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Parrondo's Paradox Calculator

Play game A and you lose slowly. Play game B and you lose slowly. Alternate between them and you win. This is not a trick of accounting: it happens because game B's odds depend on your current capital, so the two games spend different amounts of time in game B's bad state. Mixing in a game that does nothing on its own changes how often you land there, and that alone flips the sign.

What this generator does

Solves how often your capital sits in each of game B's three states in the long run, under game B alone and under the alternation, and works out the rate of gain that follows from each.

How to use this tool

  1. Set the chance of winning game A.
  2. Set game B's two chances — the bad one when capital divides by three, the good one otherwise.
  3. Work it out and compare the three rates of gain.
  4. Look at the long-run shares to see where the effect comes from.

Understanding the controls

Game A's chance of winning
The same on every play, regardless of capital. Just below half makes it a slow loser.
Game B when capital divides by three
Game B's bad state, usually set low. This is where the losses come from.
Game B otherwise
Game B's good state. Set generously, so that B is only a loser because of how often it lands in the bad state.

Common use cases

  • Demonstrating that combining strategies is not the same as averaging them
  • Teaching Markov chains with a result that surprises people
  • Illustrating why a strategy's performance depends on the state it leaves behind
  • A worked counterexample to the idea that two bad options cannot combine well
  • Exploring how sensitive the effect is to the numbers chosen

How this generator works

Game B's odds depend on capital, which makes it a three-state chain. The long-run share of time in each state is found by settling the chain rather than by playing it, and the rate of gain follows from those shares. The check recomputes the rate from the shares independently and confirms the shares add up to one, which is what a set of long-run shares must do.

Randomness and fairness

Nothing is random. The long-run behaviour is solved directly, so no simulation is run and no result depends on a draw.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

  • The alternation here is a strict A, B, A, B. Random switching gives a different, usually smaller, effect.
  • Game B's dependence is on capital divisible by three, which is the classic form. Other dependences behave differently.
  • This is the long run. Any short run can go either way, and the gains are tiny per play.
  • It is a demonstration of a mechanism, not a betting strategy — no real game has these rules.

Privacy and your data

Everything is worked out in your browser and nothing you enter is transmitted or stored.

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