**WARNING: if you are going to attend a lecture by Juan M.R. Parrondo do not read any further, if your interested in the maths contine.**

The man who everyone came to see is Juan M.R. Parrondo was introduced by Noel-Ann Bradshaw and Eduardo Cuntin. This problem is apparently 10 years old. The idea is there is 2 games which if played individually will end up with you losing money (gambling is easier to visualise) but if you alternate between the 2 games randomly then you will win. Confused? read on.

He gave an example, Game A is a simple game you win with probability p1 and lose with p2. The second game Game B is more complicated if your capital is divisible by 3 then you win with prob p3 and lose with p4, if your capital is not divisible by 3 then you win with prob p5 and lose with p6. This should help visualise it

This is what happens when you play the games randomly or the games separate

As you can see playing randomly turns to be profitable (The above was the game being played 5000 times).

Apparently the trick to working out this paradox comes from something called Brownian motor and the probability of each game.

Game A has a small positive gradient and Game B is a mixture of 2 gradients 1 big positive gradient and one shallow negative gradient. Switching between the 2 games means you increase your profits. This website may explain the Brownian motor better.

There was some criticism of the paradox in 2004. They questioned the paper and said if it was not random but they got to pick which game they chose then their odds and profit would go up. This is true but if you deal with more than one person choosing between which game to play i.e. democracy then you will end up losing but random still triumphs.

The lecture ended.

Parrondo's paradoxical games homepage