# Series over fat partitions: matrix models and discrete ensembles

A.Yu. Orlov Institute of Oceanology, Nahimovskii Prospekt 36, Moscow 117997,
Russia; National Research University Higher School of Economics email:
<EMAIL_ADDRESS>
###### Abstract

We consider series over Young diagrams of products of Schur functions
$s_{\lambda\cup\lambda}$, marked with “fat partitions” $\lambda\cup\lambda$,
which appear in matrix models associated with ensembles of symplectic and
orthogonal matrices and quaternion Ginibre ensemble. We consider mixed matrix
models that also contain complex Ginibre ensembles labeled by graphs and the
three ensembles mentioned above. Cases are identified when the series of
perturbations in coupling constants turn out to be tau functions of the DKP
hierarchy introduced by the Kyoto school. This topic relates matrix models to
random partitions - discrete symplectic ensemble and its modifications.

##  1 Introduction

This work is a remark about special series over partitions of the Schur
functions $s_{\lambda\cup\lambda}$ labeled by “fat partitions”
$\lambda\cup\lambda=(\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2},\dots)$.
It relates three topics: new matrix models, integrable systems, namely, DKP
hierarchy introduced by Kyoto school [16] and discrete symplectic ensemble
introduced in [6] and its modifications. We are going to present a number of
matrix models whoose partition function is written as

This work is a remark on special series of Schur functions
$s_{\lambda\cup\lambda}$ labeled by “fat partitions”
$\lambda\cup\lambda=(\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2},\dots)$.
It concerns three topics: new matrix models, integrable systems, namely the
DKP hierarchy introduced by Kyoto school [16], and the discrete symplectic
ensemble introduced and studied in [6] and modifications of this ensemble. We
are going to present a series of matrix models whose partition function is
written as

$S_{E^{*}}(\mathbf{p}^{1},\dots,\mathbf{p}^{k})=\sum_{\lambda}w(\lambda)\left(s_{\lambda\cup\lambda}(1,0,0,\dots)\right)^{E^{*}-k}\prod_{i=1}^{k}s_{\lambda\cup\lambda}(\mathbf{p}^{i}),\quad
E^{*}=1,0,-1,\dots$ (1)

Here $\mathbf{p}^{i}=(p^{(i)}_{1},p^{(i)}_{2},\dots)$ are sets of power sum
variables [25] and each $s_{\lambda\cup\lambda}(\mathbf{p}^{i})$ is the Schur
function labeled with the special (fat) partition $\lambda\cup\lambda$. The
Schur function is a polynomial function of its variables, see (7) below. The
prefactor $w(\lambda)$ is the product

$w(\lambda)=\prod_{i=1}^{\ell(\lambda)}f(\lambda_{i}-i)$ (2)

We will consider new matrix models whose partition functions are presented in
form (1) where $w(\lambda)$ is a product of the Pochhammers Symbols related to
Young diagram $\lambda$. An example of such $w(\lambda)$ is written in (24)
below.

Let us choose all parameters $\mathbf{p}^{i}$ equal to $(1,0,0,\dots)$. Put
$\mu=\lambda\cup\lambda,\,h_{i}=\mu_{i}-i+2N$, $N=\ell(\lambda)$. The
following property:

$s_{\lambda\cup\lambda}(1,0,0,\dots)=\frac{\prod_{i<j\leq
N}\left(x_{i}-x_{j}\right)^{2}\left((x_{i}-x_{j})^{2}-1\right)}{\prod_{i=1}^{N}x_{i}!(x_{i}-1)!},\quad
x_{i}=h_{2i-1}=\lambda_{i}-2i+1+2N,$ (3)

relates series (1) with discrete symplectic ensembles studied in [6],[15] in
the case of $E^{*}=k=1$; with a different choice of numbers $E^{*}$ and $k$,
this leads to modifications of discrete symplectic ensembles, which are not
studied yet. This discrete ensemble can be described as a one-dimensional log
gas of interacting pairs of particles (“dumbbells”) on a one-dimensional semi-
infinite lattice. For $E^{*}=1$ this is repulsion according to Coulomb’s law
for bound pairs of particles with charge $1$, studied in [6]. For $E^{*}<0$,
the series (1) can be interpreted as the partition function of a system of
particles connected in a dumbbell with mass $m=-2E^{*}$ and with the
attraction according to the law of two-dimensional gravity. In the case of a
zero external potential $\log w$, the state with the minimum energy is the
dense packing. The limiting form for an infinite number of particles in an
external potential at a finite temperature should be obtained from the
variational principle, as shown in [18] for Coulomb lattice gas; a similar
problem was solved, for example, in [23]). This will be discussed in more
detail in the text. Here we note only the appearance such discrete ensembles
in the context of matrix models.

Commonly, the series

$G_{E}(\mathbf{p}^{1},\dots,\mathbf{p}^{k})=\sum_{\lambda}w(\lambda)\left(s_{\lambda}(1,0,0,\dots)\right)^{E-k}\prod_{i=1}^{k}s_{\lambda}(\mathbf{p}^{i}),\quad
E=2,1,0,\dots$ (4)

obtained in the form of perturbation series for many studied matrix models,
including the famous one- and two-matrix models and many others. The series
(4) have also been studied as generating series for the Hurwitz numbers:
firstly in the case of $E=k=2$ in a number of works [30],[2],[3],[3],[13], and
also in the case $E=k=1$ in [27]. These two cases are associated with the
classical integrable hierarchies, which is a two-component KP hierarchy and
its various reductions. In fact, the series (4) formally contains the series
(1) for the special selection of $w$, however this is a singular case of (4)
which can not be obtained from the studied matrix models and we prefer
consider the series (1) and (4) separately.

For us, the key fact is that the series (4) are equal to the partition
function of the multi-matrix complex Ginibre ensemble with the sources
previously discussed in [31],[28],[29],[4]. Such models are labeled using
graphs drawn on an orientable surface with Euler characteristic equal to $E$.
By “mating” of any such model, labeled with a graph, with any of the ensembles
(i), (ii), (iii) considered in the next Section we obtain a series of type (1)
with $E^{*}\leq E-1$ as a perturbation series for this new mixed matrix model
in terms of coupling constants.

The series (4) have the following property: being rewritten as the Taylor
series in the variables $\mathbf{p}^{i},\,i=1,\dots,n$

$G_{E}(\mathbf{p}^{1},\dots,\mathbf{p}^{k})=\sum_{\mu^{1},\dots,\mu^{n}}H_{E}(\mu^{1},\dots,\mu^{n})\prod_{i}\mathbf{p}^{i}_{\mu_{i}}$
(5)

where
$\mathbf{p}^{i}_{\mu^{i}}=\prod_{j=1}^{\infty}p^{(i)}_{\mu^{i}_{j}},\,p^{(i)}_{0}\equiv
1$ and $\mu^{i}=(\mu^{i}_{1},\mu^{i}_{1},\dots),\,i=1,\dots,n$ is the set of
$n$ partitions, we get prefactors $H_{E}(\mu^{1},\dots,\mu^{n})$ which are
related to the Hurwitz numbers in the problem of covering of a closed
oreintable surface of Euler characteristic $E$. In case $E=k=2$ series (4) are
related to various versions of the Schur measure on random partitions (the
works of G.Olshaski and his school), see also [1], and, in case
$\mathbf{p}=(1,0,0,\dots)$, series (4) can be related to discrete Coulomb log
gas [14] problem. The case $E<0$ corresponds to the attraction between
particles, the same as in 2D gravity for particles on the line, see (3) with
the mass equal to $-E$ each. It can be treated as a discrete version of
$\beta$ ensemble with negative $\beta=E$.

If we write

$S_{E^{*}}(\mathbf{p}^{1},\dots,\mathbf{p}^{k})=\sum_{\mu^{1},\dots,\mu^{n}}{\cal{H}}_{E^{*}}(\mu^{1},\dots,\mu^{n})\prod_{i}\mathbf{p}^{i}_{\mu_{i}}$
(6)

then in the case of a mixed ensemble consisting of complex and quaternion
matrices, there are combinatorial interpretation of factors
${\cal{H}}_{E^{*}}$ in terms of gluing polygons. In the case of (4) which
describes only complex ensembles, this approach generates a natural
combinatorial meaning of $H_{E}$ as Hurwitz numbers [28], however, such an
interpretation of ${\cal{H}}_{E^{*}}$ is not yet known.

I am sure that the series (1) will find their useful applications. At present
it can be argued that in the case $E=1,k=1$ the series (1) are examples tau
function of the DCT type discussed in [34].

My task is to present all matrix models known to me that can be associated
with (1). Note that in matrix models it is often natural to scale parameters:
$\mathbf{p}^{i}\to N\mathbf{p}^{i}=(Np^{(i)}_{1},Np^{(i)}_{2},\dots)$.

##  2 Random matrix ensembles and matrix models

We remind that the Schur function $s_{\lambda}$ is the polynomial function
defined by the partition $\lambda=(\lambda_{1},\lambda_{2},\dots)$ as follows

$s_{\lambda}(\mathbf{p})=\mathrm{det}\left[s_{\lambda_{i}-i+j}(\mathbf{p})\right]_{i,j\geq
0},\quad e^{\sum_{m>0}\frac{1}{m}p_{m}z^{m}}=\sum_{m\geq
0}z^{m}s_{(m)}(\mathbf{p})$ (7)

In case $\mathbf{p}$ is chosen in the following special way:

$\mathbf{p}=\mathbf{p}(X)=(\mathrm{tr}X,\mathrm{tr}\left(X^{2}\right),\mathrm{tr}\left(X^{3}\right),\dots)$
(8)

where $X$ is an $N\times N$ matrix, we write
$s_{\lambda}(\mathbf{p}(X))=s_{\lambda}(X)$, using bold and capital letters to
make the difference. The Schur function $s_{\lambda}(X)$ is known also as the
character of the linear group $\mathbb{GL}_{N}$ evaluated at
$X\in\mathbb{GL}_{N}$ in representation $\lambda$. It is the symmetric
polynomial in the eigenvalues $x_{1},\dots,x_{N}$ of $X$:

$s_{\lambda}(X)=\frac{\mathrm{det}\left[x_{j}^{\lambda_{i}-i+N}\right]_{i,j}}{\mathrm{det}\left[x_{j}^{-i+N}\right]_{i,j}}$
(9)

The important fact is that the Schur function $s_{\lambda}(X)$ is also a
polynomial of degree $|\lambda|$ in the entries of matrix $X$, see Appendix A,
and therefore it can be integrated over matrix elements, see (12), (13) and
(14) below.

