# Apparent pathologies in stochastic entropy production in the thermalisation
of an open two-level quantum system

Jonathan Dexter and Ian J. Ford  Department of Physics and Astronomy,
University College London, Gower Street, London WC1E 6BT, U.K.

###### Abstract

We investigate the entropic consequences of the relaxation of an open two-
level quantum system towards a thermalised statistical state, using a
framework of quantum state diffusion with a minimal set of raising and
lowering Lindblad operators. We demonstrate that thermalisation starting from
a general state is accompanied by a persistent non-zero mean rate of change of
the environmental component of stochastic entropy production. This
thermodynamic signature can be associated with the purification of the reduced
density matrix $\rho$ of the randomly evolving state, to be contrasted with
the impurity of the more often considered ensemble average of $\rho$. The
system adopts stationary statistics, with zero stochastic entropy production,
once purity has been achieved. However, we show that apparent pathological
mathematical difficulties in the computation of stochastic entropy production
emerge if $\rho$ is represented using a certain set of coordinates, though
these can be removed by choosing a different set. We conclude that frameworks
for modelling open quantum systems must be carefully selected to provide
satisfactory thermodynamic as well as dynamic behaviour.

##  I Introduction

The second law of thermodynamics emerges when the world is described at a
coarse grained level, but governed by underlying equations of motion with a
sufficient degree of deterministic chaos [1]. Entropy increase corresponds to
a growth in subjective uncertainty regarding the adopted configuration of the
world in such a situation, and this applies both to classical and to quantum
systems [2, 3]. To illustrate the latter, we study the evolution of an open
two-level quantum system using a minimal framework of stochastic dynamics that
represents the effects of coupling to an underspecified environment. We treat
the reduced density matrix of the system as a set of physical coordinates that
undergo Brownian motion, and concern ourselves with the mechanical and
thermodynamic approach to a thermalised state. The stochastic entropy
production that accompanies the evolution is a measure of its thermodynamic
irreversibility and identifies an arrow of time [4].

We employ a framework of quantum state diffusion [5] where the system dynamics
are modelled using Itô processes: Markovian stochastic differential equations
(SDEs) with continuous solutions. The associated stochastic entropy production
can also be described using an SDE [6, 7]. We find several apparently
pathological features in the stochastic entropy production and resolve them.
In Section II we encounter persistent stochastic entropy production
accompanying thermalisation of the system by the environment, suggesting that
a stationary statistical state is reached only in infinite time. We link this
behaviour with asymptotic purification under the chosen dynamics. Furthermore,
relaxation of a pure system towards the thermalised state can appear to be
burdened with mathematical difficulties in the form of singularities in
stochastic entropy production. We investigate and resolve such peculiarities
in Section III. Our conclusions are given in Section IV.

Appendix A provides a discussion of the basis for computing stochastic entropy
production, and Appendix B is a summary of how to do so in situations
characterised by restricted diffusion [8]. Appendix C explores some unexpected
mathematical features that arise in the averaging of SDEs, and how they can
lead to additional terms in the mean system component of stochastic entropy
production.

##  II Thermalising a two-level open quantum system

###  II.1 System dynamics

A stochastic Lindblad equation for the evolution of the reduced density matrix
$\rho$ of an open system can be constructed using Lindblad operators that
describe the pseudorandom influence of the environment on the system [9, 2,
3]. Such an approach allows us to model the evolution of $\rho$ given an
underspecified state of the environment: a concept familiar in classical
statistical mechanics but less so in quantum mechanics. No projective
measurement is imposed. Environmental interactions drive quantum state
diffusion such that $\rho$ follows a continuous Brownian trajectory.

We consider a two-level quantum system with environmental couplings
characterised by the raising and lowering Lindblad operators given by

$c_{+}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},\qquad
c_{-}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},$ (1)

expressed in a basis of eigenstates of the $\sigma_{z}$ Pauli matrix. The
stochastic Lindblad equation is

$\begin{split}d\rho&=\left(c_{+}\rho c_{+}^{\dagger}-\frac{1}{2}\rho
c_{+}^{\dagger}c_{+}-\frac{1}{2}c_{+}^{\dagger}c_{+}\rho\right)dt\\\
&+\left(\rho c_{+}^{\dagger}+c_{+}\rho-{\rm
Tr}\left[\left(c_{+}+c_{+}^{\dagger}\right)\rho\right]\rho\right)dW_{1}\\\
&+\left(c_{-}\rho c_{-}^{\dagger}-\frac{1}{2}\rho
c_{-}^{\dagger}c_{-}-\frac{1}{2}c_{-}^{\dagger}c_{-}\rho\right)dt\\\
&+\left(\rho c_{-}^{\dagger}+c_{-}\rho-{\rm
Tr}\left[\left(c_{-}+c_{-}^{\dagger}\right)\rho\right]\rho\right)dW_{2},\end{split}$
(2)

where $dW_{1,2}$ are independent Wiener increments [2, 3, 8]. Expressing
$\rho$ in terms of components of the coherence or Bloch vector
$\boldsymbol{r}=(r_{x},r_{y},r_{z})\equiv(x,y,z)$, we write

$\rho=\frac{1}{2}\begin{pmatrix}1+z&x-iy\\\ x+iy&1-z\end{pmatrix},$ (3)

namely
$\rho=\frac{1}{2}\left(\mathbb{I}+\boldsymbol{r}\cdot\boldsymbol{\sigma}\right)$
in terms of Pauli matrices $\sigma_{i}$. After some manipulation we obtain
three SDEs describing the dynamics of the system:

$\begin{pmatrix}dx\\\ dy\\\ dz\end{pmatrix}=\begin{pmatrix}-x\\\ -y\\\
-2z\end{pmatrix}dt+\begin{pmatrix}1-x^{2}-z&1-x^{2}+z\\\ -xy&-xy\\\
x\left(1-z\right)&-x\left(1+z\right)\end{pmatrix}\begin{pmatrix}dW_{1}\\\
dW_{2}\end{pmatrix},$ (4)

which take the form

$dr_{i}=A_{i}(\boldsymbol{r},t)dt+\sum_{j}B_{ij}(\boldsymbol{r},t)dW_{j},$ (5)

in terms of a vector $\boldsymbol{A}$ and matrix $\boldsymbol{B}$.

The number of coordinates (three) is larger than the number of noise terms
(two), suggesting that a reduced description exists involving only two
coordinates. Indeed an example of the evolution of $\boldsymbol{r}$ in its
phase space in Fig. 1 suggests the motion is constrained to lie on the surface
of an ellipsoid lying inside the Bloch sphere defined by
$|\boldsymbol{r}|=r=1$. The purity of the system $P={\rm
Tr}\rho^{2}=\frac{1}{2}\left(1+r^{2}\right)$ evolves according to

$dP=4\left(1-P\right)\left(1-x^{2}\right)dt+4x\left(1-P\right)\left(dW_{1}+dW_{2}\right),$
(6)

suggesting that purification $P\to 1$ takes place as $t\to\infty$. This is
demonstrated in Fig. 2.

Figure 1: Example evolution of the coherence vector $\boldsymbol{r}=(x,y,z)$
governed by Eq. (4). The trajectory was based on 10000 steps with timestep
$dt=10^{-3}$ and initiated at the point $x=y=z=0.5$. The coherence vector
appears to be confined to an ellipsoid and to be driven towards a circle of
unit radius in the $x$-$z$ plane, corresponding to evolution towards purity.
Figure 2: Evolution of the magnitude of the coherence vector $\boldsymbol{r}$
under the dynamics described by Eq. (4). We see that $|\boldsymbol{r}|^{2}$ is
driven towards unity, which is synonymous with evolution towards purity,
$P=1$. The trajectory was initiated at the point $x=y=z=0.5$ and constructed
using a timestep of $dt=10^{-5}$.

###  II.2 Stochastic entropy production

The incremental stochastic entropy production associated with the stochastic
dynamics, which measures the irreversibility of the evolution, is usually
separated into system and environmental components as follows:

$d\Delta s_{{\rm tot}}=d\Delta s_{\text{sys}}+d\Delta s_{\text{env}},$ (7)

with $d\Delta s_{\text{sys}}=-d\ln p(\boldsymbol{r},t)$ where
$p(\boldsymbol{r},t)$ is the probability density function (pdf) characterising
the stochastic dynamical variables, governed by an appropriate Fokker-Planck
equation. The average of the system component $d\Delta s_{\text{sys}}$ over
all possible trajectories is related to the incremental change in Gibbs
entropy of the system: $d\langle\Delta s_{{\rm sys}}\rangle=dS_{G}$, to which
boundary terms should be added in certain circumstances [2], as shown in
Appendix C. We shall encounter a case where boundary contributions are
important in Section III.

An SDE for the evolution of the environmental component $d\Delta
s_{\text{env}}$ of stochastic entropy production [6] is given in Appendix A. A
significant problem in computing $d\Delta s_{\text{env}}$ using this SDE is
that the determinant of the diffusion matrix in the Fokker-Planck equation can
be zero, making it impossible to define an inverse matrix needed in Eq. (31).
This pathology, however, can be overcome, as we now demonstrate using
procedures summarised in Appendix B.

