[1]Bastian Boll

1]Institute for Mathematics, Heidelberg University, Im Neuenheimer Feld 205,
Heidelberg, 69120, Baden-Württemberg, Germany

2]Institute for Mathematics, Augsburg University, Universitätsstraße 2,
Augsburg, 86159, Bavaria, Germany

# A Geometric Embedding Approach to Multiple Games and Multiple Populations

<EMAIL_ADDRESS>Jonas Cassel  Peter Albers  Stefania Petra
Christoph Schnörr  [  [

###### Abstract

This paper studies a meta-simplex concept and geometric embedding framework
for multi-population replicator dynamics. Central results are two embedding
theorems which constitute a formal reduction of multi-population replicator
dynamics to single-population ones. In conjunction with a robust mathematical
formalism, this provides a toolset for analyzing complex multi-population
models. Our framework provides a unifying perspective on different population
dynamics in the literature which in particular enables to establish a formal
link between multi-population and multi-game dynamics.

###### keywords:

Replicator dynamics, Assignment Flows, Manifold Embedding, Information
Geometry

###### Contents

  1. 1 Introduction
    1. 1.1 Overview, Contribution
    2. 1.2 Organization
    3. 1.3 Basic Notation.
  2. 2 Preliminaries
    1. 2.1 Fisher-Rao Geometry, Replicator Dynamics
    2. 2.2 Data Labeling and Assignment Flows
  3. 3 Embedding the Assignment Manifold
  4. 4 Multiple Populations and Multiple Games
  5. 5 Tangent Space Parameterization
  6. 6 Learning Replicator Dynamics from Data
  7. 7 Asymptotic Behavior
  8. 8 Conclusion
  9. A Additional Lemmata
  10. B Proofs
    1. B.1 Proof of Theorem 3.1
    2. B.2 Proof of Proposition 3.2
    3. B.3 Proof of Lemma 3.3
    4. B.4 Proof of Theorem 3.5

##  1 Introduction

###  1.1 Overview, Contribution

Evolutionary game theory [24, 39] is an established framework for modeling
problems in diverse areas ranging from mathematical biology [50, 49, 20, 35,
30] to economics [16, 37]. It assumes a dynamic perspective on games played by
a large and well-mixed population of agents. In this context, the earliest
dynamical model of population state is the _replicator equation_ [52, 46],
which has since been generalized in several ways [6, 14] to accommodate more
complex situations.

This paper provides an embedding approach for studying the following two
classes of scenarios within a single framework.

  * •

_Multi-population dynamics_ model multiple interacting populations or species.
The state space is a product manifold of multiple simplices, and payoff may
depend on the state of all populations. The resulting dynamics are multiple
coupled replicator dynamics on the product manifold of multiple simplices.

  * •

_Multi-game dynamics_ model agents that simultaneously play multiple games,
earning cumulative payoff. The state space is a single simplex with dimension
growing exponentially in the number of games. Interaction between games occurs
whenever the population state is outside of a specific submanifold.

Here, we study an embedding of multiple probability simplices into a
combinatorially large simplex of joint distributions. As further detailed in
Section 8, our approach is closely related to _Segre embeddings_ of projective
spaces [17] which play a prominent role in many areas of mathematics and
physics, such as _independence models_ in algebraic statistics [15] and
_entanglement_ in quantum mechanics [7]. Based on this Ansatz, we develop a
geometric perspective and formalism to study the relationship between
replicator dynamics of multiple populations and multi-game replicator
dynamics. In particular, we demonstrate that the multi-game dynamics of [22]
share a generic payoff structure with multi-population games.

Our work further constitutes a formal reduction of multi-population dynamics
to – a much higher-dimensional – single-population dynamics, which is helpful
for theoretical analysis. We demonstrate this by transferring two results on
the asymptotic behavior of replicator dynamics from the single-population to
the multi-population setting.

Concerning applications, our work aims to provide insight into the structure
of multi-population and multi-game dynamics, along with a robust mathematical
toolset for domain experts to analyze complex systems. Indeed, there is a
growing need for more powerful dynamical models in emerging applications. For
instance, [53] argue for the use of generalized replicator dynamics to model
interactions in nature – considering multi-player interaction in a multi-game
setting.

The present paper also extends our previous work on _assignment flows_ [3,
45]. They are dynamical systems that leverage interaction along edges of a
graph ${\mathcal{G}}$ to infer an assignment of class labels to the nodes of
${\mathcal{G}}$ from node-wise data. Applications include structured
prediction problems such as semantic image segmentation in supervised [48] and
unsupervised scenarios [58, 56].

In [8], we have shown that assignment flows can be seen as multi-population
replicator dynamics and studied how payoff is transformed by embedding the
state space into a single simplex of joint distributions. In the present work,
we generalize this analysis to nonlinear payoff functions and provide a
careful study of the involved manifolds.

We also highlight previous findings on assignment flows and their relevance to
the evolutionary game theory community. In particular

  * •

[57] present an exhaustive study of conditions under which certain assignment
flows converge to integer assignments. These are states in which only a single
played strategy remains in each population.

  * •

[54] have proposed a generically applicable framework for geometric numerical
integration which scales to large replicator dynamics.

  * •

[26, 55] have studied methods of learning the parameters that generate
replicator dynamics from data.

###  1.2 Organization

Section 2 contextualizes assignment flows as multi-population replicator
dynamics and establishes related notation. Section 3 describes the proposed
meta-simplex concept and related geometric notions as well as the embedding
theorems 3.1 and 3.5, which constitute the main results of the present work.
Section 4 gives three examples of dynamics considered in prior work and
establishes their relationship through the lens of the geometric embedding
theorems. Section 5 recapitulates a tangent space parameterization of
replicator dynamics from the literature on assignment flows and studies it in
the context of geometric embedding. Section 6 highlights previous findings on
parameter learning for assignment flows. Section 7 demonstrates how the
proposed formal reduction of multi-population to single-population dynamics
can be used as a tool for formal analysis of asymptotic behavior. Section 8
gives an outlook on current assignment flow developments and concludes the
paper.
The present work substantially extends the conference paper [8] in the
following ways:

  * •

The submanifold of embedded multi-population states is identified as a
generalized Wright manifold and its geometry is analyzed (Theorem 3.1).

  * •

The embedding theorem of multi-population replicator dynamics is generalized
to nonlinear payoff functions (Theorem 3.5).

  * •

Multi-game dynamics are studied as embedded multi-population replicator
dynamics (Section 4).

  * •

Tangent space parameterization of replicator dynamics is studied in the
context of geometric embedding (Theorem 5.2).

###  1.3 Basic Notation.

For $k\in\mathbb{N}$ we use the shorthands
$[k]:=\\{1,\ldots,k\\}\subset\mathbb{N}$ and
$\mathbf{1}_{k}:=(1,\ldots,1)^{\top}\in\mathbb{R}^{k}$. Angle brackets
$\langle\cdot,\cdot\rangle$ are used for both the standard inner product
between vectors and the Frobenius inner product between matrices. The
Kronecker product of matrices [18] is denoted by $A\otimes B$. Componentwise
multiplication of vectors $x$ and $y$ is denoted by $x\diamond y$, and by
$\frac{x}{y}$ the componentwise division of a vector $x$ by a strictly
positive vector $y$. Likewise, logarithms and exponentials of vectors apply
componentwise. For vectors $x\in\mathbb{R}^{c}$, the expression $x\geq 0$
denotes $x_{i}\geq 0$ for all $i\in[c]$.

##  2 Preliminaries

###  2.1 Fisher-Rao Geometry, Replicator Dynamics

In matrix games, players from a large population engage in two-player
interactions. For simplicity, we assume that each player chooses from a
constant set of $c$ strategies. The payoff for a two-player interaction is
then given by a $c\times c$ payoff matrix $B$. If players change their
strategy to imitate other players with more effective strategies, the overall
distribution $p\in{\mathcal{S}}_{c}$ of strategies in the population changes
over time according to the well-known _replicator dynamics_

$\dot{p}(t)={R}_{p(t)}[Bp(t)],\qquad p(0)=p_{0}\in{\mathcal{S}}_{c},$ (1)

where

${\mathcal{S}}_{c}=\\{p\in\mathbb{R}^{c}\colon\langle\mathbf{1}_{c},p\rangle=1,\;p>0\\}$
(2)

denotes the relative interior of the probability simplex with $c$ vertices and

${R}_{p}=\operatorname{Diag}(p)-pp^{\top}$ (3)

is called the _replicator operator_. We regard ${\mathcal{S}}_{c}$ as a
Riemannian manifold with trivial tangent bundle

$T{\mathcal{S}}_{c}\cong{\mathcal{S}}_{c}\times T_{0}{\mathcal{S}}_{c},\qquad
T_{0}{\mathcal{S}}_{c}=\\{v\in\mathbb{R}^{c}\colon\langle\mathbf{1}_{c},v\rangle=0\\}$
(4)

and equipped with the _Fisher-Rao metric_

$\langle\cdot,\cdot\rangle_{g}\colon T_{p}{\mathcal{S}}_{c}\times
T_{p}{\mathcal{S}}_{c}\to\mathbb{R},\qquad(u,v)\mapsto\Big{\langle}\frac{u}{p},v\Big{\rangle}.$
(5)

The barycenter of ${\mathcal{S}}_{c}$ is denoted
${\mathbf{1}_{\mathcal{S}}}=\frac{1}{c}\mathbf{1}_{c}$. Vectors in
$\mathbb{R}^{c}$ are projected onto the tangent space $T_{0}{\mathcal{S}}_{c}$
by the linear map

$\Pi_{0}\colon\mathbb{R}^{c}\to T_{0}{\mathcal{S}}_{c},\qquad
v\mapsto\Pi_{0}v=v-\frac{1}{c}\langle\mathbf{1}_{c},v\rangle\mathbf{1}_{c}.$
(6)

The manifold ${\mathcal{S}}_{c}$ has dimension $c-1$. Two coordinate charts
are particularly relevant to the following discussion. A point
$p\in{\mathcal{S}}_{c}\subseteq\mathbb{R}^{c}$ has $m$-coordinates $\mu$ with

$p=(\mu,1-\langle\mathbf{1}_{c-1},\mu\rangle),\qquad\mu\in\mathbb{R}^{c-1},\quad\mu>0,\quad\langle\mu,\mathbf{1}_{c-1}\rangle<1$
(7)

and $e$-coordinates $\theta$ with

$p=\frac{1}{Z(\theta)}\exp\begin{pmatrix}\theta\\\
-\langle\theta,\mathbf{1}_{c-1}\rangle\end{pmatrix},\qquad\theta\in\mathbb{R}^{c-1},$
(8)

where $Z(\theta)$ normalizes the vector on the right-hand side such that
$p\in{\mathcal{S}}_{c}$, as defined by (2). The $e$-coordinates $\theta$ are
unconstrained and define a _global_ chart for ${\mathcal{S}}_{c}$.

