Edward W. Piotrowski

Jan Sladkowski

English

Research Papers in Economics

Letters to the Editors
Scientific American
415 Madison Ave.
New York, NY 10017-1111 September 5 2001
The Astumian’s paradox
(RePEc:sla:eakjkl:7 5-IX-2001)
R. D. Astumian considers in a recent article Making molecules into motors
(Scientiﬁc American 7(2001)57-64) a game based on the presentedbelow diagram.
The game consist in jumping between ﬁve diﬀerent states
1
;
:
:
:
;
5:
1
2
3
4
5
1
3
5
7
2
3
1
3
2
7
2
3
;
where the numbers written above or below the arrows are the probabilities of
transitions between neighboring states (only such transitions are allowed). The
player wins if she (or he) winds up in the state
5 and loses if she reach the
state
1. If the player starts from the state
3 the probability of losing is equal
to
5
9 and winning to
4
9. This is because in games of this kind the proportion
of probabilities of defeat and success is given by the proportion of products of
the appropriate transition probabilities (
p
3
(
1
)
p
3
(
5
)
=
p
(
3
!
2
)
p
(
2
!
1
)
p
(
3
!
4
)
p
(
4
!
5
), though it is not
true that
p
3
(
1
)
=
p
(
3
!
2
)
p
(
2
!
1
) nor
p
3
(
5
)
=
p
(
3
!
4
)
p
(
4
!
5
) !). The same
numerical result might be obtained in the game
1
2
3
4
5
5
7
1
3
2
7
5
7
2
3
2
7
with diﬀerent transition probabilities. Astumian suggests that in a modiﬁed game
with the transition probabilities being arithmetic mean of the probabilities of the
both above described games that is the game given by
1
2
3
4
5
1
1
2
1
1
1
2
1
1
0
2
1
1
1
2
1
1
0
2
1
1
0
2
1
the probability of of success is greater than the probability of defeat. He calls it
the player paradox. Even a superﬁcial analysis of the above diagram suggests
that this is wrong: the transitions from the states
2
;
3
;
4 to the left are more proba-
ble than to the right. It seems that an elementary error in counting probabilities
resulted in drawing wrong conclusion by prof. Astumian.
The authors of this letter suggest the readers to calculate the probabilities
of success in games of this kind. An exemplary listing of a mini-programme
w r i t t e ni nt h el a n g u a g eMathematica 4.1 together with results they obtained
can be found below.
Edward W. Piotrowski1 and Jan Sładkowski2
1Institute of Theoretical Physics, University of Białystok, Lipowa 41, Pl 15424 Białystok, Poland,
e-mail: ep@alpha.uwb.edu.pl
2Institute of Physics, University of Silesia, Uniwersytecka 4, Pl 40007 Katowice, Poland,
e-mail: sladk@us.edu.pl
1A
= Table
[Which
[
m==
n==
1
_
m==
n==
5
;
1
;
m==
1
^
n==
2
_
m==
3
^
n==
4
;
1
3
;
m==
3
^
n==
2
_
m==
5
^
n==
4
;
2
3
;
m==
2
^
n==
3
;
5
7
;
m==
4
^
n==
3
;
2
7
;True
;
0
]
;
f
m
;
5
g
;
f
n
;
5
g
]
;
B
=
A
=
:
f
1
3
!
5
7
;
2
3
!
2
7
;
5
7
!
1
3
;
2
7
!
2
3
g
;
C
=
1
2
A
+
1
2
B
;
f
i
n
i
s
h
[
A
]
:
=Rationalize
[N
[Nest
[
(#
:#
)&
;
A
;
1
2
]
:
f
0
;
0
;
1
;
0
;
0
g
;
2
]
]
;
fMatrixForm
[
A
]
;MatrixForm
[
B
]
;MatrixForm
[
C
]
g
f
f
i
n
i
s
h
[
A
]
;
f
i
n
i
s
h
[
B
]
;
f
i
n
i
s
h
[
C
]
g
f
0
B
B
B
B
B
B
B
B
@
1
1
3
0
0
0
0
0
5
7
0
0
0
2
3
0
1
3
0
0
0
2
7
0
0
0
0
0
2
3
1
1
C
C
C
C
C
C
C
C
A
;
0
B
B
B
B
B
B
B
B
@
1
5
7
0
0
0
0
0
1
3
0
0
0
2
7
0
5
7
0
0
0
2
3
0
0
0
0
0
2
7
1
1
C
C
C
C
C
C
C
C
A
;
0
B
B
B
B
B
B
B
B
@
1
1
1
2
1
0
0
0
0
0
1
1
2
1
0
0
0
1
0
2
1
0
1
1
2
1
0
0
0
1
0
2
1
0
0
0
0
0
1
0
2
1
1
1
C
C
C
C
C
C
C
C
A
g
f
f
5
9
;
0
;
0
;
0
;
4
9
g
;
f
5
9
;
0
;
0
;
0
;
4
9
g
;
f
1
2
1
2
2
1
;
0
;
0
;
0
;
1
0
0
2
2
1
g
g
2

The Astumian's Paradox

http://alpha.uwb.edu.pl/ep/RePEc/sla/eakjkl/7.pdf

The Astumian's Paradox

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Research Papers in Economics