Let us note that $s_{\lambda}(X)=0$ in case the length of partition (i.e. the
number of it’s non-zero parts) exceeds $N$. The following relation is called
Cauchi-Littlewood identity:

$e^{\sum_{m>0}\frac{1}{m}p_{m}\mathrm{tr}\left(X^{m}\right)}=\sum_{\lambda}s_{\lambda}(X)s_{\lambda}(\mathbf{p})$
(10)

Matrix ensembles are given by a space of matrices, say $\Omega$, it may be
Hermitian, symmetric, skew-symmetric, quaterninion self-dual, anti-self-dual,
complex, real, unitary, orthogonal, symplectic) matrices and a given invariant
probabilistic measure on this space. Then the expectaion value

$Z(\mathbf{p})=\langle
e^{\sum_{m}\frac{1}{m}p_{m}\mathrm{tr}\left(M\right)}\rangle_{\Omega}=\sum_{\lambda}\langle
s_{\lambda}(M)\rangle_{\Omega}s_{\lambda}(\mathbf{p})$ (11)

is called the partition function of a matrix model or just a matrix model
related to matrix ensemble $\Omega$. It depends on the coupling constants
$\mathbf{p}$. The exponential can be replaced by another spectral invariant
function.

The are also multi-matrix models, where the ensembles can be either the same
or they can be different and the last case is in our focus.

We need the expectations of the Schur functions
$s_{\lambda}(M)\rangle_{\Omega}$ in various ensembles and there are a number
of works where such expectation values were evaluated. Let me make references
only to the very last mentionings of this problem [26].

Let me write down examples we need.

(i) Let $M\in\mathbb{S}p(2N)$. Then

$\langle
s_{\lambda}(M)\rangle_{\mathbb{S}p(2N)}=\int_{\mathbb{S}p(2N)}s_{\mu}(M)d\nu(M)=\begin{cases}1\quad\mu\in\mathbb{FP},\quad\ell(\lambda)\leq
N\\\ 0\quad{\rm otherwise}\end{cases}$ (12)

Here $d\nu(M)=d_{*}M$ is the Haar measure on $\mathbb{S}p(2N)$ (see Appendix
B), and $\mathbb{FP}$ denotes the set of all fat partitions.

(ii) Let $M\in\mathbb{O}(N)$, where $N$ is equal either to $2n$ or to $2n-1$.
Then

$\langle
s_{\lambda}(M)\rangle_{\mathbb{O}(N)}=\int_{\mathbb{O}(N)}s_{\mu}(M)d\nu(M)=\begin{cases}1\quad\mu\in
2\mathbb{P},\quad\ell(\lambda)\leq n\\\ 0\quad{\rm otherwise}\end{cases}$ (13)

Here $d\nu(M)=d_{*}M$ is the Haar measure on $\mathbb{O}(N)$ (see Appendix B),
and $2\mathbb{P}$ denotes the set of all partitions with even parts.

(iii) Let $M$ is $N\times N$ matrix with quaternion entries which belongs to
quaternion-real Ginibre ensemble $q{\mathbb{GIN}}(N)$. Then for integer $L$ we
have:

$\langle
s_{\lambda}(M)\mathrm{det}M^{L}\rangle_{q\mathbb{GIN}(N)}=\int_{q\mathbb{GIN}(N)}s_{\mu}(M)\mathrm{det}(M^{L})d\nu(M)=\begin{cases}N^{-|\lambda|}(N+L)_{\lambda}\quad\mu\in\mathbb{FP}\\\
0\quad{\rm otherwise}\end{cases}$ (14)

Then, the matrix models related to ensembles symplectic matrices are

$\langle
e^{\sum_{m>0}\frac{1}{m}p_{m}\mathrm{tr}\left(M^{m}\right)}\rangle_{\mathbb{S}p(2N)}=\sum_{\lambda\atop\ell(\lambda)\leq
N}s_{\lambda\cup\lambda}(\mathbf{p})$ (15)

and

$\langle
e^{-\sum_{m>0}\frac{1}{m}p_{m}\mathrm{tr}\left(M^{m}\right)}\rangle_{\mathbb{S}p(2N)}=\sum_{\lambda\atop\lambda_{1}\leq
n}s_{\lambda\cup\lambda}(\mathbf{p})$ (16)

which are tau DKP functions, see [34], [22]. Tau functions solve differential
and difference equations in the variables $\mathbf{p}$ and discrete variables
$N$ and $L$, see [17] and Appendix.

More about quaternion Ginibre ensemble:

### Quaternion-real Ginibre ensemble.

Quaternion Ginibre ensemble was introduced in [9]. Details can be found in
[24], Ch 15.2, [14].

The quaternion Ginibre ensemble is the ensemble of $N\times N$ matrices $M$
with real quaternionic enties and with independently destributed according to
the Gaussian measure. One can see it as $2N\times 2N$ matrix with the block
structure, each block being presented as a sum with real coeffisients over
$\sigma matrices$:

${\bf M}_{ij}=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]_{ij}=\sum_{\alpha}M_{ij}^{(\alpha)}\sigma_{\alpha}$ (17)
$\sigma_{1}=\left[\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right],\sigma_{2}=\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right],\sigma_{3}=\left[\begin{array}[]{cc}0&i\\\
i&0\end{array}\right],\sigma_{0}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right],$ (18)

The measure on the space of real quaternion matrices is chosen as

$d\nu(M)=C\left(\prod_{{k\leq
j}}^{N}dM_{kj}^{(0)}\prod_{\alpha=1}^{3}\prod_{{k\leq
j}}^{N}dM_{kj}^{(\alpha)}\right)\text{e}^{-\frac{N}{2}\mathrm{tr}\left(M^{2}\right)}$
(19)

where the normalization constant $C$ is chosen to provide $\int d\nu(M)=1$.

In [32] the following matrix model

$\tau(\mathbf{p},N)=\int
e^{\sum_{m>0}\frac{N}{m}p_{m}\mathrm{tr}\left(M^{m}\right)}\mathrm{det}M^{L}d\nu(M)$
(20)

(here $L=0,1,\dots$) was considered and it was shown that as the function of
the variables $N\mathbf{p}=(Np_{1},Np_{2},\dots)$ the integral (20) is the DKP
tau function introduced in [16] where the variables $t_{i}=\frac{1}{i}p_{i}$
and discrete $L$ plays the role of the DKP higher times.

Taking in account that the eigenvalues of $M$ enter by complex conjugated
pairs $(z_{i},\bar{z}_{i}),\,i=1,\dots,N$ and using the expression for joint
probability function (formula (15.2.10) in [24]) this integral can be re-
written in form the following integral over $n$ pairs of complex variables
$z_{i},\bar{z}_{i}$:

$\tau(N\mathbf{p},N)=\int_{\mathbb{C}^{n}}\Delta(\mathbf{z},\bar{\mathbf{z}})\prod_{i=1}^{n}e^{-N|z_{i}|^{2}+\sum_{m>0}\frac{N}{m}p_{m}\left(z_{i}^{m}+{\bar{z}}_{i}^{m}\right)}(z_{i}-{\bar{z}}_{i})|z_{i}|^{2L}d^{2}z_{i}$
(21)

where

$\Delta(\mathbf{z},\bar{\mathbf{z}})=\prod_{i<j}|z_{i}-z_{j}|^{2}\prod_{i,j}(z_{i}-\bar{z}_{j})$
(22)

Using the definition of Schur functions $s_{\lambda}$ and Cauchi-Littlewood
relation, see Appendix, one can re-write (21) as

$\tau(N\mathbf{p},N)=\sum_{\lambda\atop\ell(\lambda)\leq
N}N^{-|\lambda|}(N+L)_{\lambda}s_{\lambda\cup\lambda}(N\mathbf{p})$ (23)

where

$(N+L)_{\lambda}=\prod_{(i,j)\in\lambda}(N+L+j-i)$ (24)

is the Pochhammer symbol related to the partition
$\lambda=(\lambda_{1},\dots,\lambda_{N})$ and can be also written as
$(N)_{\lambda_{1}}(N-1)_{\lambda_{2}}\cdots(1)_{\lambda_{N}}$ where
$(N)_{k}=\frac{\Gamma(N+k)}{\Gamma(N)}$ is the ordinary Pochhammer symbol. One
can varify that it can be also written in form (2). Such series is an example
of hypergeometric DKP tau functions considered in [34].

######  Remark 1.

The series (21) for the matrix model (20) may be compared with the series

$\tau^{\rm KP}(\mathbf{p},N)=\sum_{\lambda\atop\ell(\lambda)\leq
N}(N)_{\lambda}s_{\lambda}(\mathbf{p}^{\prime})s_{\lambda}(0,1,0,0,\dots)$
(25)

obtained in [11] for the Kazakov-Brezin-Gross-Migdal [7],[10] matrix model,
where $M$ Hermitian:

$\int
e^{\sum_{m>0}\frac{1}{m}p_{m}\mathrm{tr}\left(M^{m}\right)}e^{-c\mathrm{tr}\left(M\right)^{2}}dM,\quad
dM=\prod_{i\leq j}d\Re M_{ij}\prod_{i<j}d\Im M_{ij}$ (26)

In (25) $\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$ and
$\mathbf{p}^{\prime}=(p_{1},p_{2}-\frac{1}{4},p_{3},\dots)$.

######  Remark 2.

The series in the right hand side can be either divergent or convergent in
dependence of the choice of the coupling constants
$\mathbf{p}=(p_{1},p_{2},\dots)$. In case we choose

$p_{k}=-\sum_{i=1}^{N^{\prime}}x_{i}^{k}$ (27)

with any choice of the complex parameters $x_{i},\,i=1,\dots,N^{\prime}$ and
the number $N^{\prime}$, then the right hand side is a polynomials, namely, a
polynomial of degree $NN^{\prime}$ if we put $\deg p_{k}=k$.

##  3 Mixed ensembles and series over fat partition. Selection of integrable
cases

We call series (1) integrable in case they are tau functions. Matrix models
(15),(16) and (21) are integrable. Now we will couple these models with the
multimatrix models of complex Ginibre enesmble, see Appendix E considered in
[28],[29],[4]. Such models are labeled by embedded ribbon graphs $\Gamma$
drawn of an orientable surface. The partition functions of such models has
form (4) and are defined by matrices assigned to the vertices of the graph.
Namely, some of the sets $\mathbf{p}^{i}$ has a form

$\mathbf{p}^{i}=(\mathrm{tr}M_{i},\mathrm{tr}\left(M_{i}^{2}\right),\mathrm{tr}\left(M_{i}^{3}\right),\dots),\quad
i=1,\dots,V$ (28)

where each matrix $M_{i}$ is assigned to a vertex number $i$ and belong either
to any ensemble (i),(ii) or (iii). Our goal is to get the series (1) from the
series for multimatrix complex Ginibre ensemble (4).

We consider such matrices at the vertices as members of one of ensembles
(i),(ii),(iii). Let us start with examples.

These are the cases where either $E=k=2$, then these are examples of the two-
component KP nau functions, or the cases where $E=k=1$ and these are examples
of the DKP [16] tau functions.