For general references to Riemannian geometry, we refer to [29, 28]. For
references to _information geometry_ which underlies the above definitions,
see [1, 4].

The mean payoff in a population with state $p$ is $\langle p,Bp\rangle$. In
particular, if $B$ is symmetric $Bp=\frac{1}{2}\partial_{p}\langle
p,Bp\rangle$, then it is well known that (1) is the Riemannian gradient ascent
flow of mean payoff with respect to the metric (5). With an eye toward
numerical computation, a useful object is the _lifting map_

$\exp_{p}\colon
T_{0}{\mathcal{S}}_{c}\to{\mathcal{S}}_{c},\qquad\exp_{p}(v)=\frac{p\diamond\exp(v)}{\langle
p,\exp(v)\rangle}.$ (9)

It can be shown that

$\exp_{p}=\exp_{p}\circ\,\Pi_{0},$ (10)

with the projection $\Pi_{0}$ given by (6), such that $\exp_{p}$ is well-
defined on $\mathbb{R}^{c}$. Furthermore, the mapping (9) is a first-order
approximation to the Levi-Civita geodesics on ${\mathcal{S}}_{c}$ [3]. It is
also closely related to the $e$-geodesics of information geometry [1].

###  2.2 Data Labeling and Assignment Flows

We have studied dynamics similar to (1) for _data labeling_ on graphs. Given a
graph ${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$ and data on each node, the
task is to infer node-wise classes. For example, in image segmentation, the
graph may be a grid graph of image pixels, and pixel-wise data lives in some
feature space, most basically a color space. Another example is node-wise
classification of citation graphs such as [9]. Here, nodes are academic papers
and edges between them denote citations. The task is to classify the topic of
papers from node-wise features and citations.

In each case, graph connectivity is crucial information and a natural approach
is to facilitate interaction along graph edges to inform the labeling process.
We further abstract from the raw feature space of given data by lifting to a
probability simplex ${\mathcal{S}}_{c}$ on each node. The resulting state
lives in the product manifold

${\mathcal{W}}={\mathcal{S}}_{c}\times\cdots\times{\mathcal{S}}_{c}$ (11)

containing $n=|{\mathcal{V}}|$ copies of ${\mathcal{S}}_{c}$ which we call
_assignment manifold_. For $S\in{\mathcal{W}}$, let ${\mathcal{R}}_{S}$ denote
the operator which applies ${R}_{S_{i}}$ on each node $i\in[n]$. Similarly,
let

$\exp_{S}(V),\qquad V\in T_{0}{\mathcal{W}}=(T_{0}{\mathcal{S}}_{c})^{n}$ (12)

denote the map which applies (9) separately on each node, whose domain can be
extended

$\exp_{S}\colon\mathbb{R}^{n\times c}\to\mathcal{W}.$ (13)

due to (10). We will still call these objects _replicator operator_ and
_lifting map_ , respectively. Likewise, the projection

$\Pi_{0}\colon\mathbb{R}^{n\times c}\to T_{0}\mathcal{W}$ (14)

applies separately the mapping (6) on each node and the barycenter of
${\mathcal{W}}$ reads
${\mathbf{1}_{\mathcal{W}}}=\frac{1}{c}\mathbf{1}_{n\times c}$.

By defining a payoff function

$F\colon{\mathcal{W}}\to\mathbb{R}^{n\times c}$ (15)

which has the state $S_{i},S_{j}\in{\mathcal{S}}_{c}$ of nodes $i,j\in[n]$
interact exactly if $ij\in{\mathcal{E}}$, we have found a natural inference
dynamic on ${\mathcal{G}}$ given by

$\dot{S}(t)={\mathcal{R}}_{S(t)}\big{[}F\big{(}S(t)\big{)}\big{]},\qquad
S(0)=S_{0},$ (16)

whose solution is called _assignment flow_. For labeling, payoff functions are
designed such that the state $S$ is driven towards an extremal point of the
set ${\mathcal{W}}$. These states unambiguously associate each node with a
single class. From a game-theoretical perspective, the extremal points of
${\mathcal{W}}$ are states in which only a single strategy is played in each
population. A more detailed overview can be found in the original work [3] and
the survey [45].

##  3 Embedding the Assignment Manifold

In our previous work [8], we showed that assignment flows can be seen as
multi-population replicator dynamics. Furthermore, we introduced a preliminary
formalism for embedding the state space of multi-population dynamics into a
_single_ , much higher-dimensional _meta-simplex_ of joint distributions.
Assuming again the data labeling perspective introduced in Section 2.2, one
may enumerate all $c^{n}$ possible assignments of $c$ classes to $n$ graph
nodes. This enumeration represents data labeling as a single decision between

$N=c^{n}$ (17)

alternatives which we view as pure strategies of a _single_ population game on
the meta-simplex ${\mathcal{S}_{N}}$.

Here, we describe a refined version of the embedding formalism as well as
several additional results, generalizing and expanding our earlier findings.
Note that the proposed meta-simplex ${\mathcal{S}_{N}}$ is not to be confused
with the meta-simplex concept proposed by [2]. The latter explicitly considers
the relative size of populations and has much lower dimension.

To simplify notation, we assume that agents of each population have the same
number $c$ of available pure strategies. However, the following results remain
valid in more general scenarios of variable strategy sets. In addition, we
index entries of vectors ${p}\in{\mathcal{S}_{N}}$ by multi-indices $[c]^{n}$
as opposed to integer indices in $[c^{n}]$ to improve readability. The
component $\gamma_{i}$ of a multi-index $\gamma\in[c]^{n}$ indexes a label
$\gamma_{i}\in[c]$ at vertex $i\in[n]$.

We consider the following maps, defined componentwise by

$\displaystyle T\colon$
$\displaystyle\mathcal{W}\to{\mathcal{T}}\subseteq{\mathcal{S}_{N}},$
$\displaystyle T(W)_{\gamma}:=$ $\displaystyle\prod_{i\in[n]}W_{i,\gamma_{i}}$
$\displaystyle\text{for all }\gamma\in[c]^{n}$ (18a) $\displaystyle Q\colon$
$\displaystyle\mathbb{R}^{n\times c}\to\mathbb{R}^{N},$ $\displaystyle
Q(X)_{\gamma}:=$ $\displaystyle\sum_{i\in[n]}X_{i,\gamma_{i}}$
$\displaystyle\text{for all }\gamma\in[c]^{n}$ (18b) $\displaystyle M\colon$
$\displaystyle\mathbb{R}^{N}\to\mathbb{R}^{n\times c},$ $\displaystyle
M(x)_{ij}:=$ $\displaystyle\sum_{\gamma\in[c]^{n}\;:\;\gamma_{i}=j}x_{\gamma}$
$\displaystyle\text{for all }(i,j)\in[n]\times[c].$ (18c)

The particular choice of these maps will be justified by laying out several
compatibility properties which intricately link them to each other and to the
geometries of ${\mathcal{W}}$ and ${\mathcal{S}_{N}}$. Specifically,

  * •

$T$ realizes the concept of enumerating labelings in the sense that the
extremal points of $\overline{{\mathcal{W}}}$ are bijectively mapped to the
extremal points of $\overline{{\mathcal{S}_{N}}}$.

  * •

The restriction of $M$ to $\mathcal{T}$ inverts $T$ by computing node-wise
marginals. We choose the larger domain $\mathbb{R}^{N}$ for $M$ such that it
becomes the adjoint mapping of $Q$ (ref Lemma 3.4).

######  Theorem 3.1 (assignment manifold embedding).

The map $T\colon{\mathcal{W}}\to{\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ is an
isometric embedding of ${\mathcal{W}}$ equipped with the product Fisher-Rao
geometry, into ${\mathcal{S}_{N}}$ equipped with the Fisher-Rao geometry. On
its image $T({\mathcal{W}})=:{\mathcal{T}}\subseteq{\mathcal{S}_{N}}$, the
inverse is given by marginalization

$M|_{\mathcal{T}}=T^{-1}\colon{\mathcal{T}}\to{\mathcal{W}}.$ (19)

###### Proof.

Section B.1. ∎

In view of the expression (18a), it is clear that ${\mathcal{T}}$ is precisely
the set of rank-1 tensors in
${\mathcal{S}_{N}}\subseteq\mathbb{R}^{N}\cong(\mathbb{R}^{c})^{n}$. In
addition, we have the following interpretation of points
$T(W)\in{\mathcal{T}}$ within the simplex of joint distributions
${\mathcal{S}_{N}}$.

######  Proposition 3.2 (maximum entropy property).

For every $W\in{\mathcal{W}}$, the distribution $T(W)\in{\mathcal{S}_{N}}$ has
maximum entropy among all $p\in{\mathcal{S}_{N}}$ subject to the marginal
constraint $Mp=W$, with $M$ given by (18c).

###### Proof.

Section B.2. ∎

In general, each collection of marginal distributions $S\in{\mathcal{W}}$ has
(infinitely) many possible joint distributions. Proposition 3.2 shows that $T$
precisely selects the least informative one among them. This situation is
illustrated in Figure 1.

(a) A randomly generated joint distribution of $S_{1}$ and $S_{2}$.

(b) The maximum-entropy joint distribution $T(S)$ of $S_{1}$ and $S_{2}$.

Figure 1: Marginals distributions $S=(S_{1},S_{2})$ and two possible
conforming joint distributions. Joint distribution values are scaled by a
factor of $c$ for visual clarity.

