Parrondo's Paradox

The problem
each round, pick game A or game BGame AP(lose) = 0.505slight lossGame Bcapital-statedependent lossboth games are losing in isolationcan mixing them yield a winning strategy?

Two games:

  • Game A — flip a slightly biased coin. Win $1 with probability 0.4950.495, lose $1 with probability 0.5050.505. Slightly losing.

  • Game B — your capital determines the rules:

    • If your capital is a multiple of 3: flip a bad coin, lose with probability ∼0.905\sim 0.905.
    • Otherwise: flip a good coin, lose with probability ∼0.255\sim 0.255.

    Averaged over the natural distribution of capital states, also slightly losing.

You play forever. What happens if you alternate A and B in some pattern?