###  II.3 Environmental stochastic entropy production in $x,z$ coordinates

The confinement of the trajectory to an ellipsoid in Fig. 1 is a consequence
of the existence of a constant function of the motion

$f\left(x,y,z\right)=\frac{1}{y^{2}}\left(1-x^{2}-z^{2}\right),$ (8)

which evolves by Itô’s lemma according to

$\begin{split}df&=\sum_{i}\frac{\partial f}{\partial
r_{i}}dr_{i}+\sum_{i,j}\frac{\partial^{2}f}{\partial r_{i}\partial
r_{j}}D_{ij}dt,\end{split}$ (9)

where elements of the diffusion matrix are
$D_{ij}=\frac{1}{2}\sum_{k}B_{ik}B_{jk}$ and the matrix $\boldsymbol{B}$ is
defined in Eq. (5). Using Eq. (4) we obtain

$\boldsymbol{D}=\begin{pmatrix}x^{4}-2x^{2}+z^{2}+1&xy\left(x^{2}-1\right)&xz\left(x^{2}-2\right)\\\
xy\left(x^{2}-1\right)&x^{2}y^{2}&x^{2}yz\\\
xz\left(x^{2}-2\right)&x^{2}yz&x^{2}\left(z^{2}+1\right)\end{pmatrix},$ (10)

and by insertion into Eq. (9) we find that $df=0$, demonstrating $f$ to be a
constant of the motion.

The constancy of $f$ creates a situation characterised by restricted diffusion
[8]. The dynamics in Eq. (4) express evolution in an $N=3$ dimensional phase
space driven by $M=2$ noise terms. Following the arguments summarised in
Appendix B, we identify the single $\left(N-M=1\right)$ eigenvector of
$\boldsymbol{D}$ with an eigenvalue equal to zero, finding this to be
$\boldsymbol{\alpha}=\begin{pmatrix}\frac{x}{z},&\frac{1-x^{2}-z^{2}}{yz},&1\end{pmatrix}^{\mathsf{T}}$.
The existence of a ‘null’ eigenvector allows us to remove one of the SDEs from
the dynamics. If we eliminate the evolution of coordinate $y$ the dynamics are
then expressed according to

$\begin{pmatrix}dx\\\ dz\end{pmatrix}=\begin{pmatrix}-x\\\
-2z\end{pmatrix}dt+\begin{pmatrix}1-x^{2}-z&1-x^{2}+z\\\
x\left(1-z\right)&-x\left(1+z\right)\end{pmatrix}\begin{pmatrix}dW_{1}\\\
dW_{2}\end{pmatrix},$ (11)

and the resulting reduced diffusion matrix is given by

$\boldsymbol{D}_{{\rm
red}}=\begin{pmatrix}x^{4}-2x^{2}+z^{2}+1&xz\left(x^{2}-2\right)\\\
xz\left(x^{2}-2\right)&x^{2}\left(z^{2}+1\right)\end{pmatrix},$ (12)

which is non-singular. We can now compute trajectories for $d\Delta
s_{\text{env}}$ using Eq. (31) with Eq. (11) and the reduced diffusion matrix
(12). Figure 3 illustrates 50 realisations of the dynamics together with the
average evolution.

Figure 3: Environmental component of stochastic entropy production computed
using Eq. (31) for 50 realisations of the dynamics described by Eq. (11). The
black line represents the ensemble mean. Each trajectory was initiated at the
point $x=z=0.5$ and was evaluated using a timestep of $dt=10^{-5}$.

The increase in $d\Delta s_{\text{env}}$ with time, on average, seems to be a
pathology: we might expect entropy production to cease as the system becomes
thermalised. However, the dynamics are incapable of achieving thermalisation
in finite time. The purity approaches unity only as $t\to\infty$, according to
Eq. (6), and as this evolution continues, stochastic entropy production will
persist. The situation is analogous to behaviour found in a two-level system
undergoing measurement [3]. Thermalisation is only achieved asymptotically,
and the production of stochastic entropy continues to measure progress in this
direction. This resolves the issue, but in the next Section we turn to a more
puzzling matter.

##  III Pathologies in stochastic entropy production for a pure state

Consider the situation where purity has been established such that $y\to 0$
and $x^{2}\to 1-z^{2}$. The coherence vector then evolves randomly on a circle
of unit radius in the $x$-$z$ plane and the stochastic dynamics and stochastic
entropy production for the further relaxation of the pure state involve the
evolution of a single coordinate $z$. We give this situation our attention in
order to resolve an apparent pathology in the stochastic entropy production.

Inserting the limits for $x$ and $y$ into Eq. (4) leads to an SDE

$dz=-2zdt+\left(2\left(1-z^{4}\right)\right)^{1/2}dW,$ (13)

describing the thermalisation of a pure state of the two-level system through
the Lindblads $c_{\pm}$. Note that in the absence of a system Hamiltonian the
thermal state will be characterised by equal occupational probabilities of the
two levels, each represented by coherence vectors at the north and south poles
of the Bloch sphere, or $z=\pm 1$. The mean value of $z$ will tend
asymptotically to zero as a consequence. Note that if the Lindblad operators
were unequally weighted, for example using
$\left(1-\gamma/2\right)^{1/2}c_{+}$ and $\left(1+\gamma/2\right)^{1/2}c_{-}$
where $\gamma$ is a constant, the SDE would read

$dz=-\left(\gamma+2z\right)dt+\left(2\left(1-z^{2}\right)\left(1+\gamma
z+z^{2}\right)\right)^{1/2}dW,$ (14)

which has a stationary state with $\langle z\rangle=-\gamma/2$.

However, there are difficulties in computing the stochastic entropy production
associated with the dynamics of Eq. (13), which we now identify.

###  III.1 Environmental component

Referring to Eq. (13) and Appendix A, the environmental component of
stochastic entropy production takes the form

$\displaystyle d\Delta s_{{\rm env}}$ $\displaystyle=$
$\displaystyle\frac{A^{{\rm irr}}}{D}dz+\frac{\partial A^{{\rm irr}}}{\partial
z}dt-\frac{1}{D}\frac{\partial D}{\partial z}dz$ (15)
$\displaystyle-\frac{A^{{\rm irr}}}{D}\frac{\partial D}{\partial
z}dt-\frac{\partial^{2}D}{\partial z^{2}}dt+\frac{1}{D}\left(\frac{\partial
D}{\partial z}\right)^{2}dt,\qquad$

with $A^{{\rm irr}}=-2z$ and $D=1-z^{4}$. Inserting $dD/dz=-4z^{3}$ and
$d^{2}D/dz^{2}=-12z^{2}$, we obtain

$\displaystyle d\Delta s_{{\rm env}}$
$\displaystyle=2z\frac{\left(2z^{2}-1\right)}{\left(1-z^{4}\right)^{1/2}}2^{1/2}dW$
$\displaystyle+\frac{2}{1-z^{4}}\left(-1-7z^{4}+8z^{2}+2z^{6}\right)dt,$ (16)

which has singularities at $z=\pm 1$, potentially creating difficulties in
evaluating $d\Delta s_{{\rm env}}$ numerically.

###  III.2 System component

Further problems emerge when we consider the system component of stochastic
entropy production. This is apparent from the histogram of values of $z$ in
Fig. 4, obtained from a single trajectory generated using the set of SDEs in
Eq. (4). The dynamics clearly allow $z$ to take values $\pm 1$, suggesting
that the singularities in the environmental component of stochastic entropy
production identified in Eq. (16) might be encountered.

Figure 4: Histogram illustrating the stationary pdf of $z$ generated from a
solution to Eq. (4) using $10^{7}$ timesteps of size $dt=10^{-3}$. The
trajectory was initiated at the point $x=y=z=0.5$ and the bin size is
$10^{-4}$.

We can derive the stationary pdf analytically to illustrate this further. The
Fokker-Planck equation associated with Eq. (13) is

$\frac{\partial p(z,t)}{\partial t}=-\frac{\partial J}{\partial z},$ (17)

where the current is

$J=-2zp-\frac{\partial}{\partial z}\left(\left(1-z^{4}\right)p\right).$ (18)

The stationary pdf in $-1\leq z\leq 1$ is given by

$p_{{\rm st}}(z)\propto\frac{1}{\left(1-z^{4}\right)^{1/2}}\frac{1}{1+z^{2}},$
(19)

which possesses singularities at $z=\pm 1$, consistent with the numerical
results in Fig. 4.

Note that the stationary expectation value $\int_{-1}^{1}zp_{{\rm st}}(z)dz=0$
is compatible with the asymptotic solution to the evolution $d\langle
z\rangle=-2\langle z\rangle dt$ obtained by averaging the SDE (13) over the
noise and employing $\langle dW\rangle=0$, where angled brackets denote
stochastic averages. In Appendix C we show, however, that this result need not
hold if the product of diffusion coefficient and pdf at the boundaries does
not vanish: an extra term contributes such that

$d\langle z\rangle=-2\langle z\rangle dt-\left[Dp\right]_{z=-1}^{z=1}dt.$ (20)

In this case we have $Dp_{{\rm st}}\propto(1-z^{4})^{1/2}/(1+z^{2})$, which
vanishes at $z=\pm 1$. Assuming, not unreasonably, that this feature is
preserved when the system is nonstationary, the usual outcome of the
stochastic averaging of Eq. (13) emerges.

Boundary terms also appear in the evolution of the mean system component of
stochastic entropy production. According to Eq. (51), the relationship between
this quantity and the change in Gibbs entropy $S_{G}$ is:

$\displaystyle d\langle\Delta s_{{\rm sys}}\rangle$
$\displaystyle=\frac{dS_{G}(t)}{dt}dt-\left[J\ln
p\right]_{z=-1}^{z=1}dt-\left[D\frac{\partial p}{\partial
z}\right]_{z=-1}^{z=1}dt$ $\displaystyle-$
$\displaystyle\left[D\left(\frac{\partial\ln p}{\partial z}\right)^{2}P(\Delta
s_{{\rm sys}},t)\right]_{\Delta s_{{\rm sys}}^{{\rm min}}}^{\Delta s_{{\rm
sys}}^{{\rm max}}}dt.$ (21)

The first additional term vanishes since $J=0$ at the boundaries and the third
can be simplified using $P(\Delta s_{{\rm sys}},t)d\Delta s_{{\rm sys}}=pdz$.
In the stationary state both the second and third terms depend on the
expression

$\displaystyle D\frac{dp_{{\rm st}}}{dz}$
$\displaystyle\propto\left(1-z^{4}\right)\Big{(}2z^{3}\left(1-z^{4}\right)^{-3/2}\left(1+z^{2}\right)^{-1}$
$\displaystyle-2z\left(1-z^{4}\right)^{-1/2}\left(1+z^{2}\right)^{-2}\Big{)},$
(22)

which is singular at $z=\pm 1$. We therefore expect difficulties to arise in
computing the system stochastic entropy production, in addition to those
already encountered in the evaluation of $\Delta s_{{\rm env}}$. But are these
genuine problems?

