Betting with Maxwell's Daemon
1
2
1
Michael Maitland , Rosemary J Harris , Stefan Grosskinsky
1 University of Warwick, 2 Queen Mary University of London
Introduction
A recently made analogy between the well-studied problems in gambling and
typical thermodynamic systems involving feedback allows for deeper insights
into the functioning of information based molecular machines. We consider
that the information used by a well informed gambler (or stock-trader) is
analogous to the information processing performed by a feedback mecha-
nism - a la Maxwell's Daemon. Both systems use an information source in
order to maximise some quantity, wealth in the case of the gambler; work
extraction in the case of the Daemon and his real-world counterparts. The
current research hopes to strengthen this analogy and use it to study the
role of information processing in physcics and nature.
Szilard Engine
Horse Race
Let p(X = x) be the probability that horse x wins a given race between
m horses, where X ∈ {1, . . . , m}. P
Let b(x) ≥ 0 be the fraction of wealth a
gambler invests in horse x, where b(x) = 1. If horse x wins, the gambler
receives o(x) times his investment and all other bets are lost. After each
race, the gambler's wealth is multiplied by b(X)o(X) = S(X). After n
races, his wealth is
1)
2)
(L,L)
(L,R)
(R,L)
Sn = 2nW (X)
(R,R)
where W is the `exponential doubling rate', given by
3)
W (X) = hln S(X)i =
Vf(L|L))
Vf(R|L))
Vf(L|R))
X
p(x) ln b(x)o(x)
x
Vf(R|R))
4)
= p(x)), the maximum wealth growth
Assuming `fair' odds (that is,
rate is 0 and the gambler can at best break even. The optimal bet is
then proportional gambling (also known as `Kelly Gambling') such that
b(x | y) = p(x | y).
Let Y be a second variablePwhich has a joint distribution p(X = x, Y = y)
and let b(x | y) ≥ 0, with x b(x | y) = 1 be a conditional betting strategy
depending on the value of the additional variable. The gambler seeks to
optimise his betting strategy b(x | y). Assuming proportional gambling
and fair odds, the optimal wealth growth rate is then given by:
1
o(x)
Vf(L|L))
Vf(L|R))
Vf(R|L))
Vf(R|R))
5)
Information engine
The Szilard engine is one of the rst examples of an `information engine',
and has been used to study the eects of Maxwell's Daemon. The engine
works in the follwing cycle:
W (X | Y ) = hln S(X | Y )i =
= H(X) − H(X | Y ) = I(X; Y )
which leads to
3. The particle position is measured and the result is recorded in Y ∈
{L, R}
4. The divider is moved quasistatically until the volumes on the left
and right are VfL and VFR
5. The divider is removed from the box and the particle equilibrates,
returning to 1.
From ideal gas dynamics, we can show that the work extracted by the
operation of a cycle is:
Vf (X | Y )
Vf (X | Y )
W (X | Y ) = kB T ln
= kB T ln
V0 (X)
P (X)
where kB is the Boltzmann constant. The optimal Vf is then found by:
Vf∗ (X | Y ) = argmaxhW (X | Y )i = P (X | Y )
Vf
and after n cycles, the maximum amount of work extracted will be:
maxhWn i = nkB T
Vf
P (X | Y )
ln
P (X)
= nkB T I(X; Y )
The work extracted after n cycles is directly proportional to the mutual
information between the measurement and the actual state of the system.
References
[1] D. Vinkler, H. Permuter, and N. Merhav, Analogy between gambling and
measurement-based work extraction, submitted to ISIT 2014, January 2014.
[2] Cover, Thomas M., and Joy A. Thomas. Elements of information theory.
John Wiley & Sons, 2012.
[3] Kelly, J. L. A New Interpretation of Information Rate, Bell System Technical
Journal 35 (4) 1956
p(x, y) ln o(x)b(x | y)
x
1. A particle moves freely in a box at equilibrium with a heat bath
2. The box is partitioned, separating it into two volumes. The side that
the particle is now located on is denoted X ∈ {L, R}.
X
hln Sn i = nI(X; Y )
that is in the presence of side information, the maximum wealth (obtained
by the optimal gambling strategy) is directly proportional to the mutual
information between the races and the second variable.
Gambling
Analogy
X - Winning horse
Y - Side information
P (x) - Probability that horse x wins
P (x | y) - As above, conditioned on
the state of side information
o(x) - Payout for x winning
Placing bets
b(x | y) - money gambled on each
horse given y
log Sn - Log wealth after n rounds
Convert information entropy into
wealth
Szilard Engine
X - Position of particle
Y - Measurement (possibly noisy)
P (x) - Probability that particle is
on side x
P (x | y) - As above, conditioned on
the outcome of measurement
1
- reciprocal of the initial volume
V0
Moving dividers
Vf (x | y) - Final volume of the box's
parts given y
Wn
- Work extracted after n cycles
kB T
Convert information entropy into
work
Discussion
Presented here was the analogy between the sequential Horse Race and the Szilard information-engine. The analogy is based on the inclusion of information
theoretic measures in the analysis, and a mapping between quantities. This
framework can be used use techniques from gambling and game theoretic treatments of stock markets, in the setting of thermodynamic systems. We hope to
provide a more rigorous principle on which this analogy can be founded, such as
the theory of large deviations, and nd novel applications of the analogy outside
of those already studied.