Preprint No.
A-04-07
Ehrhard Behrends
On Astumian's paradox
Abstract:
In this note we discuss some natural questions in
connection with Astumian's paradox. An
{\em Astumian game\/} is defined by a finite Markov chain with state
space $S$ with precisely two absorbing states, the {\em winning} and
the {\em losing} state. The other states are transient, one of them
is the starting position. The game is said to be {\em losing\/}
(resp.\ {\em winning\/}) if the probability to be absorbed at the
winning state is smaller than $0.5$ (resp.\
larger than $=0.5$). Astumian's paradox states that there are losing
games on the same state space $S$ a stochastic mixture of
which is winning.
(By ``stochastic mixture'' we mean that in each step one decides with the
help of a fair coin whether to use the transition probabilities
of the first or the second game.)
Most of our results concern {\em fair\/} games, these are games
where the winning probability is exactly $0.5$. Mixtures are
systematically investigated. Rather surprisingly, the winning
probability of the mixture of fair games can be arbitrarily close to
zero (or to one). Even more counter-intuitive are examples of
{\em definitely losing games\/} (this means that the winning
probablity is exactly zero) such that the winning probability of
the mixture is arbitrarily close to one. We show, however, that
such extreme examples are possible only if one tolerates huge
running times of the game.
As a natural generalization one can also consider
{\em arbitrary mixtures\/}: the fair coin is replaced by a biased one,
with probability $\lambda$ resp.\ $1-\lambda$ one plays
with the first resp.\ the second game. It turns out that
fair games exist such that -- depending on the choice of
$\lambda$ -- the $\lambda$-mixture can be fair, losing or winning.
Keywords:
random games, Astumian's paradox, Parrondo paradox, Markov chain
Mathematics Subject Classification (MSC2000):
60J10, 60J20
Language: ENG
Available: Pr-A-04-07.ps,
Pr-A-04-07.ps.gz
Contact: Ehrhard Behrends, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (behrends@math.fu-berlin.de)
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