###  III.3 Removal of pathological behaviour

The pathologies may be removed by a simple change of coordinate frame.
Defining $\theta=\cos^{-1}z$ and using Itô’s lemma, we find that the dynamics
of the pure state may also be described by the SDE

$d\theta=\frac{1}{2}\sin 2\theta
dt+\left(2\left(1+\cos^{2}\theta\right)\right)^{1/2}dW,$ (23)

instead of Eq. (13) for $z$. The diffusion coefficient is
$D=1+\cos^{2}\theta$. The stationary pdf in $\theta$ is

$p_{{\rm st}}(\theta)\propto\left(1+\cos^{2}\theta\right)^{-3/2},$ (24)

and so $Dp_{{\rm st}}\propto\left(1+\cos^{2}\theta\right)^{-1/2}$ such that
the boundary term $\left[Dp_{{\rm st}}\right]_{\theta=0}^{\theta=\pi}$
vanishes. As a consequence, the averaging of Eq. (23) proceeds along usual
lines, giving $d\langle\theta\rangle=\frac{1}{2}\langle\sin 2\theta\rangle
dt$.

Furthermore, we have

$D\frac{dp_{{\rm st}}}{d\theta}\propto\frac{\sin 2\theta}{1+\cos^{2}\theta},$
(25)

which is zero at $\theta=0$ and $\pi$. As a consequence, the boundary terms in
the evolution of mean system stochastic entropy production in the stationary
state vanish. It is not unreasonable to expect that even in a nonstationary
situation the evolution should satisfy

$\displaystyle d\langle\Delta s_{{\rm sys}}\rangle$
$\displaystyle=\frac{dS_{G}(t)}{dt}dt,$ (26)

again in line with what is typically assumed.

Finally, the evolution of environmental stochastic entropy production can be
identified using the SDE for $\theta$:

$d\Delta s_{{\rm env}}=\frac{6+18\cos
2\theta+3\sin^{2}2\theta}{4\left(1+\cos^{2}\theta\right)}dt+\frac{3\sin
2\theta}{\sqrt{2}\left(1+\cos^{2}\theta\right)}dW,$ (27)

in which there are once again no singularities at the boundaries. We conclude
that all the pathologies disappear when an appropriate coordinate system is
used to describe the dynamics and thermodynamics.

##  IV Conclusions

The main purpose in presenting this study is to identify interpretational and
mathematical issues concerning stochastic entropy production in a minimal
model of the thermalisation of a two-level quantum system, and then to resolve
them. The treatment of open system dynamics as a continuous Brownian motion of
a physical density matrix, a quantum state diffusion, is perhaps unfamiliar
and the consideration of stochastic entropy production as the outcome of an
Itô process particularly so. The mean value of stochastic entropy production
is a measure of the change in subjective uncertainty in the adopted state of
the world given the employment of a coarse grained model of its evolution. We
need to address any apparent pathologies in the calculation of stochastic
entropy production that might cast doubt on its use in this role.

Our conclusions are as follows. Thermalisation of a two-level system can be
achieved through coupling to a coarse grained environment by way of raising
and lowering operators. Such an interaction is not the same as quantum
measurement, but nevertheless the minimal model we use purifies the reduced
density matrix asymptotically in time, and this brings about a persistent
production of stochastic entropy. This is not a pathology and indeed it seems
very reasonable that thermal interactions should disentangle a system from its
environment, creating greater uncertainty in the state of the world as the
more classical, less correlated system behaviour emerges. Note that an impure
or mixed reduced density matrix, such as the Gibbs state often used to
describe a thermalised system, is to be interpreted in the framework of
quantum state diffusion as the _average_ state of an ensemble of pure reduced
density matrices. The mixed reduced density matrix does not necessarily imply
entanglement.

Furthermore, if the two-level system is thermalised starting from a
disentangled or pure state, the stochastic entropy production can appear to be
pathological in both the environmental and system components. It turns out,
however, that such a conclusion is an artefact of a particular choice of
coordinate system and a simple transformation of variables can remove the
mathematical difficulties. Stochastic entropy production should not depend on
choice of coordinates, but some are more suitable than others. We note as a
corollary that the separate system and environmental components of stochastic
entropy production _are_ coordinate frame dependent. Gibbs entropy, for
example, expresses uncertainty of adopted system state in a particular
coordinate phase space.

In conclusion, we have presented arguments to support the use of stochastic
entropy production in open quantum system dynamics in a manner similar to its
employment in classical situations. Its role and the implications are just the
same, and we suggest, just as profound.

##  Appendix A Stochastic entropy production

We consider a set of coordinates $\boldsymbol{x}\equiv(x_{1},x_{2},\cdots)$
that specify the configuration of a system, and model their evolution using
Itô processes:

$dx_{i}=A_{i}(\boldsymbol{x})dt+\sum_{j}B_{ij}(\boldsymbol{x})dW_{j},$ (28)

where the $dW_{j}$ are independent Wiener increments. The Fokker-Planck
equation for the pdf $p(\boldsymbol{x},t)$ is

$\frac{\partial p}{\partial t}=-\sum_{i}\frac{\partial}{\partial
x_{i}}\left(A_{i}p\right)+\frac{\partial}{\partial x_{i}\partial
x_{j}}\left(D_{ij}p\right),$ (29)

in terms of a diffusion matrix
$\boldsymbol{D}(\boldsymbol{x})=\frac{1}{2}\boldsymbol{B}(\boldsymbol{x})\boldsymbol{B}(\boldsymbol{x})^{\mathsf{T}}$.
The stochastic entropy production of the system, and its environment,
associated with the stochastic dynamics is given by

$d\Delta s_{{\rm tot}}=d\Delta s_{\text{sys}}+d\Delta s_{\text{env}},$ (30)

with $d\Delta s_{\text{sys}}=-d\ln p(\boldsymbol{x},t)$. If we define
$A_{i}^{{\rm
irr}}(\boldsymbol{x})=\frac{1}{2}\left[A_{i}(\boldsymbol{x})+\varepsilon_{i}A_{i}(\boldsymbol{\varepsilon}\boldsymbol{x})\right]$
and $A_{i}^{{\rm
rev}}(\boldsymbol{x})=\frac{1}{2}\left[A_{i}(\boldsymbol{x})-\varepsilon_{i}A_{i}(\boldsymbol{\varepsilon}\boldsymbol{x})\right]$
[10] where $\varepsilon_{i}=1$ for variables $x_{i}$ with even parity under
time reversal symmetry (for example position) and $\varepsilon_{i}=-1$ for
variables with odd parity (for example velocity), with
$\boldsymbol{\varepsilon}\boldsymbol{x}$ representing
$(\varepsilon_{1}x_{1},\varepsilon_{2}x_{2},\cdots)$, it may be shown that [6]

$\begin{split}&d\Delta s_{\text{env}}=-\sum_{i}\frac{\partial
A_{i}^{\text{rev}}(\boldsymbol{x})}{\partial
x_{i}}dt+\sum_{i,j}\Biggl{\\{}\frac{D_{ij}^{-1}(\boldsymbol{x})}{2}\left(A_{i}^{\text{irr}}(\boldsymbol{x})dx_{j}+A_{j}^{\text{irr}}(\boldsymbol{x})dx_{i}\right)\\\
&-\frac{D_{ij}^{-1}(\boldsymbol{x})}{2}\left(\left(\sum_{n}\frac{\partial
D_{jn}(\boldsymbol{x})}{\partial
x_{n}}\right)dx_{i}+\left(\sum_{m}\frac{\partial
D_{im}(\boldsymbol{x})}{\partial
x_{m}}\right)dx_{j}\right)-\frac{D_{ij}^{-1}(\boldsymbol{x})}{2}\left(A_{i}^{\text{rev}}(\boldsymbol{x})A_{j}^{\text{irr}}(\boldsymbol{x})+A_{j}^{\text{rev}}(\boldsymbol{x})A_{i}^{\text{irr}}(\boldsymbol{x})\right)dt\\\
&+\frac{D_{ij}^{-1}(\boldsymbol{x})}{2}\left(A_{j}^{\text{rev}}(\boldsymbol{x})\left(\sum_{m}\frac{\partial
D_{im}(\boldsymbol{x})}{\partial
x_{m}}\right)+A_{i}^{\text{rev}}(\boldsymbol{x})\left(\sum_{n}\frac{\partial
D_{jn}(\boldsymbol{x})}{\partial x_{n}}\right)\right)dt\\\
&+\frac{1}{2}\sum_{k}\Biggl{[}D_{ik}(\boldsymbol{x})\frac{\partial}{\partial
x_{k}}\left(D_{ij}^{-1}(\boldsymbol{x})A_{j}^{\text{irr}}(\boldsymbol{x})\right)+D_{jk}(\boldsymbol{x})\frac{\partial}{\partial
x_{k}}\left(D_{ij}^{-1}(\boldsymbol{x})A_{i}^{\text{irr}}(\boldsymbol{x})\right)\\\
&-D_{ik}(\boldsymbol{x})\frac{\partial}{\partial
x_{k}}\left(D_{ij}^{-1}(\boldsymbol{x})\left(\sum_{n}\frac{\partial
D_{jn}(\boldsymbol{x})}{\partial
x_{n}}\right)\right)-D_{jk}(\boldsymbol{x})\frac{\partial}{\partial
x_{k}}\left(D_{ij}^{-1}(\boldsymbol{x})\left(\sum_{m}\frac{\partial
D_{im}(\boldsymbol{x})}{\partial
x_{m}}\right)\right)\Biggr{]}dt\Biggr{\\}}.\end{split}$ (31)

