.

Furthermore, just because you enjoy a 60% win rate over a given set of trades...everyone knows that it's not a guarantee you'll see that in the future, but it has an "expectation" which is exactly why we all trade....if it was ALL truly random and unpredictable, then we'd all be fools to try to make money at trading (or we'd simply look at someone who makes money and chalk it up to blind luck).

If you have an observed edge, then string theory becomes useful. It does NOT however always result in an increase in profitability, you have to run the numbers to see....

Like I said, for a 95% win rate for a strategy that makes a small amount 19/20 times but then loses a large amount (on average), by progressively increasing your positionsize by 50% of your previous winning, you would in fact make less money than you would if you were to simply increase your positionsize each time by the full amount of profit realized on the previous trade.

Progressive strategies only increase the profitability of systems dealing with events that are not mutually exclusive.

However, what it WILL do for you is decrease risk and drawdown. A 95% strategy that goes say 5 profit/50 loss, (assuming it's still able to overcome comissions and slippage) will be able to reduce some of the drawdown experienced by increasing positionsize only 50% after each successful trade, rather than the entire 100% of profits....

Additionally, returning to the "base" or first positionsize for a loser, becomes a very smart risk management strategy for minimizing losing strings (more so for strategies that are closer to 50%, because with a 95% winner, the odds of consecutive losing strings are much less).

When you evaluate these methods based off account size, rather than positionsize, you will see that it does not increase your profitability with respect to account size (i.e. if you had enough in reserve to increase your profit amount after a loser, you'd have been better off simply putting that reserve in play for all the positions)...but again, it does help to "smooth" things out a bit. By "rat holing" 50% of your profits each time, you "ensure" that you're retaining at least SOME profit.

This becomes VERY useful when the profit/loss goals are very large with respect to account size.

Let's say for a second that your broker allows you to trade 4 CL Contracts with a $5k account (yes they are out there), that means that every 125 ticks, you earned a 100% margin (minus fees and slippage).

So if you're strategy had a profit goal of 125 ticks, a loss limit of 125 ticks and an "observed" historical edge of 60%, then what you would discover is that there were strings of events in your favor. IF that edge were to continue, and over the course of the next n sequence of trades you were to up your positionsize by 50% after each win, (and return to the original positionsize for a loser) you'd discover that it made more money.....as increasing your position size by all the profits would eventually result in you losing ALL of it back once a loss ocurred. Furthermore, a string of losses would be catastrophic. By varying positionsize, you CAN take adavantage of strings (not singular events, but STRINGS) to smooth out your equity curve and ensure you make sustainable and growing profits, rather than grow like crazy, only to see it all burn away very quickly.

I think it all hinges on whether there is any real relationship between subsequent trades, where the outcome of one trade can affect the following trades. If the trades are all independant of each other then Fat Tails is completely correct, there is no edge to be had from progressive betting or martingaling. Doesn't matter if you have a winning system or not; if there is nothing in the system that causes a relationship between trades, then there is no advantage to be had doubling down or progressive betting.

However there may be situations where there is a real relationship between the results of a trade and the results of subsequent trades. As someone pointed out, Blackjack is a game where there can be a real relationship between 'trades' because the game uses a deck of cards which gets depleted and therefore the odds change as the composition of the cards remaining in the deck changes.

In trading, especially systems that are in the market most of the time and take every single setup presented without skipping many, there can be situations where the results of one trade do affect the probabilities of subsequent trades. For example if a trend following system that takes every trade suffers a string of losses but continues to take every single trade, then the probability that each new trade will be a winner will increase somewhat. This is one big reason it is so important for these types of systems to take every single trade and not cherry pick only some of the trades.

Same with a mean reversion system that keeps taking the same trade even after a loss. Eventually , if you keep trying over and over on the same MR trade, you will finally catch that falling dagger and your odds should improve with subsequent attempts.

David Viraldi at CSS Analytics recently posted an interesting indicator idea that relates to measuring this tendency for trend following systems.

Last edited by GoldStandard; April 3rd, 2011 at 03:04 PM.
Reason: spelling & grammar

The following user says Thank You to GoldStandard for this post:

No. This is the fallacy. It only can become useful if the outcome of the current trade depends on the outcome of the prior trade. In case there is no dependency (example coins, dice, roulette) it is not useful. Progressive betting is only useful, if the trades are (positively or negatively) correlated, and if these correlations are stable.

This is exactly the point.

It is easiest to understand the difference by comparing dice and black jack.

Dice: The current event is not contingent on the prior event. Even if you had a long string of losing bets, this does not increase the probability that the current bet is a winning bet. Progressive betting cannot be applied. The idea of strings is a fallacy.

Black Jack: The current event is indeed contingent on the prior event. Cards once played cannot be played again. This is why a string of losses changes the probability that the current bet is a winning bet. It makes sense to count the cards (by attributing them a value) and use progressive betting strategies. Edward O.Thorp has described this in his book "Beat The Dealer".

It is not a question of a high or low winning probability, but a question whether consecutive bets are correlated or not.

The following user says Thank You to Fat Tails for this post:

Even if the events are mutually exclusive it follows.

here's an example.

Let's say you had a typical 6 sided die....and if you rolled a 1,2,3 or 4, that's considered a win. If you rolled a 5 or a 6, that's considered a loss. The predicted outcome would be 66% yes?

So if you were to plot a series of 100 rolls, you'd find that there were not only more wins than losses, you'd also find that the strings of wins and losses were different as well, there'd be MORE winning strings and the average length of a winning string would also be more.

So, if you know that the string of 4 wins, is much more likely than a string of 4 losers, now you can start to realize the phenomenon of progressive strategies.

If you were to increase your "bet" (positionsize) 50% for every winning trade, you'd realize MORE profits than if you were to simply wager 1 unit. Moreover, if you were to return to the original 1 unit wager each time you lost, you'd MINIMIZE the losing strings and only lose 1 unit for each loss in a successive losing string.

You have no future expectation of any single event...they are mutually exclusive, the probability of a winner on the next roll is the same as the roll before. Again, you're looking at it from a single event. I'm not approaching the strategy/progressive betting system from a single event. I'm approaching ALL events with the same strategy.

Your point would be taken, if I were to try to "cherry pick" and increase bet size once a certain number of wins or losses had been achieved. THAT would not work, as the prob is still the same for the next roll.

However, if I employ a progressively increasing bet for wins, and a return to 1 for losers, over time, it WILL yield more than simply wagering 1 unit over all rolls.

The problem here is again, in order for this to work, you have to have an equal profit/loss ratio and you either have to have reserve on hand to increase the wager sizes or the profit/loss amounts have to be significant to the size of the account.

Thus, the strategy WILL NOT increase you're overall profit margin (if you needed additional cash on hand to employ it, you'll find you'll make MORE profit by simply wagering as much as you can on every bet). However, you'll find that employing that strategy helps keep you from getting "felted" (your chips taken to zero) in the event of a large run of losing outcomes and also help you to ensure you keep at least some profits on fortunate winning strings.

It's simply a money/risk management strategy.

I cannot explain it any further than that.

The following user says Thank You to RM99 for this post:

Betting 2, or even 200 units per roll (assuming you have a system with positive expectancy) will similarly yield more than simply wagering 1 unit over all rolls. This does not mean you improved your odds. All you did was increase your position size.