Parrondo's Paradox Calculator
Adjust the win probabilities for Game A and Game B, then simulate how alternating or mixing the games produces a surprising positive expectation.
Game A: win -> +1, loss -> −1. Typically slightly losing.
Probability of winning Game B when capital mod 3 = 0.
Probability of winning when capital mod 3 ≠ 0.
Used when mixing randomly between Game A and B.
Game A (alone)
-0.0110
Average gain per turn
Game B (alone)
-0.0113
Average gain per turn
Alternating A→B
-0.0023
Average gain per turn
Random mix (0.5)
0.0143
Average gain per turn
How to Use This Calculator
- Enter the win probabilities for Game A and for Game B's two states (capital mod 3 = 0 and otherwise).
- Choose how many rounds and simulations to run. More simulations increase accuracy but take longer.
- Optionally set a random mixing probability to model a stochastic combination of games.
- Compare average gains. When both standalone games lose yet the combined strategy wins, you are observing Parrondo's paradox.
Formula
Parrondo's paradox arises from alternating two losing Markov games. Game A has a fixed negative expectation. Game B’s expectation depends on capital modulo 3: it is strongly negative when capital ≡ 0 and positive otherwise. Switching between the games alters the distribution of states, turning the combined process into a winner.
E[A] = 2 · pA − 1
E[B] depends on stationary distribution over capital mod 3
Simulation approximates the stationary distribution for complex sequences.
Full Description
Parrondo's paradox demonstrates how alternating two losing strategies can create a winning expectation. It has applications in physics, population biology, and finance, highlighting that variance and state-dependent payoffs matter as much as raw expected value. The paradox challenges intuitions about averaging and diversification.
By tuning probabilities and mixing patterns, you can explore when the paradox appears or disappears. This calculator uses Monte Carlo simulation to approximate long-run outcomes without solving Markov chains analytically.
Frequently Asked Questions
Why are there fluctuations between runs?
The simulation uses randomness. Increase the number of rounds or simulations to reduce variance and see more stable averages.
Can I model deterministic sequences?
Yes. The alternating strategy (A→B) mimics a deterministic ABAB pattern. You can modify the chooseGame function to experiment with other cycles.
What values recreate the classic paradox?
Common choices are pA = 0.495, pBBad = 0.095, and pBGood = 0.745. Both games lose alone, yet alternating wins on average.
Is the paradox real or just simulation noise?
It is mathematically proven. The simulation provides empirical evidence by sampling many trajectories, revealing the positive drift of combined strategies.