Asymptotic Dynamics of Hamiltonian Polymatrix Replicators

A new method to study the asymptotic dynamics of flows defined on polytopes was presented in [ADP2020]. This method allows us to analyze the asymptotic dynamics of flows defined on polytopes along the edge-vertex heteroclinic network. Examples of such dynamical systems arise naturally in the context of Evolutionary Game Theory (EGT) developed by J. Maynard Smith and G. R. Price [smith1973logic].

One such example is the polymatrix replicator, introduced in [AD2015, ADP2015]. This is a system of ordinary differential equations that models the time evolution of behavioral strategies of individuals in a stratified population.

The polymatrix replicator induces a flow on a prism (simple polytope) given by a finite product of simplices. These systems extend the class of the replicator and the bimatrix replicator equations studied e.g. in [TJ1978] and [schuster1981coyness, schuster1981selfregulation], respectively.

In [AD2015] the authors have introduced the subclass of conservative polymatrix replicators (see Definition 4.4) which are Hamiltonian systems with respect to appropriate Poisson structures. In [ADP2015] these Hamiltonian polymatrix replicators are used to describe the asymptotic dynamics of the larger class of dissipative polymatrix replicators.

For Hamiltonian vector fields on symplectic manifolds it is well known that the Poincaré map preserves the induced symplectic structure on any transversal section. In this paper we extend this fact to Hamiltonian systems on Poisson manifolds, showing that any transversal section inherits a Poisson structure and the Poincaré map preserves it. Using this result we study the Hamiltonian character of the asymptotic dynamics of conservative polymatrix replicators along their edge-vertex heteroclinic network. Our main result states that for conservative polymatrix replicators the map describing the asymptotic dynamics is Hamiltonian with respect to an appropriate Poisson structure (Theorem 6.17).

The paper is organized as follows. In Section 2 we recall the method in [ADP2020], outlining the construction of the asymptotic dynamics for a large class of flows on polytopes that includes the polymatrix replicators. In Section 3 we define Poincaré maps for Hamiltonian systems on Poisson manifolds. In Section 4 we provide a short introduction to polymatrix replicators, following [AD2015]. Namely, we state the basic definitions and results for the class of conservative polymatrix replicators, that we also designate as Hamiltonian polymatrix replicators. In Section 5 we review the main definitions and results for the polymatrix replicator systems regarding the construction outlined in Section 2. In Section 6 we analyze the Hamiltonian character of Poincaré maps in the case of Hamiltonian polymatrix replicators. Finally, in Section 7 we present an example of a five-dimensional Hamiltonian polymatrix replicator to illustrate the main concepts and results of this paper. The graphics of this section were produced with Wolfram Mathematica and Geogebra software.

We now outline the construction of the asymptotic dynamics for a large class of flows on polytopes that includes the polymatrix replicators. A polytope is a compact convex set in some Euclidean space obtained as the intersection of finitely many half-spaces. A polytope is called simple if the number of edges (or facets) incident with each vertex equals the polytope’s dimension. The phase space of polymatrix replicators, that are prisms given by a finite product of simplices, are examples of simple polytopes. In [ADP2020] we consider analytic vector fields defined on simple polytopes which have the property of being tangent to every face of the polytope. Such vector fields induce complete flows on the polytope which leave all faces invariant. Vertices of the polytope are singularities of the vector field, while edges without singularities, called flowing edges, consist of single orbits flowing between two end-point vertices. The vertices and flowing edges form a heteroclinic network of the vector field. The purpose of this construction is to analyze the asymptotic dynamics of the vector field along this one-dimensional skeleton. Throughout the text we assume that every vector field is non-degenerate. This means that the transversal derivative of the vector field is never identically zero along any facet of the polytope.

