Coxeter tournaments

TL;DR

This paper introduces a Coxeter group-based framework for analyzing tournaments, extending classical results with geometric, probabilistic, and algorithmic proofs, and proposes a Coxeter analogue of the Bradley--Terry model for paired comparisons.

Contribution

It develops a Coxeter group perspective on tournaments, providing new proofs of Moon's theorem and introducing a Coxeter version of the Bradley--Terry model.

Findings

Established a Coxeter version of Moon's theorem.

Provided geometric, probabilistic, and algorithmic proofs.

Introduced a Coxeter analogue of the Bradley--Terry model.

Abstract

We describe the Coxeter permutahedra, recently studied by Ardila, Castillo, Eur and Postnikov, in terms of random Coxeter tournaments, which involve cooperative and solitaire games, as well as the usual competitive games in graph tournaments. In this way, we establish a Coxeter version of Moon's theorem on random tournaments. We present a geometric proof by the Mirsky--Thompson generalized Birkhoff's theorem, a probabilistic proof by Strassen's coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel--Hakimi algorithm. These proofs have interpretations in terms of players choosing competitors/collaborators with respect to relative weakness/strength. We also introduce a natural Coxeter analogue of the Bradley--Terry model, from the statistical theory of paired comparisons.

Tables

Table 1. Table 1. Construction of a type C 7 subscript 𝐶 7 C_{7} random tournament, with mean score sequence x = ( − .4 , .5 , 2.3 , 3.4 , − 4.1 , 4.9 , − 5.2 ) 𝑥 .4 .5 2.3 3.4 4.1 4.9 5.2 x=(-.4,.5,2.3,3.4,-4.1,4.9,-5.2) , via a Coxeter analogue of the Havel–Hakimi algorithm. In this greedy algorithm, players prefer to compete/cooperate with weak/strong players, and compete/cooperate with strong/weak players only as necessary.

$-5.2$	$4.9$	$-4.1$	$3.4$	$2.3$	$.5$	$-.4$
$p_7^{\ell }=0$	$p_{76}^{-}=0$	$p_{75}^{-}=.75$	$p_{74}^{-}=0$	$p_{73}^{-}=0$	$p_{72}^{-}=0$	$p_{71}^{-}=0$
	$p_{76}^{+}=1$	$p_{75}^{+}=0$	$p_{74}^{+}=.05$	$p_{73}^{+}=0$	$p_{72}^{+}=0$	$p_{71}^{+}=0$
$-1$	$0$	$-.25$	$-.95$	$-1$	$-1$	$-1$	$-5.2$
	$3.9$	$-3.35$	$3.35$	$2.3$	$.5$	$-.4$
	$q_6^{\ell }=0$	$q_{65}^{-}=0$	$q_{64}^{-}=1$	$q_{63}^{-}=.1$	$q_{62}^{-}=0$	$q_{61}^{-}=0$
		$q_{65}^{+}=1$	$q_{64}^{+}=0$	$q_{63}^{+}=0$	$q_{62}^{+}=0$	$q_{61}^{+}=0$
	$1$	$0$	$0$	$.9$	$1$	$1$	$3.9$
		$-2.35$	$2.35$	$2.2$	$.5$	$-.4$
		$p_5^{\ell }=0$	$p_{54}^{-}=0$	$p_{53}^{-}=0$	$p_{52}^{-}=0$	$p_{51}^{-}=.275$
			$p_{54}^{+}=1$	$p_{53}^{+}=1$	$p_{52}^{+}=.375$	$p_{51}^{+}=0$
		$-1$	$0$	$0$	$-.625$	$-.725$	$-2.35$
			$1.35$	$1.2$	$.125$	$-.125$
			$q_4^{\ell }=0$	$q_{43}^{-}=1$	$q_{42}^{-}=.475$	$q_{41}^{-}=.35$
				$q_{43}^{+}=0$	$q_{42}^{+}=.35$	$q_{41}^{+}=.475$
			$1$	$0$	$.175$	$.175$	1.35
				$.2$	$0$	$0$
				$q_3^{\ell }=0$	$q_{32}^{-}=.7$	$q_{31}^{-}=.7$
					$q_{32}^{+}=.7$	$q_{31}^{+}=.7$
				$1$	$-.4$	$-.4$	.2
					$0$	$0$
					$p_2^{\ell }=.5$	$p_{21}^{-}=.5$
						$p_{21}^{+}=.5$
					$0$	$0$	0
						$0$
						$p_1^{\ell }=.5$
						$0$	0

Equations

$$v_n=(0,1,\dots ,n-1)$$

$${\Pi }_{n-1}=\mathrm{conv}\{\sigma \cdot v_n:\sigma \in S_n\},$$

$${\Pi }_{n-1}=v_n+\sum _{i<j}[0,e_i-e_j],$$

$${\Pi }_{n-1}=\{x\in {\mathbb{R}}^n:x⪯v_n\},$$

$${\Pi }_{\Phi }=\sum _{\alpha \in {\Phi }^{+}}[-\alpha /2,\alpha /2].$$

$${\Pi }_{\Phi }=\mathrm{conv}\{w\cdot \rho :w\in W\},$$

$$\rho =\sum _{\alpha \in {\Phi }^{+}}\alpha /2=\sum _{i=1}^n{\lambda }_i$$

$$x_i=\sum _{ij\in E}{p}_{ij}.$$

$$Z_{\mathcal{G}}=\sum _{\alpha \in S}[-\alpha /2,\alpha /2].$$

$$x_i=\sum _{ij\in E^{-}}(p_{ij}^{-}-1/2)+\sum _{ij\in E^{+}}(p_{ij}^{+}-1/2)+\{\begin{array}{ll}{p}_i^h-1/2& i\in H\\ 2(p_i^{\ell }-1/2)& i\in L.\end{array}$$

$$\mathcal{L}=\{w\cdot {\lambda }_k:w\in W,k\in [n]\}$$

$${\rho }_{\Phi }=v_n+{\delta }_{\Phi }{1}_n,$$

$${\delta }_{\Phi }=\{\begin{array}{ll}1/2& \Phi =B_n\\ 1& \Phi =C_n\\ 0& \Phi =D_n,\end{array}$$

$$\sum _{ij}[p_{ij}^{-}\log (p_{ij}^{-})+(1-p_{ij}^{-})\log (1-p_{ij}^{-})+p_{ij}^{+}\log (p_{ij}^{+})]+\sum _i{p}_i^{\ell }\log (p_i^{\ell }).$$

$$Z_{\mathcal{G}}^{\mathrm{tr}}=Z_{\mathcal{G}}+{\rho }_{\mathcal{G}}=\sum _{\alpha \in S}[0,\alpha ]$$

$${\rho }_{\mathcal{G}}=\sum _{\alpha \in S}\alpha /2.$$

$$s_v(x)=x-2\frac{⟨v,x⟩}{⟨v,v⟩}v.$$

$${\Phi }_1\oplus {\Phi }_2=\{(\alpha ,0):\alpha \in {\Phi }_1\}\cup \{(0,\beta ):\beta \in {\Phi }_2\}.$$

$$⟨{\lambda }_i,{\alpha }_j^{\vee }⟩=1_{i=j},$$

$$e_S=\sum _{i\in S_{+}}{e}_i-\sum _{-i\in S_{-}}{e}_i.$$

$$S\mapsto \sum _{i\in S}\overline{e_i}.$$

$$S\mapsto \{\begin{array}{ll}{e}_S& |S|<n\\ \frac{1}{2}{e}_S& |S|=n.\end{array}$$

$$S\mapsto e_S.$$

$$S\mapsto \{\begin{array}{ll}{e}_S& \text{∣S∣≤n−2}\\ \frac{1}{2}{e}_S& \text{∣S∣=n}.\end{array}$$

$$E(\mathcal{G})=E^{-}\cup E^{+}\cup H\cup L$$

$$\Gamma (\mathcal{G})=\{e_i-e_j:ij\in E^{-},i>j\},$$

$$\Gamma (\mathcal{G})=\{e_i\pm e_j:ij\in E^{\pm },i>j\}\cup \{e_i:i\in H\},$$

$$\Gamma (\mathcal{G})=\{e_i\pm e_j:ij\in E^{\pm },i>j\}\cup \{2e_i:i\in L\},$$

$$\Gamma (\mathcal{G})=\{e_i\pm e_j:ij\in E^{\pm },i>j\}.$$

$${\Pi }_{\Phi }=\sum _{\alpha \in {\Phi }^{+}}[-\alpha /2,\alpha /2].$$

Full text

We establish a Coxeter analogue of Moon’s classical theorem, regarding mean score sequences of random tournaments. We present a geometric proof by the Mirsky–Thompson generalized Birkhoff’s theorem, a probabilistic proof by Strassen’s coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel–Hakimi algorithm. These proofs have natural interpretations in terms of players seeking out potential competitors/collaborators with respect to their relative weakness/strength. We also observe that the Bradley–Terry model, from the statistical theory of paired comparisons, extends to the Coxeter setting. Finally, we show that an analogue of Landau’s classical theorem, concerning deterministic tournaments, holds for balanced Coxeter tournaments.

1. Introduction

The field of “Coxeter combinatorics,” named in honor of the influential geometer H. S. M. Coxeter, is motivated by the observation that a large variety of combinatorial objects, such as matroids, posets, graphs, and the associahedron, etc., are connected in many ways to the standard type $A_{n}$ root system and its related algebra, geometry and combinatorics. The objective is to find combinatorial objects which generalize the usual type $A_{n}$ objects to other root systems, particularly those of the nonstandard types $B_{n}$, $C_{n}$ and $D_{n}$.

For instance, this program has resulted in signed graphs (see Zaslavsky [68, 69]), Coxeter matroids which describe decomposition of flag varieties G/B (see Borovik, Gelfand and White [10]), parsets which relate to root cones in root systems (see Reiner [52]) and Coxeter associahedron which appear in the study of cluster algebras (see Hohlweg, Lange and Thomas [34]).