Theorem 3.1 expresses an intricate relationship between the product Fisher-Rao
geometry of ${\mathcal{W}}$ and the Fisher-Rao geometry of
${\mathcal{S}_{N}}$. A similar compatibility is found between the lifting map
(9) on ${\mathcal{W}}$ and its analog on ${\mathcal{S}_{N}}$.

######  Lemma 3.3 (Lifting Map Lemma).

Let $S\in{\mathcal{W}}$ and $V\in\mathbb{R}^{n\times c}$. Then the mappings
$T,Q$ given by (18) satisfy

$T\big{(}\exp_{S}(V)\big{)}=\exp_{T(S)}\big{(}Q(V)\big{)},$ (20)

where $\exp_{S}$ on the left is given by (12) and $\exp_{T(S)}$ on the right
naturally extends the mapping (9).

###### Proof.

Section B.3. ∎

We will also frequently use the following useful identity connecting $Q$ to
marginalization.

######  Lemma 3.4 ($Q$ Adjoint Lemma).

$M$ and $Q$ given by given by (18) are adjoint linear maps with respect to the
standard inner product, i.e. for each $p\in\mathbb{R}^{N}$ and each
$V\in\mathbb{R}^{n\times c}$ it holds that

$\langle p,Q(V)\rangle=\langle Mp,V\rangle.$ (21)

###### Proof.

Let ${p}\in\mathbb{R}^{N}$ and $V\in\mathbb{R}^{n\times c}$, then

$\displaystyle\langle{p},Q(V)\rangle$
$\displaystyle=\sum_{\gamma\in[c]^{n}}{p}_{\gamma}Q(V)_{\gamma}=\sum_{\gamma\in[c]^{n}}{p}_{\gamma}\sum_{l\in[n]}V_{l,\gamma_{l}}=\sum_{l\in[n]}\sum_{j\in[c]}\sum_{\gamma_{l}=j}{p}_{\gamma}V_{l,\gamma_{l}}$
(22a)
$\displaystyle=\sum_{l\in[n]}\sum_{j\in[c]}V_{l,j}\sum_{\gamma_{l}=j}{p}_{\gamma}\stackrel{{\scriptstyle\eqref{eq:def_Margin}}}{{=}}\sum_{l\in[n]}\sum_{j\in[c]}V_{l,j}(M{p})_{l,j}$
(22b) $\displaystyle=\langle M{p},V\rangle.$ (22c)

∎

Our main result stated next is that the embedding
$T\colon{\mathcal{W}}\to{\mathcal{S}_{N}}$ maps multi-population replicator
dynamics on ${\mathcal{W}}$ to single-population replicator dynamics on
${\mathcal{S}_{N}}$ by a transformation of payoff functions. This generalizes
our earlier finding [8] to arbitrary nonlinear payoffs.

######  Theorem 3.5 (Multi-Population Embedding Theorem).

For any payoff function $F\colon{\mathcal{W}}\to\mathbb{R}^{n\times c}$, the
multi-population replicator dynamics

$\dot{W}={\mathcal{R}}_{W}[F(W)],\quad W(0)=W_{0}$ (23)

on ${\mathcal{W}}$ is pushed forward by $T$ to the replicator dynamics

$\dot{p}(t)={R}_{p(t)}\widehat{F}\big{(}p(t)\big{)},\quad
p(0)=T(W_{0}),\qquad\widehat{F}=Q\circ F\circ M,$ (24)

on ${\mathcal{S}_{N}}$ and the map $T$ satisfies

$dT|_{W}[{\mathcal{R}}_{W}[X]]=R_{T(W)}Q[X],\quad\text{for all
}X\in\mathbb{R}^{n\times c}\text{ and }W\in{\mathcal{W}}.$ (25)

###### Proof.

Section B.4. ∎

Intuitively, the structure of $\widehat{F}$ in (24) can be seen as follows.
The joint population state ${p}\in{\mathcal{S}_{N}}$ is first marginalized and
payoff $F(M{p})$ is computed from the marginal multi-population state. Theorem
3.5 now shows that when multi-population state $W\in{\mathcal{W}}$ is seen as
factorizing joint population state ${p}\in{\mathcal{S}_{N}}$ according to
${p}=T(W)$, then the payoff gained in state $W$ is transformed by $Q$ to
induce replicator dynamics of the joint population state.

In the following, leading examples will be matrix games, i.e. linear payoff
functions that model two-player interactions. Note, however, that Theorem 3.5
applies to arbitrary nonlinear payoff functions, including multi-player
interactions.

Figure 2: The embedded submanifold
${\mathcal{T}}\subseteq{\mathcal{S}_{N}},\,N=4$. For two marginal
distributions, this is known as the _Wright manifold_ [24, Section 18.8],
[11].

##  4 Multiple Populations and Multiple Games

Because both $M$ and $Q$ are linear operators, generalized matrix games on
multiple populations reduce to simple matrix games of the joint population
state exactly if the payoff $F$ is a linear function of the multi-population
state. Here, we give two examples of multi-population games, one from the
assignment flow literature and one from game theory. To this end, denote by

$s:=\operatorname{vec}_{\text{row}}(S)\in\mathbb{R}^{nc}$ (26)

a vectorized multi-population state which contains all entries of
$W\in{\mathcal{W}}\subseteq\mathbb{R}^{n\times c}$ stacked row-wise. Table 1
summarizes the scenarios discussed in the following.

Table 1: Structure of payoff (29) for simple instances of different games. | S-Flow | EGN | Multi-Game
---|---|---|---
Payoff | $\overline{A}=\Omega\otimes\mathbb{I}_{c}$ | $\overline{A}=\Omega\otimes B$ | $\overline{A}=\mathbb{I}_{n}\otimes B$

_S-flows_ [43] define payoff by averaging the state $S$ according to a
weighted graph adjacency matrix $\Omega\in\mathbb{R}^{n\times n}$. The
resulting assignment flow with vectorized state (26) reads

$\dot{s}(t)={\mathcal{R}}_{s(t)}[(\Omega\otimes\mathbb{I}_{c})s(t)],\qquad
s(0)=s_{0}.$ (27)

This dynamical system promotes similarity of adjacent populations. Depending
on the initialization $s_{0}$, ‘pockets of consensus’ are formed. It has also
been shown that these dynamics converge to extremal points of ${\mathcal{W}}$
for almost all initializations under weak conditions [57].
_Evolutionary Games on Networks (EGN)_ [31, 27] are dynamics which generalize
(27) by incorporating payoff matrices for games played between players of
adjacent populations. In the simplest case, all such games have a constant
payoff matrix $B\in\mathbb{R}^{c\times c}$. Then, the multi-population
replicator dynamics of EGN read

$\dot{s}(t)={\mathcal{R}}_{s(t)}[(\Omega\otimes B^{\top})s(t)],\qquad
s(0)=s_{0}.$ (28)

Both (27) and (28) have a linear (in the vectorized state $s$) payoff
function. Let

$\overline{A}\in\mathbb{R}^{nc\times nc}$ (29)

be an arbitrary payoff matrix for the vectorized state. Then by Lemma 3.4 and
Theorem 3.5, the embedded dynamics in
${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ read

$\dot{p}(t)=R_{p(t)}[Q\overline{A}Q^{\top}p(t)],\qquad p(0)=T(s_{0}).$ (30)

The _multi-game dynamics_ of [22] can also be written as a matrix game in
${\mathcal{S}_{N}}$. Given matrices $A^{(i)}\in\mathbb{R}^{c\times c}$,
$i\in[n]$, it reads

$\dot{p}(t)=R_{p(t)}[Ap(t)],\qquad p(0)=p_{0},\qquad
A_{\alpha,\beta}=\sum_{i\in[n]}A^{(i)}_{\alpha_{i},\beta_{i}}.$ (31)

The structure of this payoff matrix has a natural shape within our formalism,
too.

######  Lemma 4.1.

The payoff matrix in (31) can be written as $A=Q\overline{A}Q^{\top}$ where
$\overline{A}$ denotes the block diagonal matrix with diagonal blocks
$A^{(i)}$.

###### Proof.

$\displaystyle(Q\overline{A}Q^{\top})_{\alpha,\beta}$ $\displaystyle=\langle
e_{\alpha},Q\overline{A}Q^{\top}e_{\beta}\rangle=\langle
Q^{\top}e_{\alpha},\overline{A}Q^{\top}e_{\beta}\rangle$ (32a)
$\displaystyle=\sum_{i\in[n]}\langle
e_{\alpha_{i}},A^{(i)}e_{\beta_{i}}\rangle=\sum_{i\in[n]}A^{(i)}_{\alpha_{i},\beta_{i}}$
(32b)

∎

In particular, if all single-game payoff submatrices are the same
$A^{(i)}=B\in\mathbb{R}^{c\times c}$, then multi-game dynamics have payoff
$\overline{A}=\mathbb{I}_{n}\otimes B$.

It was shown by [22] that the multi-game dynamics (31) do not generally
decompose into individual single-game dynamics, unless the initialization is
on the _Wright manifold_ (see Figure 2). The set
${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ defined by (18a) is a generalization
of the Wright manifold for $n>2$ and Theorem 3.5 generalizes the decomposition
of multi-game dynamics to more than two populations. For
$p(0)\in{\mathcal{T}}$, the dynamics (31) is the embedded dynamics of

$\dot{s}(t)={\mathcal{R}}_{s(t)}[\overline{A}s(t)],\qquad s(0)=Mp(0)$ (33)

by Lemma 4.1 and Theorem 3.5. Since $\overline{A}$ is block diagonal, (33) is
a collection of non-interacting single-game replicator dynamics

$\dot{W}_{i}(t)=R_{W_{i}(t)}[A^{(i)}W_{i}(t)],\qquad
W_{i}(0)=(Mp(0))_{i},\qquad i\in[n]$ (34)

in accordance with the findings of [22] for the specific case $n=2$.

##  5 Tangent Space Parameterization

Multi-population replicator dynamics evolve in the _curved_ space
${\mathcal{W}}$ and the usual parameterization in $m$-coordinates of
information geometry is subject to simplex constraints on the state. With an
eye toward numerical integration, it is desirable to instead parameterize
replicator dynamics in a _flat_ and _unconstrained_ vector space. This was
done in [54] using Lie group methods.

