Parrondo's paradox, a paradox in game theory, describes how a combination of losing strategies can become a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996.

A simple example involves two coin flip games: Game A uses a biased coin that loses 50.5% of the time, while Game B switches between two different biased coins depending on whether your current winnings are even or odd. Though both games individually favor the house, alternating between them creates a net winning strategy. This occurs because the alternation causes players to spend more time in the favorable states of Game B, while Game A's consistent bias helps reset the system into those advantageous conditions.

Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman. However, the paradox disappears when rigorously analyzed. Winning strategies consisting of various combinations of losing strategies were explored in biology before Parrondo's paradox was published.

Consider an example in which there are two points A and B having the same altitude, as shown in Figure 1. In the first case, we have a flat profile connecting them. Here, if we leave some round marbles in the middle that move back and forth in a random fashion, they will roll around randomly but towards both ends with an equal probability. Now consider the second case where we have a saw-tooth-like profile between the two points. Here also, the marbles will roll towards either end depending on the local slope. Now if we tilt the whole profile towards the right, as shown in Figure 2, it is quite clear that both these cases will become biased towards B.

Now consider the game in which we alternate the two profiles while judiciously choosing the time between alternating from one profile to the other.

When we leave a few marbles on the first profile at point E, they distribute themselves on the plane showing preferential movements towards point B. However, if we apply the second profile when some of the marbles have crossed the point C, but none have crossed point D, we will end up having most marbles back at point E (where we started from initially) but some also in the valley towards point A given sufficient time for the marbles to roll to the valley. Then we again apply the first profile and repeat the steps (points C, D and E now shifted one step to refer to the final valley closest to A). If no marbles cross point C before the first marble crosses point D, we must apply the second profile shortly before the first marble crosses point D, to start over.

It easily follows that eventually we will have marbles at point A, but none at point B. Hence if we define having marbles at point A as a win and having marbles at point B as a loss, we clearly win by alternating (at correctly chosen times) between playing two losing games.

A third example of Parrondo's paradox is drawn from the field of gambling. Consider playing two games, Game A and Game B with the following rules. For convenience, define ${\displaystyle C_{t}}$ to be our capital at time t, immediately before we play a game.