In this work, we extend the classical theory of graph tournaments to the Coxeter setting, via the geometric perspective developed in [40], and in relation to the Coxeter generalized permutahedra studied recently by Ardila, Castillo, Eur and Postnikov [5]. These polytopes have been described geometrically in terms of submodular functions. Our main contribution is to show that they can also be described in terms of Coxeter tournaments (see Section 1.5 below). Classical tournaments involve only competitive games between players. As it turns out, the Coxeter generalization is the right setting in which to incorporate collaborative and solitaire games into a tournament.

1.1. Outline

The classical permutahedron and the more recent Coxeter generalized permutahedra are discussed in Sections 1.2 and 1.3, with further background (on root systems, signed graphs, etc.) given in Section 2. Classical tournaments and their connection to classical permutahedra are discussed in Section 1.4. Coxeter tournaments are introduced in Section 1.5. Our results are presented in Sections 1.6, 1.7, 1.8 and 1.9 and proved in Sections 3 and 4.

The remainder of this introduction is organized as follows. The Coxeter analogue of Moon’s theorem is stated in full generality in Section 1.6. In the complete case (Section 1.7), our results are simpler to state, and more methods of proof are available. We present a geometric (Section 1.7.1), probabilistic (Section 1.7.2) and algorithmic (Section 1.7.3) proof of the Coxeter analogue of Moon’s theorem in the complete case. The extension of the Bradley–Terry model to the Coxeter setting is discussed in Section 1.8. Extending Landau’s theorem to the Coxeter setting is discussed in Section 1.9.

1.2. The permutahedron

The permutahedron $\Pi_{n-1}$ is a classical (type $A_{n-1}$) combinatorial object, obtained by taking the convex hull of

[TABLE]

and its permutations, that is,

[TABLE]

where $S_{n}$ is the symmetric group, which acts on ${\mathbb{R}}^{n}$ by permuting coordinates. The reason for the “$n-1$” is that $\Pi_{n-1}$ is only $(n-1)$-dimensional.

The permutahedron is also the graphical zonotope $Z_{K_{n}}$ of the complete graph $K_{n}$ on $[n]=\{1,2,\ldots n\}$ (see, e.g., Ziegler [71]). More specifically, it is the (translated) Minkowski sum of line segments

[TABLE]

where $e_{i}$ are the standard basis vectors. Note that, in this sum, we have a line segment $[0,e_{i}-e_{j}]$ for each edge $ij\in E(K_{n})$. To geometry of $\Pi_{n-1}$ is inextricably linked with $K_{n}$. Notably, the volume of $\Pi_{n-1}$ is the number of spanning trees $T\subset K_{n}$, and the number of lattice points in $\Pi_{n-1}$ is the number of spanning forests $F\subset K_{n}$ (see Stanley [58] and cf. [49] for generalizations).

Rado [50] proved that the permutahedron has hyperplane description

[TABLE]

where $x\preceq v_{n}$ means that $x$ is majorized by $v_{n}$, that is, $\sum_{i=1}^{k}(x_{\uparrow})_{i}\geq{k\choose 2}$ for all $k\in[n]$ with equality when $k=n$, where $x_{\uparrow}$ is the non-decreasing rearrangement of $x$ (see, e.g., Marshall, Olkin and Arnold [44]). Intuitively, these conditions ensure that $x$ is at most as “spread out” as $v_{n}$ and has the same total sum.

1.3. Coxeter permutahedra

Ardila, Castillo, Eur and Postnikov [5] have recently introduced the so-called Coxeter generalized permutahedra, which extend the class of generalized permutahedra (see Postnikov [49]) to the Coxeter setting. Recall that generalized permutahedra are obtained by deformations of classical permutahedra (e.g., the permutahedra, associahedron, cyclohedron, Pitman–Stanley [60] polytope, etc.) which preserve directions while relocating faces.

As discussed in [5], the permutahedron $\Pi_{n-1}$ (as are a number of other classical combinatorial objects) is related to the symmetric group $S_{n}$, see (1.2) above. However, more generally, if $\Phi\subset V$ is the root system corresponding to some reflection group $W$, then a $\Phi$-permutahedron is obtained as the convex hull of the $W$-orbit of some $v\in V$. More specifically, if $\Phi$ is a root system with an associated positive system $\Phi^{+}\subset\Phi$, then the corresponding $\Phi$-permutahedron is obtained as the Minkowski sum (cf. (1.3))

[TABLE]

Equivalently (cf. (1.2)),

[TABLE]

where

[TABLE]

is the sum of the fundamental weights. See Section 2.1 below for definitions and more details.

In [5], deformations of $\Phi$-permutahedra are studied. It is observed that a number of objects of interest, such as weight polytopes, Coxeter matroids, root cones, etc., are examples of such Coxeter generalized permutahedra.

A main result of [5] shows that hyperplane descriptions of Coxeter generalized permutahedra are described in terms of submodular functions $h$. In root systems of types $B_{n}$ and $C_{n}$, the functions $h$ are bisubmodular. In type $D_{n}$, the functions $h$ are a new class of submodular functions called disubmodular. Using these functions, analogues of Rado’s theorem (1.4) for Coxeter generalized permutahedra are derived, see Theorem 2.22 below.

1.4. Tournaments

A tournament on a graph $G=(V,E)$ on $V=[n]$ is an orientation of its edge set. Informally, we think of each vertex as a player. A game is played between each pair of vertices joined by an edge, which is then directed towards the winner. The associated score sequence $s\in{\mathbb{Z}}^{n}$ is the in-degree sequence, listing the total number of wins by each player.

More generally, a random tournament on $G$ is a collection of probabilities $p_{ij}=1-p_{ji}$ for each edge $ij\in E$. In this context, we think of $p_{ij}$ as the probability that $i$ wins against $j$. The mean score sequence $x\in{\mathbb{R}}^{n}$ lists the expected number of wins by each player, where each coordinate

[TABLE]

1.4.1. Tournaments and permutahedra

There is a strong connection between tournaments and permutahedra. In classical work, Landau [41] showed that $s\in{\mathbb{Z}}^{n}$ is the score sequence of a tournament on $G=K_{n}$ if and only if $s\preceq v_{n}$, and so, by Rado’s theorem (1.4), if and only if $s$ is a lattice point of the permutahedron $\Pi_{n-1}$. Note that $v_{n}$ is the score sequence of the tournament on $K_{n}$ in which each player $i$ wins/loses all games against other players $j\neq i$ of smaller/larger index. Also recall that, informally, $s\preceq v_{n}$ means that $s$ is at most as spread out as $v_{n}$ and has the same total sum.

Extending Landau’s result to the setting of random tournaments, Moon [46] showed that $x\in{\mathbb{R}}^{n}$ is the mean score sequence of a random tournament on $K_{n}$ if and only if $x\preceq v_{n}$.

Although Rado’s work preceded that of Landau’s, it appears that the connection with $\Pi_{n-1}$ was not recorded in the literature until Stanley [56] relayed this observation of Zaslavsky. Indeed, many proofs of these classical theorems of Landau and Moon (by a variety of methods from optimization, probability and combinatorics) have appeared (see, e.g., [41, 46, 1, 2, 4, 13, 6, 7, 8, 12, 14, 16, 17, 18, 19, 25, 26, 27, 29, 42, 36, 43, 51, 62, 53, 47, 23, 64]). However, none have fully exploited the geometric perspective given by the equivalent descriptions (1.2)–(1.4) of the permutahedron.

In recent work [40], we have extended the theorems of Landau and Moon to all multigraphs, via a consolidated, geometric proof which capitalizes on the theory of zonotopal tilings.

Theorem 1.1** ([40]).**

Let $Z_{M}$ be the graphical zonotope of the multigraph $M$ on vertex set $[n]$. Then

1. (1)

$s\in{\mathbb{Z}}^{n}$* is a score sequence of a tournament on $M$ if and only if $s\in{\mathbb{Z}}^{n}\cap Z_{M}$.* 2. (2)

$x\in{\mathbb{R}}^{n}$* is a mean score sequence of a random tournament on $M$ if and only if $x\in Z_{M}$.*

Furthermore, the zonotopal perspective allows for a refinement of these results, which shows that any score sequence can be realized by a tournament with at most a “forest’s worth of randomness,” see [40] for details.

In discussing these results, Stanley [59] asked about extending the theory of tournaments to the Coxeter setting of types $B_{n}$, $C_{n}$ and $D_{n}$. The purpose of the current work is to introduce such a theory and describe its connections with the Coxeter generalized permutahedra [5]. As it turns out, these geometric objects can be described in terms of the mean score sequences of random tournaments on signed graphs (see, e.g., Zaslavsky [68, 69]) in which there are cooperative and solitaire games, in addition to the usual (type $A_{n-1}$) competitive ones.

1.5. Coxeter tournaments

As discussed above, “Coxeter combinatorics” is concerned with extending classical combinatorial objects to the Coxeter setting. In pursuit of this, Zaslavsky [68, 69] defined the notion of a signed graph, which corresponds to the Coxeter analogue of graphs in nonstandard root systems. These objects have been used, e.g., to study subarrangements of the hyperplane arrangement associated to root systems, the class of graphical Coxeter matroids, and the volumes and Ehrhart polynomials of Coxeter permutahedra, etc. They have also found concrete applications. See [5] and references therein.

Let $\Phi$ be a root system of type $A_{n-1},B_{n},C_{n}$ or $D_{n}$. A (simple) signed graph ${\mathcal{G}}$ on vertex set $[n]$ has sets of

• •

negative edges $E^{-}\subseteq{[n]\choose 2}$,

• •

positive edges $E^{+}\subseteq{[n]\choose 2}$,

• •

half-edges $H\subseteq[n]$,

• •

loops $L\subseteq[n]$.

