May 4

3:30 pm

## Parrondo\'s Paradox: Winning Mixtures of Losing Games

### Ron Pyke (Joint with Biostatistics)

Seminar

University of Washington - Department of Mathematics

Consider playing a sequence of simple games, with *Xn* denoting the gain (or loss) from the *n*th game. If *Sn* = *X1* + ... + *Xn* is the player\'s capital after *n* games, we say that the game is winning, losing or fair according as the average capital [*S sub n*]/*n* converges to a positive, negative or zero limit. In 1997, J. Parrondo created some simple games to illustrate the apparent paradox that random or deterministic mixtures of losing games may produce winning games. These games were discussed in an article by Harmer and Abbott entitled "Parrondo's Paradox" in *Statistical Science*, (May 1999 vol. 14). Large simulations to support this paradox and some theoretical results are given in this and a subsequent series of papers by these authors. In this talk, classical cyclic random walks on the integers modulo m, a given integer, are used in a straightforward way to derive the strong law limits of a general class of games that contains the Parrondo games. We then consider the question of when random mixtures of fair games related to these walks may result in winning games. Although the context for these problems is elementary, most questions remain open. We are able to establish the desired result only for the case of random mixtures of Parrondo games themselves, and for this case the tools used are elementary facts about polynomials. An extension of the structure of these walks to a class of shift diffusions may also be presented.