Parrondo's Paradox Calculator — Two Losing Games That Win
Parrondo's paradox is a remarkable result in game theory: two games each with negative expected gain can be combined into a strategy with positive expected gain. Enter the bias parameter ε to see the individual and combined expected gains per step, and how the paradox holds across the valid range of ε.
Positive = winning despite both individual games being losing
How does this calculator work?
Parrondo's paradox: Game A has E[gain] = −2ε (slightly losing biased coin). Game B is also slightly losing (its stationary Markov distribution falls often in the bad state). Alternating ABAB gives E[gain] > 0 for ε ∈ (0, 0.1) because switching disrupts the bad-state dwell, making the favourable coin dominate on average. Original parameters: ε = 0.005.
Formula
How this is calculated
Parrondo's paradox (Juan Parrondo, 1996) uses two games. Game A is a simple biased coin: P(win) = 0.5 − ε, giving expected gain per step = −2ε (slightly losing). Game B uses two biased coins selected by the player's current capital modulo 3: if capital mod 3 = 0, play a very unfair coin with P(win) = 0.1 − ε; otherwise play a favourable coin with P(win) = 0.75 − ε. In isolation, Game B's stationary distribution over the three capital-modulo-3 states is skewed toward the losing state, making Game B also slightly losing overall.
When the games are alternated (ABAB…), the switching perturbs the Markov chain dynamics in a way that reduces how often the capital falls into the losing state of Game B. The combined game's stationary distribution has smaller weight on capital mod 3 = 0 than Game B alone, so the favourable coin is used more often, flipping the overall expectation positive. This is computed here by finding the stationary distribution of the 2-step (A then B) Markov chain via power iteration and averaging the per-step gains.
The paradox is analogous to a mechanical ratchet: individual noisy signals cancel, but their combination rectifies in one direction. It has applications in evolutionary biology, financial portfolio theory, and quantum game theory. The valid range for ε is (0, 0.1) — beyond 0.1 the probability p_bad = 0.1 − ε becomes non-positive.
Frequently asked questions
The key is the capital-dependent switching in Game B. When capital mod 3 = 0 you play a very unfair coin; other states play a favourable coin. Game A's slight bias disturbs the capital mod 3 distribution just enough to reduce time spent in the losing state, making the favourable coin dominate when averaged over both games.
All three probabilities p_A = 0.5 − ε, p_bad = 0.1 − ε, and p_good = 0.75 − ε must be positive, so ε must be less than 0.1. For ε ≥ 0.1, p_bad ≤ 0 which is not a valid probability. Parrondo originally used ε = 0.005.
No — random mixing also works. If each step you randomly play A with probability γ and B with probability 1 − γ, for many values of γ the combined expected gain is still positive. The key is that A and B must be mixed; playing only A or only B loses.
Also known as
TG we-Calculate Editorial Team. (2026). Parrondo's Paradox Calculator — Two Losing Games That Win [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/parrondo-paradox-calculator
TG we-Calculate Editorial Team. "Parrondo's Paradox Calculator — Two Losing Games That Win." TG we-Calculate. 2026. https://we-calculate.com/calculator/parrondo-paradox-calculator.
TG we-Calculate Editorial Team, "Parrondo's Paradox Calculator — Two Losing Games That Win," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/parrondo-paradox-calculator
@misc{wecalculate_parrondo_paradox_calculator, title = {Parrondo's Paradox Calculator — Two Losing Games That Win}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/parrondo-paradox-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?