##  Appendix B Dimensional reduction of a singular diffusion matrix

Consider a system evolving stochastically in an $N$ dimensional phase space
under the influence of $M$ noise terms such that the $\boldsymbol{B}$ matrix
in Eq. (28) is $N\times M$ and the diffusion matrix $\boldsymbol{D}$ in the
Fokker-Planck equation (29) is $N\times N$. The phase space is described by a
set of $N$ coordinates $\left\\{x_{1},x_{2},...,x_{N}\right\\}$, and we
conjecture that there exist $L=N-M$ functions of these coordinates that are
invariant under the dynamics, or more generally, evolve without noise. These
allow us to remove $L$ coordinates from the $N$ dimensional phase space and
consider evolution in a reduced $M$ dimensional phase space. We call the $M$
coordinates $\left(\left\\{x_{m}\right\\}\right)$ of the reduced phase space
_dynamical variables_ and the remaining $L$ coordinates
$\left(\left\\{x_{l}\right\\}\right)$ _spectator variables_. The $L$
constraints allow us to write the spectator variables as functions of the
dynamical variables:
$\left\\{x_{l}\left(\left\\{x_{m}\right\\}\right)\right\\}$. The diffusion
matrix in the reduced phase space will be $M\times M$ and non-singular
allowing us to use Eq. (31) to compute the stochastic entropy production
associated with the evolution [8].

If $f(\boldsymbol{x})$ is a constant of the motion, we can regard
$\boldsymbol{\nabla}f$ as a ‘null’ eigenvector of $\boldsymbol{D}$ with zero
eigenvalue. Furthermore, since the infinitesimal displacement
$d\boldsymbol{x}$ lies on the hypersurface of $f$ to which the evolution is
confined, we also have $\boldsymbol{\nabla}f\cdot d\boldsymbol{x}=0$.
Combining these we write

$\sum_{m=1}^{M}\alpha_{km}dx_{m}+\sum_{l=1}^{L}\alpha_{kl}dx_{l}=0,$ (32)

where $\alpha_{km}$ and $\alpha_{kl}$ are the dynamical and spectator
components of the $k$th null eigenvector of $\boldsymbol{D}$. The
$\alpha_{km}$ are elements of a rectangular $L\times M$ matrix $Q_{km}$ and
the $\alpha_{kl}$ are elements of a square $L\times L$ matrix $P_{kl}$ so that
$P_{kl}dx_{l}=-Q_{km}dx_{m}$ and

$\begin{split}dx_{l}&=-P_{lk}^{-1}Q_{km}dx_{m}=R_{lm}dx_{m},\end{split}$ (33)

where summations over repeated indices are implied and $R_{lm}$ is a component
of the $L\times M$ matrix $\boldsymbol{R}$. Next we write

$dD_{ij}=\sum_{m=1}^{M}\frac{\partial D_{ij}}{\partial
x_{m}}dx_{m}+\sum_{l=1}^{L}\frac{\partial D_{ij}}{\partial x_{l}}dx_{l},$ (34)

and substituting from Eq. (33) we obtain the derivative of $D_{ij}$ with
respect to the dynamical coordinate $x_{m}$ in the form

$\frac{dD_{ij}}{dx_{m}}=\frac{\partial D_{ij}}{\partial
x_{m}}+\sum_{l}\frac{\partial D_{ij}}{\partial x_{l}}R_{lm}.$ (35)

Equation (31) can now be used to compute stochastic entropy production based
on the dynamics of the $M$ dynamical variables with derivatives modified
according to Eq. (35). Additional discussion of the approach is given
elsewhere [8].

##  Appendix C System component of mean stochastic entropy production

It is typically assumed that the average of the $d\Delta s_{{\rm sys}}=-d\ln
p(\boldsymbol{x},t)$ term in the expression for the incremental stochastic
entropy production corresponds to the change in system Gibbs entropy, but this
is not necessarily the case, as we now show.

Consider the SDE for a function $g(\boldsymbol{x})$ of the stochastic
variables $\boldsymbol{x}$:

$dg=\hat{A}(\boldsymbol{x})dt+\hat{B}(\boldsymbol{x})dW,$ (36)

with pdf $P(g,t)$ satisfying

$\frac{\partial P(g,t)}{\partial t}=-\frac{\partial\hat{J}}{\partial g},$ (37)

in terms of a probability current $\hat{J}$ given by

$\hat{J}=\hat{A}P-\frac{1}{2}\frac{\partial}{\partial
g}\left(\hat{B}^{2}P\right).$ (38)

Defining a stochastic average of $g$ as $\langle g\rangle_{t}=\int gP(g,t)dg$,
we consider the difference $d\langle g\rangle$ between $\langle
g\rangle_{t+dt}$ and $\langle g\rangle_{t}$:

$\displaystyle d\langle g\rangle$ $\displaystyle=\int
g\left(P(g,t+dt)-P(g,t)\right)dg=\int g\frac{\partial P}{\partial t}dgdt$
$\displaystyle=-\int g\frac{\partial\hat{J}}{\partial
g}dgdt=-\left[g\hat{J}\right]dt+\int\hat{J}\,dgdt,$ (39)

where the square brackets indicate boundary terms. Since $\hat{J}=0$ at the
boundaries in the absence of probability current sources and sinks, the first
term vanishes and so

$\displaystyle d\langle g\rangle$
$\displaystyle=\int\left(\hat{A}P-\frac{1}{2}\frac{\partial}{\partial
g}\left(\hat{B}^{2}P\right)\right)dgdt$
$\displaystyle=\langle\hat{A}\rangle_{t}dt-\frac{1}{2}\left[\hat{B}^{2}P\right]dt,$
(40)

where another boundary term has appeared. Note that the average in the first
term on the right hand side can be written in two forms:
$\langle\hat{A}\rangle_{t}=\int\hat{A}(\boldsymbol{x})P(g,t)dg=\int\hat{A}(\boldsymbol{x})p(\boldsymbol{x},t)d\boldsymbol{x}$.
The second version involves the pdf $p(\boldsymbol{x},t)$ over the stochastic
variables $\boldsymbol{x}$, as specified by the Fokker-Planck equation

$\frac{\partial p}{\partial t}=-\sum_{j}\frac{\partial J_{j}}{\partial
x_{j}},$ (41)

associated with the dynamics $dx_{i}=A_{i}dt+\sum_{j}B_{ij}dW_{j}.$ The
probability current is

$J_{j}=A_{j}p-\sum_{i}\frac{\partial}{\partial x_{i}}\left(D_{ij}p\right),$
(42)

where $D_{ij}=\frac{1}{2}\sum_{k}B_{ik}B_{jk}$.

The first term on the right hand side of Eq. (40) would perhaps be expected to
be the only contribution after averaging Eq. (36), but it is augmented by a
term that depends on the values of $\hat{B}^{2}P$ at the boundaries of the
domain of $g$. This extra term could vanish but it is not guaranteed to do so.

Consider, for example, a single stochastic variable evolving as $dx=adt+bdW$:
the stochastic average satisfies $d\langle x\rangle/dt=\langle a\rangle-[Dp]$
where $D=b^{2}/2$. In many cases, the pdf vanishes at the boundaries, or for
periodic boundary conditions the contributions might cancel. It is only in
certain cases that further attention is required. An explicit case involves
$a(x)=x$ and $b(x)=x^{2}$ in the domain $0\leq x\leq\infty$. The stationary
pdf is $p_{{\rm st}}(x)=4\pi^{-1/2}x^{-4}\exp(-1/x^{2})$ and the diffusion
coefficient is $D=x^{4}/2.$ Hence $Dp_{{\rm st}}$ vanishes at $x=0$ but is
equal to $2/\pi^{1/2}$ at $x=\infty$. We should hence model the evolution of
$\langle x\rangle$ according to $d\langle x\rangle/dt=\langle
x\rangle-2/\pi^{1/2}$, which is consistent with a non-zero mean of $x$ as
$t\to\infty$, as suggested by the stationary pdf. The boundary term is crucial
here.

