Parrondo's Paradox, explained.
Parrondo's paradox shows that combining particular state-dependent losing games can produce a positive long-run drift under specified switching rules.
Why it happens
Game A wins with probability 0.495. B wins with 0.095 at capital divisible by three and 0.745 otherwise. Compare A, B or random equal mixing across a 300-round path and 100 paths. A three-state Markov iteration supplies stationary expected drift for the policy.
Parrondo's paradox shows that combining particular state-dependent losing games can produce a positive long-run drift under specified switching rules.
Read the result
Compare stationary drift with sampled capital paths. Mixing changes the distribution of remainder states, so it changes more than an arithmetic average of the two standalone outcomes.
A worked example
Two losing rules with a different mixture
Run A alone, whose win probability is 0.495, and inspect its negative stationary drift alongside the noisy realised path.
Run B alone and note its dependence on capital modulo three, then select random equal mixing of A and B.
Compare stationary drifts and the hundred-path means. A positive expectation is conditional on the exact rules and does not guarantee any one run wins.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Game A wins with probability 0.495. B wins with 0.095 at capital divisible by three and 0.745 otherwise. Compare A, B or random equal mixing across a 300-round path and 100 paths. A three-state Markov iteration supplies stationary expected drift for the policy.
Where this idea is useful
A practical use
The outcome of alternating processes can depend on the states each process leaves for the next one.
A common misconception
“Combining any two losing strategies creates a winning one.”
The paradox depends on particular state-dependent transitions and switching rules. Arbitrary losing investments or bets need not satisfy those conditions.
What this explanation leaves out
- Paths fluctuate and positive expected drift guarantees no individual gain. These fictional games depend on specific transition rules; combining arbitrary losing investments does not reproduce the result. Negative capital is allowed.
Why does the model allow negative capital?
It studies a mathematical walk on all integer states rather than a finite bankroll with absorbing ruin. Adding a stopping boundary changes the process.
What state does one process leave behind that changes the result of the next process?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.