# How strong can the Parrondo effect be?

## TL;DR

This paper investigates the potential strength of the Parrondo effect, demonstrating that with arbitrary parameters and sequences, the combined game can yield profits arbitrarily close to 100%.

## Contribution

It extends the understanding of the Parrondo effect by showing how parameter choices and game sequences can maximize the profit rate.

## Key findings

- Profit rate can approach 100% with appropriate parameters.
- Arbitrary periodic sequences can optimize the Parrondo effect.
- The effect's strength is unbounded under certain conditions.

## Abstract

If the parameters of the original Parrondo games $A$ and $B$ are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games $A$ and $B$ are played in an arbitrary periodic sequence, then the rate of profit can not only be positive, it can be arbitrarily close to 1 (i.e., 100%).