Parrondo Mixed Game Stationary Drift

Concept

Let the capital be an integer. In each game, the capital changes by +1 on a win and -1 on a loss.

• Game A: The probability of winning is 1/2 - ε.
• Game B: The probability of winning depends on the capital modulo 3. It is 1/10 - ε if the capital is divisible by 3, and 3/4 - ε otherwise.

For a small positive ε, both Game A and Game B have a negative expected drift. However, when we mix them by choosing Game A with probability γ and Game B with probability 1 - γ, the combined game alters the stationary distribution of the states (the residues 0, 1, and 2 modulo 3). This change in occupancy means the "bad" state (0 mod 3) in Game B is visited less frequently, allowing the "good" states to dominate and result in a positive drift.

The drift is calculated by finding the normalized stationary distribution π = π P for the 3x3 transition matrix on capital modulo 3, and computing the sum ∑ \pi_i (2p_i - 1), where p_i is the probability of winning in state i.

Worked Example

Consider mixing Game A and Game B with ε = 0.005 and a mix probability γ = 0.5. The winning probabilities for each state in the mixed game become:

1. For state 0: p_0 = 0.5(0.495) + 0.5(0.095) = 0.295
2. For states 1 and 2: p_1 = p_2 = 0.5(0.495) + 0.5(0.745) = 0.62

We construct the transition matrix P for states 0, 1, and 2. Solving π = π P gives the stationary residue occupancies:

pi_0 ≈ 0.345\npi_1 ≈ 0.254\npi_2 ≈ 0.401

Using these probabilities, the drift of the mixed game is 0.345(2(0.295)-1) + 0.254(2(0.62)-1) + 0.401(2(0.62)-1) ≈ 0.0157, which is positive, showing that the mixed game is a winning game!

Experiment

In this experiment, you can interact with the mixed game to see how different parameters affect the stationary drift.

1. Open the interactive module at https://riano.app/module/parrondo-mixed-game-stationary-drift.
2. Set Epsilon to 0.005 and Mix A probability to 0.5.
3. Observe that while both Game A and Game B individual drifts are negative, the Mixed Drift is positive (~0.0157).
4. Change Mix A probability to 0 or 1 to see the individual losing games. The stationary residues shift, explaining the paradox.

Limitations

This educational finite-state paradox is purely a mathematical curiosity and is not investment advice. Real-world financial systems do not exhibit such stationary Markov behavior without incurring transaction costs, non-stationary dynamics, or risk of ruin that this simple modulo-three model ignores.

References

For more information on Parrondo's Paradox and its applications, consult the original literature:

• Parrondo's paradoxical games (Nature)