Note that $E^{+}$ and $E^{-}$ are not necessarily disjoint.

Next, we define a (signed) $\Phi$-graph as a certain type of signed graph in one of the root systems $\Phi=A_{n-1}$, $B_{n}$, $C_{n}$ or $D_{n}$. If $\Phi=A_{n-1}$ then it has only negative edges (that is, $E^{+}=H=L=\emptyset$) in which case ${\mathcal{G}}$ is (in bijection with) a classical graph. If $\Phi=B_{n}$ then it has no loops, $L=\emptyset$. If $\Phi=C_{n}$ then it has no half-edges, $H=\emptyset$. If $\Phi=D_{n}$ then it has neither half-edges nor loops, $H=L=\emptyset$.

Let us emphasize here that what we call negative/positive edges are usually instead called positive/negative edges in the literature (see, e.g., [68]). However, the above reversed choice of terminology is more natural in the current context, as negative/positive edges will turn out to represent competitive/collaborative games in the corresponding tournament in which, e.g., it will be natural to think about substracting/adding “strengths,” etc.

The edges in ${\mathcal{G}}$ are associated with a subset $S\subseteq\Phi^{+}$ of a positive system $\Phi^{+}\subset\Phi$. The Coxeter graphical zonotope of $\mathcal{G}$ is the zonotope

[TABLE]

Extending the definition in Section 1.4, we define a random Coxeter tournament on a $\Phi$-graph ${\mathcal{G}}$ as a collection of probabilities:

• •

$p_{ij}^{-}=1-p_{ji}^{-}$ for each negative edge $ij\in E^{-}$,

• •

$p_{ij}^{+}=p_{ji}^{+}$ for each positive edge $ij\in E^{+}$,

• •

$p_{i}^{h}$ for each half-edge $i\in H$,

• •

$p_{i}^{\ell}$ for each loop $i\in L$.

As above, we view the vertices as players and each edge as a certain type of game. Negative edges correspond to the usual competitive games where one player wins and the other loses. Positive edges correspond to cooperative games where both players can win by working together. Half-edges and loops correspond to solitaire games which an individual player can win on their own (the only difference between that loops count for twice as many points). More specifically, in Coxeter tournaments of types $B_{n}$, $C_{n}$ and $D_{n}$, we parametrize the value of games as follows:

• •

In a competitive game (negative edge), one player wins and the other loses $1/2$ point.

• •

In a cooperative game (positive edge), both players win or lose $1/2$ point.

• •

In a half-edge (resp. loop) solitaire game, a player wins or loses $1/2$ point (resp. $1$ point) if in type $B_{n}$ (resp. $C_{n}$).

Hence, the mean score sequence of a random Coxeter tournament is the sequence $x\in{\mathbb{R}}^{n}$ with coordinates (cf. (1.5))

[TABLE]

See Proposition 3.2 below for a concise, geometric description.

It might be helpful, although rather informal, to think of a loop as a “collaborative solitaire” game, so worth twice the points. Likewise, a half-edge could be thought of as a competitive game against an “external player” whose score is not listed in the mean score sequence.

1.6. Moon’s theorem

Moon’s theorem extends to the Coxeter setting as follows, cf. Theorem 1.1 (2).

Theorem 1.2**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let ${\mathcal{G}}$ be a $\Phi$-graph with Coxeter graphical zonotope $Z_{\mathcal{G}}$. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random Coxeter tournament on $\mathcal{G}$ if and only if $x\in Z_{\mathcal{G}}$.

Using the theory of Coxeter generalized permutahedra [5] (see Section 2.1 below for definitions) we can associate to $Z_{\mathcal{G}}$ a certain function $\mu_{\mathcal{G}}$ on the set of rays of the Coxeter arrangement of $\Phi$ that gives a hyperplane description of $Z_{\mathcal{G}}$. More specifically, suppose that $\lambda_{1},\ldots,\lambda_{n}$ are the fundamental weights of $\Phi$. Then $\mu_{\mathcal{G}}:\mathcal{L}\to{\mathbb{R}}$ is the Coxeter submodular function corresponding to ${\mathcal{G}}$, where

[TABLE]

is the set of fundamental weight conjugates and $W$ is the Weyl group of $\Phi$.

Coxeter mean score sequences are equivalently classified as follows (cf. (1.4) in relation to Moon’s Theorem).

Theorem 1.3**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let ${\mathcal{G}}$ be a $\Phi$-graph. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random Coxeter tournament on $\mathcal{G}$ if and only if $\langle\lambda,x\rangle\leq\mu(\lambda)$ for all $\lambda\in\mathcal{L}$.

This second description gives explicit formulas in terms of the edges of $\mathcal{G}$, however, the details depend on the exact root system in use.

1.7. The complete case

Next, we focus on a natural case of interest, that of complete $\Phi$-graphs. The complete $\Phi$-graph $K_{\Phi}$ is the $\Phi$-graph which includes all possible edges. These graphs extend in a natural way the notion of the complete (type $A_{n-1}$) graph $K_{n}$. If $\Phi$ is of type $B_{n}$, $C_{n}$ or $D_{n}$ then all possible positive and negative edges $E^{\pm}={[n]\choose 2}$ are included. Furthermore, if $\Phi$ is of type $B_{n}$ then $H=[n]$ and $L=\emptyset$; if $C_{n}$ then $H=\emptyset$ and $L=[n]$; and if $D_{n}$ then $H=L=\emptyset$.

In the complete case ${\mathcal{G}}=K_{\Phi}$, we provide three proofs of the Coxeter analogue of Moon’s theorem. To this end, we first note (in Section 4.1) that the connection with majorization extends to the complete Coxeter setting, allowing for the following succinct version of Moon’s theorem (Theorems 1.2 and 1.3) in these cases. Recall (see [44]) that for $x,y\in{\mathbb{R}}^{n}$, we say that $x$ is weakly sub-majorized by $y$, and write $x\preceq_{w}y$ if $\sum_{i=1}^{k}(x^{\downarrow})_{i}\leq\sum_{i=1}^{k}(y^{\downarrow})_{i}$ for all $k\in[n]$, where $z^{\downarrow}$ denotes the non-increasing rearrangement of $z$. Note that, if $x\preceq_{w}y$ and moreover $\sum_{i}x_{i}=\sum_{i}y_{i}$, then $x$ is in fact majorized by $y$, written as $x\preceq y$, as discussed above.

Theorem 1.4**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence on the complete signed graph $K_{\Phi}$ if and only if $|x|=(|x_{1}|,\ldots,|x_{n}|)$ is weakly sub-majorized by

[TABLE]

where

[TABLE]

$1_{n}=(1,\ldots,1)\in{\mathbb{R}}^{n}$, and $v_{n}$ is as in (1.1).

1.7.1. Geometric proof

Our first proof of Theorem 1.4 exploits a connection between the Coxeter (signed) analogue [45, 63] of the Birkhoff polytope (the convex body of doubly stochastic matrices) and Coxeter tournaments, and is inspired by one of the proofs of Moon’s classical theorem in the recent work [4]. Roughly speaking, the proof follows by describing a specific tournament $T_{\phi}$ corresponding to each signed permutation $\phi$ which is an extreme point of the Coxeter permutahedron $\Pi_{\Phi}$ (see Definition 2.18 below). This description is intuitive, involving the interpretation of players $i$ with $\phi(i)$ positive/negative as those which are strong/weak. For instance, it type $C_{n}$, in the tournament $T_{\phi}$, player $i$ wins its solitaire game if $\phi(i)>0$. Its competitive (resp. collaborative) games with player $j$ are won if $\phi(i)>\phi(j)$ (resp. $\phi(i)+\phi(j)>0$).

1.7.2. Probabilistic proof

In Section 4.3, we give a probabilistic proof of Theorem 1.4 by Strassen’s coupling theorem [61]. The construction is similar in spirit to (but more involved than) the “football” proof of Moon’s classical theorem in [4]. Weak sub-majorization $x\preceq_{w}y$ is equivalent to inequality $\mu_{x}\preceq_{\rm inc}\mu_{y}$ with respect to the increasing stochastic order on probability measures, where $\mu_{x}$ and $\mu_{y}$ are uniform discrete measures on the multisets $\{x_{1},\ldots,x_{n}\}$ and $\{y_{1},\ldots,y_{n}\}$. For instance, in the case of $C_{n}$, Strassen’s theorem gives sub-probability measures $\mu_{i}$ on $\{1,\ldots,n\}$ with means $\sum_{j}j\mu_{i}(j)=|x_{i}|$. We then extend these to probability measures $\nu_{i}$ on $\{\pm 1,\ldots,\pm n\}$ with means $x_{i}$, in such a way that a Coxeter tournament with mean score sequence $x$ can be constructed for which in all games, the players $i$ involved (two if competitive or cooperative and one if solitaire) score an independent number of points, distributed as $\nu_{i}$.

1.7.3. Algorithmic proof

Our third and final proof of Theorem 1.4 given in Section 4.4 is fully constructive, and can be viewed as a natural continuous Coxeter analogue of the Havel–Hakimi [31, 28] algorithm. This algorithm has an intuitive description in terms of players seeking out potential competitors/collaborators with respect to their relative weakness/strength, and is inspired by an unpublished proof of Moon’s classical theorem by David Aldous [3].

1.8. Paired comparisons

In the classical (type $A_{n-1}$) setting, all irreducible mean score sequences $x$ on the complete graph $K_{n}$ can be realized by the Bradley–Terry [11] model (see, e.g., Theorem 1 in [4]). By “irreducible” we mean that $x$ is strictly majorized by $v_{n}$, written $x\prec v_{n}$, that is, all $\sum_{i=1}^{k}(x_{\uparrow})_{i}>{k\choose 2}$ with equality only when $k=n$. The Bradley–Terry model, arising from the statistical theory of paired comparisons (see, e.g., Cattelan [21]) was in fact introduced and studied in the much earlier work of Zermelo [70] (motivated by the problem of ranking chess players based on incomplete information).