The analysis of the vector field’s dynamics along its edge-vertex heteroclinic network makes use of Poincaré maps between cross sections tranversal to the flowing edges. Any Poincaré map along a heteroclinic or homoclinic orbit is a composition of two types of maps, global and local Poincaré maps. A global map, denoted by $P_{\gamma}$, is defined in a tubular neighborhood of any flowing-edge $\gamma$. It maps points between two cross sections $\Sigma_{\gamma}^{-}$ and $\Sigma_{\gamma}^{+}$ transversal to the flow along the edge $\gamma$. A local map, denoted by $P_{v}$, is defined in a neighborhood of any vertex $v$. For any pair of flowing-edges $\gamma,\gamma^{\prime}$ such that $v$ is both the ending point of $\gamma^{\prime}$ and the starting point of $\gamma$, the local map $P_{v}$ takes points from $\Sigma_{\gamma^{\prime}}^{+}$ to $\Sigma_{\gamma}^{-}$ (see Figure 1).

Asymptotically, the nonlinear character of the global Poincaré maps fade away as we approach a heteroclinic orbit. This means that these non-linearities are irrelevant for the asymptotic analysis. For regular¹¹1The reader should bare in mind that the concept of regularity used here (Definition 5.6) is more restrictive then the one in [ADP2020, Definition $6.3$]. vector fields, the skeleton character at a vertex, defined as the set of eigenvalues of the tangent map along the edge eigen-directions, completely determines the asymptotic behavior of the local Poincaré map at that vertex.

To describe the limit dynamical behavior we introduce the dual cone of a polytope where the asymptotic piecewise linear dynamics unfolds. This space lies inside ${\mathbb{R}}^{F}$, where $F$ is the set of the polytope’s facets. The dual cone of a $d$-dimensional simple polytope $\Gamma$ is the union

where for each vertex $v$, $\Pi_{v}$ is the $d$-dimensional sector consisting of points $y\in{\mathbb{R}}^{F}$ with non-negative coordinates such that $y_{\sigma}=0$ for every facet that does not contain $v$. See Figure 2.

Given a vector field $X$ on a $d$-dimensional polytope $\Gamma\subset{\mathbb{R}}^{d}$, we now describe a rescaling change of coordinates $\Psi_{\epsilon}^{X}$, depending on a blow up parameter $\epsilon$. See Figure 3.

This change of coordinates maps tubular neighborhoods of edges and vertices to the dual cone ${\cal C}^{\ast}(\Gamma)$. For instance, the tubular neighborhood $N_{v}$ of a vertex $v$ is defined as follows. Consider a system $(x_{1},\ldots,x_{d})$ of affine coordinates around $v$, which assigns coordinates $(0,\ldots,0)$ to $v$ and such that the hyperplanes $x_{j}=0$ are precisely the facets of the polytope through $v$. Then $N_{v}$ is defined by

The sets $\{x_{j}=0\}\cap N_{v}$ are called the outer facets of $N_{v}$. The remaining facets of $N_{v}$, defined by equations like $x_{i}=1$, are called the inner facets of $N_{v}$. The previous cross sections $\Sigma_{\gamma}^{\pm}$ can be chosen to match these inner facets of the neighborhoods $N_{v}$.

The rescaling change of coordinates $\Psi^{X}_{\epsilon}$ maps $N_{v}$ to the sector $\Pi_{v}$. Enumerating $F$ so that the facets through $v$ are precisely $\sigma_{1},\ldots,\sigma_{d}$, the map $\Psi^{X}_{\epsilon}$ is defined on the neighborhood $N_{v}$ by

