Winning by losing
Consider two games of the prisoner's dilemma type with different pay-off tables. The first, L1, has a Pareto optimal (green box) with pay-off 2 to each player, but a Nash equilibrium (purple box) with pay-off -1 to each player. Game theory predicts that both players would choose strategy B, hence the game is given the value -1. The second game L2 has a Pareto optimal (green box) with pay-off 0 to each player, but a Nash equilibrium (purple box) with pay-off -1/2 to each player. Both players would play strategy A and the game is therefore given the value -1/2. Players in each game L1 and L2 will hence lose. However, if we combine the playing of games L1 and L2 randomly, the pay-off matrix becomes the average of the two pay-off matrices for L1 and L2. The Nash equilibrium (purple box) of this combined game W is now Pareto optimal (green box), yielding a positive pay-off of 3/4 to each player. By combining two losing games randomly, we have created a winning game. This is an example of the so-called Parrondo effect. This effect will hold for other choices for the numerical pay-offs in games L1 and L2 as long as the two resulting pay-off tables have a similar structure to those in this example.