Compare standalone and randomly mixed capital-dependent Parrondo games using finite-state expectations and stationary drift.
How to use this tool
- Choose the event or model and review the assumptions below.
- Enter the labeled inputs. Probability inputs are percentages from 0 to 100, not decimal fractions.
- Select Calculate and read the labeled outputs.
- Change an input and calculate again; Reset restores the example inputs.
Calculation method
A wins with 0.5−ε. B wins with 0.1−ε at capital divisible by 3 and 0.75−ε otherwise. A random mixture uses qA+(1−q)B within each residue state.
Assumptions and limits
An exact finite-state expectation for the classic capital-modulo-3 model, not simulated profits or a general claim that losing options become profitable. Epsilon is restricted to −0.1 through 0.1; rounds up to 10,000. Capital can go below zero; no ruin boundary or fees are modeled.
Worked example
Use epsilon 0.005, a 50% mixture and initial capital zero. Compare the finite expected gains with the long-run drifts rather than assuming each first-step result equals the stationary behavior.
Worked reading and practical questions
- Parrondo’s Games: Track Capital Residues, Not Just Average Coins
- Transient Gains and Long-Run Drift in Parrondo’s Paradox
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Frequently asked questions
What does this calculator assume?
An exact finite-state expectation for the classic capital-modulo-3 model, not simulated profits or a general claim that losing options become profitable. Epsilon is restricted to −0.1 through 0.1; rounds up to 10,000. Capital can go below zero; no ruin boundary or fees are modeled.
How do I enter and update the inputs?
Use the labeled fields, then select Calculate. Percent fields use 0 to 100. After editing, calculate again; Reset restores the example inputs.