Let us now analyse the evolution of the system stochastic entropy production,
$g(\boldsymbol{x})=\Delta s_{{\rm sys}}=-\ln p(\boldsymbol{x},t)$. We write

$\displaystyle d\Delta s_{{\rm sys}}$ $\displaystyle=-\frac{\partial\ln
p}{\partial t}dt-\sum_{j}\frac{\partial\ln p}{\partial
x_{j}}dx_{j}-\sum_{ij}D_{ij}\frac{\partial^{2}\ln p}{\partial x_{i}\partial
x_{j}}dt$
$\displaystyle=\hat{A}dt+\sum_{ij}\hat{B}_{ij}dW_{j}=\hat{A}dt+\hat{B}dW,$
(43)

with

$\hat{A}=-\frac{\partial\ln p}{\partial t}-\sum_{j}\frac{\partial\ln
p}{\partial x_{j}}A_{j}-\sum_{ij}D_{ij}\frac{\partial^{2}\ln p}{\partial
x_{i}\partial x_{j}},$ (44)

and

$\hat{B}_{ij}=-\frac{\partial\ln p}{\partial x_{i}}B_{ij},$ (45)

such that

$\displaystyle\hat{B}$
$\displaystyle=\left(\sum_{ijk}\hat{B}_{kj}\hat{B}_{ij}\right)^{1/2}=\left(\sum_{ijk}\frac{\partial\ln
p}{\partial x_{k}}\frac{\partial\ln p}{\partial
x_{i}}B_{kj}B_{ij}\right)^{1/2}$
$\displaystyle=\left(2\sum_{ik}\frac{\partial\ln p}{\partial
x_{k}}\frac{\partial\ln p}{\partial x_{i}}D_{ki}\right)^{1/2}.$ (46)

The mean system stochastic entropy production therefore satisfies

$d\langle\Delta s_{{\rm
sys}}\rangle=\langle\hat{A}\rangle_{t}dt-\frac{1}{2}\left[\hat{B}^{2}P(\Delta
s_{{\rm sys}},t)\right]dt,$ (47)

where $P$ is the pdf of $\Delta s_{{\rm sys}}$. $\hat{B}$ is a function of the
stochastic variables $\boldsymbol{x}$ and the second term is evaluated at the
boundary values of $\Delta s_{{\rm sys}}$ using the appropriate values of
$\boldsymbol{x}$.

We further develop the first term on the right hand side of Eq. (47), writing

$\langle\hat{A}\rangle_{t}=\int\hat{A}(\boldsymbol{x}(\Delta s_{{\rm
sys}}))P(\Delta s_{{\rm sys}},t)d\Delta s_{{\rm sys}},$ (48)

and noting that for a given $\Delta s_{{\rm sys}}$ there are potentially
several corresponding values of $\boldsymbol{x}$, each of which contributes to
the integrand. It is better to cast the integral in terms of the
$\boldsymbol{x}$ instead of $\Delta s_{{\rm sys}}$, namely
$\langle\hat{A}\rangle_{t}=\int\hat{A}(\boldsymbol{x})p(\boldsymbol{x},t)d\boldsymbol{x}$
and employing integration limits for $\boldsymbol{x}$ instead of $\Delta
s_{{\rm sys}}$. Continuing the development we find:

$\displaystyle\langle\hat{A}\rangle_{t}$
$\displaystyle=\int\left(-\frac{\partial\ln p}{\partial
t}-\sum_{j}\frac{\partial\ln p}{\partial
x_{j}}A_{j}-\sum_{ij}D_{ij}\frac{\partial^{2}\ln p}{\partial x_{i}\partial
x_{j}}\right)pd\boldsymbol{x}$ $\displaystyle=$
$\displaystyle\int\left(-\frac{\partial\ln p}{\partial
t}p-\sum_{j}\left(A_{j}p-\frac{\partial\left(D_{ij}p\right)}{\partial
x_{i}}\right)\frac{\partial\ln p}{\partial x_{j}}\right)d\boldsymbol{x}$
$\displaystyle-\sum_{ij}\left[D_{ij}\frac{\partial\ln p}{\partial
x_{j}}p\right]$ $\displaystyle=$ $\displaystyle\int\left(-\frac{\partial\ln
p}{\partial t}p+\sum_{j}\frac{\partial J_{j}}{\partial x_{j}}\ln
p\right)d\boldsymbol{x}$ $\displaystyle-\sum_{j}\left[J_{j}\ln
p\right]-\sum_{ij}\left[D_{ij}\frac{\partial p}{\partial x_{j}}\right],$ (49)

reducing to

$\displaystyle\langle\hat{A}\rangle_{t}$
$\displaystyle=\int\left(-\frac{\partial\ln p}{\partial t}p-\frac{\partial
p}{\partial t}\ln p\right)d\boldsymbol{x}$
$\displaystyle-\sum_{j}\left[J_{j}\ln
p\right]-\sum_{ij}\left[D_{ij}\frac{\partial p}{\partial x_{j}}\right]$
$\displaystyle=$ $\displaystyle-\frac{d}{dt}\int p\ln
p\,d\boldsymbol{x}-\sum_{j}\left[J_{j}\ln
p\right]-\sum_{ij}\left[D_{ij}\frac{\partial p}{\partial x_{j}}\right].$ (50)

Combining Eqs. (47) and (50), and inserting the Gibbs entropy $S_{G}(t)=-\int
p(\boldsymbol{x},t)\ln p(\boldsymbol{x},t)\,d\boldsymbol{x}$, we arrive at

$\displaystyle d\langle\Delta s_{{\rm sys}}\rangle$
$\displaystyle=\frac{dS_{G}(t)}{dt}dt-\sum_{j}\left[J_{j}\ln p\right]dt$
$\displaystyle-\sum_{ij}\left[D_{ij}\frac{\partial p}{\partial
x_{j}}\right]dt-\frac{1}{2}\left[\hat{B}^{2}P(\Delta s_{{\rm
sys}},t)\right]dt.$ (51)

The increment in mean system stochastic entropy production corresponds in part
to the increment in Gibbs entropy $dS_{G}=\frac{dS_{G}}{dt}dt$, but three
boundary terms also contribute, the first two involving the limits of the
phase space of the $\boldsymbol{x}$ coordinates, and the third the limits of
the range of $\Delta s_{{\rm sys}}$.

## References

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  * Matos  _et al._ [2022] D. Matos, L. Kantorovich, and I. J. Ford, Stochastic entropy production for continuous measurements of an open quantum system, Journal of Physics Communications 6, 125003 (2022).
  * Clarke and Ford [2023] C. L. Clarke and I. J. Ford, Stochastic entropy production associated with quantum measurement in a framework of Markovian quantum state diffusion, arXiv:2301.08197 [quant-ph] (2023).
  * Seifert [2008] U. Seifert, Stochastic thermodynamics: principles and perspectives, The European Physical Journal B 64, 423 (2008).
  * Percival [1998] I. Percival,  _Quantum State Diffusion_ (Cambridge University Press, 1998).
  * Spinney [2012] R. E. Spinney,  _The use of stochastic methods to explore the thermal equilibrium distribution and define entropy production out of equilibrium_ , Ph.D. thesis, UCL (University College London) (2012).
  * Spinney and Ford [2012] R. E. Spinney and I. J. Ford, Entropy production in full phase space for continuous stochastic dynamics, Physical Review E 85, 051113 (2012).
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  * Jacobs [2014] K. Jacobs,  _Quantum Measurement Theory and its Applications_ (Cambridge University Press, 2014).
  * R.E. Spinney and I.J. Ford [2012] R.E. Spinney and I.J. Ford, Entropy production in full phase space for continuous stochastic dynamics, Phys. Rev. E 85, 051113 (2012).


# Bertrand’s analysis of baccarat

Stewart N. Ethier

###### Abstract

Joseph Bertrand [1822–1900], who is often credited with a model of duopoly
that has a unique Nash equilibrium, made another significant contribution to
game theory. Specifically, his 1888 analysis of baccarat was the starting
point for Borel’s investigation of strategic games in the 1920s. In this paper
we show, with near certainty, that Bertrand’s results on baccarat were
borrowed, without attribution, from an 1881 paper of Albert Badoureau
[1853–1923]. In addition, we discuss Borel’s criticisms of Bertrand’s
analysis, one of which helps to explain why Badoureau’s contribution was
overlooked until now.

Keywords: baccarat chemin de fer, noncooperative game, pure strategy, best
response.

##  1 Introduction

In The History of Game Theory, Volume I: From the Beginnings to 1945 (Dimand
and Dimand, 1996, p. 132), we find, “As Borel (1924, p. 101) noted, the
starting-point for his investigation of strategic games was the analysis of
baccarat by Bertrand.” In Borel’s (1924) own words,111This is an edited
version of Savage’s (1953) translation.

> The only author who has studied a particular case of the problem we envisage
> is Joseph Bertrand, in the passage of his calculus of probabilities that he
> devotes to the draw at five in the game of baccarat; there he clearly
> distinguishes the purely mathematical side of the problem from the
> psychological side, because he asks, on the one hand, if the punter has an
> advantage in drawing at five when the banker knows the punter’s manner of
> play and, on the other hand, if the punter has an advantage in drawing at
> five by letting the banker believe, if he can, that it is not his custom; he
> also poses the same questions for the non-draw at five. But, as we will see
> from what follows, this study is extremely incomplete; on the one hand,
> Bertrand does not investigate what would happen if the punter would draw at
> five in a certain fraction of the total number of coups and, on the other
> hand, he does not investigate whether the banker’s opinion on the draw at
> five of the punter does not affect the way the banker plays in some cases
> where the punter does not have five, which may result in an advantage for
> either player in the whole game. We see, by these brief indications, how
> complex is the problem approached by Bertrand, in spite of the rudimentary
> simplicity of the game of baccarat; […].

To describe Bertrand’s results, we must first explain the rules of the two-
person zero-sum game of baccarat (more precisely, baccarat chemin de fer).