######  Theorem 5.1 (Proposition 3.1 in [54]).

The solution for multi-population replicator dynamics

$\dot{W}(t)={\mathcal{R}}_{W(t)}[F(W(t))],\quad W(0)=W_{0}$ (35)

in ${\mathcal{W}}$ admits the parameterization

$\displaystyle W(t)$ $\displaystyle=\exp_{{\mathbf{1}_{\mathcal{W}}}}(V(t))$
(36a) $\displaystyle\dot{V}(t)$
$\displaystyle=\Pi_{0}F(\exp_{{\mathbf{1}_{\mathcal{W}}}}(V(t))),\qquad
V(0)=\Pi_{0}\log W_{0}$ (36b)

in the tangent space $V(t)\in T_{0}{\mathcal{W}}$.

With regard to the Embedding Theorem 3.5, it turns out that while $T$ maps
assignment matrices $W\in{\mathcal{W}}$ to joint states
$p\in{\mathcal{S}_{N}}$, $Q$ assumes a corresponding role for tangent vectors
in $T_{0}{\mathcal{W}}$.

######  Theorem 5.2 (Tangent Space Embedding Theorem).

The multi-population tangent space replicator dynamics

$\dot{V}=\Pi_{0}F\big{(}\exp_{{\mathbf{1}_{\mathcal{W}}}}(V)\big{)},\quad
V(0)=V_{0}$ (37)

on $T_{0}{\mathcal{W}}$ is pushed forward by $Q$ to the tangent space
replicator dynamics

$\dot{U}=\Pi_{0}\widehat{F}\big{(}\exp_{\mathbf{1}_{N}}(U)\big{)},\quad
U(0)=Q(V_{0}),\qquad\widehat{F}=Q\circ F\circ M$ (38)

on $T_{0}{\mathcal{S}_{N}}$.

###### Proof.

Denoting $U=QV$ and using the lifting map (Lemma 3.3), we directly compute

$\displaystyle\dot{U}$
$\displaystyle=Q\dot{V}=Q\Pi_{0}F\big{(}\exp_{{\mathbf{1}_{\mathcal{W}}}}(V)\big{)}$
(39a)
$\displaystyle=\Pi_{0}QF\big{(}\exp_{{\mathbf{1}_{\mathcal{W}}}}(V)\big{)}$ by
Lemma A.3 (39b) $\displaystyle=\Pi_{0}QF\big{(}(M\circ
T)\big{(}\exp_{{\mathbf{1}_{\mathcal{W}}}}(V)\big{)}\big{)}$ by (19) (39c)
$\displaystyle=\Pi_{0}QF\big{(}M\exp_{\mathbf{1}_{N}}(QV)\big{)}$ by Lemma 3.3
(39d) $\displaystyle=\Pi_{0}QF\big{(}M\exp_{\mathbf{1}_{N}}(U)\big{)}$ (39e)
$\displaystyle=\Pi_{0}\widehat{F}\big{(}\exp_{\mathbf{1}_{N}}(U)\big{)}.$
(39f)

∎

Pushforward via $Q$ thus preserves the shape of (37) up to the same fitness
function transformation $\widehat{F}=Q\circ F\circ T^{-1}$ from Theorem 3.5.

The set $\operatorname{img}Q\subseteq T_{0}{\mathcal{S}_{N}}$ contains exactly
those tangent vectors corresponding to assignments
${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ via lifting, because
$T(W)=T(\exp_{{\mathbf{1}_{\mathcal{W}}}})(V)=\exp_{\mathbf{1}_{N}}(QV)$ for
any $W\in{\mathcal{W}}$ and $V=\exp_{{\mathbf{1}_{\mathcal{W}}}}^{-1}(W)$ by
Lemma 3.3. In particular, the set $\operatorname{img}Q$ in which $U$ evolves,
is a linear subspace of $T_{0}{\mathcal{W}}$. This is a reason to study the
tangent space flow (38) rather than the corresponding replicator dynamics if
applicable, because ${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ is the (curved)
set of rank-1 tensors in ${\mathcal{S}_{N}}$.

##  6 Learning Replicator Dynamics from Data

Figure 3: _Left_ : Noisy input assignment of $c=47$ colors to the pixels of an
image. _Center_ : Limit of an EGN flow (28) with learned interaction in
$3\times 3$ pixel neighborhoods. _Right_ : Ground truth noise-free color
assignment.

Several applications have been proposed for the replicator dynamics of Section
4 including as a model of human brain functioning [32], collective learning
[41], epileptic seizure onset detection [19], task mapping [33] and collective
adaptation [42]. Assignment flows have been applied recently to the
segmentation of digitized volume data under layer ordering constraints [48]
(which reflect prior knowledge about tissues and anatomical structure) as well
as for unsupervised image labeling tasks, employing spatial regularization
[58, 56].

This small sample of examples illustrates that replicator dynamics can act as
powerful data models in diverse applications. In situations where only partial
knowledge about the system is available, system parameters may also be learned
from data. To this end, [25] have studied the use of _adjoint integration_ to
compute the model sensitivity of assignment flows, i.e. the gradient of system
state with respect to parameters generating the flow.

Suppose we integrate a general dynamical system generated by parameters
$\mathfrak{p}$ and wish for the final state $v(T)$ to minimize some loss
function $\mathcal{L}$. The parameter learning problem for a fixed time
horizon $T>0$ then reads

$\displaystyle\min_{\mathfrak{p}}\,\mathcal{L}($ $\displaystyle
v(T,\mathfrak{p}))$ (40a) $\displaystyle\text{subject to}\quad\dot{v}(t)$
$\displaystyle=f(v(t),\mathfrak{p},t),\quad t\in[0,T],$ (40b) $\displaystyle
v(0)$ $\displaystyle=v_{0},$ (40c)

and a central quantity of interest is the gradient
$\partial_{\mathfrak{p}}\mathcal{L}(v(T,\mathfrak{p}))$. It can be
approximated in a _discretize-then-optimize_ fashion by first choosing a
discretization of the ODE (40b) on $[0,T]$ and subsequently computing the
gradient of the discrete scheme used for computing
$\mathcal{L}(v(T,\mathfrak{p}))$. This approach is easy to implement by using
automatic differentiation software [5]. However, it entails a large memory
footprint in practical applications because system state $v(t_{i})$ needs to
be saved for all discretization points. To circumvent this issue, one may
instead proceed in an _optimize-then-discretize_ fashion as follows.

######  Theorem 6.1 (Theorem 6 of [25]).

The gradient of (40) is given by

$\partial_{\mathfrak{p}}\mathcal{L}(v(T,\mathfrak{p}))=\int_{0}^{T}\textup{d}_{\mathfrak{p}}f(v(t),\mathfrak{p},t)^{\top}\lambda(t)\textup{d}t$
(41)

where $\textup{d}_{\mathfrak{p}}f$ denotes the differential of $f$ with
respect to $\mathfrak{p}$ and $x(t)$ and $\lambda(t)$ solve

$\displaystyle\dot{v}(t)$ $\displaystyle=f(v(t),\mathfrak{p},t),\qquad
v(0)=v_{0},$ (42a) $\displaystyle\dot{\lambda}(t)$
$\displaystyle=-\textup{d}_{v}f(v(t),\mathfrak{p},t)^{\top}\lambda(t),\qquad\lambda(T)=\partial\mathcal{L}(v(T)).$
(42b)

By choosing a quadrature for the integral (41), Theorem 6.1 allows to compute
the desired gradient without the need to save system state at all
discretization points. Moreover, it has been shown [25, 40] that for
particular _symplectic_ integrators, discretization commutes with
optimization, i.e. both orders of operation yield the same gradient.

Since the tangent space parameterization (36) evolves on the unconstrained
flat space ${T_{0}{\mathcal{W}}}$, Theorem 6.1 is directly applicable to it.
An image labeling example is shown in Figure 3. Here, the data is modeled by
EGN dynamics (28) with graph adjacency matrix $\Omega$ representing pixel
neighborhoods ($3\times 3$). Starting from a noisy, high-entropy assignment of
pixels to color prototypes ($c=47$), the goal is to learn a label interaction
matrix $B\in\mathbb{R}^{c\times c}$ such that EGN dynamics (28) drive the
state to a given noise-free assignment after the fixed integration time
$T=15$. We initialized $B$ as identity matrix and performed $100$ steps of the
Adam optimizer to minimize cross-entropy between the ground truth assignment
and the assignment state reached by EGN dynamics. This training procedure is
highly scalable – for the $256\times 256$ pixel image in Figure 3, training
takes less than a minute on a laptop computer and requires around 1.3GB of
vRAM.

##  7 Asymptotic Behavior

A central topic in population dynamics is the study of how the properties of
the underlying game characterized by the payoff function relate to steady
states of the dynamical model. In this section, we describe how

  * •

_Nash equilibria (NE)_ and

  * •

_Evolutionarily stable states (ESS)_

of multi-population games and their replicator dynamics behave under the
embedding (18a). Nash equilibria for multi-population games are population
states at which no agent (in any population) has payoff to gain from
unilaterally switching strategies.

######  Definition 7.1 (Nash Equilibrium).

Let $\overline{{\mathcal{W}}}$, the closure of ${\mathcal{W}}$ be the set of
multi-population states ($n$ populations, $c$ strategies) and let
$F\colon\overline{{\mathcal{W}}}\to\mathbb{R}^{n\times c}$ be the payoff for a
multi-population game. The set of Nash equilibria of $F$ is defined as

$\operatorname{NE}(F)=\\{W\in\overline{{\mathcal{W}}}\;|\;\forall
i\in[n],\,\forall j\in\operatorname{supp}(W_{i}),\,\forall k\in[c]\,\colon
F(W)_{i,j}\geq F(W)_{i,k}\\}$ (43)

Definition 7.1 naturally extends the classic notion of Nash equilibrium to
multi-population games. Nash equilibria are preserved if the multi-population
game is embedded as specified by Theorem 3.5.

