Parrondo's Paradox: A Study

Since hearing about Parrondo’s Paradox (referred to as PP from here on in) and seeing the attempts at making it work for trading I decided to check it out. SMJones appears to have come up with an EA that implements PP in his thread titled Modified Parrondo’s Paradox but without also testing the individual games alongside the paradox then the test is actually incomplete. I know this because you can still use PP and make positive returns because it will at worst average the three games (ie a positive return for a PP system does not mean the other games are individually worse – its more likely that some of them are better).

I decided to create a simple and quick MS Access application (which I will post here later on) that runs the Original and Coop Parrondo’s Paradox simulations to whatever setting you would like to give them. I did create a Excel version but I wasn’t happy with it (mainly because the performance was crap). I haven’t created a proper .NET version because unfortunately I don’t see its worth – yet.

Over the next days I will run numerous tests using my app and post the results and graphs so I can show those of you who have PM’ed me questions regarding PP and also those who are thinking of trying to create an EA with it.

To kick things off I will explain how the simulator works - and this is the Original PP for the time being.

Firstly the overall parameters:
Spread: this is set to 1 pip
Cycles: these are the number of cycles to run in one go before resetting some of tracking variables (including the randomiser)
Iterations: these are the number of complete cycles to run.
Mod: 3 (set at 3 for the classic Original PP but it can be anything from 2 to whatever)

Game Parameters:
There are 3 games, named A, B1 and B2. Each game has its own:
-Win and Loss amounts in pips (in both pre-spread and post spread)
-Win and Loss amounts in steps (defaults to 1 – this is self explanatory in the sim test).
-Probability
-Edge. The edge gives each game its own um edge.

Please seek other sources for more depth regarding the games.

The Outputs
The simulator outputs all the cycle and iteration results to a table (for study purposes) and also outputs three different charts. Chart 1 shows only the “step” results for all of the following five games and combinations whilst Chart 2 shows the pip results for these:
1)PP
2)Game A only
3)Game B only (B1 and B2 combined using same mod rules as the PP)
4)Game B1 only
5)Game B2 only
Chart 3 only shows the step results for PP, Game A and Game B (namely for clarity purposes because game B2 is always sky rocketing)

Because I am using VBA’s semi-broken random number generator I have included several controls to monitor the actually randomness of the values used.

Important things to note:
1)Games B1 and B2 must be “fair” or PP does not work. The formula for making B1 and B2 fair is:
Prob(B2) = (1 - Prob(B1) - Sqr(Prob(B1) - (Prob(B1) * Prob(B1)))) / (1 - (2 *Prob(B1)))
2)The making and breaking of using PP for trading is the tracking of the Steps and Pips individually. The classic coin toss is equivalent to the steps and pips being the same, ie 1 for win, -1 for a loss. This is not suitable for trading.

Comments:
Something to keep in mind regarding PP. By definition the original PP was discovered to show two losing games (Game A and Game B – which is game B1 and B2 together) could be combined together to create an overall winning strategy. And that’s it. As far as I can see this will not work in trading because there will always be a Game (mostly B2) that will always perform better than the PP. I think the trick is to see if rules of PP can be bent such a combination of Games will make the performance of PP much better than any of the individual games. So far, in my opinion, PP will not work for trading but I have achieved some interesting results using parameters very different to the original PP’s parameters

Overall that’s about it, I will be updating this daily with new sim results and some explanations etc.
Here is a sample of Chart’s 1 and 2:
Quite clearly you can see the yellow PP line which positive gains in the Step chart (Chart 1) but it loses in the pips chart (Chart 2)

More later …


http://i573.photobucket.com/albums/s...ep_090727A.jpg


http://i573.photobucket.com/albums/s...ps_090727A.jpg

I'm always a fan at the attempt to gain positive expectancy method with math. I've been pondering PP for quite sometime too. My early conclusion is: in order forparadox to happen, the individual B1 or B2 game needs to be a winning game. And this winning game, if played alone would yield much better return than any combination of it with the other losing games. But I still haven't found how smjones modified PP works (and excellent at that). Tell me if you need any help to research on this.