The game is dealt from a shoe comprising six 52-card decks mixed together.
Denominations A, 2–9, 10, J, Q, K have values 1, 2–9, 0, 0, 0, 0,
respectively, and suits are irrelevant. The total of a hand, comprising two or
three cards, is the sum of the values of the cards, modulo 10. In other words,
only the final digit of the sum is used to evaluate a hand. Two cards are
dealt face down to Player and two face down to Banker, and each looks only at
his own hand. The object of the game is to have the higher total (closer to 9)
at the end of play. A two-card total of 8 or 9 is a natural. If either hand is
a natural, the game is over. If neither hand is a natural, Player then has the
option of drawing a third card. If he exercises this option, his third card is
dealt face up. Next, Banker, observing Player’s third card, if any, has the
option of drawing a third card. This completes the game, and the higher total
wins. Winning bets on Player’s hand are paid at even odds, with Banker, as the
name suggests, playing the role of the bank. Losing bets on Player’s hand are
collected by Banker. Hands of equal total result in a tie (no money is
exchanged). Since multiple players can bet on Player’s hand, Player’s strategy
is restricted. He must draw on a two-card total of 4 or less and stand on a
two-card total of 6 or 7. When his two-card total is 5, he is free to stand or
draw as he chooses. (The decision is usually made by the player with the
largest bet.) In much of the French baccarat literature, a Player who always
draws at 5 is called a tireur; likewise, a Player who always stands at 5 is
called a non-tireur. Banker, on whose hand no one can bet, has no constraints
on his strategy in the classical version of the game considered here.

Bertrand (1888, pp. 39–40) states his problem precisely:

> We must solve four problems:
>
> The player having 5 and not requesting a card, what is the probability for
> him to win and what is that of a tie, when the banker, unaware that he has
> the point 5, knows that he has the custom, when he has this point, of not
> requesting a card?
>
> The player having 5 and not requesting a card, what are the probabilities
> for him to win or to tie, when the banker, unaware that he has the point 5,
> knows that he has the custom, when he has this point, of requesting a card?
>
> The player having 5 and requesting a card, what are the probabilities for
> him to win or to tie, when the banker knows his custom of requesting a card
> in this circumstance or when he thinks he knows that he does not request
> one?

Keep in mind that, in the late 19th century, game theory did not yet exist, so
terms such as “pure strategy” and “best response,” which we use below, were
not yet in common usage.

In the next section we show how to solve these four problems. In Section 3 we
compare our results with those of Bertrand and those of Bertrand’s
predecessors in the analysis of baccarat, Dormoy (1872) and Badoureau (1881).
Finally, in Section 4 we discuss Borel’s criticisms of Bertrand’s analysis.

The principal characters in the story include:

  * •

Joseph Bertrand [1822–1900], professor of mathematics at École Polytechnique.
He contributed to several areas of mathematics, including probability, in
which he is best known for the ballot theorem and “Bertrand’s paradox”
(concerning a random chord of a circle).

  * •

Émile Dormoy [1829–1891], a French mining engineer and actuary. He published
numerous books, including three on card games (baccarat, bouillotte, and
écarté). His best-known work is the 1878 book, Théorie mathématique des
assurances sur la vie, in two volumes.

  * •

Albert Badoureau [1853–1923], a French mining engineer and amateur
mathematician. He is best known for his 1878 discovery of 37 of the 75 non-
prismatic uniform polyhedra. He was also scientific advisor to Jules Verne and
wrote a technical appendix to Topsy Turvy.

  * •

Émile Borel [1871–1956], professor of mathematics at the Sorbonne and one of
the great French mathematicians. He was one of the founders of the theory of
functions of a real variable, and he was an early contributor to game theory.

Despite their 49-year age difference, Borel knew Bertrand personally (Bru,
2006, note 12); in fact, Borel’s wife was granddaughter of Bertrand’s elder
brother. Moreover, it is very likely that Badoureau and Bertrand also knew
each other. Badoureau, having been ranked first in his class at École
Polytechnique, was responsible for writing the course handouts for his fellow
students. This would have included the first-year analysis course, which was
offered in alternate years by Bertrand and Charles Hermite [1822–1901].
Records show that Bertrand taught the 1878 class and Badoureau was in the 1874
class. This suggests that Badoureau’s professor for this subject was Bertrand,
not Hermite (Bernard Bru, personal communication, 2021; Jacques Crovisier,
personal communication, 2021). It is also possible that Dormoy and Bertrand
knew each other, for Dormoy ranked fourth in his class at École Polytechnique,
though this was while Bertrand was still a tutor in analysis, before being
promoted to professor (Bru and Jongmans, 2001).

##  2 Solution of Bertrand’s problems

There are four scenarios. Player has a two-card total of 5 and can stand or
draw. Banker replies with a best response (unaware of Player’s two-card
total), assuming either that Player is a non-tireur or that Player is a
tireur. So the first step is to derive Banker’s best response to each of
Player’s two pure strategies. We assume, as is conventional, that cards are
drawn with replacement (or, equivalently, that the shoe has infinitely many
decks). We can summarize these best responses in terms of two $8\times 11$
incidence matrices,

$\bm{D}_{0}=\left(\begin{array}[]{ccccccccccc}1&1&1&1&1&1&1&1&1&1&1\\\
1&1&1&1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1&0&0&1\\\
0&0&1&1&1&1&1&1&0&0&1\\\ 0&0&0&0&0&1&1&1&0&0&1\\\ 0&0&0&0&0&0&1&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0\\\
\end{array}\right),\qquad\bm{D}_{1}=\left(\begin{array}[]{ccccccccccc}1&1&1&1&1&1&1&1&1&1&1\\\
1&1&1&1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1&0&1&1\\\
0&1&1&1&1&1&1&1&0&0&1\\\ 0&0&0&0&1&1&1&1&0&0&1\\\ 0&0&0&0&0&0&1&1&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&0\\\ \end{array}\right),$

in which rows are labeled by Banker’s two-card total, 0–7, and columns are
labeled by Player’s third card, 0–9 and $\varnothing$, where $\varnothing$
means that Player stands. Entries are 0 for stand and 1 for draw, with
$\bm{D}_{0}$ giving Banker’s best response when Player is known to be a non-
tireur, and $\bm{D}_{1}$ giving Banker’s best response when Player is known to
be a tireur.

Notice that $\bm{D}_{0}$ and $\bm{D}_{1}$ differ at four entries: At $(3,9)$,
$(4,1)$, $(5,4)$, and $(6,\varnothing)$, $\bm{D}_{0}$ has a 0 and $\bm{D}_{1}$
has a 1.

How are $\bm{D}_{0}$ and $\bm{D}_{1}$ derived? Let
$p(i):=(1+3\delta_{i,0})/13$ ($i=0,1,\ldots,9$) be the distribution of the
value of a third card, and $q(j):=(16+9\delta_{j,0})/(13)^{2}$
($j=0,1,\ldots,9$) be the distribution of the value of a two-card hand. Define
the function $M$ on the set of nonnegative integers by
$M(i):=\text{mod}(i,10)$. Then, for $u\in\\{0,1\\}$, the $(j,k)$ entry of
$\bm{D}_{u}$ is 1 if, when Banker’s two-card total is $j$ and Player’s third
card is $k$, Banker’s expected profit by drawing is greater than Banker’s
expected profit by standing, that is, if $b_{u,1}(j,k)>b_{u,0}(j,k)$, where

$\begin{split}b_{u,0}(j,k)&:=\sum_{i=0}^{4+u}\text{sgn}(j-M(i+k))q(i)\bigg{/}\sum_{i=0}^{4+u}q(i),\\\
b_{u,1}(j,k)&:=\sum_{i=0}^{4+u}\sum_{l=0}^{9}\text{sgn}(M(j+l)-M(i+k))q(i)p(l)\bigg{/}\sum_{i=0}^{4+u}q(i),\end{split}$
(1)

when $j\in\\{0,1,\ldots,7\\}$ and $k\in\\{0,1,\ldots,9\\}$, and

$\begin{split}b_{u,0}(j,\varnothing)&:=\sum_{i=5+u}^{7}\text{sgn}(j-i)q(i)\bigg{/}\\!\\!\\!\sum_{i=5+u}^{7}q(i),\\\
b_{u,1}(j,\varnothing)&:=\sum_{i=5+u}^{7}\sum_{l=0}^{9}\text{sgn}(M(j+l)-i)q(i)p(l)\bigg{/}\\!\\!\\!\sum_{i=5+u}^{7}q(i),\end{split}$
(2)

when $j\in\\{0,1,\ldots,7\\}$.

One can go on to express the $2\times 2^{88}$ payoff matrix for baccarat in
terms of the conditional expectations (1) and (2) and then eliminate strictly
dominated pure strategies and solve the game. See Ethier (2010, Chapter 5) for
full details.

In each of the equations in (1) and (2), Player’s two-card total is random, so
these quantities will not help us solve Bertrand’s problems, in which Player’s
two-card total is 5. But we do need $\bm{D}_{0}$ and $\bm{D}_{1}$ to address
these problems. Let $f$ be a function on the set of integers and define

$\displaystyle A_{0,v}$ $\displaystyle:=\sum_{\begin{subarray}{c}0\leq j\leq
7:\\\
D_{v}(j,\varnothing)=1\end{subarray}}\sum_{l=0}^{9}f(5-M(j+l))q(j)p(l)\bigg{/}\sum_{j=0}^{7}q(j)$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}0\leq j\leq 7:\\\
D_{v}(j,\varnothing)=0\end{subarray}}f(5-j)q(j)\bigg{/}\sum_{j=0}^{7}q(j)$ (3)

and

$\displaystyle A_{1,v}$ $\displaystyle:=\sum_{\begin{subarray}{c}0\leq j\leq
7,\,0\leq k\leq 9:\\\
D_{v}(j,k)=1\end{subarray}}\;\sum_{l=0}^{9}f(M(5+k)-M(j+l))q(j)p(k)p(l)\bigg{/}\sum_{j=0}^{7}q(j)$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}0\leq j\leq 7,\,0\leq k\leq
9:\\\
D_{v}(j,k)=0\end{subarray}}f(M(5+k)-j)q(j)p(k)\bigg{/}\sum_{j=0}^{7}q(j).$ (4)

Then, with $f$ equal to the indicator of the set of positive integers,
$A_{u,v}$ is $W_{u,v}$, the probability of a Player win; with $f$ equal to the
indicator of $\\{0\\}$, $A_{u,v}$ is $T_{u,v}$, the probability of a Player
tie; and with $f(x):=\text{sgn}(x)$, $A_{u,v}$ is $E_{u,v}$, Player’s expected
profit. In all cases, $u=0$ (resp., $u=1$) if Player has a two-card total of 5
(with Banker unaware of this) and stands (resp., draws), and $v=0$ (resp.,
$v=1$) if Banker assumes that Player is a non-tireur (resp., tireur) and makes
a best response.