Let $Z$ be a complex matrix and $Z^{\dagger}$ it’s Hermitian conjugate, then
let $M$ is matrix which belongs to an ensemble $\Omega$ and $J_{l}$ is a
matrix which has $n$ non-zero eignevalues equal to $1$ each, so for $l=N$ it
is the identity matrix $I_{N}$. Consider the integral

$S(N\mathbf{p},N|\Gamma_{1})=\int_{\Omega}\int_{\mathbb{GL}_{N}}e^{\sum_{m>0}\frac{N}{m}p_{m}\mathrm{tr}\left((Z^{\dagger}MZJ_{l})^{m}\right)}d\mu(Z)d\nu(M)$
(29)

where the notation $\Gamma_{1}$ points out the type of the integral, here it
is the graph which is the interval, it will be explained in Appendix E. In our
example it is as follows. $\Gamma_{1}$ has one face, one ribbon edge and two
vertices. Let us go around the boundary of the face in the positive direction
(which is clockwise direction around the interval drawn on the sphere): we
assign the matrix $Z^{\dagger}$ to the side of the ribbon edge we meet first,
then we assign the matrix $M$ to the vertex we meet, then the matrix $Z$ to
the other side of the ribbon and, finaly, the matrix $J_{l}$ to the second
vertex. We get the monodromy of the face which is $Z^{\dagger}MZJ_{l}$. We a
spectral invariant function of the face monodrome chosen in our example to be
the exponential function with the “coupling constants”
$\mathbf{p}=(p_{1},p_{2},\dots)$.

Here $\Omega$ is one of three ensembles pointed out in (i),(ii),(iii) in the
previous section. For these three ensemble this example generalize the
integral considered in Theorem 2 in [31]. and one of the integrals (15),
(16),(20) above and in this sense it is a coupling of complex Ginibre ensemble
with one of ensembles (15), (16),(20).

In case $M\in\mathbb{S}p(N),\,N=2n$ after evaluation we get

$S(N\mathbf{p},2n|\Gamma_{1})=\sum_{\lambda\atop\ell(\lambda)\leq
n}(l)_{\lambda\cup\lambda}\left((2n)_{\lambda\cup\lambda}\right)s_{\lambda\cup\lambda}(N\mathbf{p})N^{-2|\lambda|}$
(30)

which is also an example of the DKP tau function.

The similar answer we obtain for $M\in\mathbb{O}(N)$, say $N=2n$,

$S(N\mathbf{p},2n|\Gamma_{1})=\sum_{\lambda\atop\ell(\lambda)\leq
2n,\,\lambda_{1}\leq
n}(-l)_{\lambda\cup\lambda}(-2n)_{\lambda\cup\lambda}s_{\lambda\cup\lambda}(-N\mathbf{p})N^{-2|\lambda|}$
(31)

In case $M\in q{\mathbb{GIN}}(2n)$ after evaluation we get

$S(N\mathbf{p},2n|\Gamma_{1})=\sum_{\lambda\atop\ell(\lambda)\leq
2n}(n)_{\lambda}(l)_{\lambda\cup\lambda}(2n)_{\lambda\cup\lambda}s_{\lambda\cup\lambda}(N\mathbf{p})N^{-3|\lambda|}$
(32)

The right hand sides of (30),(31),(32) are examples of the DKP tau function.

The full version of the integral (29) related to the graph $\Gamma_{1}$ is the
following modification of the integral (49) in [31]:

$S_{\rm
full}(N\mathbf{p},N|\Gamma_{1})=\int_{\Omega_{1}}\int_{\Omega_{2}}\int_{\mathbb{GL}_{N}}e^{\sum_{m>0}\frac{N}{m}p_{m}\mathrm{tr}\left((Z^{\dagger}M_{1}ZM_{2})^{m}\right)}d\mu(Z)d\nu(M_{1})d\nu(M_{2})$
(33) $=\sum_{\mu\atop\ell(\mu)\leq
N}(N)_{\mu}\frac{s_{\mu}(N\mathbf{p})\langle
s_{\mu}(M_{1})\rangle_{\Omega_{1}}\langle
s_{\mu}(M_{2})\rangle_{\Omega_{2}}}{s_{\mu}(1,0,0,\dots)}N^{-|\mu|}$

Here the face monodromy is $Z^{\dagger}M_{1}ZM_{2}$. In case we choose any of
(i),(ii),(iii) for ensembles $\Omega_{1},\Omega_{2}$ we obtain the series (1)
with $E=0,\,k=1$.

The other variant is not to use an expectaion value of $M_{2}$ instead to fix
it as $M_{2}=J_{l}$ as it was done in (30), (31) and (32).

Next, consider another generalization of (15),(16),(21):

$S(\mathbf{p},N|\Gamma_{2})=\int_{\Omega}\int_{\mathbb{GL}_{N}}e^{N\mathrm{tr}Z^{\dagger}+\sum_{m>0}\frac{N}{m}p_{m}\mathrm{tr}\left((ZM)^{m}\right)}d\mu(Z)d\nu(M)$
(34)

This integral can be viewed as a modification of the intgeral of Theorem 4 in
[31].

Here $\Gamma_{2}$ is the following graph: it is a circle with a vertex (two
faces, one edge, one vertex). Here we have two face monodromies: one is equal
to $Z^{\dagger}$, which is $Z^{\dagger}$ assigned to the side of the ribbon
edge times the identity matrix assigned to the segment of the vertex (the face
corner). And the monodromy of the second face is $ZM$ which $Z$ assigned to
the second side of the ribbon edge times the source matrix $M_{1}$ assigned to
the second segment of the vertex. As the result the monodromy of the vertex is
$M\cdot I_{N}=M$ and it enter the right hand side the following expression:

After evaluation we obtain

$S(N\mathbf{p},N|\Gamma_{2})=S(N\mathbf{p},N|\Gamma_{1})|_{l=N}$ (35)

Third example:

$S_{1}(N\mathbf{p},N|\Gamma_{2})=\int_{\Omega}\int_{\mathbb{GL}_{2N}}e^{\mathrm{tr}\left(NMZ^{\dagger}\right)+\sum_{m>0}\frac{N}{m}p_{m}\mathrm{tr}\left((Z)^{m}\right)}d\mu(Z)d\nu(M)$
(36)

This integral is based on the same graph $\Gamma_{2}$ but it is differently
decorated. $\Gamma_{2}$ has two two faces, each has two corners. We assign $Z$
and $Z^{\dagger}$ to the different sides of ribbon edge. And we assign $M$ to
different corners. Actually, we deal with the integral

$\int_{\Omega}\int_{\mathbb{GL}_{2N}}e^{\sum_{m>0}\frac{N}{m}p^{(1)}\mathrm{tr}\left((Z^{\dagger}M_{1})^{m}\right)+\sum_{m>0}\frac{N}{m}p^{(2)}_{m}\mathrm{tr}\left((ZM_{2})^{m}\right)}d\mu(Z)d\nu(M)=\sum_{\lambda}(N)_{\mu}\frac{s_{\mu}(N\mathbf{p}^{1})s_{\mu}(N\mathbf{p}^{2})\langle
s_{\mu}(M)\rangle_{\Omega}}{s_{\lambda}(1,0,0,\dots)}(N)^{-|\mu|}$ (37)

where $M=M_{1}M_{2}$. However, until we put either
$N\mathbf{p}^{1}=(1,0,0,\dots)$ or $N\mathbf{p}^{1}=(a,a,a,\dots)$ this is not
a tau function (namely, the DKP tau function). Anyway, it is of type (1) where
$E=1,\,k=2$.

Next example is the modified integral considered in [8] coupled with one of
ensembles $\Omega$ considered above, namely

$S(N\mathbf{p},N|\Gamma_{3})=\int_{\mathbb{GL}(N)\times\mathbb{GL}(N)}\int_{\Omega}e^{\sum_{m>0}\frac{N}{m}p_{m}\left(Z_{1}\cdots
Z_{p-1}Z_{p}MZ^{\dagger}_{p}J_{l_{p}}\cdots
Z^{\dagger}_{1}J_{l_{1}}\right)^{m}}d\nu(M)\prod_{i=1}^{p}d\mu(Z_{i})$ (38)

Here $J_{l_{i}}=\mathrm{diag}(1,1,\dots,1,0,0,\dots 0)$, ${\rm
rank}I_{l_{i}}=l_{i}$. This matrix model is related to the graph $\Gamma_{3}$
which is the interval with $p+1$ vertices. We go around the boundary of the
face it in the positive direction (in a clockwise direction around the
interval) assigning $Z_{1}$ to the segments 1, then the identity matrix to the
segment of the second vertex and so on getting the product $Z_{1}Z_{2}\cdots
z_{p}$ until the end vertex where we assign $M$ then continue the path around
the interval getting $Z^{\dagger}_{p}J_{l_{p}}\cdots
Z^{\dagger}_{1}J_{l_{1}}$, where we assign $J_{l_{i}}$ to the $i$-th vertex.
As a result we get the face monodromy $Z_{1}\cdots
Z_{p-1}Z_{p}MZ^{\dagger}_{p}J_{l_{p}}\cdots Z^{\dagger}_{1}J_{l_{1}}$ and $p$
vertex monodromies, which are $J_{l_{1}},\dots,J_{l_{p}}$ and $M$ which enter
the answer.

If $M\in\mathbb{S}p(N)$ we obtain

$S(N\mathbf{p},N|\Gamma_{3})=\sum_{\lambda\atop\ell(\lambda)\leq
N}\prod_{i=1}^{p}(l_{i})_{\lambda\cup\lambda}s_{\lambda\cup\lambda}(N\mathbf{p})N^{-p|\lambda|}$
(39)

and so on for other ensembles.

The idea of these mixed ensembles on graphs is that we assign complex matrices
to it’s edges, and matrices from either ensembles $\Omega$ or put them to be
$J_{l_{i}}$.

To get DKP tau function we need to take a graph drawn on the sphere and some
additional requiments on vertex monodromies and the the coupling constants
$\mathbf{p}^{i}$ of the f monodromy of the face $i$.

The full version of the integral (38) is as follows:

$S_{\rm
full}(N\mathbf{p},N|\Gamma_{3})=\int_{\mathbb{GL}(N)\times\cdots\mathbb{GL}(N)}\int_{\Omega_{1},\dots,\Omega_{2p}}e^{\sum_{m>0}\frac{N}{m}p_{m}\left((Z_{1}M_{1})\cdots(Z_{p}M_{p})(Z^{\dagger}_{p}{M^{\prime}_{p}}\cdots
Z^{\dagger}_{1}M^{\prime}_{1}\right)^{m}}\prod_{i}d\nu(M_{i})d\nu(M^{\prime}_{i})\prod_{i=1}^{p}d\mu(Z_{i})$
(40)

It gives rise to the series (1) in case any of $M_{i},M^{\prime}_{i}$ belong
to one of (i),(ii),(iii). In this case $E^{*}=2-2p$, $k=1$. However we get
negative $E^{*}$ and it is in no way a tau function.