Similarly, given an edge $\gamma$, $\Psi^{X}_{\epsilon}$ maps a tubular neighborhood $N_{\gamma}$ of $\gamma$ to the facet sector $\Pi_{\gamma}:=\Pi_{v}\cap\Pi_{v^{\prime}}$ of $\Pi_{v}$ where $v^{\prime}$ is the other end-point of $\gamma$. The map $\Psi^{X}_{\epsilon}$ sends interior facets of $N_{v}$ and $N_{\gamma}$ respectively to boundary facets of $\Pi_{v}$ and $\Pi_{\gamma}$ while it maps outer facets of $N_{v}$ and $N_{\gamma}$ to infinity. As the rescaling parameter $\epsilon$ tends to $0$, the rescaled push-forward $\epsilon^{-2}\,(\Psi^{X}_{\epsilon})_{\ast}X$ of the vector field $X$ converges to a constant vector field $\chi^{v}$ on each sector $\Pi_{v}$. This means that asymptotically, as $\epsilon\to 0$, trajectories become lines in the coordinates $(y_{\sigma})_{\sigma\in F}=\Psi^{X}_{\epsilon}$. Given a flowing-edge $\gamma$ between vertices $v$ and $v^{\prime}$, the map $\Psi^{X}_{\epsilon}$ over $N_{\gamma}$ depends only on the coordinates transversal to $\gamma$. Moreover, as $\epsilon\to 0$ the global Poincaré map $P_{\gamma}$ converges to the identity map in the coordinates $(y_{\sigma})_{\sigma\in F}=\Psi^{X}_{\epsilon}$. Hence the sector $\Pi_{\gamma}$ is naturally identified as the common facet between the sectors $\Pi_{v}$ and $\Pi_{v^{\prime}}$. Hence the asymptotic dynamics along the edge-vertex heteroclinic network is completely determined by the vector field’s geometry at the vertex singularities and can be described by a piecewise constant vector field $\chi$ on the dual cone, whose components are precisely those of the skeleton character of $X$. We refer to this piecewise constant vector field as the skeleton vector field of $X$. This vector field $\chi$ induces a piecewise linear flow on the dual cone whose dynamics can be computationally explored.

We use Poincaré maps for a global analysis of the asymptotic dynamics of the flow of $X$. We consider a subset $S$ of flowing-edges with the property that every heteroclinic cycle goes through at least one edge in $S$. Such sets are called structural sets. The flow of $X$ induces a Poincaré map $P_{S}$ on the system of cross sections $\Sigma_{S}:=\cup_{\gamma\in S}\Sigma^{+}_{\gamma}$. Each branch of the Poincaré map $P_{S}$ is associated with a heteroclinic path starting with an edge in $S$ and ending at its first return to another edge in $S$. These heteroclinic paths are the branches of $S$. The flow of the skeleton vector field $\chi$ also induces a first return map $\pi_{S}:D_{S}\subset\Pi_{S}\to\Pi_{S}$ on the system of cross sections $\Pi_{S}:=\cup_{\gamma\in S}\Pi_{\gamma}$. This map $\pi_{S}$, called the skeleton flow map, is piecewise linear and its domain is a finite union of open convex cones. In some cases, see Proposition 5.18, the map $\pi_{S}$ becomes a closed dynamical system.

We can now recall the main result in [ADP2020], Theorem 5.20 below, which says that under the rescaling change of coordinates $\Psi^{X}_{\epsilon}$, the Poincaré map $P_{S}$ converges in the $C^{\infty}$ topology to the skeleton flow map $\pi_{S}$, in the sense that the following limit holds

with uniform convergence of the map and its derivatives over any compact set contained in the domain $D_{S}\subset\Pi_{S}$.

Consider now, for each facet $\sigma$ of the polytope, an affine function ${\mathbb{R}}^{d}\ni q\mapsto x_{\sigma}(q)\in{\mathbb{R}}$ which vanishes on $\sigma$ and is strictly positive on the rest of the polytope. With this family of affine functions we can present the polytope as $\Gamma^{d}=\cap_{\sigma\in F}\{x_{\sigma}\geq 0\}$. Any function function $h:{\rm int}(\Gamma^{d})\to{\mathbb{R}}$ of the form

rescales to the following piecewise linear function on the dual cone

in the sense that $\eta=\lim_{\epsilon\to 0}\epsilon^{-2}\,(h\circ(\Psi^{X}_{\epsilon})^{-1})$. When all coefficients $c_{\sigma}$ have the same sign then $\eta$ is a proper function on the dual cone and all levels of $\eta$ are compact sets. If the function $h$ is invariant under the flow of $X$, i.e. $h\circ P_{S}=h$, then the piecewise linear function $\eta$ is also invariant under the skeleton flow, i.e. $\eta\circ\pi_{S}=\eta$. Thus integrals of motion (of vector fields on polytopes) of the previous form give rise to (asymptotic) piecewise linear integrals of motion for the skeleton flow.