In [4] it is argued that Bradley–Terry is in some sense the Gaussian analogue in the context of tournaments. Indeed, they both arise when maximizing Shannon [54] entropy. Moreover, just as $x\prec v_{n}$ if and only if $x=Av_{n}$ for some positive definite doubly stochastic matrix $A$ (see Chao and Wong [22] and Brualdi, Hwang and Pyo [15]), Gaussian densities are those associated with positive definite covariance matrices.

We say that $x$ is strictly weakly sub-majorized by $y$, and write $x\prec_{w}y$, if for all $k\in[n]$, we have that $\sum_{i=1}^{k}(x^{\downarrow})_{i}<\sum_{i=1}^{k}(y^{\downarrow})_{i}$.

Theorem 1.5**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let $|x|\prec_{w}\rho_{\Phi}$ be an irreducible mean score sequence on the complete signed graph $K_{\Phi}$. Then $x$ can be realized by a random Coxeter tournament of Bradley–Terry form, with $p_{ij}^{\pm}=\varphi(\lambda_{i}\pm\lambda_{j})$ and also $p_{i}^{h}=\varphi(\lambda_{i})$ (resp. $p_{i}^{\ell}=\varphi(\lambda_{i})$) if in type $B_{n}$ (resp. $C_{n}$), where $\varphi(u)=e^{u}/(1+e^{u})$ is the standard logistic function and where $\lambda_{i}\in{\mathbb{R}}$ is the “strength” of player $i$.

Theorem 1.5 essentially follows by maximizing the entropy of a Coxeter tournament, subject to the contraints (1.7) that each player $i$ has mean score $x_{i}$. For instance, in type $C_{n}$, the entropy is

[TABLE]

As it turns out, the strengths $\lambda_{i}$ are the Lagrange multipliers in this optimization problem. Hence the closure of the set of all mean score sequences of Coxeter random tournaments of Bradley–Terry form is the set of all possible mean score sequences, that is, the Coxeter permutahedron $\Pi_{\Phi}$ (see Definition 2.18 below). We omit the proof, since the details are similar to the proof of Theorem 1 in [4].

Finally, let us note that the logistic function is not the only possible choice above. For instance, replacing $\varphi$ with the Gaussian cumulative distribution function leads to the Thurstone–Mosteller model [65, 48]. See, e.g., the discussion following Theorem 2.7 in Joe [37] for more details.

1.9. Landau’s theorem

Recall (see Theorem 1.1 (1) above) that, for classical (type $A_{n-1}$) graphs $G$, all possible integer score sequences $s\in{\mathbb{Z}}^{n}$ can be realized deterministically. That is, $s\in{\mathbb{Z}}^{n}\cap Z_{G}$ if and only if there is a tournament with all $p_{ij}\in\{0,1\}$ and score sequence $s$. In this section, we discuss an extension of Landau’s theorem for Coxeter tournaments that holds when the $\Phi$-graph ${\mathcal{G}}$ is, in a certain sense, balanced.

Before stating our results, let us discuss some of the subtleties involved with distinguishing between integer and deterministic score sequence in the Coxeter setting. One might expect a classification of score sequences of deterministic Coxeter tournaments on $\mathcal{G}$ in terms of the weight lattice points in $Z_{\mathcal{G}}$. The first issue with this idea is that $Z_{\mathcal{G}}$ is not a lattice polytope in the root lattice. However, this is easily remedied by considering a translation (cf. (1.6))

[TABLE]

of $Z_{\mathcal{G}}$ by

[TABLE]

Then it follows directly that $x+\rho_{\mathcal{G}}$ is an integer lattice point of $Z_{\mathcal{G}}^{\rm tr}$ whenever $x$ is a deterministic score sequence on $\mathcal{G}$. However, even with this modification, the converse fails in general, as the following examples show.

Example 1.6**.**

Consider ${\mathcal{G}}$ with no solitaire games and a competitive and cooperative game between a pair of players. There are four deterministic tournaments obtained by setting each of $p_{12}^{\pm}$ to [math] or $1$. After translation by $\rho=(1/2,1/2)$, the score sequences are $(0,1)$, $(1,0)$, $(1,2)$ and $(2,1)$. However, notice that $Z_{\mathcal{G}}+\rho$ also contains the lattice point $(1,1)$, corresponding to the random tournament with $p_{12}^{\pm}=1/2$.

As another example, consider ${\mathcal{G}}$ consisting of a single solitaire loop. After translation by $\rho=(1)$, the two score sequences are $(0)$ and $(2)$. However, the random tournament with $p_{11}=1/2$ has mean score sequence $(1)$.

As these examples suggest, loops and cycles with an odd number of positive edges can prevent Landau’s theorem from extending. The case of half-edges, as we will see, is more subtle.

As in [68, 69], we make the following definition.

Definition 1.7**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. A $\Phi$-graph ${\mathcal{G}}$ is balanced if it contains no half-edges, loops and all cycles have an even number of positive edges.

In particular, if ${\mathcal{G}}$ is balanced then $E^{-}\cap E^{+}=\emptyset$ are disjoint. (Also recall that, in this work, we call negative/positive edges what are usually referred to as positive/negative edges in the literature. As such, in the literature, “balanced” is usually defined to mean that the product of signs along any cycle is positive.)

Part (1) in our next result shows that Landau’s theorem extends when ${\mathcal{G}}$ is balanced. In fact, in type $B_{n}$, it is possible to add half-edges. Part (2) is a partial converse, which shows that balanced is necessary when there are no half-edges.

Theorem 1.8**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let ${\mathcal{G}}$ be a $\Phi$-graph and let $Z_{\mathcal{G}}^{\rm tr}$ be the translated Coxeter graphical zonotope of $\mathcal{G}$ as in (1.9) above. Let ${\mathcal{G}}^{\prime}$ be the subgraph of ${\mathcal{G}}$ obtained by removing any half-edges.

1. (1)

If ${\mathcal{G}}^{\prime}$ is balanced then $s$ is a score sequence of a deterministic Coxeter tournament on ${\mathcal{G}}$ if and only if $s+\rho_{\mathcal{G}}\in{\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$. 2. (2)

On the other hand, if ${\mathcal{G}}={\mathcal{G}}^{\prime}$ is unbalanced (that is, if ${\mathcal{G}}$ has no half-edges and at least one loop or cycle with an odd number of positive edges), then there are integer vectors $t\in{\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$ which can only be realized randomly, that is, as $t=x+\rho_{\mathcal{G}}$ for some mean score sequence $x$ of a random Coxeter tournament on ${\mathcal{G}}$.

For a $\Phi$-graph ${\mathcal{G}}$ of type $B_{n}$, $C_{n}$ or $D_{n}$, let $S_{\mathcal{G}}$ denote the set of deterministic score sequences on ${\mathcal{G}}$. Then $S_{\mathcal{G}}+\rho_{\mathcal{G}}\subseteq{\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$. By Theorem 1.8, in types $C_{n}$ and $D_{n}$, we have that $S_{\mathcal{G}}+\rho_{\mathcal{G}}={\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$ if and only if ${\mathcal{G}}$ is balanced (since these types have no half-edges, and so ${\mathcal{G}}={\mathcal{G}}^{\prime}$). On the other hand, in type $B_{n}$, it is possible to have $S_{\mathcal{G}}+\rho_{\mathcal{G}}={\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$ even when ${\mathcal{G}}^{\prime}$ is unbalanced. For instance, consider the two-player Coxeter tournament involving a competitive, cooperative and half-edge solitaire game.

The following questions remain open.

Problem 1.9**.**

Let ${\mathcal{G}}$ be a $\Phi$-graph of type $B_{n}$, $C_{n}$ or $D_{n}$.

1. (1)

Describe the set $S_{\mathcal{G}}$ of deterministic score sequences on ${\mathcal{G}}$. 2. (2)

Determine when $S_{\mathcal{G}}+\rho_{\mathcal{G}}={\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$.

The answer to (2) is “balanced” in types $C_{n}$ and $D_{n}$, but in type $B_{n}$ the situation is less clear. Part (1) seems interesting even in the complete case ${\mathcal{G}}=K_{\Phi}$.

1.10. Notation

As usual, we let $[n]=\{1,2,\ldots,n\}$. Similarly, we put $[-n]=-[n]=\{-1,-2,\ldots,-n\}$ and $[\pm n]=[n]\cup[-n]=\{\pm 1,\pm 2,\ldots,\pm n\}$.

1.11. Acknowledgements

We thank David Aldous, Federico Ardila and Richard Stanley for useful discussions. MS was partially supported by the National Science Foundation under Award No. DMS-2103391

2. Background

In this section, we briefly discuss some background information used in this work. In Section 2.1 we recall some basic facts about root systems. We refer the reader to, e.g., Humphreys [35] for proofs and more details. In Section 2.2 we discuss Zaslavsky’s [68, 69] theory of signed graphs, which are the natural setting in which to extend the classical theory of graph tournaments. Finally, in Section 2.3 we discuss the Coxeter generalized permutahedra developed recently by Ardila, Castillo, Eur and Postnikov [5], which describe the geometry of Coxeter tournaments.

2.1. Root systems

Throughout, we let $V$ be a Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$. We usually take $V$ to be ${\mathbb{R}}^{n}$ with the standard orthonormal basis. Any vector $v\in V$ determines an automorphism $s_{v}$ of $V$ given by

[TABLE]

Definition 2.1**.**

A (crystallographic) root system is a finite collection of vectors $\Phi\subset V$ such that the following properties hold:

1. (1)

$\operatorname{span}(\Phi)=V$, 2. (2)

if $\alpha\in\Phi$ then the only other multiple of $\alpha$ in $\Phi$ is $-\alpha$, 3. (3)

$\Phi$ is closed under all automorphisms $s_{\alpha}$, $\alpha\in\Phi$, and 4. (4)

for all $\alpha,\beta\in\Phi$, we have that $2\frac{\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\in{\mathbb{Z}}$.