######  Theorem 7.2 (Embedded Nash Equilibria).

Let $F\colon\overline{{\mathcal{W}}}\to\mathbb{R}^{n\times c}$ be a multi-
population game on $\overline{{\mathcal{W}}}$ and $\widehat{F}=Q\circ F\circ
M$ be the related population game on ${\mathcal{S}_{N}}$. Then

$T\big{(}\operatorname{NE}(F)\big{)}=\operatorname{NE}(\widehat{F})\cap\overline{{\mathcal{T}}}.$
(44)

###### Proof.

Let $W\in\operatorname{NE}(F)$ and let $\alpha\in\operatorname{supp}(T(W))$ be
arbitrary. Then

$\displaystyle\widehat{F}(T(W))_{\alpha}=(QF(W))_{\alpha}$
$\displaystyle=\sum_{l\in[n]}F_{l,\alpha_{l}}(W)$ (45a)
$\displaystyle\geq\sum_{l\in[n]}F_{l,\beta_{l}}(W)=\widehat{F}(T(W))_{\beta},\quad\forall\,\beta\in[c]^{n},$
(45b)

because $\alpha_{l}\in\operatorname{supp}(W_{l}),\;\forall l\in[n]$, by Lemma
A.2 and $W$ is a Nash equilibrium of $F$. This implies
$T(\operatorname{NE}(F))\subseteq\operatorname{NE}(\widehat{F})\cap\overline{{\mathcal{T}}}$.
Conversely, let ${p}\in\operatorname{NE}(\widehat{F})\cap{\mathcal{T}}$ have
shape ${p}=T(W)$ and let $\alpha_{l}\in\operatorname{supp}(W_{l}),\;\forall
l\in[n]$. Then $\alpha\in\operatorname{supp}({p})$ by Lemma A.2 and

$\sum_{l\in[n]}F_{l,\alpha_{l}}(W)=\widehat{F}({p})_{\alpha}\geq\widehat{F}({p})_{\beta}=\sum_{l\in[n]}F_{l,\beta_{l}}(W),\quad\forall\,\beta\in[c]^{n},$
(46)

because ${p}$ is a Nash equilibrium. Choose $\beta\in[c]^{n}$ such that it
matches $\alpha$ at all positions but $i\in[n]$. Then (46) implies
$F_{i,\alpha_{i}}(W)\geq F_{i,\beta_{i}}(W)$ for arbitrary $\beta_{i}\in[c]$
which shows
$\operatorname{NE}(\widehat{F})\cap\overline{{\mathcal{T}}}\subseteq
T(\operatorname{NE}(F))$. ∎

######  Definition 7.3 (Evolutionarily Stable State (ESS)).

A multi-population state $W^{\ast}\in{\mathcal{W}}$ is called an
evolutionarily stable state (ESS) of a game
$F\colon{\mathcal{W}}\to\mathbb{R}^{n\times c}$, if there is an environment
$U\subseteq{\mathcal{W}}$ of $W^{\ast}$ such that

$\langle W-W^{\ast},F(W)\rangle<0,\qquad\forall\,W\in
U\setminus\\{W^{\ast}\\}.$ (47)

This generalization of the classic ESS [50] to multi-population settings is
called _Taylor ESS_ by [39]. Within our embedding framework, an apparent
reason recommends Definition 7.3 over the weaker notion of _monomorphic ESS_
[13].

######  Theorem 7.4 (Embedded ESS).

Let $F\colon{\mathcal{W}}\to\mathbb{R}^{n\times c}$ be a multi-population
game. Then $W^{\ast}$ is an ESS of $F$ exactly if there exists an environment
$U\subseteq{\mathcal{T}}$ of $T(W^{\ast})$ such that

$\langle p-T(W^{\ast}),\widehat{F}(p)\rangle<0,\qquad\forall\,p\in
U\setminus\\{T(W^{\ast})\\},$ (48)

where $\widehat{F}=Q\circ F\circ M$ denotes the embedded single-population
game on ${\mathcal{S}_{N}}$ as specified by Theorem 3.5.

###### Proof.

Since $U\subseteq{\mathcal{T}}$, we may write $p=T(W)$ for $W$ in an
environment $M(U)\subseteq{\mathcal{W}}$ of $W^{\ast}$. (48) then reads

$\displaystyle\langle T(W)-T(W^{\ast}),\widehat{F}(T(W))\rangle$
$\displaystyle=\langle T(W)-T(W^{\ast}),QF(MT(W))\rangle$ (49a)
$\displaystyle=\langle M(T(W)-T(W^{\ast})),F(W)\rangle\text{
(Lemma~{}\ref{lem:Q_adjoint})}$ (49b) $\displaystyle=\langle
W-W^{\ast},F(W)\rangle$ (49c)

and the last row is strictly smaller than $0$ for all $W\in
M(U)\setminus\\{W^{\ast}\\}$ exactly if $W^{\ast}$ is an ESS of $F$ according
to Definition 7.3. ∎

One useful aspect of Theorem 3.5 is that it formally reduces multi-population
replicator dynamics to single-population ones. This enables us to transfer
analysis of e.g. asymptotic behavior from the single-population to the multi-
population setting. We first summarize standard results on the asymptotic
behavior of replicator dynamics derived from a potential function and refer to
[39] for a comprehensive overview.

######  Theorem 7.5 (Replicators converge to NE).

Let $\widehat{J}\colon\mathcal{S}_{c}\to\mathbb{R}$ be a $C^{1}$ potential
such that the induced payoff function $\widehat{F}=\Pi_{0}\nabla\widehat{J}$
is Lipschitz on $\mathcal{S}_{c}$. Then for any internal point
$p_{0}\in\mathcal{S}_{c}$, the replicator dynamics

$\dot{W}(t)=R_{p(t)}[\widehat{F}(p)],\qquad p(0)=p_{0}$ (50)

converge to a Nash equilibrium.

###### Proof.

Because $F$ is Lipschitz, the forward trajectories of the dynamics (50) are
unique by the Picard-Lindelöf theorem. The potential $\widehat{J}$ is a strict
Lyapunov function for replicator dynamics and unique forward trajectories
converge to restricted equilibria [23, 38]. Since replicator dynamics do not
satisfy Nash stationarity, there may be restricted equilibria which are not
Nash equilibria. However, no internal trajectory converges to any of these
points [10]. The solution trajectories of (50) are internal trajectories
because $p_{0}$ is an internal point and $\mathcal{S}_{c}$ is invariant under
all replicator dynamics with Lipschitz payoff function for finite time, as is
clear from e.g. the tangent space parameterization (36). ∎

There is a simple relationship between potential functions in the multi-
population and single-population settings.

######  Lemma 7.6 (Potential Embedding).

If $F\colon{\mathcal{W}}\to T_{0}{\mathcal{W}}$ has potential $J$, then
$\widehat{F}=Q\circ F\circ M$ has potential $\widehat{J}=J\circ M$.

###### Proof.

For $\widehat{J}(p)=(J\circ M)(p)$, we directly compute

$\nabla\widehat{J}(p)=(DM(p))^{\top}\nabla J(W)=(M)^{\top}\circ\nabla
J(W)=(Q\circ\nabla J\circ M)(p)$ (51)

by denoting $W=M(p)$ and using Lemma 3.4. ∎

We can now use the embedded potential of Lemma 7.6 and embedded Nash
equilibria of Theorem 7.2 to generalize the findings of Theorem 7.5 to
multiple populations.

######  Theorem 7.7 (Multi-Population Replicators converge to NE).

Let $J\colon{\mathcal{W}}\to\mathbb{R}$ be a $C^{1}$ potential such that the
induced payoff function $F=\Pi_{0}\nabla J$ is Lipschitz on ${\mathcal{W}}$.
Then, for any internal point $W_{0}\in{\mathcal{W}}$, the multi-population
replicator dynamics

$\dot{W}(t)={\mathcal{R}}_{W(t)}[F(W)],\qquad W(0)=W_{0}$ (52)

converge to a Nash equilibrium.

###### Proof.

Let $p(t)=T(W(t))$. Then $p(t)$ follows the single-population replicator
dynamics (24) by Theorem 3.5 which are induced by the embedded potential
$\widehat{J}(p)=J\circ M$ due to Lemma 7.6 and start at the interior point
$T(W_{0})$ of ${\mathcal{S}_{N}}$. By Theorem 7.5, $p(t)$ converges to a NE of
$\widehat{F}=\Pi_{0}\circ Q\circ F\circ M$ on ${\mathcal{S}_{N}}$. Since
$p(t)=T(W(t))\in{\mathcal{T}}$ for all times $t$, the limit point necessarily
lies in the closure of ${\mathcal{T}}$. Theorem 7.2 then shows the assertion.
∎

By [38, Proposition 3.1] all Nash equilibria satisfy the KKT optimality
conditions for maximizing $J$ subject to simplex constraints. If $J$ is
concave, the KKT conditions are sufficient optimality conditions and thus (50)
converges to a local maximizer. In addition, $W$ is a Nash equilibrium exactly
if $F(W)$ lies in the normal cone of the state space at $W$ [21, 34]. Thus,
convergence of (50) to a boundary point which is not an extremal point only
occurs if the trajectory reaches the boundary exactly perpendicularly. For
assignment flows, it has been known that convergence to a non-extremal point
of ${\mathcal{W}}$ is an unusual occurrence. In fact, this behavior is not
observed at all in the numerical solution of labeling problems for real-world
data. [3] thus conjectured that convergence to a non-extremal point only
occurs for a null set of initial population states. This was shown to be true
for non-negative, linear fitness functions derived from a quadratic potential
[57]. From a game-theoretical perspective, only extremal points can be ESS
under the posed conditions.

Note that the content of Theorem 7.7 is likely known to experts. We present it
here to illustrate the power of the proposed formalism around Theorem 3.5
which provides a mathematical toolset for reducing the analysis of multi-
population replicator dynamics to single-population ones.

##  8 Conclusion

The proposed embedding framework for multi-population replicator dynamics
provides a robust mathematical toolset for modeling complex population
interactions. It formally reduces the complex multi-population case to a
single-population one, simplifying subsequent analysis. Current developments
in the framework of assignment flows suggest multiple extensions of the
present work.

In [44], assignment flows are characterized as critical points of an action
functional within a geometric formalism of mechanics. An analogous
characterization was previously suggested for single-population replicator
dynamics [36] under assumptions which are valid only in the special case $n=2$
[44, Section 4.4]. In light of the present paper, a natural question is
whether both perspectives are equivalent under embedding of the multi-
population case.