In this section we will define Poincaré map for Hamiltonian systems on Poisson manifolds. For Hamiltonian vector fields on symplectic manifolds it is well known that the Poincaré map preserves the induced symplectic structure on any transversal section (see [MR2554208]*Theorem 1.8.). We extend this fact to Hamiltonian systems on Poisson manifolds, showing that any transversal section inherits a Poisson structure and the Poincaré map preserves this structure.

A Poisson manifold is a pair $(M,\pi)$ where $M$ is a smooth manifold without boundary and ${\pi}$ a Poisson structure on $M$. Recall that a Poisson structure is a smooth bivector field ${\pi}$ with the property that $[{\pi},{\pi}]=0$, where $[\cdot,\cdot]$ is the Schouten bracket (cf. e.g. [MR2178041]). The bivector field $\pi$ defines a vector bundle map

The image of this map is an integrable singular distribution which integrates to a symplectic foliation, i.e., a foliation whose leaves have a symplectic structure induced by the Poisson structure.

Notice that a Poisson structure can also be defined as a Lie bracket $\{\cdot,\cdot\}$ on $C^{\infty}(M)\times C^{\infty}(M)$ satisfying the Leibniz rule

These two descriptions are related by ${\pi}(\mathrm{d}f,\mathrm{d}g)=\{f,g\}$. In a local coordinate chart $(U,x_{1},..,x_{n})$, or equivalently when $M={\mathbb{R}}^{n}$, a Poisson bracket takes the form

where $\pi(x)=\left[\,{\pi_{ij}(x)}\right]_{{ij}}=\left[\,{\{x_{i},x_{j}\}(x)}\right]_{{ij}}$ is a skew symmetric matrix valued smooth function satisfying

As in the symplectic case, to define the Poincaré map we will consider the traversal sections inside the level set of the Hamiltonian. We will show that such a transversal section is a cosymplectic submanifolds of the ambient Poisson manifold and naturally inherits a Poisson structure. For more details on cosymplectic submanifolds see [CALVO2010259]*Section $5.1$.

Every, cosympletic submanifold is naturally equipped with a Poisson bracket called Dirac bracket. Paul Dirac, [paul-dirac-book], developed this bracket to treat classical systems with second class constraints in Hamiltonian mechanics.

Dirac bracket is actually a Poisson bracket on the open submanifold $U$, see [CALVO2010259]. It takes an easy calculation to see that constraint functions $G_{i},\,i=1,\ldots,2k$ are Casimirs of Dirac bracket. This fact allows the restriction of Dirac bracket to the cosymplectic submanifold $N$. Note that, in general, restricting (pulling back) a Poisson structure to an arbitrary submanifold is not straightforward. Actually, the decomposition

that holds for every point $x\in N$ and a strait forward calculation yield the independence of the extension in the following definition. In Equation (3.5), the term $T_{x}N^{\circ}$ is the annihilator of $T_{x}N$ in $T^{\ast}_{x}M$. We will use this notation in the rest of the paper. Equation (3.5) can be used as definition of a cosymplectic submanifold, see [CALVO2010259], but for our propose it suits better to use the second class constraints to define cosymplectic submanifolds.

We consider a Hamiltonian $H$ on the $m$-dimensional Poisson manifold $(M,{\pi})$ and its associated Hamiltonian vector field defined by $X_{H}=\{H,.\}=\pi^{\sharp}(dH)$. For a given point $x_{0}\in M$ let $U$ be a neighborhood around it such that $X_{H}(x)\neq 0\quad\forall x\in U$, and $\mathcal{E}_{x_{0}}$ be the energy surface passing through $x_{0}$, i.e., the connected component of $H^{-1}(H(x_{0}))$ containing $x_{0}$. We call level transversal section to $X_{H}$ at a regular point $x_{0}\in M$ any $(m-2)$-dimensional transversal section $\Sigma\subset\mathcal{E}_{x_{0}}\cap U$ through $x_{0}$.

The following lemma shows that $\Sigma$ is a cosymplectic submanifold.

We will use the same notation $f$ for arbitrary extension and reserve the notation $\tilde{f}$ for extension that their differentials vanishes on $X_{H}$ at every point $x\in\Sigma$. To avoid any possible confusion, we observe that in [MR2128714]*Section 8 and [MR2128714]*Section 8 the notation $\tilde{f}$ is used in a slightly different sense.