I think your heading in the right direction but I don't think its that simple. As I mentioned before SMJones' modified PP testing does not test the individual games (as far as I am aware) and is not complete. Combining 3 systems that have an overall positive expectancy will give you a positive result no matter what but combining these 3 systems may actually return you less than just using the highest expectancy system. That is why SMJones' version needs to be compared down to the individual game level.

Anyways, hopefully I explain myself a bit clearer here:

Ok now I want to explain some of the differences in adjustable parameters available for trading compared against the coin flipping exercise as the original PP defines.

Steps vs Pips
The classic original PP method, as we know, uses coin flips and so we monitor the success of the games by adding their wins and losses which I will call “steps”. A win is +1 and a loss is -1. The tracking of the steps is important for Game B only because the tracking of the steps and the mod (which is usually 3) determines whether Game B1 or B2 is used. Apart from that, the tracking gives us the final score for the number of iterations the PP is run for. Now this is fine to show the concept because in the coin flipping exercise a win and loss are both worth the same in value (ie $1 each). In trading this is no good because if we had Games A, B1 and B2 all with 1:1 RRs then we would throw away Games A and B1 and keep B2 (this is of course using the example hit rates of 49% for Game A, 10% Game B1 and 75% for game B2).

So where does that leave us? What is the point of investigating PP for use in trading. Well interestingly enough there are a lot of PP parameters that can be adjusted and tested. I will go through these now and compare them to the coin flipping settings.

Trading lots
In the coin flips each win and loss is the same for each game but in trading we can these. Eg Game A 1 lot, Game B1 0.1 lot and Game B2 3.0 lots. In my sim I enter them as dollars.

Win / Loss Steps
The coin flipping exercise only allows for +1 and -1 for wins and losses respectively but we can change these to whatever we want, eg Game A Win +1, -1 Loss, Game B1 Win +2, -1 Loss, Game B2 Win +1, -3 Loss. These Steps will impact the choice of mod and the switching between B1 and B2. It will not impact the actual hit rates.

Edge and Spreads
With coin flipping there are no spreads. But as we all know in trading there are spreads and commissions which immediately draws us closer to negative expectancy. In this sim we can specify any spread (I chose 1 pip) but we can also give any of the games an edge we want. The edge is given in dollars and this parameter is used to adjust the expectancy of each game.

Sequence, Cycles and Iterations
These are the same as in the coin flipping exercise but are worth a brief mention. In the sim we can select a recurring sequence of Games A and B to a maximum of 5 (Eg AAAAB, ABBAB and so on). The cycles are set around 1000 and each iteration consists of the completed number of cycles. The tracking variables are reset for each iteration and the values are saved for the charts. Most of the sims I run will have a minimum of 1000 cycles and 500 iterations which equates to 500 000 games.

Possible Aims
There was a slight hope that when I started researching PP that using a combination of negative expectancy games could be turned into a positive result but this has not been evident as yet (and I’m not holding my breathe for that one). But what about only one of the games having negative expectancy or two or none? Can the combination of 3 games all with positive expectancy produce better results than any of the game individually (or even Game B in total).

As far as my expectations are towards this little exercise I have none. I will post as many worthwhile charts and results as I can and as quick as I can and I will post the sim itself once I clean it up and make it more user friendly.

There's a simple way to gain certain winning rate but not necessarily has a positive expectancy. Anyone that has fiddling with optimization of fire and forget trading setup may spot this.

Lets say I had randomized setup with 100 pips TP and SL. Put it on any random currency pair. The winning rate probably will go around 50%.

On the same randomized setup, try 100 pips TP and 50 pips SL. the winning rate probably will reduced to 33%.

Or try with high risk low reward like 10 pips tp and 100 pips sl. Win rate prolly goes around 80-90%.

But this skewed RR could interfere with the sequence timing, since b1 or b2 games played based on equity modulus. Judging from the equity curve generated maybe it should not be a problem, or maybe he used a coop PP so the sequence doesn't dependent on equity.

I fear you are wasting your time, and the paradox is a mere quirk of nature (maths) much like primes or the magic number squares.

Whilst a hard factual explanation as to why these anominalies exist or work, and whether such an answer could be useful in trading or any other real world application is debatable, the following remains true about PP.