Notice another distinction between (1)–(2) and (2)–(2). In the former, the $u$
variable signifies Player’s pure strategy, and the $v$ variable signifies
Banker’s stand-or-draw decision based on the available information. In the
latter, the $u$ variable signifies Player’s stand-or-draw decision with a two-
card total of 5, and the $v$ variable signifies what Banker assumes is
Player’s pure strategy, to which Banker makes a best response.

Computations show that

$\displaystyle W_{0,0}$ $\displaystyle=\frac{792}{1781},$ $\displaystyle\quad
T_{0,0}$ $\displaystyle=\frac{153}{1781},$ $\displaystyle\quad E_{0,0}$
$\displaystyle=-\frac{44}{1781},$ (5a) $\displaystyle W_{0,1}$
$\displaystyle=\frac{872}{1781},$ $\displaystyle\quad T_{0,1}$
$\displaystyle=\frac{169}{1781},$ $\displaystyle\quad E_{0,1}$
$\displaystyle=\frac{132}{1781},$ (5b) $\displaystyle W_{1,0}$
$\displaystyle=\frac{10352}{23153},$ $\displaystyle\quad T_{1,0}$
$\displaystyle=\frac{2928}{23153},$ $\displaystyle\quad E_{1,0}$
$\displaystyle=\frac{479}{23153},$ (5c) $\displaystyle W_{1,1}$
$\displaystyle=\frac{10176}{23153},$ $\displaystyle\quad T_{1,1}$
$\displaystyle=\frac{2976}{23153},$ $\displaystyle\quad E_{1,1}$
$\displaystyle=\frac{175}{23153}.$ (5d)

Alternatively,

$\displaystyle W_{0,0}$ $\displaystyle\approx 0.444694,$ $\displaystyle\quad
T_{0,0}$ $\displaystyle\approx 0.085907,$ $\displaystyle\quad E_{0,0}$
$\displaystyle\approx-0.024705,$ (6a) $\displaystyle W_{0,1}$
$\displaystyle\approx 0.489613,$ $\displaystyle\quad T_{0,1}$
$\displaystyle\approx 0.094891,$ $\displaystyle\quad E_{0,1}$
$\displaystyle\approx 0.074116,$ (6b) $\displaystyle W_{1,0}$
$\displaystyle\approx 0.447113,$ $\displaystyle\quad T_{1,0}$
$\displaystyle\approx 0.126463,$ $\displaystyle\quad E_{1,0}$
$\displaystyle\approx 0.020688,$ (6c) $\displaystyle W_{1,1}$
$\displaystyle\approx 0.439511,$ $\displaystyle\quad T_{1,1}$
$\displaystyle\approx 0.128536,$ $\displaystyle\quad E_{1,1}$
$\displaystyle\approx 0.007558.$ (6d)

Although the formulas (2) and (2) are not difficult to evaluate, they depend
crucially on $\bm{D}_{0}$ and $\bm{D}_{1}$, whose evaluation was a daunting
task prior to the computer age.222None of the 19th-century baccarat authors
got $\bm{D}_{0}$ and $\bm{D}_{1}$ exactly right. This includes Billard (1883),
Bertezène (1889, 1896, 1897), Hoffmann (1891a,b), and Laun (1891), in addition
to Dormoy (1872), Badoureau (1881), and Bertrand (1888). We can add 20th-
century baccarat authors Savigny (1906), Lafaye and Krauss (1927), and Huitte
(1928) to this list. (To be fair, Lafaye and Krauss assumed a different model
of card distribution.) Thus, the first to get $\bm{D}_{0}$ and $\bm{D}_{1}$
exactly right were Lafrogne (1927), Le Myre (1935), and Boll (1936). Jules
Louis Henry Lafrogne [1867–1933] was a Rear Admiral in the French Navy. As far
as we know, his baccarat book is his only publication. “Georges Le Myre” was
said (in a 1937 review of his book) to be the pseudonym of a professor of
mathematics well known in Paris, but his identity is unknown today. Marcel
Boll [1886–1971] was professor of Chemistry and Electricity at l’École des
Hautes Études Commerciales in Paris. This may explain why a solution of the
game of baccarat, promised by von Neumann (1928), would have to wait for
Kemeny and Snell (1957), but even they did not use a computer but rather “one
of the old calculators.”333From a 3 August 2000 email from J. Laurie Snell to
Régis Deloche. Kemeny, incidentally, would later co-develop the BASIC
programming language.

##  3 Bertrand and his baccarat predecessors

Bertrand had at least two predecessors in baccarat analysis.

Dormoy’s (1872) article (and nearly identical book (1873)) was the first
mathematical study of baccarat chemin de fer, though he referred to it as
“baccarat tournant” (rotating baccarat). He obtained the equivalent of
$\bm{D}_{0}$ and $\bm{D}_{1}$, in the form of two tables, with only one error
in each. He argued that it is indifferent whether Banker stands or draws at
$(5,4)$ when Player is known to be a non-tireur. This is not really an error,
just a consequence of rounding expectations to two decimal places (the
standing expectation is $-299/1157$, and the drawing expectation is
$-300/1157$; both round to $-0.26$). He also had an error in $\bm{D}_{1}$ at
$(6,6)$.

Dormoy addressed three of Bertrand’s four problems, at least in terms of
expectations, but rounded them to two or three decimal places. He obtained
$E_{0,0}\approx-0.024$, $E_{1,0}\approx 0.012$, and $E_{1,1}\approx 0.02$
(Section 79); these numbers should have been $-0.025$, $0.021$, and $0.01$. He
provided enough detail to allow one to check that his methodology was sound,
so the errors were only computational.

The second mathematical study of baccarat chemin de fer was by Badoureau
(1881), who was aware of Dormoy’s work. He had $\bm{D}_{0}$ exactly correct
but had three errors in $\bm{D}_{1}$, at $(4,1)$, $(4,9)$, and $(6,6)$.
Badoureau expressed $\bm{D}_{0}$, for example, in terms of a list of rules for
Banker:

>   1. 1.
>
> Always draw at baccarat [i.e., zero], one or two;
>
>   2. 2.
>
> Always stand at seven;
>
>   3. 3.
>
> Before a player who stands, draw at 3, at 4 or at 5 and stand at 6;
>
>   4. 4.
>
> Before a player who has taken a card, always draw at 3 unless one has given
> 8 or 9; always draw at 4, unless one has given 8, 9, 0 or 1; only draw at 5
> if one has given 5, 6 or 7; only draw at 6 if one has given 6 or 7.
>
>

Badoureau addressed all four of Bertrand’s problems and expressed his answers
as fractions:

$\displaystyle W_{0,0}$ $\displaystyle=\frac{792}{1781},$ $\displaystyle\quad
T_{0,0}$ $\displaystyle=\frac{153}{1781},$ $\displaystyle\quad E_{0,0}$
$\displaystyle=-\frac{44}{1781},$ (7a) $\displaystyle W_{0,1}$
$\displaystyle=\frac{872}{1781},$ $\displaystyle\quad T_{0,1}$
$\displaystyle=\frac{169}{1781},$ $\displaystyle\quad E_{0,1}$
$\displaystyle=\frac{132}{1781},$ (7b) $\displaystyle W_{1,0}$
$\displaystyle=\frac{10352}{23153},$ $\displaystyle\quad T_{1,0}$
$\displaystyle=\frac{2928}{23153},$ $\displaystyle\quad E_{1,0}$
$\displaystyle=\frac{479}{23153},$ (7c) $\displaystyle W_{1,1}$
$\displaystyle=\frac{10288}{23153},$ $\displaystyle\quad T_{1,1}$
$\displaystyle=\frac{2800}{23153},$ $\displaystyle\quad E_{1,1}$
$\displaystyle=\frac{223}{23153}.$ (7d)

Actually, instead of finding $E_{u,v}$, Badoureau evaluated what he called
Player’s chances, $C_{u,v}:=W_{u,v}+\frac{1}{2}T_{u,v}$. Then
$E_{u,v}=2C_{u,v}-1$, so $E_{u,v}$ and $C_{u,v}$ are effectively equivalent.

Finally, Bertrand (1888), who did not reveal $\bm{D}_{0}$ or $\bm{D}_{1}$ and
cited no prior work, expressed his results to six decimal places:

$\displaystyle W_{0,0}$ $\displaystyle\approx 0.444694,$ $\displaystyle\quad
T_{0,0}$ $\displaystyle\approx 0.085907,$ $\displaystyle\quad E_{0,0}$
$\displaystyle\approx-0.024706,$ (8a) $\displaystyle W_{0,1}$
$\displaystyle\approx 0.489612,$ $\displaystyle\quad T_{0,1}$
$\displaystyle\approx 0.094890,$ $\displaystyle\quad E_{0,1}$
$\displaystyle\approx 0.074115,$ (8b) $\displaystyle W_{1,0}$
$\displaystyle\approx 0.447113,$ $\displaystyle\quad T_{1,0}$
$\displaystyle\approx 0.126463,$ $\displaystyle\quad E_{1,0}$
$\displaystyle\approx 0.020689,$ (8c) $\displaystyle W_{1,1}$
$\displaystyle\approx 0.444348,$ $\displaystyle\quad T_{1,1}$
$\displaystyle\approx 0.120935,$ $\displaystyle\quad E_{1,1}$
$\displaystyle\approx 0.009631.$ (8d)

Actually, Bertrand found $L_{u,v}=1-W_{u,v}-T_{u,v}$, the probability of a
Player loss, instead of $E_{u,v}=W_{u,v}-L_{u,v}=2W_{u,v}+T_{u,v}-1$.