For a fixed time $t_{0}$, let $x_{1}=\phi_{H}(t_{0},x_{0})$, where $\phi_{H}$ is the flow of the Hamiltonian vector field $X_{H}$, and $\Sigma_{0},\Sigma_{1}$ be level transversal sections at $x_{0}$ and $x_{1}$, respectively. As usual, a Poincaré map ${P}=\phi_{H}(\tau(x),x)$ can be defined from an appropriate neighborhood of $x_{0}$ in $\Sigma_{0}$ to a neighborhood of $x_{1}$ in $\Sigma_{1}$. The existence of the smooth function $\tau(x)$ is guaranteed by the Implicit Function Theorem. We replace $\Sigma_{0}$ and $\Sigma_{1}$ by the domain and the image of the Poincaré map $P$.

By Lemma 3.8 both $\Sigma_{i}$, $i=0,1$, are cosymplectic submanifolds equipped with Dirac brackets $\{.,.\}_{{\rm Dirac}_{i}},\,i=0,1$. We will show that the Poincaré map ${P}$ is a Poisson map (see Definition 3.1).

In this section we provide a short introduction to polymatrix replicators, following [AD2015]. In particular we will focus on the class of conservative polymatrix replicators that we designate as Hamiltonian polymatrix replicators.

Consider a population divided in $p$ groups where each group is labeled by an integer $\alpha\in\{1,\dots,p\}$, and the individuals of each group $\alpha$ have exactly $n_{\alpha}$ strategies to interact with other members of the population (including of the same group). In total we have $n=\sum_{\alpha=1}^{p}n_{\alpha}$ strategies that we label by the integers $i\in\{1,\ldots,n\}$, denoting by

the set (interval) of strategies of group $\alpha$.

Given $\alpha,\beta\in\{1,\ldots,p\}$, consider a real $n_{\alpha}\times n_{\beta}$ matrix, say $A^{\alpha,\beta}$, whose entries $a_{ij}^{\alpha,\beta}$, with $i\in[\alpha]$ and $j\in[\beta]$, represent the payoff of an individual of the group $\alpha$ using the $i^{\textrm{th}}$ strategy when interacting with an individual of the group $\beta$ using the $j^{\textrm{th}}$ strategy. Thus the matrix $A$ with entries $a_{ij}^{\alpha,\beta}$, where $\alpha,\beta\in\{1,\ldots,p\}$, $i\in[\alpha]$ and $j\in[\beta]$, is a square matrix of order $n=n_{1}+\ldots+n_{p}$, consisting of the block matrices $A^{\alpha,\beta}$.

Let ${\underline{n}}=(n_{1},\ldots,n_{p})$. The state of the population is described by a point $x=(x^{\alpha})_{1\leq\alpha\leq p}$ in the polytope

where $\Delta^{n_{\alpha}-1}=\{x\in{\mathbb{R}}_{+}^{[\alpha]}:\sum_{i\in[\alpha]}x_{i}^{\alpha}=1\}$, $x^{\alpha}=(x_{i}^{\alpha})_{i\in[\alpha]}$ and the entry $x_{i}^{\alpha}$ represents the usage frequency of the $i^{\textrm{th}}$ strategy within the group $\alpha$. We denote by $\partial\Gamma_{{\underline{n}}}$ the boundary of $\Gamma_{{\underline{n}}}$.

Assuming random encounters between individuals, for each group $\alpha\in\{1,\dots,p\}$, the average payoff of a strategy $i\in[\alpha]$ within a population with state $x$ is given by

where the overall average payoff of group $\alpha$ is given by

Demanding that the logarithmic growth rate of the frequency of each strategy $i\in[\alpha]$, $\alpha\in\{1,\dots,p\}$, is equal to the payoff difference between strategy $i$ and the overall average payoff of group $\alpha$ yields the system of ordinary differential equations defined on the polytope $\Gamma_{{\underline{n}}}$,

that will be designated as a polymatrix replicator.

If $p=1$ equation (4.1) becomes the usual replicator equation with payoff matrix $A$. When $p=2$ and $A^{11}=A^{22}=0$ are null matrices, equation (4.1) becomes the bimatrix replicator equation with payoff matrices $A^{12}$ and $(A^{21})^{t}$.