Vectors $\alpha\in\Phi$ are called roots.

The direct sum of two root systems $\Phi_{1}$ and $\Phi_{2}$ is defined as

[TABLE]

A root system is irreducible if it is not the direct sum of root systems.

Theorem 2.2** (Killing [39], Cartan [20]).**

The irreducible (crystallographic) root systems are classified (up to isomorphism) as the infinite families $A_{n-1}$, $B_{n}$, $C_{n}$ and $D_{n}$ and the exceptional types $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ and $G_{2}$.

Example 2.3**.**

The infinite families are:

• •

$A_{n-1}=\{e_{i}-e_{j}:i\not=j\in[n]\}$,

• •

$B_{n}=\{\pm e_{i}\pm e_{j}:i\not=j\in[n]\}\cup\{\pm e_{i}:i\in[n]\}$,

• •

$C_{n}=\{\pm e_{i}\pm e_{j}:i\not=j\in[n]\}\cup\{\pm 2e_{i}:i\in[n]\}$,

• •

$D_{n}=\{\pm e_{i}\pm e_{j}:i\not=j\in[n]\}$.

where the $e_{i}$ form the standard orthonormal basis of ${\mathbb{R}}^{n}$. One difficulty arises in the definition of the root system $A_{n-1}$. Since all of the vectors of $A_{n-1}$ lie in the subspace ${\mathbb{R}}_{0}^{n}=\{x\in{\mathbb{R}}^{n}:\sum_{i}x_{i}=0\}$, we must choose $V$ to be this subspace in order for condition (1) in Definition 2.1 to hold.

Definition 2.4**.**

The Weyl group $W$ of $\Phi$ is the group generated by the reflections $\{s_{\alpha}:\alpha\in\Phi\}$.

Example 2.5**.**

Examples of Weyl groups are:

• •

In type $A_{n-1}$, the Weyl group is isomorphic to $S_{n}$.

• •

In types $B_{n}$ and $C_{n}$, the Weyl groups are isomorphic to the group of signed permutations $S_{n}^{\pm}$ of $[n]$. Recall that elements of $S_{n}^{\pm}$ are bijections of $[\pm n]$ (see Section 1.10) such that $\phi(-i)=-\phi(i)$.

• •

In type $D_{n}$, the Weyl group is isomorphic to a subgroup of $S_{n}^{\pm}$ consisting of the signed permutations such that $|\{i\in[n]:\phi(i)<0\}|$ is even.

Definition 2.6**.**

Let $\Phi\subset V$ be a root system. A positive system of $\Phi$ is a subset of $\Phi^{+}\subset\Phi$ with the property that there exists a linear functional $h\in V^{*}$ (where $V^{*}$ is the dual space of $V$) such that $h(\alpha)>0$ for all $\alpha\in\Phi^{+}$.

For any given root system, all choices of positive systems are equivalent up to conjugation by elements of the Weyl group. For this reason, it usually suffices to consider one choice of positive system.

Example 2.7**.**

We will use the following choices of $\Phi^{+}$ for the root systems in Example 2.3 are as follows:

• •

$A_{n-1}^{+}=\{e_{i}-e_{j}:i>j\in[n]\}$,

• •

$B_{n}^{+}=\{e_{i}\pm e_{j}:i>j\in[n]\}\cup\{e_{i}:i\in[n]\}$,

• •

$C_{n}^{+}=\{e_{i}\pm e_{j}:i>j\in[n]\}\cup\{2e_{i}:i\in[n]\}$,

• •

$D_{n}^{+}=\{e_{i}\pm e_{j}:i>j\in[n]\}$.

Definition 2.8**.**

Let $\Phi\subset V$ be a root system and $\Phi^{+}\subset\Phi$ a positive system. The simple system $\Delta$ of $\Phi^{+}$ is the minimal collection of vectors such that every $\alpha\in\Phi^{+}$ is a positive linear combination of vectors in $\Delta$. A simple system of $\Phi$ is a subset of roots which is a simple system of some positive system $\Phi^{+}\subset\Phi$.

As with positive systems, all simple systems of $\Phi$ are equivalent up to conjugation by elements of the Weyl group.

Definition 2.9**.**

Let $\Phi\subset V$ be a root system and $\Delta=\{\alpha_{1},\ldots,\alpha_{n}\}$ a simple system of $\Phi$. The fundamental weights of $\Phi$ associated with $\Delta$ are the elements $\lambda_{1},\ldots\lambda_{n}$ of (the dual space) $V^{*}$ defined by

[TABLE]

where $\alpha_{j}^{\vee}=\frac{2}{\langle\alpha_{j},\alpha_{j}\rangle}\alpha_{j}$ is the coroot of $\alpha_{j}$. The weight lattice of $\Phi$ is the lattice generated by integer combinations of the fundamental weights.

Example 2.10**.**

Once we identify $({\mathbb{R}}^{n})^{*}$ with ${\mathbb{R}}^{n}$, the fundamental weights of the infinite families are:

• •

$\overline{e_{1}},\overline{e_{1}}+\overline{e_{2}},\ldots,\overline{e_{1}}+\cdots+\overline{e_{n}}$ in $A_{n-1}$,

• •

$e_{1},e_{1}+e_{2},\ldots,e_{1}+\cdots+e_{n-1},(e_{1}+\cdots+e_{n})/2$ in $B_{n}$,

• •

$e_{1},e_{1}+e_{2},\ldots,e_{1}+\cdots+e_{n-1},e_{1}+\cdots+e_{n}$ in $C_{n}$,

• •

$e_{1},e_{1}+e_{2},\ldots,e_{1}+\cdots+e_{n-2},(e_{1}+\cdots+e_{n-1}\pm e_{n})/2$ in $D_{n}$.

For type $A_{n-1}$, the $\overline{e_{i}}$ are the representatives of $e_{i}$ in the quotient ${\mathbb{R}}^{n}/{\mathbb{R}}1_{n}$ where $1_{n}=(1,\ldots,1)\in{\mathbb{R}}^{n}$. Notice that the roots of type $A_{n-1}$ all lie in the subspace of ${\mathbb{R}}^{n}$ where the coordinates sum to [math]. Hence, the span is not full dimensional. Likewise, the weights live in the dual space to $V$, where $V$ is the ambient space of the root system. In the type $A_{n-1}$ case, the dual is ${\mathbb{R}}^{n}/{\mathbb{R}}1_{n}$.

Definition 2.11**.**

Let $\Phi$ be a root system with fundamental weights $\lambda_{1},\ldots,\lambda_{n}$. Let ${\mathcal{L}}$ as in (1.8) above denote the set of fundamental weight conjugates.

Definition 2.12**.**

We say that a subset $S\subseteq[\pm n]$ is admissible if $\{i,-i\}\not\subseteq S$ for all $i\in[n]$. For such a set, we let $S_{+}=S\cap[n]$ and $S_{-}=S\cap[-n]$. Note that $S_{+}\cap(-S_{-})=\emptyset$.

Example 2.13**.**

Continuing with Example 2.10, in these cases $\mathcal{L}$ can be understood in terms of certain types of subsets. For any signed subset $S\subseteq[\pm n]$ let

[TABLE]

• •

For type $A_{n-1}$, the fundamental weight conjugates are in bijection with subsets $S\subseteq[n]$ as follows:

[TABLE]

• •

For type $B_{n}$, the fundamental weight conjugates are in bijection with admissible subsets $S\subseteq[\pm n]$ as follows:

[TABLE]

• •

For type $C_{n}$, the fundamental weight conjugates are in bijection with admissible subsets $S\subseteq[\pm n]$ as follows:

[TABLE]

• •

For type $D_{n}$, the fundamental weight conjugates are in bijection with admissible subsets $S\subseteq[\pm n]$ with $|S|\not=n-1$ as follows:

[TABLE]

2.2. Signed graphs

Zaslavsky’s theory of signed graphs [68, 69] extends the classical theory of graphs to the Coxeter setting.

Definition 2.14**.**

A (simple) signed graph ${\mathcal{G}}$ on a vertex set $[n]$ is a tuple ${\mathcal{G}}=([n],E^{-},E^{+},H,L)$, where

• •

$E^{\pm}\subseteq{[n]\choose 2}$ are sets of positive and negative edges,

• •

$H\subseteq[n]$ is a set of half-edges, and

• •

$L\subseteq[n]$ is a set of loops.

We will be interested in particular bijections between signed graphs and positive roots in a root system.

Definition 2.15**.**

Let $\Phi$ be a root system of type $A_{n-1}$, $B_{n}$, $C_{n}$ or $D_{n}$ with the canonical choice of positive systems in Example 2.7. If a $\Phi$-signed graph ${\mathcal{G}}$ is of type $B_{n}$ it has no loops; if of type $C_{n}$ it has no half-edges; and if of type $D_{n}$ it has neither half-edges nor loops. If ${\mathcal{G}}$ is of type $A_{n-1}$ then it has no positive edges, half-edges nor loops, and so is (in bijection with) a simple graph.

For a $\Phi$-signed graph $\mathcal{G}$, let $\Gamma$ denote the bijection between the edges

[TABLE]

and its associated subset of the positive system $\Phi^{+}\subset\Phi$ given by:

• •

in type $A_{n-1}$,

[TABLE]

• •

in type $B_{n}$,

[TABLE]

• •

in type $C_{n}$,

[TABLE]

• •

in type $D_{n}$,

[TABLE]

We note here that, although the $\Phi$-graphs of the root systems of types $B_{n}$ and $C_{n}$ are in bijection, the map $\Gamma$ is different for these objects, since they are subsets of different sets of vectors. This difference will further manifest itself in the theory of tournaments on these signed graphs.