A generalized perspective on assignment flows was proposed by [47]. The
authors study a dynamical system on a product of density matrix manifolds
called _Quantum State Assignment Flow_. Although density matrices can
represent entangled states and constitute a strict generalization of discrete
probability measures, the underlying information geometric framework is
broadly analogous. This suggests that generalized, quantum embedding results,
along the lines introduced in the present paper, should be achievable.

We briefly elaborate this point. Quantum mechanics is formulated on complex
projective space $\mathbb{P}(\mathbb{C}^{c})=\mathbb{P}^{c-1}$, i.e. the state
of an $c$-dimensional quantum system lives in the $(c-1)$-dimensional
projective space. The states of a composite quantum system with $n$ components
comprise the space
$\mathbb{P}\big{(}\bigotimes_{i=1}^{n}\mathbb{C}^{c}\big{)}\cong\mathbb{P}^{N-1}$,
the projective space of the tensor products and $N=c^{n}$. The Segre embedding
$\sigma$ [51] is an analytic isometric embedding of products of projective
spaces into higher dimensional projective space [12], i.e.

$\sigma:\mathbb{P}^{c-1}\times\dots\times\mathbb{P}^{c-1}\hookrightarrow\mathbb{P}^{N-1},$
(53)

where the product contains $n$ copies of $\mathbb{P}^{c-1}$. The map $\sigma$
is an isometry if the product of projective spaces is equipped with the
product of Fubini-Study metrics and the projective space of the tensor product
with the high dimensional Fubini-Study metric. The _separable (unentangled)
quantum states_ of the composite system are precisely the image of the Segre
embedding. Furthermore, $\mathbb{P}^{c-1}$ admits a description as a toric
variety with base $\triangle_{c}$, the simplex with boundary [7]. Exploiting
this structure makes it possible to choose compatible smooth embeddings

$\iota:\mathcal{S}_{c}\times\dots\times\mathcal{S}_{c}\hookrightarrow\mathbb{P}^{c-1}\times\dots\times\mathbb{P}^{c-1},$
(54)

and to define a projection map $\pi_{c}:\mathbb{P}^{c-1}\to\triangle_{c}$. The
embedding $T$ given by (18a) is compatible with the Segre embedding $\sigma$
in the sense that the diagram

${\mathbb{P}^{c-1}\times\dots\times\mathbb{P}^{c-1}}$${\mathbb{P}^{N-1}}$${\mathcal{S}_{c}\times\dots\times\mathcal{S}_{c}}$${\mathcal{S}_{N}}$$\scriptstyle{\sigma}$$\scriptstyle{\pi_{N}}$$\scriptstyle{\iota}$$\scriptstyle{T}$
(55)

is well-defined and commutes. Well-defined refers here to the fact that
$\pi_{N}\circ\sigma\circ\iota$ maps to $\mathcal{S}_{N}\subset\triangle_{N}$.
Additionally, similar compatibility relations remain valid when quantum states
are described in terms of density matrices.

Elaborating the consequences of the results in this paper in connection with
the more general quantum state assignment flow approach is an attractive
research problem for future work.

Acknowledgments

The original artwork used in Figure 3 was designed by dgim-studio / Freepik.

The authors thank Fabrizio Savarino for many fruitful discussions and
suggestions.

## Declarations

#### Funding

This work is funded by the Deutsche Forschungsgemeinschaft (DFG), grant SCHN
457/17-1, within the priority programme SPP 2298: “Theoretical Foundations of
Deep Learning”. This work is funded by the Deutsche Forschungsgemeinschaft
(DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the
Heidelberg STRUCTURES Excellence Cluster).

#### Competing interests

The authors have no relevant financial or non-financial interests to disclose.

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##  Appendix A Additional Lemmata

######  Lemma A.1.

The mapping $T\colon{\mathcal{W}}\to{\mathcal{T}}$ defined by (18a) is
injective.

###### Proof.

Let $W^{(1)},W^{(2)}\in{\mathcal{W}}$ satisfy $T(W^{(1)})=T(W^{(2)})$. Let
$\gamma\in[c]^{n}$ be an arbitrary multi-index. Fix an arbitrary vertex
$i\in[n]$ and let $\alpha\in[c]^{n}$ match $\gamma$ at all vertices
$k\in[n]\setminus\\{i\\}$. Then $T(W^{(1)})=T(W^{(2)})$ implies both
$T(W^{(1)})_{\gamma}=T(W^{(2)})_{\gamma}$ and
$T(W^{(1)})_{\alpha}=T(W^{(2)})_{\alpha}$. Division thus gives

$W^{(1)}_{i,\alpha_{i}}W^{(2)}_{i,\gamma_{i}}=W^{(1)}_{i,\gamma_{i}}W^{(2)}_{i,\alpha_{i}}.$
(56)

Since $W^{(1)},W^{(2)}\in{\mathcal{W}}$, the entries of row $i$ sum to 1.
Using this and the fact that $\alpha_{i}\in[c]$ is arbitrary, we find

$W^{(2)}_{i,\gamma_{i}}=\sum_{j\in[c]}W^{(1)}_{i,j}W^{(2)}_{i,\gamma_{i}}\stackrel{{\scriptstyle\eqref{eq:t_inj_proof1}}}{{=}}\sum_{j\in[c]}W^{(1)}_{i,\gamma_{i}}W^{(2)}_{i,j}=W^{(1)}_{i,\gamma_{i}}.$
(57)

Since $\gamma_{i}\in[c]$ was arbitrary, this shows $W^{(1)}=W^{(2)}$. ∎

######  Lemma A.2.

For every $W\in\overline{{\mathcal{W}}}$ one has
$\gamma\in\operatorname{supp}(T(W))$ if and only if
$\gamma_{i}\in\operatorname{supp}(W_{i})$ for all $i\in[n]$.

###### Proof.

We directly compute

$\displaystyle\operatorname{supp}(T(W))$
$\displaystyle=\\{\gamma\in[c]^{n}\colon T(W)_{\gamma}>0\\}$ (58)
$\displaystyle=\\{\gamma\in[c]^{n}\colon\prod_{i\in[n]}W_{i,\gamma_{i}}>0\\}$
(59) $\displaystyle=\\{\gamma\in[c]^{n}\colon W_{i,\gamma_{i}}>0,\;\forall
i\in[n]\\}$ (60)
$\displaystyle=\\{\gamma\in[c]^{n}\colon\gamma_{i}\in\operatorname{supp}(W_{i}),\;\forall
i\in[n]\\}.$ (61)

∎

######  Lemma A.3.

For any $V\in\mathbb{R}^{n\times c}$ it holds $Q\Pi_{0}V=\Pi_{0}QV$ for the
mappings $Q,\Pi_{0}$ given by (18b) and (14).

###### Proof.

For arbitrary $\gamma\in[c]^{n}$, we compute

$(Q\Pi_{0}V)_{\gamma}=\sum_{i\in[n]}(\Pi_{0}V_{i})_{\gamma_{i}}=\sum_{i\in[n]}\big{(}V_{i,\gamma_{i}}-\langle
V_{i},\frac{1}{c}\mathbf{1}_{c}\rangle\big{)}=(QV)_{\gamma}-\langle
V,\underbrace{\frac{1}{c}\mathbf{1}_{n\times
c}}_{=M{\mathbf{1}_{\mathcal{S}}}}\rangle$ (62)

and thus, by Lemma 3.4 ,

$Q\Pi_{0}V=QV-\langle
QV,\frac{1}{N}\mathbf{1}_{N}\rangle\mathbf{1}_{N}=\Pi_{0}QV,$ (63)

which was the assertion. ∎

######  Lemma A.4.

Let $\gamma\in[c]^{n}$. The differential of $T$ at $W\in{\mathcal{W}}$ in
direction $V\in T_{0}{\mathcal{W}}$ is given by

$dT|_{W}[V]=\big{(}dT_{\gamma}|_{W}[V]\big{)}_{\gamma\in[c]^{n}}=T(W)\diamond
Q\Big{[}\frac{V}{W}\Big{]}.$ (64)

###### Proof.

Suppose $\eta\colon(-\varepsilon,\varepsilon)\to{\mathcal{W}}$ is a smooth
curve with $\eta(0)=W$ and $\dot{\eta}(0)=V$, for some $\varepsilon>0$. Let
$\gamma\in[c]^{n}$ be arbitrary and consider the component $T_{\gamma}$. Then

$\displaystyle dT_{\gamma}|_{W}[V]$
$\displaystyle=\tfrac{d}{dt}T_{\gamma}(\eta(t))\big{|}_{t=0}=\tfrac{d}{dt}\prod_{i\in[n]}\eta_{i,\gamma_{i}}(t)\big{|}_{t=0}$
(65a)
$\displaystyle=\sum_{k\in[n]}\dot{\eta}_{k,\gamma_{k}}(0)\prod_{i\in[n]\setminus\\{k\\}}\eta_{i,\gamma_{i}}(0)=\sum_{k\in[n]}V_{k,\gamma_{k}}\prod_{i\in[n]\setminus\\{k\\}}W_{i,\gamma_{i}}$
(65b)
$\displaystyle=\sum_{k\in[n]}\frac{V_{k,\gamma_{k}}}{W_{k,\gamma_{k}}}T_{\gamma}(W)\overset{\eqref{eq:def_Q}}{=}T_{\gamma}(W)Q_{\gamma}\Big{(}\frac{V}{W}\Big{)}.$
(65c)

Because of $dT|_{W}[V]=(dT_{\gamma}|_{W}[V])_{\gamma\in[c]^{n}}$ the
expression in (64) directly follows. ∎

######  Lemma A.5.

It holds $\operatorname{ker}Q=\\{\operatorname{Diag}(d)\mathbf{1}_{n\times
c}\colon d\in\mathbb{R}^{n},\;\langle d,\mathbf{1}_{n}\rangle=0\\}$ as well as
$\operatorname{rank}Q=nc-(n-1)$.