The flow $\phi_{{\underline{n}},A}^{t}$ of this equation leaves the polytope $\Gamma_{{\underline{n}}}$ invariant. The proof of this fact is analogous to that for the bimatrix replicator equation, see [HS1998, Section 10.3]. Hence, by compactness of $\Gamma_{{\underline{n}}}$, the flow $\phi_{{\underline{n}},A}^{t}$ is complete. From now on the term polymatrix replicator will also refer to the flow $\phi_{{\underline{n}},A}^{t}$ and the underlying vector field on $\Gamma_{{\underline{n}}}$, denoted by $X_{{\underline{n}},A}$.

Given ${\underline{n}}=(n_{1},\ldots,n_{p})$, let

A set $I\in\mathscr{I}_{{\underline{n}}}$ determines the facet $\sigma_{I}:=\{\,x\in\Gamma_{{\underline{n}}}\,:\,x_{j}=0,\,\forall\,j\notin I\,\}$ of $\Gamma_{{\underline{n}}}$. The correspondence between labels in $\mathscr{I}_{{\underline{n}}}$ and facets of $\Gamma_{{\underline{n}}}$ is bijective.

For a set $I\in\mathscr{I}_{{\underline{n}}}$ consider the pair $({\underline{n}}^{I},A_{I})$, where ${{\underline{n}}^{I}=(n_{1}^{I},\ldots,n_{p}^{I})}$ with $n_{\alpha}^{I}=\#(I\cap[\alpha])$, and $A_{I}=\left[\,{a_{ij}}\right]_{{i,j\in I}}$.

For a fixed ${\underline{n}}=(n_{1},\ldots,n_{p})$ the correspondence $A\mapsto X_{{\underline{n}},A}$ is linear and its kernel consists of the matrices $C=\left(C^{\alpha,\beta}\right)_{1\leq\alpha,\beta\leq p}$ where each block $C^{\alpha,\beta}$ has equal rows, i.e., has the form

Thus ${X_{{\underline{n}},A_{1}}=X_{{\underline{n}},A_{2}}}$ if and only if for every $\alpha,\beta\in\{1,\dots,p\}$ the matrix $A^{\alpha,\beta}_{1}-A^{\alpha,\beta}_{2}$ has equal rows (see [AD2015, Proposition 1]).

We have now the following characterization of the interior equilibria.

In [ADP2015], another characterization of conservative polymatrix replicators, using quadratic forms, is provided. Furthermore, in [hassan-JGM-2020] the concept of conservative replicator equations (where $p=1$) is generalized using Dirac structures.

In what follows, the vectors in ${\mathbb{R}}^{n}$, or ${\mathbb{R}}^{[\alpha]}$, are identified with column vectors. Let $\mathbbm{1}_{n}=(1,..,1)^{t}\in{\mathbb{R}}^{n}$. We will omit the subscript $n$ whenever the dimension of this vector is clear from the context. Similarly, we write $I=I_{n}$ for the $n\times n$ identity matrix. Given $x\in{\mathbb{R}}^{n}$, we denote by $D_{x}$ the $n\times n$ diagonal matrix $D_{x}:=\operatorname{diag}(x_{1},\dots,x_{n})$. For each $\alpha\in\{1,\ldots,p\}$ we define the $n_{\alpha}\times n_{\alpha}$ matrix

and $T_{x}$ the $n\times n$ block diagonal matrix $T_{x}:=\operatorname{diag}(T^{1}_{x},\dots,T^{p}_{x})$.

Given an anti-symmetric matrix $A_{0}$, we define the skew symmetric matrix valued mapping $\pi_{A_{0}}:{\mathbb{R}}^{n}\to{\rm Mat}_{n\times n}({\mathbb{R}})$

The interior of the polytope $\Gamma_{\underline{n}}$, denoted by ${\rm int}(\Gamma_{\underline{n}})$, equipped with $\pi_{A_{0}}$ is a Poisson manifold, see [AD2015]*Theorem 3.5. Furthermore, we have the following theorem.