More examples will be considered in a more detailed version of the text.

See Appendix E for more examples.

Acknowledgements.

This work is an output of a research project implemented as part of the Basic
Research Program at the National Research University Higher School of
Economics (HSE University).

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##  Appendix A Partitions. The Schur polynomials [25]

We recall that a nonincreasing set of nonnegative integers
$\lambda_{1}\geq\cdots\geq\lambda_{k}\geq 0$, we call partition
$\lambda=(\lambda_{1},\dots,\lambda_{l})$, and $\lambda_{i}$ are called parts
of $\lambda$. The sum of parts is called the weight $|\lambda|$ of $\lambda$.
The number of nonzero parts of $\lambda$ is called the length of $\lambda$, it
will be denoted $\ell(\lambda)$. See [25] for details. Partitions will be
denoted by Greek letters: $\lambda,\mu,\dots$. The set of all partitions is
denoted by $\mathop{\mathrm{missing}}{P}\nolimits$. The set of all partitions
with odd parts is denoted $\mathop{\mathrm{missing}}{OP}\nolimits$. Partitions
with distinct parts are called strict partitions, we prefer letters
$\alpha,\beta$ to denote them. The set of all strict partitions will be
denoted by $\mathop{\mathrm{missing}}{DP}\nolimits$. The Frobenius coordinated
$\alpha,\beta$ for partitions
$(\alpha|\beta)=\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$ are of
usenames (let me recall that the coordinates
$\alpha=(\alpha_{1},\dots,\alpha_{k})\in\mathop{\mathrm{missing}}{DP}\nolimits$
consists of the lengths of arms counted from the main diagonal of the Young
diagram of $\lambda$ while
$\beta=(\beta_{1},\dots,\beta_{k})\in\mathop{\mathrm{missing}}{DP}\nolimits$
consists of the lengths of legs counted from the main diagonal of the Young
diagram of $\lambda$, $k$ is the length of the main diagonal of $\lambda$, see
[25] for details).

To define the Schur function $s_{\lambda}$,
$\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$ at the first step we
introduce the set of elementary Schur functions $s_{(m)}$ by

$e^{\sum_{m>0}\frac{1}{m}p_{m}x^{m}}=\sum_{m\geq 0}x^{m}s_{(m)}(\mathbf{p})$

where the variables $\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$ are called power
sum variables. For
$\lambda=(\lambda_{1},\lambda_{2},\dots)\in\mathop{\mathrm{missing}}{P}\nolimits$
we define

$s_{\lambda}(\mathbf{p})=\mathrm{det}\left[s_{(\lambda_{i}-i+j)}(\mathbf{p})\right]_{i,j>0}$
(41)

If we put $p_{m}=p_{m}(X)=\mathrm{tr}X^{m}$ where $X$ is a matrix we write
$s_{\lambda}(\mathbf{p}(X))=s_{\lambda}(X)$. The Schur functions play the role
of the character of the linear group.

The following relation is known as the character map relation

$s_{\lambda}(X)=s_{\lambda}(1,0,0,\dots)\sum_{\mu\atop|\mu|=|\lambda|}\varphi_{\lambda}(\mu)\prod_{i=1}^{\ell(\mu)}\mathrm{tr}\left(X^{\mu_{i}}\right)$
(42)

where $\varphi_{\lambda}(\mu)$ are rational numbers which are expressed in
terms of the characters of the symmetric group $S_{d},\,d=|\lambda|$, see
[25]. It follows from (42) that $s_{\lambda}(X)$ is a polynomial of degree
$d=|\mu|=|\lambda|$ in entries of $X$.

##  Appendix B Matrix ensembles

### Symplectic matrices.

The Haar measure on $\mathbb{S}p_{2N}$ is

$d_{*}M=\frac{2^{(N)^{2}}}{\pi^{N}}\prod_{i<j}^{N}\left(\cos(\theta_{i})-\cos(\theta_{j})^{2}\right)\prod_{i=1}^{N}\sin^{2}\theta_{i}d\theta_{i},$
(43)

where $d\theta_{i},\quad 0\leq\theta_{i}<\cdots<\theta_{N}\leq\pi$. The
prefactors are chosen to provide $\int_{\mathbb{S}p_{2N}}d_{*}M=1$.

### Orthogonal matrices.

The Haar measure on $\mathbb{O}_{N}$ is

$d_{*}M=\begin{cases}\frac{2^{(n-1)^{2}}}{\pi^{n}}\prod_{i<j}^{n}\left(\cos(\theta_{i})-\cos(\theta_{j})^{2}\right)\prod_{i=1}^{n}d\theta_{i},\quad
N=2n\\\
\frac{2^{n^{2}}}{\pi^{n}}\prod_{i<j}^{n}\left(\cos(\theta_{i})-\cos(\theta_{j})^{2}\right)\prod_{i=1}^{n}\sin^{2}\frac{\theta_{i}}{2}d\theta_{i},\quad
N=2n+1\end{cases}$ (44)

where $d\theta_{i},\quad 0\leq\theta_{i}<\cdots<\theta_{n}\leq\pi$. The
prefactors are chosen to provide $\int_{\mathbb{O}_{N}}d_{*}M=1$.

### Complex matrices.

The measure on the space of complex matrices is defined as

$d\Omega(Z)=c_{N}\prod_{a,b=1}^{N}d\Re Z_{ab}d\Im
Z_{ab}\text{e}^{-N|Z_{ab}|^{2}}$ (45)

Consider integrals over $N\times N$ complex matrices $Z_{1},\dots,Z_{n}$ where
the measure is defined as

$d\Omega(Z_{1},\dots,Z_{n})=c_{N}^{n}\prod_{i=1}^{n}\prod_{a,b=1}^{N}d\Re(Z_{i})_{ab}d\Im(Z_{i})_{ab}\text{e}^{-N|(Z_{i})_{ab}|^{2}}$
(46)

where the integration domain is
$\mathbb{C}^{N^{2}}\times\cdots\times\mathbb{C}^{N^{2}}$ and where $c_{N}^{n}$
is the normalization constant defined via $\int d\Omega(Z_{1},\dots,Z_{n})=1$.

##  Appendix C Tau functions as fermionic expectation value

Let $\psi_{i},\psi^{\dagger}_{i},\,i\in\mathbb{Z}$ be Fermi modes which obey

$\psi_{i}|L\rangle=0=\psi^{\dagger}_{-i-1}|L\rangle,\quad i<L$ (47)

where $L$ is the charge of the vacuum (or, the same, the level of Dirac sea),
and

$[\psi_{i},\psi_{j}]_{+}=[\psi^{\dagger}_{i},\psi^{\dagger}_{j}]_{=}0,\quad[\psi_{i},\psi^{\dagger}_{j}]_{+}=\delta_{i,j}$
(48)

The fermionic current modes are
$J_{m}=\sum_{i\in\mathbb{Z}}\psi_{i}\psi^{\dagger}_{i+m},\,m>0$.

The DKP tau functions related to series (1) have the following expression in
terms of free fermion theory:

$\langle
L+2N|e^{\sum_{m>0}\frac{1}{m}p_{m}J_{m}}e^{\sum_{i<0}U_{i}\psi^{\dagger}_{i}\psi_{i}-\sum_{i\geq
0}U_{i}\psi_{i}\psi^{\dagger}_{i}}e^{\sum_{i}\psi_{i+1}\psi_{i}}|L\rangle=c_{L}\sum_{\lambda\atop\ell(\lambda)\leq
N}w(\lambda)s_{\lambda\cup\lambda}(\mathbf{p})$ (49)
$w(\lambda)=\prod_{i=1}^{\ell(\lambda)}e^{-U_{\lambda_{i}-i+L+1}-U_{\lambda_{i}-i+L}}$
(50)

where $\\{U_{i},\,i\in\mathbb{Z}\\}$ is a set of parameters and
$c_{L}=e^{-U_{L}-\cdots-U_{L+2N}}$. The vacuum expectation value (49) is an
example of the DKP tau function. It has the form (1) where $E=k=1$.
Expressions (15),(16) and (21) up to the multiplier $c_{L}$ have form
(49)-(50). DKP tau functions were introduced in [16].

The two-component KP tau function is constructed in terms of the two-component
Fermi modes. The following one:

$\psi^{(1)}_{i}|L,*\rangle=\psi^{(2)}_{i}|*,L\rangle=0=\psi^{{\dagger}(1)}_{-i-1}|L,*\rangle=\psi^{{\dagger}(2)}_{-i-1}|*,L\rangle,\quad
i<L$ (51)

and

$[\psi^{(a)}_{i},\psi^{(b)}_{j}]_{+}=[\psi^{{\dagger}(a)}_{i},\psi^{{\dagger}(b)}_{j}]_{=}0,\quad[\psi^{(a)}_{i},\psi^{{\dagger}(b)}_{j}]_{+}=\delta_{i,j}\delta_{a,b},\quad
a,b=1,2$ (52)

The fermionic currents are
$J^{(a)}_{m}=\sum_{i\in\mathbb{Z}}\psi^{(a)}_{i}\psi^{{\dagger}(a)}_{i+m},\,a=1,2$.

Let us present following two-component tau function:

$\langle
N+L,L-N|e^{\alpha\sum_{a=1,2}\sum_{m>0}\frac{1}{m}p^{(a)}_{m}J_{m}}e^{\sum_{i\geq
0}f_{i}\psi^{(1)}_{i}\psi^{{\dagger}(2)}_{-i-1}}|L,L\rangle=\sum_{\lambda\atop\ell(\lambda)\leq
N}s_{\lambda}(\alpha\mathbf{p}^{1})s_{\lambda}(\alpha\mathbf{p}^{2})\prod_{i=1}^{\ell(\lambda)}f_{i+L}$
(53)

In case $\alpha=1$ the series in the right hand side coinsides with the series
(4) where $E=k=2$. In case we put $\alpha=-1$ and additionaly we have the
condition $f_{2i+1}=0$ if $N+L$ even and $f_{2i}=0$ if $N+L$ odd, it coinsides
with the series (1) where $E=k=2$.

##  Appendix D DKP Hirota equations

These equations were written down in [17].

##  Appendix E Multi-matrix models of complex matrices [28]

Let us remind in short the construction in [28], [4].