For a positive expectancy to occur you must employ a game (in the PP example B2) that has a winning expectancy of greater than 67%. Just tinkering with the win rates for 20 minutes will confirm this.

Once this is realised the primary problem that needs solving is defining a trade setup that has >= 67% chance of winning over the long term.

If this is achieved then it should be quickly apparent that it would be futile to place a trade with a 49% win rate, and that you could be certified clinically insane to place a trade with only a 10% chance of winning, since playing only the >67% game results in greater profits than playing any combination of PP game settings.

Spreads, sequences, bet size, and bet win amount will not change the outcome longterm (over and above random distribution) and certainly wont change the fact that you need a trade setup that has >70% win rate over the long term.

Rangebound
yes you are correct when thinking about the classical PP setup. But with regards to trading, winning rates have nothing to do with the equity side of PP, it will only affect the step curve (ie Chart 1 in my previous posts).

If anyone thinks of applying PP and using their actual equity for the mod relationship for B1 and B2 then good luck. If you use your equity for the mod calc then B1 would almost never be used and B2 (which is usually the 70%+ game) would always be used.

In my sim the steps and equity are always split and tracked separately and this allows the parameters to be varied accordingly. For example in the classic PP a win and loss is +1, -1 respectively but what if its +2 and -1 or 0 and +1? Now the actual win and loss amounts can be anything, in classic PP its +$1 and -$1 but for trading it could +100 pips and -100 pips or it could +$30 and -$30 (they dont have to be the same amounts as well).

The most important aspect of PP in trading is the systems' expectancies. Games A and B1 can low hit rates but they can also have positive expectancies whilst B2 might a 70%+ hit rate but have a negative expectancy. The aim of this thread to try mixing up the games' expectancies, hitrates and sequences because definitely the classic PP will not work in trading.

Despite there being so many variations I am approaching this as a bit fun because I do not expect it to work, the aim of it rather is to maybe save others time in testing this on actual trading systems. I am only hold a glimmer of hope something might pop up.

Newborn: I have tried the Coop PP using a coin sim but not put it in this trading sim (yet).

Ok I’ve been busy with other stuff and so without posting other results I am posting the simulator now. It is built in Access 2003 and that is all I have tested it in. I have given it minimal time for testing and making it idiot proof but it is quite stable and fast enough.

A recap on its purpose: mainly to save anyone else time on trying to test actual trading systems and to also highlight the importance of testing all the Games separately and in tangent with the Parrondo’s Paradox sequence. Like I have said before I don’t expect anything exciting to come out of it but this does not mean its impossible for something “worthwhile” to be produced.

Very brief Instructions
The default settings are for the standard classical Parrondo’s Paradox with a 1 pip spread applied.
1)Units: these are set to one and can not be changed.
2)Pre-Spread Win and Lose amounts: You can set these as you like. IF you hit the little recalc_A the post-spread lose amount and expectancy will be automatically updated. It will also make the B1 and B2 games “FAIR” as per the standard “Fair” equation used for PP (you can google this).
3)HitRates: these can be manually adjusted to be anything. The Recalc_B button will adjust the pre and post Spread Lose amounts, and also the expectancies.
4)Edge: this is the edge you can give each game. EG a game with a pre spread win of +100 pips and -100 pips lose amounts with zero edge should give a hitrate of 50%. If you give the game an edge of say 1 pip then the win amount is 100 and the lose amount is 99 and for the sim the hitrate stays at 50%. You will see the expectancy field reflect the edge for the games.
5)Post-Spread Win and Lose amounts: although these fields can be auto adjusted using the Recalc buttons you can also manually set these to anything. The two fields are the pip fields used in the PP calcs and not the pre-spread fields.
6)Expectancy: this is each game’s expectancy. They can not directly be adjusted but can indirectly be adjusted as indicated above. This field is perhaps the most important field in the sim … running sims will make it apparent why.
7)W Step, L Step: these stand for Win step and Lose Step and they represent the step amounts for each win and loss. In the classic coin toss PP these parameters are always set to +1 and -1 respectively. In this sim you can adjust these to any number.
8)Sequence: You can set the game sequence using the drop downs. The max single sequence is 5 places long but you can also set any of the sequence placements to zero which means all games will be skipped for that placement (ie no results will change for that iteration in the sim).
9)Spread: this is the spread in pips. Straight forward.
10)Mod: this is the mod used between games B1 and B2
11)Cycles: this is the number of passes the sim will do after which the results are averaged.
12)Iterations: this is how many times the cycles will be performed. The charts reflect the results of the iterations not each run inside the cycles.
Chart Instructions:
1)Chart 1 : Step Results, these show the running sum of the averages of each iterations step calcs. Ie if W step = +1 and L Step = -1 then for each win and loss the +1 and -1s are averaged across the cycles and summed up for each iteration.
2)Chart 2 : Pip Results shows the results in pips. This is the most important chart because it shows profitability for each game individually and also game B as a whole and also the PP result.
3)Chart 3: Step Results all games, very similar to Chart 1 except it includes all the games. Chart 1 only includes PP, Game A and Game B (combined). Usually Game B2 is the high hitrate game and this makes it difficult to view as can be seen in Chart 3
4)Chart 4 : Binary Results, these show the running sum of the averages of each iteration’s calcs where the step sizes are always +1 and -1 for Wins and losses respectively. Very similar to Chart 1.
You can double the charts to get the data grid if you want to copy and paste the data to excel … not sure why you would want to but you can.