We find that (8a)–(8c), (7a)–(7c), and (5a)–(5c) (equiv., (6a)–(6c)) match
exactly, ignoring rounding errors in the sixth decimal place. But (8d) matches
(7d), which differs from (5d) (equiv., (6d)). How did Badoureau arrive at
$W_{1,1}=10288/23153$, for example? When we recompute (2)–(2) for the
appropriate $f$ with $\bm{D}_{1}$ changed in three places (at $(4,1)$,
$(4,9)$, and $(6,6)$) to account for Badoureau’s errors, we obtain Badoureau’s
numbers exactly.

For Bertrand to have derived his results independently of Badoureau, he would
have had to have made no errors in the 88 entries of $\bm{D}_{0}$ and exactly
the same three errors as Badoureau made in the 88 entries of $\bm{D}_{1}$.
This seems extremely unlikely, and perhaps there is no need to quantify it. We
conclude that Bertrand borrowed Badoureau’s results without confirming them
and without attribution. As Sheynin (1994) pointed out, Bertrand had the
custom of not fully acknowledging prior work.

Was Bertrand’s interpretation of the results more insightful than Badoureau’s?
Bertrand (1888, p. 42) wrote:

> If, without trying to outwit the other, from the beginning of the game he
> frankly declares his customs, he should draw at 5.
>
> If the conventions of the game allow deception, he should stand at 5, making
> the banker think, if he can, that he has the custom of drawing.

Compare this with Badoureau (1881):

> If one were required to make known in advance the rule of conduct that one
> proposes to follow, it would be better to draw than to stand, […].
>
> The player must therefore stand at 5 while letting the banker believe the
> opposite.

The conclusions are similar. See Guilbaud (1961) for discussion.

##  4 Borel’s criticism of Bertrand’s analyis

As we saw, the first reason that Borel characterized Bertrand’s study of
baccarat as “extremely incomplete” was its lack of consideration of mixed
strategies. But both Dormoy (1872) and, to a lesser extent, Badoureau (1881)
did consider a Player $(\frac{1}{2},\frac{1}{2})$ mixed strategy, although not
entirely correctly. A Banker best response to this mixed strategy was
regarded, before game theory, as the appropriate response when Banker was
unaware of Player’s customary behavior. In effect, Dormoy found Banker’s best
move at $(j,k)\in\\{0,1,\ldots,7\\}\times\\{0,1,\ldots,9,\varnothing\\}$ by
determining which of

$\frac{b_{0,v}(j,k)+b_{1,v}(j,k)}{2},\qquad v\in\\{0,1\\},$

is larger. A more accurate approach would have compared the weighted averages

$\frac{89\,b_{0,v}(j,k)+105\,b_{1,v}(j,k)}{89+105},\qquad v\in\\{0,1\\},$

if $k\neq\varnothing$, or

$\frac{48\,b_{0,v}(j,\varnothing)+32\,b_{1,v}(j,\varnothing)}{48+32},\qquad
v\in\\{0,1\\},$

where the factor 89 (resp., 105, 48, and 32), when divided by 137, is the
conditional probability that Player’s two-card total belongs to
$\\{0,1,2,3,4\\}$ (resp., $\\{0,1,2,3,4,5\\}$, $\\{5,6,7\\}$, and
$\\{6,7\\}$), given that it is neither 8 nor 9.

Badoureau, on the other hand, made a more egregious error by using in effect
the averages

$\frac{E_{0,0}+E_{0,1}}{2}=\frac{44}{1781}\approx
0.024705\;\;\text{and}\;\;\frac{E_{1,0}+E_{1,1}}{2}=\frac{351}{23153}\approx
0.015160$

(or $327/23153\approx 0.014123$ for the second expression if the error in
$E_{1,1}$ is corrected). This mistake was said by Boll (1936, p. 234) to be
common:

> The first idea that would come to mind would be that the advantage would be
> the average of the advantages […].
>
> This simplistic idea, which is accepted without reflection by most players,
> is completely wrong: it must take into account all the details of the
> tactic.

Indeed, one should use (2)–(2) with $f(x):=\text{sgn}(x)$ and $\bm{D}_{v}$
replaced by $\bm{D}_{1/2}$, defined to be equal to $\bm{D}_{0}$ except for 1s
at $(3,9)$ and $(5,4)$. The latter approach yields

$E_{0,1/2}=-\frac{44}{1781}\approx-0.024705\quad\text{and}\quad
E_{1,1/2}=\frac{287}{23153}\approx 0.012396$

for Player’s standing and drawing expectations in this scenario.

With these numbers, Badoureau might not have come to the conclusion that “it
is almost indifferent to draw or to stand at 5.” Incidentally, $\bm{D}_{1/2}$
is Banker’s mandatory strategy in today’s nonstrategic form of baccarat, often
called baccarat punto banco. We have argued elsewhere (Ethier and Lee, 2015)
that this is not a coincidence.

It appears that Lasker (1929) was the first author to consider a mixed
strategy other than $(\frac{1}{2},\frac{1}{2})$ in the baccarat setting, and
his analysis was correct, albeit for a simplified version of the game. See
Bewersdorff (2020) for discussion.

The second of Borel’s criticisms of Bertrand’s study is arguably even more
fundamental. Can one answer the question of whether Player should draw at 5 by
examining only coups in which Player has a two-card total of 5, or must all
Player two-card totals be included in the analysis? We would argue for the
latter approach.

For example, the first of Bertrand’s four problems was the following:

> The player having 5 and not requesting a card, what is the probability for
> him to win and what is that of a tie, when the banker, unaware that he has
> the point 5, knows that he has the custom, when he has this point, of not
> requesting a card?

We believe that the problem would have been better formulated as follows:

> What is the probability for the player to win and what is that of a tie,
> under the following two assumptions: (a) If the player has 5, he does not
> request a card. (b) The banker knows that the player has the custom, when he
> has the point 5, of not requesting a card.

As we have seen, the solution to the original problem is

$W_{0,0}=\frac{792}{1781},\quad T_{0,0}=\frac{153}{1781},\quad
E_{0,0}=-\frac{44}{1781},$ (9)

whereas the solution of the reformulated problem is

$\overline{W}_{0,0}=\frac{2152648}{4826809},\quad\overline{T}_{0,0}=\frac{447337}{4826809},\quad\overline{E}_{0,0}=-\frac{74176}{4826809}.$
(10)

The distinction is clear: Both sets of statistics assume that Player is a non-
tireur and Banker knows it. But $\overline{E}_{0,0}$ in (10) is Player’s mean
profit from an arbitrary coup, whereas $E_{0,0}$ in (9) is Player’s mean
profit on only those coups in which he starts with a two-card total of 5. The
latter statistic, having been taken out of context, is of dubious value.

Another way to look at this is to consider the $2\times 2$ matrix game with
payoff matrix

$\bordermatrix{&\text{B: assumes P stands at 5}&\text{B: assumes P draws at
5}\cr\text{P: stands at 5}&E_{0,0}&E_{0,1}\cr\text{P: draws at
5}&E_{1,0}&E_{1,1}}$ (11)

Player is the row player, has a two-card total of 5, and stands or draws.
Banker is the column player, is unaware that Player has a two-card total of 5,
and makes a best response to one of Player’s two pure strategies. The
equilibrium for this game is not meaningful because after a few plays Banker
will realize that Player always has a two-card total of 5 and adjust his
strategy accordingly.

On the other hand, if (11) were replaced by

$\bordermatrix{&\text{B: assumes P stands at 5}&\text{B: assumes P draws at
5}\cr\text{P: stands at 5}&\overline{E}_{0,0}&\overline{E}_{0,1}\cr\text{P:
draws at 5}&\overline{E}_{1,0}&\overline{E}_{1,1}},$

with Player not assumed to have a two-card total of 5, the game could be
played repeatedly and the equilibrium would be meaningful, even if not exactly
the game-theoretic solution of baccarat. It is, in fact, essentially the
solution found by Kendall and Murchland (1964), who were unaware of Kemeny and
Snell (1957).

It is for this reason, we suspect, that, with one exception,444Lafaye and
Krauss (1927, Chap. XI) correctly solved two of Bertrand’s four problems,
including the one he got wrong. Yet they stated, not entirely accurately, that
their results were in conformity with those of Bertrand. no subsequent
researcher attempted to reproduce Bertrand’s numbers, and therefore no one
noticed the identical errors in Bertrand’s and Badoureau’s work.

Above all, we want to emphasize that Borel (1924) was mistaken in writing,
“The only author who has studied a particular case of the problem we envisage
is Joseph Bertrand, […].” Game theorists might point to Charles Waldegrave,
who solved the game of le her in 1713, as a counter-example (Guilbaud, 1961;
Dimand and Dimand, 1996, Chapter 7; Bellhouse and Fillion, 2015), but Albert
Badoureau is a more contemporaneous counter-example. Badoureau died in 1923 so
did not have the opportunity to correct the record. Therefore, nearly a
century later, we do so here.

## Acknowledgments

We thank Bernard Bru, Jacques Crovisier, Régis Deloche, Persi Diaconis, and
Don Schlesinger for their help.

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