Consider an embedded graph $\Gamma$ with F faced, $n$ edges and $V$ vertices.
We number all ribbon edges by $1,\dots,n$ and the sides of each edge $i$ we
denote by $i$ and $-i$, no matter what side is numbered by $i$ and what side
is numbered by $-i$. We assign matrices $Z_{i}$ to the side $i$, and we impose
$Z_{i}=Z^{\dagger}_{-i}$. We number each face of $\Gamma$ and conside each
boarder side of a face as positevely oriented. Let’s go around the edge of the
face in the positive direction. We number each corner of the face which follow
the side $i$ by the same number. We assign to a corner $i$ an $N\times N$
matrix $M_{i}$. We introduce the monodromy of the face as a consequent product
of pairs $Z_{i}M_{i}$ when we go around the boarder of the face. Let us number
each face by $i=1,\dots,F$, the monodromy of the face $i$ is denoted by
${\cal{F}}_{i}$. Next, let us number the vertices of $\Gamma$ by $1,\dots,V$
and consider a vertex $i$. Let us introduce the monodromy of the vertex $i$ as
the consequent product of adjacent corner matrices when we go around it in
when we go around it in negatice (clockwise) direction. Let us denote the
monodromy of the vertex $i$ by ${\cal{V}}_{i}$. Both monodromies are defined
up to a cyclic permutation.

Thus, we have the set of complex matrices $Z_{\pm 1},\dots,Z_{\pm n}$ and a
set of corner matrices $M_{\pm 1},\dots,M_{\pm n}$ (we [28],[29],[4] we call
them “source matrices”).

We have the following wonderful relation [29]:

$\int_{\mathbb{GL}_{N}\times\cdots\times\mathbb{GL}_{N}}\prod_{a=1}^{F}s_{\lambda^{a}}\left({\cal
F}_{a}(X)\right)dX=\delta_{\lambda}N^{-nd}\left(s_{\lambda}(\mathbf{p}_{\infty})\right)^{-n}\prod_{a=1}^{V}s_{\lambda^{a}}\left({\cal
V}_{a}\right)$ (54)

where $\delta_{\lambda}$ is equal to 1 in case
$\lambda^{1}=\cdots=\lambda^{F}$ and where we denote $\lambda^{1}$ by
$\lambda$. And $\delta_{\lambda}=0$ otherwise. Here

$s_{\lambda}(\mathbf{p}_{\infty})=\frac{d_{\lambda}}{d!},\quad
d_{\lambda}=\frac{\prod_{i<j\leq
N}(\lambda_{i}-\lambda_{j}-i+j)}{\prod_{i=1}^{N}(\lambda_{i}-i+N)!},\quad
d=|\lambda|$ (55)

(where $d_{\lambda}$ is the dimension of the representation $\lambda$ of the
symmetric group $S_{d}$).

### Matrix models.

Now we consider the matrix model related to an embedded graph $\Gamma$ as

$Z(\mathbf{t},{\cal{M}})=\int_{\mathbb{GL}_{N}\times\cdots\times\mathbb{GL}_{N}}\prod_{i=1}^{F}e^{\sum_{m>0}\frac{1}{m}p^{(i)}_{m}\mathrm{tr}\left({\cal
F}_{i}(X)\right)^{m}}dX=$ $\sum_{\lambda\atop\ell(\lambda)\leq
N}N^{-n|\lambda|}\left(s_{\lambda}(\mathbf{p}_{\infty})\right)^{-n}\prod_{i=1}^{F}s_{\lambda}(N\mathbf{p}^{i})\prod_{i=1}^{V}s_{\lambda}({\cal
V}_{i})$ (56)

where $\mathbf{t}$ denotes the collection of coupling constants
$\mathbf{p}^{i},\,i=1,\dots,F$ and ${\cal M}$ denotes the collection of the
source matrices $M_{\pm i},\,i=1,\dots,n$. This model has $F$ sets of coupling
constants $\mathbf{p}^{i}=(p^{(i)}_{1},p^{(i)}_{2},\dots)$ and the set of $V$
independent combinations ${\cal V}_{i}$ constructed from the source matrices
(or, the same, from the corner matrices of the graph $\Gamma$.)


# Nash equilibrium selection by eigenvalue control

Wang Zhijian
Experimental social science laboratory, Zhejiang University, China

###### Abstract

People choose their strategies through a trial-and-error learning process in
which they gradually discover that some strategies work better than others.
The process can be modelled as an evolutionary game dynamics system, which may
be controllable. In modern control theory, eigenvalue (pole) assignment is a
basic approach to designing a full-state feedback controller, which can
influence the outcome of a game. This study shows that, in a game with two
Nash equilibria, the long-running strategy distribution can be controlled by
pole assignment. We illustrate a theoretical workflow to design and evaluate
the controller. To our knowledge, this is the first realisation of the control
of equilibrium selection by design in the game dynamics theory paradigm. We
hope the controller can be verified in a laboratory human subject game
experiment.

###### Contents

  1. 1 Introduction
  2. 2 Workflow for controller design
  3. 3 An example
    1. 3.1 Step 0: Game and its equilibrium
    2. 3.2 Step 1: The original eigensystem
    3. 3.3 Step 2: Assign control goal
    4. 3.4 Step 3: Controller design
    5. 3.5 Step 4.1 Dynamics analysis results
    6. 3.6 Step 4.2: ABED Simulation
    7. 3.7 Step 5: Evaluate the controller
      1. 3.7.1 Statistical results
      2. 3.7.2 Explanation for the statistical result
  4. 4 Discussion
    1. 4.1 Related works on the control the game dynamics
    2. 4.2 Further directions of the controller design
  5. 5 Appendix
    1. 5.1 Pole assignment
    2. 5.2 Algorithm for the controller design
      1. 5.2.1 Basic
      2. 5.2.2 Construct ${J^{c}}$
      3. 5.2.3 Explain $B$
    3. 5.3 Agent-based evolutionary dynamics (ABED) simulation
    4. 5.4 Abbreviations
    5. 5.5 Data and code

##  1 Introduction

##### Motivation

Game theory study the strategy interaction between intelligent species, e.g.,
human. Its statics paradigm predicts equilibrium, and its dynamics paradigm
predicts motion. Equilibrium selection is an open question both in the statics
[9] and the dynamics [20].

In this study, using game dynamics paradigm [8, 22], we show how to control
game equilibrium selection by exploiting the predictability of the game
motion. In another words, instead of statics [9] or existed dynamics [20]
consideration, we design a mechanism to control the velocity field to
influence the equilibrium selection. The controller is designed by the pole
assignment approach in modern control theory [17, 2]. We will provide an
overview of the workflow for designing the controller and illustrate it with
an example.

##### Background

The research goal is primarily based on the two factors listed below:

  * •

Human strategy behaviour motion can be predicted by the eigensystem of the
game dynamics equations.  — Data from game experiments show that human
dynamics behaviour is governed by dynamics equations (for example, replicator
dynamics) and its linearization at the Nash equilibrium. Previous results have
shown that, even in discrete time and discrete strategy games, the
linearization approximation works. Examples come from the 3-strategy game (the
rock-paper-scissors game [5, 32, 10]), the 4-strategy one-population games
([42]), the 5-strategy one-population games (Yao 2021 [40]), and 4-strategy
two-role zero-sum asymmetry game of the O’Neill 1987 game (O’Neill [15], Wang
and Yao 2020 [33]). These findings demonstrate that the game dynamics theory
can accurately capture human subject experimental dynamics behaviours based on
distribution, cycle, and converge speed measurements. This logic chain can be
seen in Figure 2(a). The chain begins with the game and dynamics equations to
obtain the rest point (equilibrium), then proceeds to the Jacobian $Jo$, and
finally to the eigensystem (which includes eigenvalues, eigenvectors, and
eigencycles).

  * •

A linear dynamics system is controllable — Applied mathematics shows that the
invariant manifold concept provides a clear picture for dynamic process
description. This concept has roots in dynamical systems theory[35], a solid
branch of mathematics. Dynamical systems deal with the study of the solutions
to the equations of motion of systems that are primarily mechanical in nature.
For linear dynamic systems, state-space feedback controller design is a well-
studied and applied engineering field, namely modern control theory and
application [7, 14, 2].

##### Main logic

Table 1: The game matrix | x1 | x2 | x3 | x4 | x5
---|---|---|---|---|---
x1 | 0 | 0 | 2 | 0 | -2
x2 | 2 | 0 | 0 | -2 | 0
x3 | 0 | 2 | 0 | 2 | -1
x4 |  $-$2 | 0 | 1 | 0 | 1
x5 | 0 |  $-$2 |  $-$2 | 1 | 0

As the linear dynamics system can be controlled, we turn to equilibrium
selection. We show how to design a mechanism for equilibrium selection by
using an example of a symmetric 5-strategy one population game, whose payoff
matrix is shown in Table 1. The game has and only has two equilibrium (Nash_1
and Nash_2) as shown in Fig. 1. By using the pole assignment approach, we can
control the eigenvalue at equilibrium, equilibrium Nash_1. In other words, we
use pole assignment to control the stability of Nash_1. As a result, the long-
run distribution will remain at Nash_1 or move to Nash_2. This is the main
logic.

In fact, in the view of game velocity field: (1) In game dynamics theory,
dynamics equations (e.g., replicator dynamics) describe the velocity field.
(2) The velocity field really exists in the human game experiments of von
1947’s two elementary games [29], the standard rock-paper-scissors game and
the matching pennies game. (3) The velocity field can be controlled by the
pole assignment approach. Thus, using the pole assignment to influence human
dynamics behaviours in a game experiment is logical.

Figure 1: Conceptual figure: In a five strategy game, the two equilibrium
(Nash_1 and Nash_2) locate in the two sub space $S_{1}(x_{1},x_{2},x_{3})$ and
$S_{2}(x_{4},x_{5})$, respectively. The pole assignment can be designed to
make the Nash_1 being unstable. As consequence, the long run trajectory will
converge to Nash_2, which means that the equilibrium Nash_2 is selected.

##### Outline

The main technical point of this work is the workflow for the controller
design, which is introduced in Section 2. In Section 3, we will practically
show an example to realise the controller design and verify the theoretical
prediction of the controller by agent-based simulations. In Section 4, we
summarise the results and point out the related concepts and further
directions.

##  2 Workflow for controller design

In this study, our control-by-design (mechanism design) approach comes from
single-input pole assignment approach for linear system in the modern control
theory, which has being applied on control the dynamics structure [41].

Figure 2: Workflow: 1,$J^{o}$; 2, PA; 3, $J^{C}$; 4: DA; 5: Simulation; 6;
Evaluation (a: Distribution, b: Cycle, and c: Convergence speed).

As shown in Figure 2, the workflow includes follow steps:

  1. 1.

Solve the dynamics equation for the original eigen-system including Jacobian
${J^{o}}$, eigenvalue $\lambda^{o}$, and eigenvector $v^{o}$;

  2. 2.

Assign the desired pole $\lambda^{c}$;

  3. 3.

Solve the gain matrix ${K}$ for the given ${B}$ for the controlled Jacobian
${J^{c}}$ from the original Jacobian ${J^{o}}$ (see Eq. 6 referring to [41]).

  4. 4.

This step has two parallel parts:

    1. 4.1

Derive the theoretical dynamics analysis results on the observation set,
$O^{T}$.

    2. 4.2

Conduct agent-based simulation and report the observation set, $O^{S}$.

At this step, it is necessary to clarify the measurements of the observation.

