A technique succeeds in mathematical physics, not by a clever trick, or a happy accident, but because it expresses some aspect of physical truth (O. G. Sutton)

Imagine three unbalanced coins:

**Coin 1:**Probability of*head*=0.495 and probability of*tail*=0.505**Coin 2:**Probability of*head*=0.745 and probability of*tail*=0.255**Coin 3:**Probability of*head*=0.095 and probability of*tail*=0.905

Now let’s define two games using these coins:

**Game A:**You toss coin 1 and if it comes up head you receive 1€ but if not, you lose 1€**Game B:**If your present capital is a multiple of 3, you toss coin 2. If not, you toss coin 3. In both cases, you receive 1€ if coin comes up head and lose 1€ if not.

Played separately, both games are quite unfavorable. Now let’s define **Game A+B** in which you toss a balanced coin and if it comes up head, you play Game A and play Game B otherwise. In other words, in Game A+B you decide between playing Game A or Game B randomly.

Starting with 0€, it is easy to simulate the three games along 500 plays. This is an example of one of these simulations:

Resulting profit of Game A+B after 500 plays is +52€ and is -9€ and -3€ for Games A and B respectively. Let’s do some more simulations (I removed legends and titles but colors of games are the same):

As you can see, Game A+B is the most profitable in almost all the previous simulations. Coincidence? Not at all. This is a consequence of the stunning Parrondo’s Paradox which states that **two losing games can combine into a winning one**.

If you still don’t believe in this *brain-crashing* paradox, following you can see the empirical distributions of final profits of three games after 1.000 plays:

After 1000 plays, mean profit of Game A is -13€, is -7€ for Game B and 17€ for Game A+B.

This paradox was discovered in the last nineties by the Spanish physicist Juan Parrondo and can help to explain, among other things, why investing in *losing shares* can result in obtaining big profits. Amazing:

require(ggplot2) require(scales) library(gridExtra) opts=theme( legend.position = "bottom", legend.background = element_rect(colour = "black"), panel.background = element_rect(fill="gray98"), panel.border = element_rect(colour="black", fill=NA), axis.line = element_line(size = 0.5, colour = "black"), axis.ticks = element_line(colour="black"), panel.grid.major = element_line(colour="gray75", linetype = 2), panel.grid.minor = element_blank(), axis.text.y = element_text(colour="gray25", size=15), axis.text.x = element_text(colour="gray25", size=15), text = element_text(size=20), plot.title = element_text(size = 35)) PlayGameA = function(profit, x, c) {if (runif(1) < c-x) profit+1 else profit-1} PlayGameB = function(profit, x1, c1, x2, c2) {if (profit%%3>0) PlayGameA(profit, x=x1, c=c1) else PlayGameA(profit, x=x2, c=c2)} #################################################################### #EVOLUTION #################################################################### noplays=500 alpha=0.005 profit0=0 results=data.frame(Play=0, ProfitA=profit0, ProfitB=profit0, ProfitAB=profit0) for (i in 1:noplays) {results=rbind(results, c(i, PlayGameA(profit=results[results$Play==(i-1),2], x =alpha, c =0.5), PlayGameB(profit=results[results$Play==(i-1),3], x1=alpha, c1=0.75, x2=alpha, c2=0.1), if (runif(1)<0.5) PlayGameA(profit=results[results$Play==(i-1),4], x =alpha, c =0.5) else PlayGameB(profit=results[results$Play==(i-1),4], x1=alpha, c1=0.75, x2=alpha, c2=0.1) ))} results=rbind(data.frame(Play=results$Play, Game="A", Profit=results$ProfitA), data.frame(Play=results$Play, Game="B", Profit=results$ProfitB), data.frame(Play=results$Play, Game="A+B", Profit=results$ProfitAB)) ggplot(results, aes(Profit, x=Play, y=Profit, color = Game)) + scale_x_continuous(limits=c(0,noplays), "Plays")+ scale_y_continuous(limits=c(-75,75), expand = c(0, 0), "Profit")+ labs(title="Evolution of profit games along 500 plays")+ geom_line(size=3)+opts #################################################################### #DISTRIBUTION #################################################################### noplays=1000 alpha=0.005 profit0=0 results2=data.frame(Play=numeric(0), ProfitA=numeric(0), ProfitB=numeric(0), ProfitAB=numeric(0)) for (j in 1:100) {results=data.frame(Play=0, ProfitA=profit0, ProfitB=profit0, ProfitAB=profit0) for (i in 1:noplays) {results=rbind(results, c(i, PlayGameA(profit=results[results$Play==(i-1),2], x =alpha, c =0.5), PlayGameB(profit=results[results$Play==(i-1),3], x1=alpha, c1=0.75, x2=alpha, c2=0.1), if (runif(1)<0.5) PlayGameA(profit=results[results$Play==(i-1),4], x =alpha, c =0.5) else PlayGameB(profit=results[results$Play==(i-1),4], x1=alpha, c1=0.75, x2=alpha, c2=0.1)))} results2=rbind(results2, results[results$Play==noplays, ])} results2=rbind(data.frame(Game="A", Profit=results2$ProfitA), data.frame(Game="B", Profit=results2$ProfitB), data.frame(Game="A+B", Profit=results2$ProfitAB)) ggplot(results2, aes(Profit, fill = Game)) + scale_x_continuous(limits=c(-150,150), "Profit")+ scale_y_continuous(limits=c(0,0.02), expand = c(0, 0), "Density", labels = percent)+ labs(title=paste("Parrondo's Paradox (",as.character(noplays)," plays)",sep=""))+ geom_density(alpha=.75)+opts

— investing in losing shares can result in obtaining big profits

I’d be careful here. Profits are the result of humans gaming a system which abides by fungible rules; fungible by these same humans. The rules aren’t God, Nature, physics, mathematical physics, Newton, or Einstein. The Great Recession is the most recent example of how rule changing/breaking determines the outcome.

The implementation of Game B (Line 18) tosses the second coin (biased towards heads) more (2/3 of the time), which makes the result less paradoxical.

Game B only tosses the coin 2 when profit is multiple of three. Otherwise, tosses coin 3 which is extremely biased to tail. Game B tosses more times an extremely ‘bad’ coin rather than a significantly ‘good’ one. This is why (qualitatively) Game B results unfavorable.

Nice game. I wrote about it some time ago from a different perspective. See http://www.datanalytics.com/2012/01/12/cosa-prodigiosa-sin-palabras-i/ and the continuation, http://www.datanalytics.com/2012/01/19/cosa-prodigiosa-ahora-con-palabras-ii/

This paradox is wonderful. I enjoyed your links. Thank you.