Parrondo's Paradox Calculator

Your details

Choose how you alternate between Game A and Game B. Mixed strategies are the ones that trigger the paradox.
Win probability for each flip of Game A. The classic Parrondo value is 0.495 (just below fair).
Win probability for Game B when your capital is exactly divisible by 3. This is the "trap" state with very low odds.
Win probability for Game B when your capital is NOT divisible by 3. This is the "good" state with favorable odds.
Your initial capital. Affects the chart but not the long-run drift.
$
Number of game rounds to show in the capital trajectory chart.
rounds
Selected strategy win probabilityWinning strategy
0.508%

Long-run probability of winning each round under the chosen strategy

Expected drift per round0.0157$
Game A win probability0.495%
Game B win probability (Markov)0.496%
Game B drift per round-0.0087$
Game A drift per round-0.01$
Steady-state: P(capital mod 3 = 0)0.35%
Steady-state: P(capital mod 3 = 1)0.25%
Steady-state: P(capital mod 3 = 2)0.4%
Game A drift ($/round)-0.01
Game B drift ($/round)-0.0087
Strategy drift ($/round)0.0157

The paradox works: randomly mixing A and B 50/50 wins at 50.785% despite both games losing alone.

  • Game A alone has a 49.500% win rate - below 50%, so it loses in the long run (drift: -0.0100 $/round).
  • Game B alone has a 49.565% win rate via Markov analysis - also below 50% (drift: -0.0087 $/round).
  • The mixed strategy (randomly mixing A and B 50/50) achieves 50.785% - above 50%! After 200 rounds, expected capital gain: $3.14.
  • This is Parrondo's Paradox in action: mixing two losing games creates a winning strategy by shifting the system away from the capital-mod-3 trap state.

Next stepTry adjusting p1 lower and p2 higher while keeping both games individually losing to see the paradox emerge more strongly. The effect is maximized when epsilon (the bias away from 0.5) is small.

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