  5. 5.

Check the controller’s valuation by comparing $O^{T}$ and $O^{S}$. That is to
answer whether theoretical expectations are archived, meanwhile, whether the
constrain conditions are satisfied.

We will show how to realise the control by 5-strategy game practically in an
example.

##  3 An example

###  3.1 Step 0: Game and its equilibrium

We employ a five strategy symmetry game to show how to observe the eigensystem
and how to control the dynamics structure in experiment. The game is an one
population 5$\times$5 symmetric game. In strategy state space, its evolution
trajectory is of a 5 dimensional trajectory. Table 1 shows the 5-strategy
payoff matrix. The game has and only has two Nash equilibrium [1]. The first
Nash equilibrium, denoted as Nash_1, is

$\rho_{\text{Nash\\_1}}=\frac{1}{3}(1,1,1,0,0),$ (1)

in which the game falls into a 3-dimensional subspace, appearing as a rock-
paper-scissors game, and its expected payoff is 2/3. Intuitively, there should
exist persistent endogenous cycles along $x_{1}\rightarrow x_{2}\rightarrow
x_{3}\rightarrow x_{1}...$ when Nash_1 is selected. The second, denoted as
Nash_2, is

$\rho_{\text{Nash\\_2}}=\frac{1}{2}(0,0,0,1,1),$ (2)

in which the game falls into a 2-dimensional subspace, appearing as an anti-
coordination game, and its expected payoff is 1/2. Intuitively, there exist no
persistent endogenous cycles when Nash_2 is selected.

###  3.2 Step 1: The original eigensystem

To illustrate the dynamics controller, we use the replicator dynamics [22],
which is the original velocity vector field of the game, as the original
system.

##### Original dynamics system

The dynamics system can be expressed as follow:

$\dot{x}_{j}=x_{j}\Big{(}U_{j}-{\overline{U}}\Big{)},$ (3)

in which $x_{j}$ is the $j$th strategy player’s probability in the population
with the $j$th strategy player included, and $\dot{x}_{j}$ is the evolution
velocity of the probability; $U_{j}$ the payoff of the $j$th strategy player;
And $overline{U}_{X}$ is the population’s average payoff, which equals
$\sum_{k=1}^{5}x_{k}U_{k}$. This 5-dimensional space has one concentration,
$\sum_{j}x_{i}=1\cap x_{i}\geq 0\leavevmode\nobreak\ (i\in\\{1,2,...,5\\})$,
implying that the system must be in a 5-dimensional simplex space at all
times. The Nash_1 and Nash_2 are the original velocity field’s rest points.

##### Eigen system

The Jacobian matrix ${J^{o}}$ evaluated at Nash_1 is

$\small
J^{o}_{\text{Nash\\_1}}=\left[\begin{array}[]{rrrrr}-\frac{4}{9}&-\frac{4}{9}&\frac{2}{9}&\frac{1}{9}&\frac{1}{9}\\\
\frac{2}{9}&-\frac{4}{9}&-\frac{4}{9}&-\frac{5}{9}&\frac{7}{9}\\\
-\frac{4}{9}&\frac{2}{9}&-\frac{4}{9}&\frac{7}{9}&\frac{4}{9}\\\ 0&0&0&-1&0\\\
0&0&0&0&-2\\\ \end{array}\right].$

And its eigenvalues $\lambda^{o}_{\text{Nash\\_1}}$ are

$\lambda^{o}_{\text{Nash\\_1}}==\left[\begin{array}[]{rrrrr}-\frac{1}{3}+\frac{\sqrt{3}}{3}i&-\frac{1}{3}-\frac{\sqrt{3}}{3}i&-\frac{2}{3}&-1&-2\\\
\end{array}\right].$ (4)

We choose the $\lambda^{o}_{\text{Nash\\_1}}$ as the original pole in this
study. As all the real part of the eigenvalues are negative, the system is
stable at Nash_1 locally. The first two column of the eigenvector matrix is
the original complex eigenvector which represents the dynamics structure. The
third column represents the equilibrium distribution, because its related left
eigenvector is the unit vector, [1 1 1 1 1], and its related eigenvalue is the
oppose of the expected payoff at the equilibrium. In this example, the complex
eigenvalue is the object of control-by-design. In the same way, we can
calculate the eigensystem at $\rho_{\text{Nash\\_2}}$. As results, the
eigenvalues are

$\lambda^{o}_{\text{Nash\\_2}}=\left[\begin{array}[]{ccccc}-3/2&-3/2&-1/2&-1/2&0\\\
\end{array}\right].$

As all of the eigenvalues are negative real, the system is also stable at
Nash_2 locally; Meanwhile, all the eigenvectors are real, and no cyclic motion
is expected around Nash_2. In general, each eqilibrium can be an object to
control, but, in this study case, we do not try to control Nash_2.

###  3.3 Step 2: Assign control goal

Our goal is to control (plus or minus a small real number) the real parts of
the complex eigenvalues shown in Eq. 4. The assign pool can be expressed as,

$\leavevmode\nobreak\ {\lambda^{c}}=\leavevmode\nobreak\
{\lambda^{o}}+b\leavevmode\nobreak\ [0\leavevmode\nobreak\
0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1\leavevmode\nobreak\ 1]$ (5)

Here, the $\lambda^{o}$ is the original pool at Nash_1, and $\lambda^{c}$ is
the controlled (optimal) pool. That is, for each optimal goal $b$, in
$\lambda^{0}$ of the original system matrix $J^{o}$, a pair of poles
(eigenvalue) is to be shifted. We set $b$ form $-1$ to 1 by step 0.2,
$b\in[-1,-0.8,-0.6,...,1]$, respectively. Such control will change the
stability and the attraction of the equilibrium Nash_1.

As results, in theory, the equilibrium point Nash_1 will be more attractive
when $b\longrightarrow-1$, and Nash_1 will be more likely being selected;
alternatively, the equilibrium point Nash_1 will more repelled when
$b\longrightarrow+1$, and as a consequence, Nash_2 will be more likely being
selected.

###  3.4 Step 3: Controller design

The main philosophy of the controller design is an inverse solution procedure,
which recasts the controller design task to a optimization problem with
constraints [2]. In this study, we set two constraint conditions, equilibrium
conservation and payoff conservation. The definition of the controller is
shown in Section 5.2.2.

Denote $\mathbf{x}=[x_{1}\leavevmode\nobreak\ x_{2}\leavevmode\nobreak\
x_{3}\leavevmode\nobreak\ x_{4}\leavevmode\nobreak\ x_{5}]^{T}$, referring to
the definition of $J^{c}$, the controlled velocity field is

$\displaystyle\mathbf{\dot{x}}$ $\displaystyle=$
$\displaystyle{J^{c}}\mathbf{x}$ (6) $\displaystyle=$
$\displaystyle{(J^{o}+BK-\sum_{i=1}^{5}{k_{i}x_{i}})}\mathbf{x}.$

Notice that, as the relation — $\sum_{i=1}^{5}{k_{i}x_{i}^{*}}=0$ ($x^{*}$ is
the Nash_1) must hold. It is important, because the controller is required to
preserve the equilibrium. The theory of optimal control is concerned with
operating a dynamic system at minimum cost. At this setting, the cost is zero.
Having the ${J^{c}}$ defined in Eq 6, we can solve pole assignment problem,
shown in Eq. 5, for the gain matrix $K$. In this study case, we use the pole
assignment algorithm [13] which is name as place function in Matlab as the
tool to compute the matrices $K$ to achieve the desired closed-loop pole
locations. The results are shown in Table 2.

$b$ | $k_{1}$ | $k_{2}$ | $k_{3}$ | $k_{4}$ | $k_{5}$
---|---|---|---|---|---
$-$0.8 | 0.5247 |  0.9485 |  $-$1.4732 |  $-$1.8335 |  0.2335
$-$0.6 | 0.4843 | 0.5524 |  $-$1.0368 |  $-$1.3232 | 0.1232
$-$0.4 | 0.3834 | 0.2623 |  $-$0.6458 |  $-$0.8476 | 0.0476
$-$0.2 | 0.2220 | 0.0782 |  $-$0.3002 |  $-$0.4065 | 0.0065
0 | 0 | 0 | 0 | 0 | 0
0.2 |  $-$0.2825 | 0.0277 | 0.2548 | 0.3719 | 0.0281
0.4 |  $-$0.6256 | 0.1614 | 0.4641 | 0.7092 | 0.0908
0.6 |  $-$1.0292 | 0.4011 | 0.6281 | 1.0119 | 0.1881
0.8 |  $-$1.4933 | 0.7467 | 0.7467 | 1.2800 | 0.3200
Table 2: Controller gain matrix $K$’s value when
${B}=\big{[}0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0\leavevmode\nobreak\
1\leavevmode\nobreak\ 1\big{]}^{T}.$

###  3.5 Step 4.1 Dynamics analysis results

Now we need to evaluate the dynamics property of the controlled the velocity
field. Comparing to the original system, the controlled has observable
consequence. Due to the design aim, some of observation is invariant, and some
changed. The verifiable theoretical predictions of the controller are listed
as following.

  1. 1.

Distribution. by changing the pole assignmant parameter $b$. The strategy
distribution of the controlled game should be identical to that of the
original game. The theoretical expectation is

$\displaystyle\rho^{T}\\!$ $\displaystyle\rightarrow$
$\displaystyle{\text{Nash\\_1}}=\\!\frac{1}{3}(1,1,1,0,0)\leavevmode\nobreak\
\text{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ when\leavevmode\nobreak\ }\leavevmode\nobreak\
b\rightarrow-1,$ (7) $\displaystyle\rho^{T}\\!$ $\displaystyle\rightarrow$
$\displaystyle{\text{Nash\\_2}}=\\!\frac{1}{2}(0,0,0,1,1)\leavevmode\nobreak\
\text{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ when\leavevmode\nobreak\ }\leavevmode\nobreak\
b\\!\rightarrow\\!1.$ (8)

Wherein the superscript indicates the signal of the parameter. This is the
prediction on equilibrium selection.

In measurement, the prediction of equilibrium section can be verified by
Euclidean distance ($d$) between $\rho^{T}$ and $\rho^{E}$, where $\rho^{E}$
is the strategy proposition vector in the time series from computer simulation
or human experiment. In details,

$\displaystyle\bar{\rho}^{E}_{i}=\lim_{t\rightarrow\infty}\frac{1}{T}\sum_{t=0}^{T}\rho_{i}(t),$
(9)

where $\rho_{i}(t)$ is the proportion of $i$-th strategy used at time
$t\in[0,T]$ . We can measure the time dependent Euclidean distance $d(t)$ from
Nash_1 and Nash_2 respectively,

$\displaystyle d_{\text{Nash\\_1}}(t)$ $\displaystyle=$
$\displaystyle\big{|}\rho(t)-{\text{Nash\\_1}}\big{|}$ (10) $\displaystyle
d_{\text{Nash\\_2}}(t)$ $\displaystyle=$
$\displaystyle\big{|}\rho(t)-{\text{Nash\\_2}}\big{|}.$ (11)

We will see that,

$\displaystyle d_{\text{Nash\\_1}}(t)$ $\displaystyle=$
$\displaystyle\longrightarrow 0\leavevmode\nobreak\ \text{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
when\leavevmode\nobreak\ }\leavevmode\nobreak\ b\rightarrow-1,$ (12)
$\displaystyle d_{\text{Nash\\_2}}(t)$ $\displaystyle=$
$\displaystyle\longrightarrow 0\leavevmode\nobreak\ \text{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
when\leavevmode\nobreak\ }\leavevmode\nobreak\ b\rightarrow 1.$ (13)

  2. 2.