Given a polymatrix replicator $X_{{\underline{n}},A}$, the edges and vertices of the polytope $\Gamma_{\underline{n}}$ form a (edge-vertex) heteroclinic network for the associated flow. In this section we recall the technique developed in [ADP2020] to analyze the asymptotic dynamics of a flow on a polytope along its heteroclinic edge network. In particular we review the main definitions and results for the polymatrix replicator $X_{{\underline{n}},A}$ on the polytope $\Gamma_{\underline{n}}$.

The affine support of $\Gamma_{\underline{n}}$ is the smallest affine subspace of ${\mathbb{R}}^{n}$ that contains $\Gamma_{{\underline{n}}}$. It is the subspace $E=E_{1}\times\ldots\times E_{p}$ where for $\alpha=1,\ldots,p$,

Following [ADP2020, Definition 3.1] we introduce a defining family for the polytope $\Gamma_{{\underline{n}}}$. The affine functions ${\{f_{i}:E\to\mathbb{R}\}_{1\leq i\leq n}}$ where $f_{i}(x)=x_{i}$, form a defining family for $\Gamma_{{\underline{n}}}$ because they satisfy:

• (a)

  $\displaystyle\Gamma_{{\underline{n}}}=\bigcap_{i\in I}f_{i}^{-1}([0,+\infty[)$,

• (b)

  $\displaystyle\Gamma_{{\underline{n}}}\cap f_{i}^{-1}(0)\neq\emptyset$ for all $i\in\{1,\dots,n\}$, and

• (c)

  given $J\subseteq\{1,\ldots,n\}$ such that $\displaystyle\Gamma_{{\underline{n}}}\cap\left(\bigcap_{j\in J}f_{j}^{-1}(0)\right)\neq\emptyset$, the linear $1$-forms $(\mathrm{d}f_{j})_{p}$ are linearly independent at every point $\displaystyle p\in\cap_{j\in J}f_{j}^{-1}(0)$.

Next we introduce convenient labels for vertices, facets and edges of $\Gamma_{{\underline{n}}}$. Let $(e_{1},\ldots,e_{n})$ be the canonical basis of ${\mathbb{R}}^{n}$ and denote by $\mathcal{V}_{{\underline{n}}}$ the Cartesian product $\mathcal{V}_{{\underline{n}}}:=\prod_{\alpha=1}^{p}[\alpha]$ which contains $\prod_{\alpha=1}^{p}n_{\alpha}$ elements. Each label $\operatorname{j{}}=(j_{1},\ldots,j_{p})\in\mathcal{V}_{{\underline{n}}}$ determines the vertex $v_{\operatorname{j{}}}:=(e_{j_{1}},...,e_{j_{p}})$ of $\Gamma_{{\underline{n}}}$. This labeling is one-to-one. The set $\mathcal{F}_{{\underline{n}}}:=\{1,2,\ldots,n\}$ can be used to label the $n$ facets of $\Gamma_{{\underline{n}}}$. Each integer $i\in\mathcal{F}_{{\underline{n}}}$ labels the facet $\sigma_{i}:=\Gamma_{\underline{n}}\cap\{x_{i}=0\}$ of $\Gamma_{{\underline{n}}}$. Edges can be labeled by the set $\mathcal{E}_{{\underline{n}}}:=\left\{J\in\mathscr{I}_{{\underline{n}}}\,\colon\;\#J=p+1\;\right\}$. Given $J\in\mathcal{E}_{{\underline{n}}}$ there exists a unique (unordered) pair of labels $\operatorname{j{}}_{1},\operatorname{j{}}_{2}\in\mathcal{V}_{{\underline{n}}}$ such that $J$ is the union of the strategies in $\operatorname{j{}}_{1}$ and $\operatorname{j{}}_{2}$. The label $J$ determines the edge $\gamma_{J}:=\{tv_{\operatorname{j{}}_{1}}+(1-t)v_{\operatorname{j{}}_{2}}\colon 0\leq t\leq 1\}$. Again the correspondence $J\mapsto\gamma_{J}$ between labels $J\in\mathcal{E}_{{\underline{n}}}$ and edges of $\Gamma_{{\underline{n}}}$ is one-to-one.