Definition 2.16**.**

The complete $\Phi$-graph $K_{\Phi}$ is the $\Phi$-graph where ${\mathcal{G}}=\Phi^{+}$.

In particular, note that $K_{A_{n-1}}=K_{n}$ is the usual complete graph.

2.3. Coxeter generalized permutahedra

As discussed in Section 1.3 above, the notion of generalized permutahedra [49] has recently been extended to other root systems [5].

Definition 2.17**.**

Let $\Phi\subset V$ be a root system. A $\Phi$-generalized permutahedron is a polytope whose edge directions are parallel to roots in $\Phi$.

One of the main examples of Coxeter generalized permutahedra are the Coxeter permutahedra.

Definition 2.18**.**

Let $\Phi$ be a root system with a positive system $\Phi^{+}$. Then the corresponding $\Phi$-permutahedron is the Minkowski sum

[TABLE]

Equivalently, the $\Phi$-permutahedron is the polytope

[TABLE]

where

[TABLE]

is the sum of the fundamental weights.

Notice that in type $A_{n-1}$ the $\Phi$-permutahedron is equal (up to a translation) to the permutahedron $\Pi_{n-1}$ defined above in (1.2)–(1.4).

The relevant examples of a Coxeter generalized permutahedra for this paper are the polytopes associated to the $\Phi$-graphs first studied by Zaslavsky in the context of signed graphs.

Definition 2.19**.**

Let ${\mathcal{G}}$ be a $\Phi$-graph. The Coxeter graphical zonotope of ${\mathcal{G}}$ is the $\Phi$-generalized permutahedron $Z_{\mathcal{G}}$ given by the Minkowski sum

[TABLE]

recalling (see Definition 2.15) that $\Gamma$ denotes the bijection from ${\mathcal{G}}$ to the positive roots of $\Phi$.

Notice that the Coxeter graphical zonotope of the complete $\Phi$-graph $K_{\Phi}$ is the $\Phi$-permutahedron $\Pi_{\Phi}$.

An important aspect of Coxeter generalized permutahedra is that their hyperplane descriptions are given in terms of submodular functions.

Definition 2.20**.**

For a root $\alpha\in\Phi$, let

[TABLE]

be the hyperplane defined by $\alpha$. The collection of hyperplanes $H_{\alpha}$ is called the Coxeter arrangement. It defines a simplicial fan $\Sigma_{\Phi}$.

The rays of the fan $\Sigma_{\Phi}$ are exactly the fundamental weight conjugates ${\mathcal{L}}$ as in (1.8) above. Any function on the rays defines a function on the cones of $\Sigma_{\Phi}$ by extending it linearly on each cone. This gives a piecewise linear function.

Definition 2.21**.**

Let $\Phi$ be a root system and $\mathcal{L}$ denote the set of fundamental weight conjugates. A $\Phi$-submodular function is a function $h:\mathcal{L}\to{\mathbb{R}}$ such that

[TABLE]

where we consider $h$ as a piecewise linear function on the cones.

In type $A_{n-1}$ the functions $h$ corresponds to usual submodular functions $f$ on $[n]$ such that $f([n])=0$. On the other hand, in types $B_{n}$ and $C_{n}$ they correspond to “bisubmodular” functions, and in type $D_{n}$ to a new type called “disubmodular.” See Section 5.2 in [5] for more information.

The most important consequence for us is that the Coxeter submodular function $h$ gives a hyperplane description of its corresponding Coxeter generalized permutahedron.

Theorem 2.22** ([5], Section 5.1).**

If $h$ is a $\Phi$-submodular function, then the polytope

[TABLE]

where ${\mathcal{L}}$ as in (1.8) is the set of fundamental weight conjugates, is a $\Phi$-generalized permutahedron. On the other hand, if $P$ is a $\Phi$-generalized permutahedron, then

[TABLE]

is a $\Phi$-submodular function. Furthermore, the assignments $h\mapsto P_{h}$ and $P\mapsto h_{P}$ are inverses and thus bijections.

Example 2.23**.**

For instance, the submodular function $h$ associated to the type $C_{n}$ permutahedron $\Pi_{C_{n}}$ is the function given by

[TABLE]

In order words, $x\in\Pi_{C_{n}}$ if and only if for any $k$ distinct indices we have

[TABLE]

that is, if and only if $|x|$ is weakly sub-majorized by

[TABLE]

as in Theorem 1.4. Note that the right hand side in (2.1) is equal to

[TABLE]

which is the maximum number of points that any given set $S\subset[n]$ of $k$ players can win in total. Indeed, there are $n-k$ competitive games (worth $1/2$ point) with exactly one player in $S$. Points from competitive games between players in $S$ cancel. There are $n-k$ cooperative games (worth $1/2$ point) with exactly one player in $S$, and ${k\choose 2}$ cooperative games (worth $2\cdot(1/2)=1$ point) with both players in $S$. Finally, there are $k$ solitaire games (worth $1$ point) in $S$.

3. Coxeter tournaments

In this section, we extend the theory of graph tournaments to the setting of signed graphs. In Section 3.1, we prove an analogue of Moon’s theorem, which extends the connection between graphical zonotopes and tournaments established by Theorem 1.1 in the classical setting. In Section 3.2, we discuss the issues with extending Landau’s theorem to the Coxeter setting.

As discussed in Section 1.5, we proceed as follows.

Definition 3.1**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let $\mathcal{G}$ be a $\Phi$-graph. A random $\Phi$-tournament $T$ on ${\mathcal{G}}$ is a collection of probabilities

• •

$p^{-}_{ij}=1-p^{-}_{ji}$ for each negative edge $ij\in E^{-}$,

• •

$p^{+}_{ij}=p^{+}_{ji}$ for each positive edge $ij\in E^{+}$,

• •

$p_{i}^{h}$ for each half-edge $i\in H$ (only in type $B_{n}$),

• •

$p_{i}^{\ell}$ for each loop $i\in L$ (only in type $C_{n}$).

The corresponding mean score sequence of $T$ is the vector $x\in{\mathbb{R}}^{n}$ with coordinates

[TABLE]

Intuitively, $\Phi$-tournaments have three different type of games. In types $B_{n}$, $C_{n}$ and $D_{n}$ each negative edge represents a competitive game, in which one player wins and the other loses $1/2$ point. On the other hand, each positive edge represents a cooperative game in both players win or lose $1/2$ point. In types $B_{n}$ and $C_{n}$ there are also solitaire games in which a player wins or loses, $1/2$ point in type $B_{n}$ and $1$ point in type $C_{n}$.

A direct calculation gives the following interpretation in terms of the underlying root system. Recall that $\Gamma$ is the bijection between $E({\mathcal{G}})$ and its associated subset of the positive system $\Phi^{+}$.

Proposition 3.2**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let ${\mathcal{G}}$ be a $\Phi$-graph. Then the mean score sequence $x$ of $T$ is given by

[TABLE]

3.1. Extending Moon’s theorem

One of the classical results in the theory of tournaments is Moon’s theorem [46], which classifies the set of $x\in{\mathbb{R}}^{n}$ which are mean score sequences of tournaments on the complete graph $K_{n}$. We now give a generalization of this result in the setting of Coxeter tournaments.

Theorem 3.3**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let $\mathcal{G}$ be a $\Phi$-graph. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random $\Phi$-tournament on $\mathcal{G}$ if and only if $\langle\lambda,x\rangle\leq h_{\mathcal{G}}(\lambda)$ for all $\lambda\in\mathcal{L}$, where $h_{\mathcal{G}}=h_{Z_{\mathcal{G}}}$ is the $\Phi$-submodular function corresponding to the Coxeter graphical zonotope $Z_{\mathcal{G}}$.

Proof.

Let $C_{\mathcal{G}}=[0,1]^{|{\mathcal{G}}|}$ be the $|{\mathcal{G}}|$-dimensional unit cube indexed by $\alpha\in{\mathcal{G}}$. The zonotope $Z_{\mathcal{G}}$ is the image of $C_{\mathcal{G}}$ under the map ${\mathbb{E}}:C_{\mathcal{G}}\to{\mathbb{R}}^{n}$ given by

[TABLE]

The points of $C_{\mathcal{G}}$ are in bijection with random $\Phi$-tournaments on ${\mathcal{G}}$, and ${\mathbb{E}}(\{p_{e}\})$ is precisely the mean score sequence of the $\Phi$-tournament $T=\{p_{e}\}$. Therefore, $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random $\Phi$-tournament on ${\mathcal{G}}$ if and only if it is in the image of $C_{\mathcal{G}}$ under ${\mathbb{E}}$ (that is, in $Z_{\mathcal{G}}$). Hence, applying Theorem 2.22, the result follows. ∎

Next, we derive a “signed” version of this result, by identifying the $\Phi$-submodular function $h_{\mathcal{G}}$ corresponding to the Coxeter graphical zonotope $Z_{\mathcal{G}}$. To do this, recall (see Example 2.10) that the fundamental weight conjugates $\mathcal{L}$ can be viewed as admissible subsets $S\subseteq[\pm n]$. Recall that (see Definition 2.12) that for such a set $S$, we let $S_{+}=S\cap[n]$ and $S_{-}=S\cap[-n]$.

Definition 3.4**.**

For any admissible subset $S\subset[\pm n]$, let $S^{||}\subseteq[n]$ denote the set given by $S_{+}\cup(-S_{-})$.

The following result is obtained by direct calculations. We omit the proof.

