Project III (MATH3382) 2024-25

Parrondo's paradox

O Hryniv

Description

Imagine you have £k and can play two games, Game A and Game B with the following rules.
  • in the coin-flipping Game A, you win £1 with probability p and lose £1 with probability 1-p.
  • in the coin-flipping Game B,
    • if your current capital £k is a multiple of an integer M>1, you win £1 with probability p1 and lose £1 with probability 1-p1;
    • otherwise, you flip another coin and win £1 with probability p2 and lose £1 with probability 1-p2.
Suppose that p=.495, p1=.095, p2=.745, and M=3. Game A is obviously a losing game, and a straightforward Markov Chain computation shows that Game B is a losing game as well.

Yet, various combinations of Game A and Game B result in a winning game. For example, playing Game A with probability q and Game B with probability 1-q, where q=.4 or q=.5, is a winning combination. Similarly, following a deterministic pattern like AABB, where one plays two A games, followed by two B games, followed by two A games, etc., is also a winning combination. The same applies to other deterministic combinations, e.g., ABBAB.

The aim of the project is to explore the exciting mathematics of Parrondo's paradox, various modifications of the above scenario as well as their applications.

Prerequisites

2H Markov Chains is essential; 3H Stochastic Processes is desirable.

Resources

Do a web search for "Parrondo's paradox" or check some sources cited on the Wiki page. Further references might be suggested once the project is underway.

Get in touch, if have any questions and/or if you would be interested in doing simulations!

email: O Hryniv