###### Proof.

Let $V\in\operatorname{ker}Q$ and let $\gamma,\tilde{\gamma}$ be two multi-
indices which differ exactly at position $k$ but are otherwise arbitrary. We
have $(QV)_{\gamma}=(QV)_{\tilde{\gamma}}=0$ because
$V\in\operatorname{ker}Q$. Thus

$(QV)_{\tilde{\gamma}}=V_{k,\tilde{\gamma}_{k}}+\sum_{i\in[n]\setminus\\{k\\}}V_{i,\tilde{\gamma}_{i}}=(QV)_{\gamma}=V_{k,\gamma_{k}}+\sum_{i\in[n]\setminus\\{k\\}}V_{i,\gamma_{i}}$
(66)

which implies $V_{k,\tilde{\gamma}_{k}}=V_{k,\gamma_{k}}$, i.e.
$V=\operatorname{Diag}(d)\mathbf{1}_{n\times c}$ for some $d\in\mathbb{R}^{n}$
since $k$ was arbitrary. Further, it holds

$0=(QV)_{\gamma}=\sum_{i\in[n]}V_{i,\gamma_{i}}=\sum_{i\in[n]}d_{i}=\langle
d,\mathbf{1}_{n}\rangle.$ (67)

Thus, we have shown

$\operatorname{ker}Q\subseteq\\{\operatorname{Diag}(d)\mathbf{1}_{n\times
c}\colon d\in\mathbb{R}^{n},\;\langle d,\mathbf{1}_{n}\rangle=0\\}.$ (68)

Conversely, let $V$ be in the right-hand set. Then

$(QV)_{\gamma}=\sum_{i\in[n]}V_{i,\gamma_{i}}=\sum_{i\in[n]}d_{i}=\langle
d,\mathbf{1}_{n}\rangle=0$ (69)

for all $\gamma\in[c]^{n}$ which shows that (68) is an equation. There are
$(n-1)$ linearly independent vectors $d\in\mathbb{R}^{n}$ with $\langle
d,\mathbf{1}_{n}\rangle=0$, therefore $Q$ has the specified rank. ∎

##  Appendix B Proofs

###  B.1 Proof of Theorem 3.1

######  Theorem B.1 (Theorem 3.1 in the main text).

The map $T\colon{\mathcal{W}}\to{\mathcal{T}}$ is an isometric embedding of
${\mathcal{W}}$ equipped with product Fisher-Rao geometry into
${\mathcal{S}_{N}}$ equipped with the Fisher-Rao geometry. On its image
$T({\mathcal{W}})=:{\mathcal{T}}\subseteq{\mathcal{S}_{N}}$, the inverse is
given by marginalization

$M|_{\mathcal{T}}=T^{-1}\colon{\mathcal{T}}\to{\mathcal{W}}.$ (70)

###### Proof.

A standard argument (Lemma A.1) shows that
$T\colon{\mathcal{W}}\to{\mathcal{T}}$ is injective. We check that the inverse
of $T$ has the shape (19).

$\displaystyle(MT(W))_{i,j}$
$\displaystyle=\sum_{\gamma\;:\;\gamma_{i}=j}\prod_{r\in[n]}W_{r,\gamma_{r}}=\sum_{\gamma\;:\;\gamma_{i}=j}W_{i,j}\prod_{r\in[n]\setminus\\{i\\}}W_{r,\gamma_{r}}$
(71)
$\displaystyle=\sum_{l\in[n]\setminus\\{i\\}}\sum_{\gamma_{l}\in[c]}\prod_{r\in[n]\setminus\\{i\\}}W_{r,\gamma_{r}}$
(72)
$\displaystyle=W_{i,j}\sum_{k_{1}\in[c]}W_{1,k_{1}}\sum_{k_{2}\in[c]}W_{2,k_{2}}\dotsc\sum_{k_{n}\in[c]}W_{n,k_{n}}$
(73)
$\displaystyle=W_{i,j}\prod_{r\in[n]\setminus\\{i\\}}\underbrace{\sum_{\gamma_{r}\in[c]}W_{r,\gamma_{r}}}_{=1}=W_{i,j}.$
(74)

Clearly, all component functions of $T$ and $T^{-1}$ are smooth. We will now
show that $T$ is a topological embedding, i.e. a homeomorphism with respect to
the subspace topology of ${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$. Let

$\mathcal{Q}=Q(T_{0}{\mathcal{W}})$ (75)

denote the image of $T_{0}{\mathcal{W}}$ under $Q$. $\mathcal{Q}$ is a linear
subspace of $T_{0}{\mathcal{S}_{N}}$ because, for any $V\in
T_{0}{\mathcal{W}}$, we have

$QV=Q\Pi_{0}V=\Pi_{0}QV\in T_{0}{\mathcal{S}_{N}}$ (76)

by Lemma A.3. In addition, Lemma A.5 shows $\operatorname{ker}Q\cap
T_{0}{\mathcal{W}}=\\{0\\}$, since any matrix in $\ker Q$ has constant row
vectors. Thus, the restriction of $Q$ to $T_{0}{\mathcal{W}}$ is injective and
since $T_{0}{\mathcal{W}}$ and $\mathcal{Q}$ have finite dimension,
$Q|_{T_{0}{\mathcal{W}}}$ is a homeomorphism. The lifting map at the
barycenter is the inverse of the _global_ $e$-coordinate chart of information
geometry up to a change of basis. In particular,
$\exp_{{\mathbf{1}_{\mathcal{W}}}}\colon T_{0}{\mathcal{W}}\to{\mathcal{W}}$
and $\exp_{\mathbf{1}_{\mathcal{S}_{N}}}\colon
T_{0}{\mathcal{S}_{N}}\to{\mathcal{S}_{N}}$ are homeomorphisms. Now let

$\psi\colon{\mathcal{T}}\to\mathcal{Q},\qquad
p\mapsto\psi(p)=\exp_{\mathbf{1}_{\mathcal{S}_{N}}}^{-1}(p)$ (77)

which is well-defined due to Lemma 3.3 and denote the initial topology of
${\mathcal{T}}$ with respect to $\psi^{-1}$ by $\mathcal{A}$. Then $T$ is a
homeomorphism of ${\mathcal{W}}$ and ${\mathcal{T}}$ equipped with the
topology $\mathcal{A}$ because

$T=\exp_{\mathbf{1}_{\mathcal{S}_{N}}}\circ
Q|_{T_{0}{\mathcal{W}}}\circ\psi^{-1}$ (78)

by Lemma 3.3. It remains to show that $\mathcal{A}$ coincides with the
subspace topology of ${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$. Note that the
topology of $\mathcal{Q}$ is the subspace topology of $\mathcal{Q}\subseteq
T_{0}{\mathcal{S}_{N}}$ and recall that
$\exp_{\mathbf{1}_{\mathcal{S}_{N}}}\colon
T_{0}{\mathcal{S}_{N}}\to{\mathcal{S}_{N}}$ is a homeomorphism. For a subset
$A\subseteq\mathcal{Q}$ we thus have

$\displaystyle A\in\mathcal{A}\,$ $\displaystyle\Leftrightarrow\,\psi(A)\text{
is open in }\mathcal{Q}$ (79a)
$\displaystyle\Leftrightarrow\,\exp_{\mathbf{1}_{\mathcal{S}_{N}}}^{-1}(A)=B\cap\mathcal{Q}\text{
for an open set }B\subseteq T_{0}{\mathcal{S}_{N}}$ (79b)
$\displaystyle\Leftrightarrow\,\exp_{\mathbf{1}_{\mathcal{S}_{N}}}^{-1}(A)=\exp_{\mathbf{1}_{\mathcal{S}_{N}}}^{-1}(\overline{A})\cap\mathcal{Q}\text{
for an open set }\overline{A}\subseteq{\mathcal{S}_{N}}$ (79c)
$\displaystyle\Leftrightarrow\,A=\overline{A}\cap\exp_{\mathbf{1}_{\mathcal{S}_{N}}}(\mathcal{Q})\text{
for an open set }\overline{A}\subseteq{\mathcal{S}_{N}}$ (79d)
$\displaystyle\Leftrightarrow\,A=\overline{A}\cap{\mathcal{T}}\text{ for an
open set }\overline{A}\subseteq{\mathcal{S}_{N}}.$ (79e)

This shows that $\mathcal{A}$ is the subspace topology of
${\mathcal{T}}\subseteq{\mathcal{S}_{N}}$ and thus, $T$ is a topological
embedding of ${\mathcal{W}}$ into ${\mathcal{S}_{N}}$.

We compute the rank of $T$ by applying Lemma A.4. Let $W\in\mathcal{W}$ and
$V\in T_{0}\mathcal{W}$ be in the kernel of $dT|_{W}$. Then

$0=dT|_{W}[V]=T(W)\diamond Q\Big{[}\frac{V}{W}\Big{]}$ (80)

which implies $\frac{V}{W}\in\operatorname{ker}Q$ because $T(W)_{\gamma}\neq
0$ for all $\gamma\in[c]^{n}$. By Lemma A.5 this implies

$V=W\diamond(\operatorname{Diag}(d)\mathbf{1}_{n\times
c})=\operatorname{Diag}(d)W$ (81)

for some $d\in\mathbb{R}^{n}$ with $\langle d,\mathbf{1}_{n}\rangle=0$. From
$V\in T_{0}\mathcal{W}$ we find

$0=\langle V_{i},\mathbf{1}_{c}\rangle=d_{i}\langle
W_{i},\mathbf{1}_{c}\rangle=d_{i},\qquad\forall i\in[n]$ (82)

which shows $V=0$ by (81), i.e. $dT|_{W}$ has full rank. Thus, $T$ is an
injective immersion.