Proposition 3.5**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let $\mathcal{G}$ be a $\Phi$-graph. For any subset $S\subseteq[n]$, let

• •

${\mathcal{E}}_{k}^{\pm}(S)$* denote the number positive/negative edges in $E^{\pm}$ with exactly $k\in\{1,2\}$ endpoints in $S$.*

• •

${\mathcal{H}}(S)=|S\cap H|$* denote the number of half-edges in $S$.*

• •

${\mathcal{L}}(S)=|S\cap L|$* denote the number of loops in $S$.*

The $\Phi$-submodular function $h_{\mathcal{G}}=h_{Z_{\mathcal{G}}}$ acts on admissible sets $S\subset[\pm n]$ as follows. We have that $h_{\mathcal{G}}(S)$ is equal to

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S^{||})+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S^{||})+{\mathcal{E}}_{2}^{+}(S^{||})+\frac{1}{2}{\mathcal{H}}(S^{||})$* in $B_{n}$,*

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S^{||})+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S^{||})+{\mathcal{E}}_{2}^{+}(S^{||})+{\mathcal{L}}(S^{||})$* in $C_{n}$,*

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S^{||})+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S^{||})+{\mathcal{E}}_{2}^{+}(S^{||})$* in $D_{n}$.*

As a result, we obtain the following equivalent version of Theorem 3.3 (cf. Example 2.23).

Corollary 3.6**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Let $\mathcal{G}$ be a $\Phi$-graph. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random $\Phi$-tournament on $\mathcal{G}$ if and only if for any subset $S=\{i_{1},\ldots,i_{k}\}\subseteq[n]$ we have that $\sum_{j=1}^{k}|x_{i_{j}}|$ is at most

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S)+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S)+{\mathcal{E}}_{2}^{+}(S)+\frac{1}{2}{\mathcal{H}}(S)$* in $B_{n}$,*

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S)+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S)+{\mathcal{E}}_{2}^{+}(S)+{\mathcal{L}}(S)$* in $C_{n}$,*

• •

$\frac{1}{2}{\mathcal{E}}_{1}^{-}(S)+\frac{1}{2}{\mathcal{E}}_{1}^{+}(S)+{\mathcal{E}}_{2}^{+}(S)$* in $D_{n}$.*

3.2. Extending Landau’s theorem

In this section, we prove Theorem 1.8, which gives a partial description of deterministic score sequences in the Coxeter setting. Recall the definitions in Section 1.9 above.

We begin by proving Theorem 1.8 (2), which states that if ${\mathcal{G}}$ has no half-edges, then Landau’s theorem does not extend when ${\mathcal{G}}$ is unbalanced.

Proof of Theorem 1.8 (2).

Suppose that ${\mathcal{G}}$ has no half-edges and a loop or cycle ${\mathcal{C}}\subset{\mathcal{G}}$ with an odd number of positive edges. Consider the Coxeter random tournament which assigns probabilities $1/2$ on ${\mathcal{C}}$, and puts probability 1 everywhere else in ${\mathcal{G}}$. We claim that there is no deterministic tournament with the same mean score sequence $x$. In fact, there is no such tournament whose score sequence has the same total sum as $x$.

To see this, first note that changing the probability of any negative edge in ${\mathcal{G}}$ has no affect on the total sum of the mean score sequence, since this only shifts points between the endpoints. On the other hand, changing the probability of a positive edge in ${\mathcal{C}}$ from $1/2$ to [math] (resp. $1$) decreases (resp. increases) the total sum by $1$. On the other hand, changing the probability of a loop or positive edge outside of ${\mathcal{C}}$ from $1$ to [math] decreases the total sum by $2$.

Since ${\mathcal{C}}$ has an odd number of positive edges, any reassignment of probabilities along ${\mathcal{C}}$ to make it deterministic will result in increasing or decreasing the total sum of the mean score sequence by an odd number. There is no way to compensate for this by changing some probabilities from 1 to 0 on loops and positive edges outside of ${\mathcal{C}}$, since any such reassignment results in decreasing the total sum by an even number. ∎

The rest of this section is devoted to the proof of Theorem 1.8 (1). Recall that we let ${\mathcal{G}}^{\prime}$ denote the subgraph of ${\mathcal{G}}$ obtained by removing any half-edges. We will show that if ${\mathcal{G}}^{\prime}$ is balanced, then any integer lattice point in the translated graphical zonotope $t\in{\mathbb{Z}}^{n}\cap Z_{\mathcal{G}}^{\rm tr}$ can be realized as $t=s+\rho_{\mathcal{G}}$, for some score sequence $s$ of a deterministic Coxeter tournament on ${\mathcal{G}}$.

We recall that the notion of a balanced signed graph appears in Definition 1.7 above, and that there are many other characterizations. For us, the main utility is the following result of Heller and Tompkins [32] and Hoffman and Gale [33] regarding the incidence matrix of a signed graph. For a signed graph $\mathcal{G}$ on vertex set $[n]$, let $I(\mathcal{G})$ be a matrix whose columns are the roots corresponding to each of the edges of $\mathcal{G}$. We have only defined this matrix up to a reordering of the columns, but that will be immaterial for us. Recall that a matrix is totally unimodular if every minor has determinant $\pm 1$.

Theorem 3.7** ([32, 33]).**

Let $\mathcal{G}$ be a signed graph and ${\mathcal{G}}^{\prime}$ the subgraph of ${\mathcal{G}}$ obtained by removing any half-edges. Then ${\mathcal{G}}^{\prime}$ is balanced if and only if $I({\mathcal{G}})$ is totally unimodular.

We will prove Theorem 1.8 (1) by studying the mixed subdivisions of the Coxeter graphical zonotope $Z_{\mathcal{G}}$. This is a natural extension of the proof given in [40].

Definition 3.8**.**

A zonotopal subdivision of a zonotope $P$ is a collection of zonotopes $\{P_{i}\}$ such that $\bigcup_{i}P_{i}=P$ and any two zonotopes $P_{i}$ and $P_{j}$ intersect properly (that is, $P_{i}$ and $P_{j}$ intersect at a face of both, or else not at all) and their intersection is also in the collection $\{P_{i}\}$. We call the zonotopes $P_{i}$ the tiles of the subdivision.

The following is a classical result of Shepard. Recall that

[TABLE]

is the zonotope generated by the collection of vectors $v_{1},\ldots,v_{k}$.

Theorem 3.9** (Shepard [55]).**

Let $Z=Z(v_{1},\ldots,v_{k})$ be a zonotope generated by the vectors $v_{1},\ldots,v_{k}$. For any subset $S$ of these vectors, let $Z_{S}$ denote the zonotope generated by the vectors in $S$. Then, there exists a zonotopal subdivision of $Z$ where the tiles are the zonotopes $Z_{S}$, where $S$ is a linearly independent subset of $\{v_{1},\ldots,v_{k}\}$.

Stanley calculated the Ehrhart polynomial of any zonotope. In particular, this describes the number of lattice points in a lattice zonotope.

Theorem 3.10** (Stanley [57]).**

Let $Z=Z(v_{1},\ldots,v_{k})$ be the zonotope generated by the integer vectors $v_{1},\ldots,v_{k}$. Then the number of integer lattice points in $Z$ is given by the sum

[TABLE]

where $S$ ranges over all linearly independent subsets of $\{v_{1},\ldots,v_{k}\}$ and $m(S)$ is the absolute value of the greatest common divisor of all minors of size $|S|$ in the matrix whose columns are the vectors in $S$.

Therefore, in particular, if the matrix with column vectors $v_{1},\ldots,v_{k}$ is totally unimodular, then every tile in the zonotopal subdivision has no interior lattice points.

Definition 3.11**.**

Let $P=P_{1}+\cdots+P_{k}$ be the Minkowski sum of polytopes. A mixed cell (or Minkowski cell) $\sum_{i}B_{i}$ is a Minkowski sum of polytopes, where the vertices of $B_{i}$ are contained in the vertices of $P_{i}$. A mixed subdivision of $P$ is a collection of mixed cells which cover $P$ and intersect properly (that is, for any two mixed cells $\sum B_{i}$ and $\sum B_{i}^{\prime}$ the polytopes $B_{i}$ and $B_{j}^{\prime}$ intersect at a face of both, or else not at all).

For zonotopes, every zonotopal subdivision is a mixed subdivision, and vice-versa (see, e.g., De Loera, Rambau and Santos [24, Lemma 9.2.10]). This means that every tile of a zonotopal subdivision of $Z^{\rm tr}_{\mathcal{G}}$ is a Minkowski sum of the faces of the segments $[0,\alpha_{i}]$ where $\alpha_{i}$ are the roots that correspond to the edges of $\mathcal{G}$. The faces of these segments are either the points $\{0\}$ and $\{\alpha_{i}\}$, or else the entire segment $[0,\alpha_{i}]$.

Let $Z_{S_{1}},\ldots,Z_{S_{k}}$ be the tiles in a mixed subdivision of $Z^{\rm tr}_{\mathcal{G}}$, where each $S_{i}$ is a linearly independent subset of the roots corresponding to the edges of $\mathcal{G}$. Then, by the previous argument, for every $S_{i}$, there exists a partition $U_{i}\cup V_{i}\cup S_{i}=\Gamma(\mathcal{G})$ such that

[TABLE]

We can now complete the proof of Theorem 1.8.

Proof of Theorem 1.8 (1).

Let $s$ be a mean score sequence of $\mathcal{G}$ and put $t=s+\rho_{\mathcal{G}}$. By Theorem 3.3, we have $s\in Z_{\mathcal{G}}$ and so $t\in Z^{\rm tr}_{\mathcal{G}}$. Consider a zonotopal subdivison of $Z^{\rm tr}_{\mathcal{G}}$ and let $Z_{S}$ be one of the tiles containing the point $t$. As noted above, there is a partition $U\cup V\cup S=\Gamma(\mathcal{G})$ such that

[TABLE]

As such, $t=0+\sum_{v\in B}v+r$, for some $r\in\sum_{v\in S}[0,v]$. Notice that $t$ is an integer point if and only if $r$ is an integer vector.