Converge speed. The converge speed to desired equilibrium is impacted by the
pole assignment parameter, $b$. The theoretical prediction is that,
disregarding which equilibrium selected, convergence speed will be faster when
$|b|$ increase.

In measurement, the convergence speed is defined as the time cost of time
(denote as $\tau_{1/2}$) when $d(\tau_{1/2})=d^{0}/2$, in which $d_{0}$ is the
Euclidean distance from the full randomly initial distribution [1, 1, 1, 1,
1]/5 to equilibrium (Nash_1 or Nash_2). In details, $d^{0}_{\text{Nash\\_1}}$
= 0.184 and $d^{0}_{\text{Nash\\_2}}$ = 0.273. So, $\tau_{1/2}(b)$ can be
obtained in the time series.

  3. 3.

Cycle. In theory, the eigencycles in the 2-d subspace of the game can be
calculated from the complex eigenvector associated to the eigenvalue shown in
Eq. 4, referring to [42]. This can be verified by time average of the angular
momentum measurement along the time series, referring to [42],

$\displaystyle\bar{L}_{mn}$ $\displaystyle=$
$\displaystyle\frac{1}{t^{\prime}}\sum_{t=0}^{t^{\prime}}x_{mn}(t)\times
x_{mn}(t+1)$ (14)

Herein, $x_{mn}(t)$ is the strategy vector in the $mn$ subspace (2-d
subspace), $t^{\prime}$ is the length of time series. For a 5 strategy game,
the identical 2-d subspace number is 10, and the observer sample is 10 [42,
40, 33]. The equivalent of the theoretical eigencycle and the observed angular
momentum $L$ in is proved referring to [40, 33].

In measurement, the cycle strength $|\bar{L}|$ is defined as

$|\bar{L}|=\big{(}\sum_{mn}L_{mn}^{2}\big{)}^{1/2}$ (15)

in the time series to verify the theoretical expectation of the strength of
the cycles.

###  3.6 Step 4.2: ABED Simulation

Agent-based evolutionary dynamics (abed) simulation (see appendix 5.3) are
carried out for various control parameter $b$. Having the time series data
from the abed simulation, we can evaluate whether the theoretical expectation
is supported by data.

###  3.7 Step 5: Evaluate the controller

####  3.7.1 Statistical results

The three figures associate to the three theoretical predictions are shown,
respectively.

  1. 1.

Figure 3(a) shows the distribution of the long run average
($\bar{\rho_{1}},\bar{\rho_{2}},\bar{\rho_{3}},\bar{\rho_{4}},\bar{\rho_{5}}$)
as the function of the pole assignment parameter $b$. As expected, when $b$
being $0\rightarrow-1$, the trend is to select equilibrium Nash_1;
Alternatively, when $b$ being $0\rightarrow 1$, the trend is to select
equilibrium Nash_2.

  2. 2.

Figure 3(b) shows the changing of the convergence time depends on the changing
of the parameter $b$. Convergence speed depends on pole assignment parameter
$b$. When $b\rightarrow-1$, observed $\tau_{1/2}\rightarrow 0$, which means
the convergence to Nash_1 is faster. Alternatively, when $b\rightarrow 1$,
observed $\tau_{1/2}\rightarrow 0$ too, the convergence to Nash_2 is faster.

  3. 3.

Figure 3(c) shows the eigencycle set by changing the parameter $b$. Each curve
presents an observation of $L_{mn}$ of the 10 eigencycle, referring to the
pole assignment parameter $b$ shown in Eq. 5. It is obviously that, when the
equilibrium remained at Nash_1 the cycle in the $(x_{1},x_{2},x_{3})$ subspace
is significant; Alternatively, when the equilibrium shift to the Nash_2
equilibrium, the cycles disappear.

Figure 3:  (a) Distribution of the long run average $\rho^{S}(b)$. when
$b\rightarrow-1$, the trend is Nash_1 selected; alternatively, when
$b\rightarrow 1$, Nash_2 selected. (b) Speed of convergence depends $b$. When
$b\rightarrow-1$, the convergence to Nash_1 is faster. Alternatively, when
$b\rightarrow 1$, the convergence to Nash_2 is faster (c) cycle strength
$L(b)$.

####  3.7.2 Explanation for the statistical result

Referring to the statistical results, we can reach the following conclusion:

  1. 1.

By pole assignment, the equilibrium section is realised as expect.

  2. 2.

As a association phenomena,the measured cycle exists or not meet the
theoretical expectation well.

  3. 3.

As a association phenomena, the measured cycle exists or not meet the
theoretical expectation well.

Base on these results, we can say, in a game with multi equilibrium, pole
assignment can be apply to influence the equilibrium selection. At the same
time, the associated observation relates to the equilibrium selection by pole
assignment meet the general dynamics theory well.

Importantly, our results suggest that, game dynamics system is not an
exception of the long existed dynamics system which has been extensively
studied and applied in engineering. Meanwhile it is not an expectation of the
long existed modern control theory.

##  4 Discussion

Equilibrium selection is a critical concern in game theory. This study
demonstrates a method of controlling equilibrium selection using a dynamic
control approach. Through a straightforward example game and a workflow of
state feedback control, the study shows that it is possible to control
equilibrium selection and achieve a desired outcome. This is the first
instance, to the best of our knowledge, where equilibrium selection has been
controlled using the controllability of the velocity field.

###  4.1 Related works on the control the game dynamics

In recent years, there has been extensive research on game-environment
interaction, led by [34]. One approach is to incorporate the feedback
mechanism from the environment into replicator dynamics, whereby the feedback
changes the pay-off structure and influences the evolution of strategies.
Several studies have explored this idea, such as [34, 28, 30].

Our method distinguishes itself from the conventional approach of treating
game-environment feedback as state-dependent feedback within the game dynamics
process. Instead, our feedback controller utilizes pole assignment techniques
based on the eigensystem, which is grounded in modern control theory.
Essentially, the core of our design strategy is to regulate the velocity
field, which in turn governs the selection of equilibrium.

###  4.2 Further directions of the controller design

We have noticed the two limitation of the approach for equilibrium selection
as follow.

  1. 1.

The general condition for the approach being applied for equilibrium selection
is not know. In this study, we have show only a special game case 1 to
illustrate the ability of pole assignment for equilibrium selection. In this
case, the left-up 3 by 3 related to rock-paper scissors, and the right-down 2
by 2 is an anti coordination game, and is a special case. Referring to [37],
this question is about pole assignability in game dynamics.

  2. 2.

For various game dynamics model, whether there is general method to design the
equilibrium selection is not known. The algorithm designed for the controller
shown in Eq. 19, especially the third term (the financial balance term $T$) is
specified for the replicator dynamics shown in Eq. (16).

It is important to emphasize the need for careful evaluation of the
consequences of the control-by-design approach (as demonstrated in Equation 6)
when using it for equilibrium selection. We also hope to see the development
of more general solutions in the future.

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##  5 Appendix

###  5.1 Pole assignment

Pole assignment (called also as pole placement) is a key approach in modern
control theory. It is design methodology wherein the objective is to place the
eigenvalues of the closed-loop system in desired regions of the complex plane
[6]. It is a method employed in feedback control system theory to place the
closed-loop poles (or eigenvalues) of a plant in desired locations. Placing
poles is desirable because the location of the poles corresponds directly to
the eigenvalues of the system, which control the characteristics of the
response of the system. After more than 50 years study, the pole assignment
has becomes basic approach in modern control theory textbook in engineering
field [3, 23, 17].

It is not surprise that the pole assignment is a important role when we facing
dynamics system. This is due to our knowledge system to handle complex
dynamics system. It is well known that [26], in dynamical systems, in general,
do not have closed-form solutions, and its behavior is hard to be predicted
and controlled. However, linear dynamical systems can be solved exactly.
Linear dynamical systems have a rich set of mathematical properties [12].
Linear systems can also be used to understand the qualitative behavior of
general dynamical systems, by calculating the equilibrium points of the system
and approximating it as a linear dynamics behavior around each such points.
The central mathematical concept set around the equilibrium is the eigen
system. The concept set includes eigenvalue, eigenvector, eigen space,
invariant manifold and so on. In eigen system, eigenvalue (pole) play the
cruel role because the pole determines the stability of the equilibrium. So,
if we are able to assign the pole in theory, we are able to control the
stability of the equilibrium. This is the logic line explaining why pole
assignment is important for dynamics system.

It is not surprise, too, that the pole placement approach has been extensively
applied in real system since 1970s. Examples are to helicopter stabilization
systems [25], to the spacecraft motion control [43], to electro-hydraulic
servosystem design [18], to active control of car longitudinal oscillations
[19]. Game theory study multi agent strategy interactions and has widely
application. However, surprisingly, the pole assignment approach has rare
appear in the field covered by game theory.

###  5.2 Algorithm for the controller design

####  5.2.1 Basic

Assume that, an one population $n$ strategy game whose evolutionary dynamics
can be described by the replicator dynamics,

$\dot{x_{i}}=F(x).$ (16)

This is a time-invariant systems, because time $t$ does not appear as a
variable in the rhs terms. Assume again that, the game system can be
linearized and expanded as

$\dot{x}={J^{o}}x,$ (17)

in which, ${J^{o}}$ is the Jacobian matrix at equilibrium (rest point), and
$x$ is a small deviation from the equilibrium. If $J^{o}$ is diagonalizable,
we call its eigenvalue set as original pole vector, $\lambda^{o}$. In
mathematics, the game dynamics is specified on the $n$-simplex by an ordinary
differential equation (ODE). In control engineering, this is a continuous-time
LTI (linear time-invariant) system, which is a fairly common situation to
carry out feedback control (see wiki: State-space_representation).

The original pole assignment approach states that, given the single- or multi-
input system

$\dot{x}=J^{o}x+Bu$ (18)

and a vector $\lambda^{c}$ of desired self-conjugate closed-loop pole
locations, place computes a gain matrix $K$ such that the state feedback
$u=-Kx$ places the closed-loop poles at the locations $\lambda^{c}$. In other
words, the eigenvalues of $J^{o}-BK$ match the entries of $\lambda^{c}$.