It remains to show that $T$ is metric compatible. Suppose $W\in{\mathcal{W}}$
and $U,V\in T_{0}{\mathcal{W}}$ are arbitrary. Denoting the Fisher-Rao metric
on ${\mathcal{S}_{N}}$ by $g^{{\mathcal{S}}_{N}}$ we get

$\displaystyle(T^{*}g^{{\mathcal{S}}_{N}}\big{)}_{W}(U,V)$
$\displaystyle\overset{\phantom{\eqref{eq:dT_closed_form}}}{=}g^{{\mathcal{S}}_{N}}_{T(W)}(dT|_{W}[U],dT|_{W}[V])$
(83a)
$\displaystyle\overset{\phantom{\eqref{eq:dT_closed_form}}}{=}\big{\langle}dT|_{W}[U],\tfrac{\mathbf{1}}{T(W)}\diamond
dT|_{W}[V]\big{\rangle}$ (83b)
$\displaystyle\overset{\eqref{eq:dT_closed_form}}{=}\Big{\langle}dT|_{W}[U],Q\Big{[}\frac{V}{W}\Big{]}\Big{\rangle}$
(83c)
$\displaystyle\overset{\phantom{\eqref{eq:dT_closed_form}}}{=}\Big{\langle}MdT|_{W}[U],\frac{V}{W}\Big{\rangle}.$
(83d)

Note that $M$ is linear, implying $dM|_{p}=M$ for every
$p\in{\mathcal{S}}_{N}$. Since $M$ restricted to
${\mathcal{T}}=T({\mathcal{W}})$ is the inverse of $T$, one ahs $M\circ
T=\mathrm{id}_{{\mathcal{W}}}$. These two facts imply

$M\big{[}dT|_{W}[U]\big{]}=dM|_{T(W)}\big{[}dT|_{W}[U]\big{]}=d\big{(}M\circ
T\big{)}|_{W}[U]=d(\mathrm{id}_{{\mathcal{W}}})|_{W}[U]=U.$ (84)

Plugging this result back into (83d) gives

$(T^{*}g^{{\mathcal{S}}_{N}}\big{)}_{W}(U,V)=\Big{\langle}U,\frac{V}{W}\Big{\rangle}=g^{\mathcal{W}}_{W}(U,V)$
(85)

which shows the assertion. ∎

###  B.2 Proof of Proposition 3.2

######  Proposition B.2 (Proposition 3.2 in the main text).

For every $W\in{\mathcal{W}}$, the distribution $T(W)\in{\mathcal{S}_{N}}$ has
maximum entropy among all $p\in{\mathcal{S}_{N}}$ subject to the marginal
constraint $Mp=W$.

###### Proof.

We use the concepts of m-flat and e-flat submanifolds of information geometry,
which justify applying the Pythagorean relation of information geometry. For
details, we refer to [1]. The feasible set of all distributions with the
prescribed marginals reads

$\\{T(W)+u\colon Mu=0\\}\cap{\mathcal{S}_{N}}$ (86)

which is an m-flat submanifold of ${\mathcal{S}_{N}}$. In addition, Lemma 3.3
shows that ${\mathcal{T}}$ is an e-flat submanifold of ${\mathcal{S}_{N}}$.
Let $p=T(W)+u$ denote an arbitrary feasible point. By (86) and Lemma 3.4 we
have

$\langle u,QV\rangle=\langle Mu,V\rangle=0$ (87)

for all $V\in\mathbb{R}^{n\times c}$. Consider the $m$-geodesic connecting $p$
with $T(W)$. It intersects ${\mathcal{T}}$ at $T(W)$ and we find

$\langle dT|_{W}[V],u\rangle=\langle T(W)\diamond
Q\Big{[}\frac{V}{W}\Big{]},u\rangle_{T(W)}=\langle
Q\Big{[}\frac{V}{W}\Big{]},u\rangle=\langle\frac{V}{W},Mu\rangle=0$ (88)

by using Lemma A.4. With (88), m-flatness of (86) and e-flatness of
${\mathcal{T}}$ the prerequisites for the Pythagorean relation of information
geometry [1, Theorem 3.8] are met. Using the cross-entropy $H(p,q)=-\langle
p,\log q\rangle$ as well as the relative entropy
$\operatorname{KL}(p,q)=\langle p,\log\frac{p}{q}\rangle$ and barycenter
$\mathbf{1}_{\mathcal{S}_{N}}=\frac{1}{N}\mathbf{1}$, we find

$\displaystyle H(T(W))$
$\displaystyle=H(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})-\operatorname{KL}(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})$
$\displaystyle=\log N-\operatorname{KL}(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})$
(89)

and consequently

$\displaystyle H(p)$
$\displaystyle=H(p,\mathbf{1}_{{\mathcal{S}_{N}}})-\operatorname{KL}(p,\mathbf{1}_{{\mathcal{S}_{N}}})$
(90) $\displaystyle=\log
N-\operatorname{KL}(p,\mathbf{1}_{{\mathcal{S}_{N}}})$ (91)
$\displaystyle=H(T(W))+\operatorname{KL}(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})-\operatorname{KL}(p,\mathbf{1}_{{\mathcal{S}_{N}}})$
(92)
$\displaystyle\stackrel{{\scriptstyle(\ast)}}{{=}}H(T(W))+\operatorname{KL}(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})-\operatorname{KL}(p,T(W))-\operatorname{KL}(T(W),\mathbf{1}_{{\mathcal{S}_{N}}})$
(93) $\displaystyle=H(T(W))-\operatorname{KL}(p,T(W))$ (94)

by the Pythagorean relation $(\ast)$. Therefore $H(p)\leq H(T(W))$ with
equality only for $p=T(W)$ which shows the assertion. ∎

###  B.3 Proof of Lemma 3.3

######  Lemma B.3 (Lifting Map Lemma).

Let $S\in{\mathcal{W}}$ and $V\in\mathbb{R}^{n\times c}$. Then

$T(\exp_{S}(V))=\exp_{T(S)}(Q(V)).$ (95)

###### Proof.

We have $T(\exp(V))=\exp(Q(V))$ (without subscripts, i.e. applying the
exponential function componentwise), because for any multi-index $\gamma$

$\displaystyle\exp(Q(V))_{\gamma}$
$\displaystyle=\exp\big{(}Q(V)_{\gamma}\big{)}=\exp\Big{(}\sum_{i\in[n]}V_{i,\gamma_{i}}\Big{)}$
(96a)
$\displaystyle=\prod_{i\in[n]}\exp\Big{(}V_{i,\gamma_{i}}\Big{)}=\prod_{i\in[n]}\big{(}\exp(V)\big{)}_{i,\gamma_{i}}$
(96b) $\displaystyle=T(\exp(V))_{\gamma}.$ (96c)

Let $D\in\mathbb{R}^{n\times n}$ be a diagonal matrix with nonzero diagonal
entries. Then $T(DR)\propto T(R)$ for any $R\in\mathbb{R}^{n\times c}$ because

$T(DR)_{\gamma}=\prod_{i\in[n]}(DR)_{i,\gamma_{i}}=\Big{(}\prod_{i\in[n]}D_{ii}\Big{)}\Big{(}\prod_{i\in[n]}R_{i,\gamma_{i}}\Big{)}\propto
T(R)_{\gamma}.$ (97)

It follows that

$T(\exp_{S}(V))\propto
T(S\diamond\exp(V))\stackrel{{\scriptstyle\eqref{eq:lem_lifting_Qexp}}}{{=}}T(S)\diamond\exp(Q(V))\propto\exp_{T(S)}(Q(V))$
(98)

Because both the first and last term in (98) are clearly elements of
${\mathcal{S}_{N}}$, i.e. strictly positive vectors summing up ot 1, this
implies the assertion. ∎

###  B.4 Proof of Theorem 3.5

######  Theorem B.4 (Multi-Population Embedding Theorem).

For any payoff function $F\colon{\mathcal{W}}\to\mathbb{R}^{n\times c}$, the
multi-population replicator dynamics

$\dot{W}={\mathcal{R}}_{W}[F(W)],\quad W(0)=W_{0}$ (99)

on ${\mathcal{W}}$ is pushed forward by $T$ to the replicator dynamics

$\dot{p}(t)={R}_{p}(t)\widehat{F}(p(t)),\quad
p(0)=T(W_{0}),\quad\widehat{F}=Q\circ F\circ M$ (100)

on ${\mathcal{S}_{N}}$ and the map $T$ satisfies

$dT|_{W}[{\mathcal{R}}_{W}[X]]=R_{T(W)}Q[X],\quad\text{for all
}X\in\mathbb{R}^{n\times c}\text{ and }W\in{\mathcal{W}}.$ (101)

###### Proof.

We first show that, for any $W\in{\mathcal{W}}$, the differential of $T$ and
the replicator operator are related by (101). Let $\gamma\in[c]^{n}$ be an
arbitrary multi-index. Because of $\mathcal{R}_{W}[X]\in{T_{0}{\mathcal{W}}}$,
Lemma A.4 implies

$dT_{\gamma}|_{W}[{\mathcal{R}}_{W}[X]]=T_{\gamma}(W)Q_{\gamma}\bigg{[}\frac{{\mathcal{R}}_{W}[X]}{W}\bigg{]}=T_{\gamma}(W)\sum_{i\in[n]}\frac{({\mathcal{R}}_{W}[X])_{i,\gamma_{i}}}{W_{i,\gamma_{i}}}.$
(102)

Due to
$({\mathcal{R}}_{W}[X])_{i,\gamma_{i}}=W_{i,\gamma_{i}}(X_{i,\gamma_{i}}-\langle
X_{i},W_{i}\rangle)$, the sum can be written as

$\sum_{i\in[n]}\frac{({\mathcal{R}}_{W}[X])_{i,\gamma_{i}}}{W_{i,\gamma_{i}}}=\sum_{i\in[n]}\big{(}X_{i,\gamma_{i}}-\langle
X_{i},W_{i}\rangle\big{)}=Q_{\gamma}[X]-\langle X,W\rangle.$ (103)

Additionally using the relation $W=M[T(W)]$ due to (70), and applying Lemma
3.4 gives

$\langle X,W\rangle=\langle X,M[T(W)]\rangle=\langle Q[X],T(W)\rangle.$ (104)

Collecting all expressions, we have

$dT_{\gamma}|_{W}[{\mathcal{R}}_{W}[X]]=T_{\gamma}(W)\big{(}Q_{\gamma}[X]-\langle
Q[X],T(W)\rangle\big{)}=\big{(}R_{T(W)}Q[X]\big{)}_{\gamma}$ (105)

which shows (101). Now, denoting ${p}=T(W)\in{\mathcal{S}_{N}}$ we directly
establish (100)

$\dot{p}=dT(W)[{\mathcal{R}}_{W}[(F\circ M)({p})]]={R}_{p}[(Q\circ F\circ
M)({p})]={R}_{p}[\widehat{F}({p})].$ (106)

∎