Hong Life CalculatorsUnderstanding the Parrondo's paradox calculator — concept and principle
This calculator shows Parrondo's paradox: two games that each lose money on average can win when you switch between them at random. It computes the exact long-run gain per game for game A, game B and a random mix, and the expected change over a chosen number of games. Use it to explore probability, Markov chains and why intuition about "adding losing strategies" can fail.
How it is calculated
The capital changes by +1 or −1 each game.
- Game A: win probability 1/2 − ε.
- Game B: if the capital is a multiple of 3, win 1/10 − ε; otherwise win 3/4 − ε.
- Mix: game A with probability γ, otherwise game B, so each state's win chance is γ(1/2 − ε) + (1 − γ)·(B's chance).
The capital modulo 3 is a three-state Markov chain. The calculator solves for its stationary distribution π and gives the expected gain per game Σ πᵢ(2wᵢ − 1), where wᵢ is the win chance in state i. Total expected change = gain per game × number of games.
The principle
Game B looks favourable because two of its three states use the good coin. But a loss from state 1 drops the capital into state 0, the bad state, so B alone spends more than a third of its time there. That extra time on the bad coin makes B lose. Mixing in game A, which ignores the capital, shakes the chain out of that pattern and lowers the share of time in state 0. The good coin then gets used more often, enough to outweigh A's small loss. The paradox depends on B's rule being tied to the capital.
Worked example
Default inputs: ε = 0.005, γ = 50%, 100 games.
- Game A: win 0.495, gain per game 2(0.495) − 1 = −0.01.
- Game B: win 0.095 in state 0 and 0.745 otherwise. Solving the chain gives 38.36% of time in state 0 and a gain of −0.008695.
- Mix: win 0.5(0.495) + 0.5(0.095) = 0.295 in state 0 and 0.5(0.495) + 0.5(0.745) = 0.62 otherwise. Time in state 0 falls to 34.51%.
- Gain per game for the mix = 0.0157.
- Over 100 games: mix +1.57, game A −1, game B −0.8695. The paradox is shown.
| Chance of playing game A in the mix (%) | Expected gain per game: random mix |
|---|---|
| 12.5 | 0.005332 |
| 25 | 0.01337 |
| 37.5 | 0.01651 |
| 50 | 0.0157 |
| 62.5 | 0.01185 |
| 75 | 0.005763 |
| 100 | -0.01 |
How to read the result
The per-game figures are long-run averages. A positive number for the mix with negative numbers for A and B confirms the paradox. Larger ε makes all games worse, and above roughly ε ≈ 0.013 the mix also loses. Over a short run the actual capital fluctuates far more than these expected changes.
Common mistakes
- Thinking the paradox means any two losing bets combine into a win. It needs the capital-dependent rule of game B.
- Treating the expected change as a guaranteed result.
- Ignoring that alternating in a fixed pattern (e.g. AABB) gives a different result than random switching.
Frequently asked questions
Does Parrondo's paradox break the laws of probability?
No. Game B is not a fixed game; its odds depend on the state, and mixing changes how often each state occurs.
Who discovered it?
Physicist Juan Parrondo described it in 1996, inspired by the flashing Brownian ratchet.
Sources: S. N. Ethier and J. Lee, "Limit theorems for Parrondo's paradox", Electronic Journal of Probability (2009)