Admittedly, the original pole assignment approach does not fit the game
dynamics system. Because, for a given $B$, for example
$B=[0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0\leavevmode\nobreak\
0\leavevmode\nobreak\ 1]$, once $Bu$ term is not zero, the sum of rhs of Eq.
(18) is not zero. This deviates from the simplex concentration,
$\sum_{i}x_{i}=1$. So, the original approach has to be improved.

####  5.2.2 Construct ${J^{c}}$

We use the linear control approach, pole assignment, to design the control
term. The control terms includes gain matrix $K$ control vector (an $1\times
n$ vector, a row vector) on the channel $B$ (a $n\times 1$ vector, a column
vector), and a tax term $T$ for the payoff conservation. The payoff’s
conservation can be understood as budget balance. Then, the controlled
velocity field can be expressed as

$\dot{x}={J^{o}}\cdot x+{B}\cdot{K}\cdot x+T\cdot x$ (19)

Here, $B\cdot K$ is the ordinary linear state dependent control term (see
standard text book of modern control system), which is called as reward term.
The financial balance term

$T:=-\sum_{i=1}^{n}B_{i}\cdot\sum_{j=1}^{n}K_{j}x_{j}$ (20)

is a constant value defined at each given state $x$. It can be verified that,
at each period in the game, each agent is taxed the constant value, $T$. The
tax term makes the controller to be budget balanced—the total amount of tax
collected from the system at a given period is exactly equal to the total
amount of reward given to the system at that time.

The controlled Jacobian ${J^{c}}$ can be expressed as

${J^{c}}={J^{o}}+{B}\cdot{K}+T.$ (21)

Then for an assigned pool $\lambda^{c}$ and a given $B$, we can obtain $K$.
Once ${K}$ is given , ${J^{c}}$ as well as its the eigen system is determined.

##### On Constraint condition of the controller design

  1. 1.

Equilibrium conservation: The equilibrium (rest point) should be preserved and
not be shifted; This can be satisfied by setting $K\cdot x_{\text{Nash}}=0$
when solving the character equations for $J^{c}$.

  2. 2.

The controller, $J^{c}$, does not change the expected payoff. Or saying, the
equilibrium payoff remains an invariant by $J^{c}$.
Proof: Denote $H_{0}=\sum_{j=1}^{n}k_{j}x_{j}$, referring to the definition of
$T$ in Eq. (20), we have

$T=-H_{0}\sum_{m=1}^{n}B_{m},$ (22)

which is a scalar at given $x$. Recall Eq. (19), applying the definition of a
element of Jacobian $J_{ij}=\partial\dot{x_{i}}/\partial x_{j}$, we have

$J_{ij}^{c}=J_{ij}^{0}+B_{i}\cdot k_{j}-k_{j}\sum_{m=1}^{n}B_{m}\cdot x_{i}$
(23)

Sum over each column, we can reach that,

$\displaystyle\sum_{i=1}^{n}J_{ij}^{c}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}J_{ij}^{0}+\sum_{i=1}^{n}B_{i}k_{j}-k_{j}\sum_{m=1}^{n}B_{m}\sum_{i=1}^{n}x_{i}$
(24) $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}J_{ij}^{0}$ (25)

It is known that [8], there exists a left eigenvector (1, 1, 1, …, 1) for $J$
and its related right eigenvector is the equilibrium distribution, and the sum
of the column value of $J$ is the equilibrium payoff. Having the controller,
the system earning (expected payoff) is remained.

  3. 3.

Payoff conservation: There is no additional payoff (financial support) adding
to the game system during the controlling processes, which naturally can be
realised by the definition of the tax term $T$ is of financial balance.

    1. (a)

$K$ rewards the strategy at the channel $j$ determined by $B$, of which the
reward value is $({BKx})_{j}$. As a result, for each agent using strategy $j$,
its reward (additional payoff) is $({BKx})_{j}/x_{j}$.

    2. (b)

$T$ taxes each strategy equally. The tax value is a constant, $T$, depending
on state $x$. For each individual agent its tax value is $T$. $T$ term balance
the additional pay due to the gain matrix $K$ above to each agent in game.

####  5.2.3 Explain $B$

In this study, we limit ourselves to illustrate an example of the equilibrium
selection by a controller designed by pole assignment. We choose
${B}=[0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0\leavevmode\nobreak\
1\leavevmode\nobreak\ 1]^{T}$ for the illustration the controllablity. We hope
to emphasise that, for a multi equilibrium system, the choosing of $B$ relates
to whether a controller having a solution for its desired goal. For game
dynamics, the general answer on $B$ choosing is out of the scope of this
study.

###  5.3 Agent-based evolutionary dynamics (ABED) simulation

The simulation is introduced as following.

  1. 1.

Select simulation platform: We use ABED (Agent-based evolutionary dynamic)
simulator [11], which is widely used in the field to study evolutionary game
dynamics. The platform has integrated various learning rules and matching
rules. The platform is of agent-based long-running repeated game setting. The
simulator is implemented in the open-source platform NetLogo 6.0.4.

  2. 2.

Setting parameters: The simulation is under imitative protocols, in which
candidates are agents; meanwhile, the decision method is of pairwise-
comparison of the strategy payoff. The complete-matching is set. These setting
are follow the user guide of the platform, which system will performs like
replicator dynamics shown in Eq. (16) in large population (1000 agents) and
low reversion probability 5%) limit.

  3. 3.

Add control modular to the platform: add a modular to control the agents
payoff referring to the algorithm shown in Eq. (19).

  4. 4.

Specify controller parameter: the modular is a state-depend feedback control
’device’. The device is specified according to the parameter vector $B$ and
$K$. Various parameter assign can archive various goal. In this case, $B$ and
$K$ are shown in Table 2.

  5. 5.

Conduct the simulation: In our study case, time cost for each simulation of a
given parameter $b$, for 6000 rounds, is about 1 minute in a desktop personal
computer, which CPU is 3.70 GHz and the memory is 8 GB.

  6. 6.

Analysis the time series: Main outcome of the simulator is the time series of
the strategies density and related variables.

Table 3: The parameter setting for the ABED simulations Parameter | Replicator $[S_{1}]$
---|---
payoff-matrix | [[ 0 0 2 0 -2]
| [ 2 0 0 -2 0]
| [ 0 2 0 2 -1]
| [ -2 0 1 0 1]
| [0 -2 -2 1 0]]
n-of-agents | 1000
random-initial-condition? | TRUE
initial-condition | [200 200 200 200 200]
candidate-selection | imitative
n-of-candidates | 2
decision-method | pairwise-
| difference
complete-matching? | TRUE
n-of-trials | 999
single-sample? | TRUE
tie-breaker | uniform
log-noise-level | 0.031
use-prob-revision? | TRUE
prob-revision | 0.2
n-of-revisions-per-tick | 10
prob-mutation | 0.05
trials-with- replacement? | FALSE
self-matching? | FALSE
imitatees-with-replacement? | FALSE
consider-imitating-self? | FALSE
plot-every-?-secs | 0.2
duration-of-recent | 10
show-recent-history? | TRUE
show-complete-history? | TRUE
K | [-1.49 0.75 0.75 1.28 0.32]
B_Channel | [0 0 0 1 1]

###  5.4 Abbreviations

The abbreviations and mathematical symbols, shown in Table 4, are used in the
main text, this appendix and the supplementary information.

$b$ |  | pole assignment parameter, $b\in[-1,1]$
---|---|---
$B$ |  | channel matrix, an $5\times 1$ vector, a column vector
$K$ |  | gain matrix, an $1\times 5$ vector, a row vector
$B\cdot K$ |  | Reward term, a matrix with real number elements.
|  | It is independent of the state vector.
$T$ |  | Tax term, a constant, a real number.
|  | It is dependent of the state vector.
PTDE |  | Predictable temporary deviation from equilibrium
ABED |  | Agent-based evolutionary dynamics simulation
$\rho$ |  | The proportion vector of strategies used
| $\rho_{i}$ | The $i$-th component of $\rho$,
| $\rho_{i}(t)$ | The $\rho_{i}$ observed at time (round) $t$
Nash_1 |  | The Nash equilibrium, $\rho=(1/3,\leavevmode\nobreak\ 1/3,\leavevmode\nobreak\ 1/3,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 0)$
Nash_2 |  | The Nash equilibrium, $\rho=(0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 1/2,\leavevmode\nobreak\ 1/2)$
$d$ |  | Euclidean distance between two distribution
|  | vectors, used to evaluate their difference.
| $d_{\text{Nash\\_1}}$ | The $d$ between the observed distribution to Nash_1
| $d_{\text{Nash\\_2}}$ | The $d$ between the observed distribution to Nash_2
$o$ |  | A fixed point, rest point, or zero velocity point, equilibrium
$\lambda$ |  | Eigenvalue
| $\lambda^{o}$ | Original pool
| $\lambda^{c}$ | Controlled pool, assign pole
$\xi$ |  | Eigenvector
$\eta_{i}$ |  | The $i$-th component of a given eigenvector $\xi$
$L_{mn}$ |  | The eigencycle set in the 2D subspace set of a game
|  | space, describing the cyclic motion strength
| $L_{(m,n)}$ | The $L$ at dimension (1,2):=($\rho_{m},\rho_{n}$)
Table 4: Abbreviations and mathematical symbols. Figure 4:  Eigencycle
strength $L_{mn}(b)$.

Figure 5:  Speed of convergence depends on pole assignment parameter $b$. When
$b\rightarrow-1$, the convergence to Nash_1 is faster. Alternatively, data
shows that when $b\rightarrow 1$, the convergence to Nash_2 is faster.

    K>> t_184(1:5)
    ans =  10    12    15    24    72
    K>> t_273(6:9)
    ans =  300    54    32    23

###  5.5 Data and code

The documents containing the data and code are listed in Table 18 in SI and
are accessible at http:github.com-xxxxxx-yyyyyy.

file name |  description |  note
---|---|---
E5C.nlogo | ABED simulator code | Simulation
E5Cdata.rar | data generated by E5C.nlogo |
| x_parameter.csv parameter |
| x_A_s.csv strateg |
| x_A_w.csv payoff |
T24simAll.m | mean $\rho$ |
| mean $L_{mn}(b)$ |
| $d_{\text{Nash\\_}i}(t)$ |
SLXMAT2.mat | data generated by T24simAll.m |
| used by T24sim3Fig.m |
T24sim3Fig.m | Fig 3(a) | $\bar{\rho}_{i}(b)$
| Fig 3(b) | $\tau_{1/2}$
| Fig 3(c) | $|L|$
| Fig 4 | $L_{m,n}$
| Fig 5(a) | $d_{\text{Nash\\_1}}(t)$
| Fig 5(b) | $d_{\text{Nash\\_2}}(t)$
read_5angl_forN3E.m | function called by T24simAll.m |
Table 5: Data and code