Parrondo's Paradox Calculator

Enter Game A win probability, Game B state probabilities, a mixing ratio, and number of rounds to see Parrondo's Paradox in action. The calculator shows how combining two individually losing games produces a winning expected outcome — with average capital change, per-game expected values, and a chart comparing all three strategies.

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Probability of winning Game A on each play. Classic Parrondo uses ~49% (slightly losing).

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Win probability in Game B when your capital is a multiple of 3 (bad state). Classic value ~9.5%.

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Win probability in Game B when your capital is NOT a multiple of 3 (good state). Classic value ~74%.

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Percentage of rounds played as Game A. Remaining rounds use Game B. Try 50% for classic alternating mix.

Total number of game rounds to simulate analytically.

Results

Mixed Strategy Expected Gain

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Game A Only Expected Gain

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Game B Only Expected Gain

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Game A Expected Value per Round

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Game B Expected Value per Round

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Mixed Strategy Expected Value per Round

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Paradox Active?

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Expected Gain Comparison: Game A vs Game B vs Mixed Strategy

Results Table

Frequently Asked Questions

What is Parrondo's paradox?

Parrondo's paradox describes the counterintuitive situation where two individually losing games, when combined or alternated, can produce a winning outcome in the long run. It was discovered by Spanish physicist Juan Parrondo while studying the Brownian Ratchet thought experiment. The key insight is that the games interact through a shared state (like your current capital), creating a constructive interference that flips the overall expectation positive.

How does Game B work in Parrondo's paradox?

Game B is a capital-dependent game with two states. If your current capital is a multiple of 3, you play with a low win probability (the 'bad state', ~9.5%). Otherwise, you play with a high win probability (the 'good state', ~74%). Although Game B is still slightly losing on average due to the bad state occurring often enough, mixing it with Game A changes how frequently each state is visited.

Why do we win by mixing two losing games?

When you play only Game B, you spend too much time in the losing state (capital divisible by 3). When you introduce Game A, it subtly shifts your capital distribution so you visit the bad state less often and the good state more often. This is best understood through Markov chain analysis — the stationary distribution of states changes when the games are mixed, boosting the overall expected value above zero.

How do I replicate Parrondo's paradox?

Use Game A win probability of ~49%, Game B State 1 (bad) probability of ~9.5%, and Game B State 2 (good) probability of ~74.5%. Alternate between the two games (50/50 mix). Both games individually have a negative expected value, but the alternating strategy produces a small positive expected gain per round. This calculator lets you verify this analytically.

Is Parrondo's paradox applicable in real casinos?

No — Parrondo's paradox does not apply to standard casino games. Casino games are independent and do not share a capital-dependent state that can be exploited by switching between them. The paradox requires a very specific mathematical arrangement where game outcomes depend on the player's current state, which casino designs deliberately avoid.

Can Parrondo's paradox be modeled with games other than coin-tossing?

Yes, Parrondo's paradox has been demonstrated in various contexts including evolutionary biology, population dynamics, financial models, and quantum game theory. The coin-tossing model is simply the most accessible illustration. Any system where two losing strategies interact through a shared state variable can potentially exhibit the paradox.

What is the expected value per round for the classic parameters?

With classic parameters (Game A: 49%, Game B State 1: 9.5%, State 2: 74.5%, 50/50 mix), Game A has an EV of about −0.01 per round and Game B has an EV of roughly −0.003 per round. The mixed strategy, however, produces a positive EV of around +0.02 per round, demonstrating the paradox clearly over many rounds.

Does the mixing ratio matter for the paradox to work?

Yes, the mixing ratio affects how strongly the paradox appears. A 50/50 alternation (or random mix) is a common setup, but the paradox can emerge at other ratios too. Extreme ratios (playing almost entirely one game) will reduce or eliminate the paradox effect. This calculator lets you experiment with different mix shares to find the range where the paradox is active.

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