The yellow (PP) lines are the most important lines and the yellow PP line on Chart 2 is especially so (this is the actual pips chart). The goal is for the yellow PP line to be the highest (ie make the most money) out of all the lines.
Keep one thing in mind when using the sim, even though you may get the yellow PP line above all lines it may be a lucky strike and so repeat the run using the same parameters to see if it keeps on happening.

Final Comments:
This sim was a quick attempt to show PP applied to trading using a random number generator. These instructions were thrown together even quicker and upon feedback I will try to update areas which appear to be too difficult to use etc. there are some things I can probably do to the app like upgrading it to accept tick data (hook into SQL server) or OHLC but I will only do that if there’s enough interest.

Now I don’t assume anyone will try to sell or redress this app but if you do intend to then don’t. Its not worth anything – simple.

That’s all … have a play …

I believe it is almost impossible to apply this into trading. PP game is played on the condition that the winning/lossing rate of game A, B1 and B2 are guaranteed consistent. A combination of sequerence is just play around to take advantage of the odd of high win rate of Game B2 inorder to get overall winning at the end. Of course it will work, it all come down to positive profit factor.

The actual market is quite different from the theory of fixed win rate. No strategy can maintain the constant win rate. It sometimes perform better, some time worse. Therefore, the win/lose is not predictable.

I saw smjone's PP system, from his result, you will notice that his overall win rate (or buy or sell) are much higher. This make me wonder that his game A winning rate is actually performed better than the standard PP system. smjone is so incredibly good that he suck at picking the loosing game. LOL. But whatever system he is using, he did extemely well with individual system, and the overall combined PP system is therefore outperform. That is my guess, unless smjone can clarify otherwise.

As rangebound said, if you have a higher win rate game, and lower win rate one, it would be foolish to play the later one. The problem is that our high win rate one often underperform, due to market change. So the only worthwile pursuit is to open trade that has higher chance of winning, based on the fact that your previous trade is a losing trade.

So maybe there is some merit in experimment match two or three different system. Use the highest win rate system as the game A, but use rescue/recover stradegy (ratchet effect) after the a losing game. I think hedging, or at least open the opposite position of existing trade, is quite useful with this. which mean you need a broker outside of usa.

What about having a set up that would indicate the following;

1) Signal for long eurusd
2) You would go long on eurusd and usdchf (negatively correlated)
3) The RR is 1:2

There has been paper on this, dated September 2023 (not the method, but the study of the paradox in the Financial Markets) - and I see this has been a talk for some time here.

hi aus8fx I checked this thread because - well no reason - totally random... can you elaborate a bit more. The three items you listed appear sound .. I assume the negative correlation is close to -1. Not sure if this is applicable to PP?