Note that $\sum_{v\in S}[0,v]$ is the zonotope generated by the set of vectors $v\in S$. Since $\mathcal{G}^{\prime}$ is balanced, the matrix $I(\mathcal{G})$ is totally unimodular by Theorem 3.7. This is the same matrix that appears in the lattice point count of the zonotope, as in Theorem 3.10. Therefore, every tile, including $Z_{S}$, has no interior lattice points. Altogether, this means that if $t$ is an integer vector, then $r$ is an integer vector. Since the only lattice points of $\sum_{\alpha\in S}[0,v]$ are the vertices, this means that $r=\sum_{v\in S}c_{v}v$, for some constants $c_{v}\in\{0,1\}$, and this completes the proof. ∎

Let us remark that if there are no half-edges (that is, ${\mathcal{G}}={\mathcal{G}}^{\prime}$) then one can alternatively prove this result by relying on the classical, type $A_{n-1}$ result. Recall that two signed graphs are sign-switching equivalent (see, e.g., [68, 69]) if and only if one graph can be obtained from the other by a sequence of operations which flip the sign of every edge incident to a vertex. For a (signed) $\Phi$-graph ${\mathcal{G}}$ of type $B_{n}$, $C_{n}$ or $D_{n}$ we have that ${\mathcal{G}}^{\prime}$ is balanced if and only if it is sign-switching equivalent to a type $A_{n-1}$ graph. From the polytope perspective, switching the signs of edges at a vertex corresponds to a reflection of a coordinate hyperplane. In this case $Z_{\mathcal{G}}$ can be reflected across coordinate hyperplanes until it is the graphical zonotope of type $A_{n-1}$ graph. Since these reflections map integer lattice vectors to integer lattice vectors, Landau’s classical theorem gives the result. However, the proof we have given above in fact reproves the type $A_{n-1}$ case, rather than relying on it, and also allows for half-edges.

Example 3.12**.**

The root lattice for the type $C_{n}$ root system consists of integer vectors whose coordinates sum to an even number. One might wonder if there is an analogue of Landau’s theorem for type $C_{n}$ graphs stating that deterministic score sequences correspond to integer vectors whose coordinates sum to an even number. However, this is false. To see this, consider the Coxeter tournament consisting of two loops. The tournament which assigns probability $1/2$ to each loop has mean score sequence $(1,1)$, which cannot be achieved deterministically.

4. The complete case

Classical (type $A_{n-1}$) tournaments have primarily been studied on the complete graph $K_{n}$. In this situation, one obtains nicer results and connections with other areas of combinatorics, probability and optimization. In this section, we generalize some of these connections to the complete $\Phi$-graph $K_{\Phi}$ case.

4.1. $W$-majorization

As discussed in Section 1.4, when $G=K_{n}$, Moon’s theorem can be stated succinctly in the language of majorization. That is, an $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a random tournament on $K_{n}$ if and only if $x$ is majorized by $v_{n}$, written as $x\preceq v_{n}$. In this section, we note that this statement generalizes to the Coxeter setting, via the language of $G$-majorization.

Definition 4.1** ([44, Section C]).**

Let $G$ be a group and $V$ a representation of $G$. We say that $v\in V$ is $G$-majorized by $u\in V$, denoted by $v\preceq_{G}u$ if $v$ is in $\operatorname{conv}\{g\cdot u:g\in G\}$, that is, if $v$ is the convex hull of the orbit of $u$.

When $G=W$ is the Weyl group of type $A_{n-1}$ then $W$-majorization is the same as the usual notion of majorization. In types $B_{n},C_{n},$ and $D_{n}$, direct calculations show that $W$-majorization $\preceq_{W}$ is the same as weak sub-majorization $\preceq_{w}$, as defined in Section 1.7 above. As a result, we obtain the following, by which Theorem 1.4 above follows.

Proposition 4.2**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$. Then $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a $\Phi$-tournament on the complete $\Phi$-graph $K_{\Phi}$ if and only if $s$ is $W$-majorized by $\rho_{\Phi}=\sum_{\alpha\in\Phi^{+}}\alpha/2=v_{n}+\delta_{\Phi}1_{n}$, as defined in Theorem 1.4.

Proof.

By definition, $x$ is $W$-majorized by $\rho_{\Phi}$ if

[TABLE]

Using the hyperplane description of $\Pi_{\Phi}$ given by Theorem 2.22, this holds if and only if $\langle\lambda,x\rangle\leq h(\lambda)$ for all $\lambda\in\mathcal{L}$, where $h$ is the Coxeter submodular function associated to $\Pi_{\Phi}$. The result then follows by Theorem 3.3. ∎

4.2. Geometric proof

In this section, inspired by one of the proofs of Moon’s classical (type $A_{n-1}$) theorem in [4], we prove Theorem 1.4 using the Coxeter analogue of the Birkhoff’s theorem [9] (cf. von Neumann [67]). For simplicity we will prove this for the type $C_{n}$ root system, but our arguments can be adapted to types $B_{n}$ and $D_{n}$.

Recall that Birkhoff’s theorem states that every doubly stochastic matrix (non-negative with all row and column sums equal to 1) is a mixture of permutation matrices (0/1 matrices with exactly one 1 in each row and column). That is, the Birkhoff polytope $\operatorname{Birk}_{n}$ of doubly stochastic matrices $P\in{\mathbb{R}}^{n\times n}$ is the convex hull of the set ${\rm Perm}_{n}$ of permutation matrices of the same size.

The proof in [4] which we are generalizing is probabilistic. However, the strategy can be described combinatorially, as taking the following three steps:

1. (1)

First, note that a vector $x\preceq v_{n}$ (the conditions in Moon’s theorem) if and only if there is a doubly stochastic matrix such that $x=Pv_{n}$. This is a classical result of Hardy, Littlewood and Pólya [30] (cf. [44, Section 2]). 2. (2)

Second, by Birkhoff’s theorem, note that any such $P$ is a convex combination of permutation matrices in the set $\{M_{\sigma}:\sigma\in S_{n}\}$. 3. (3)

Third, construct a tournament associated with each permutation $\sigma\in S_{n}$, with mean score sequence equal to $M_{\sigma}v_{n}$.

Since mean score sequences are closed under convex combinations, this gives a proof of Moon’s theorem.

We note here that in [4], in step (1) above, instead of appealing to [30], Strassen’s coupling theorem [61] is used to obtain a probabilistic proof. In this context, majorization $\preceq$ can be viewed as inequality in the convex order (often also denoted by $\preceq$) of uniform probability distributions on discrete multisets.

Just as signed graphs are the natural setting for Coxeter tournaments, signed permutations play a key role in extending the proof of Moon’s theorem to the Coxeter setting. Recall (see Example 2.5 above) that a signed permutation $\phi\in S_{n}^{\pm}$ is a bijection of $[\pm n]$ (see Section 1.10) such that $\phi(-i)=-\phi(i)$.

Definition 4.3**.**

For a signed permutation $\phi\in S_{n}^{\pm}$, the corresponding signed permutation matrix $A_{\phi}$ is the matrix that represents the standard action of $\phi$ on ${\mathbb{R}}^{n}$. That is, its entries are $(A_{\phi})_{ij}=\pm 1$ if $\phi(i)=\pm j$, and 0 otherwise. We let ${\rm Perm}_{n}^{\pm}$ denote the set of all such matrices.

Definition 4.4**.**

A matrix $A=\{a_{ij}\}\in{\mathbb{R}}^{n\times n}$ is absolutely doubly sub-stochastic if and only if its absolute value ${\rm abs}(A)=\{|a_{ij}|\}$ is doubly sub-stochastic (non-negative with all row and column sums at most $1$). We let $\operatorname{Birk}_{n}^{\pm}$ denote the signed Birkhoff polytope of all such matrices.

Birkhoff’s theorem generalizes as follows, allowing us to generalize step (2). See Mirsky [45] and Thompson [63] (cf. [44, Section 2.C]).

Theorem 4.5** ([63, Theorem 4]).**

The signed Birkhoff polytope $\operatorname{Birk}_{n}^{\pm}$ of absolutely doubly sub-stochastic matrices is the convex hull of the set ${\rm Perm}_{n}^{\pm}$ of signed permutation matrices.

The following fact allows us to generalize step (1).

Theorem 4.6** ([44, Section 2.C.4]).**

Let $x,y\in{\mathbb{R}}^{n}$ and suppose that $y$ is non-negative. Then $|x|\preceq_{w}y$ if and only if $|x|=Sy$ for some doubly sub-stochastic matrix $S$, in which case $x=Ay$ for some absolutely doubly sub-stochastic matrix $A$.

By these results, we obtain the following.

Corollary 4.7**.**

Let $\ell$ be the linear map from ${\mathbb{R}}^{n\times n}\to{\mathbb{R}}^{n}$ which sends matrices $M\mapsto M\rho_{C_{n}}$, where

[TABLE]

Then the image of $\operatorname{Birk}_{n}^{\pm}$ under $\ell$ is Coxeter permutahedron $\Pi_{C_{n}}$ of type $C_{n}$.

Proof.

Recall that $\Pi_{\Phi}$ is the convex hull of the orbit of the point $\rho_{C_{n}}$ under the natural action of $S_{n}^{\pm}$ on ${\mathbb{R}}^{n}$. Since $\ell$ is linear, the image of $\operatorname{Birk}_{n}^{\pm}$ under $\ell$ is the convex hull of the images of the vertices of $\operatorname{Birk}_{n}^{\pm}$. The image of these vertices is the orbit of $\rho_{C_{n}}$ under the action of $S_{n}^{\pm}$. The result follows. ∎

Finally, we generalize step (3).

Definition 4.8**.**

Let $\phi\in S_{n}^{\pm}$ be a signed permutation. The tournament corresponding to $\phi$, denoted by $T_{\phi}$, on $K_{C_{n}}$ is defined as follows:

• •

for negative edges (competitive games),

[TABLE]

• •

for positive edges (cooperative games),

[TABLE]

• •

for loops (solitaire games),

[TABLE]

Naturally, we interpret $\phi(i)$ as the “ability” of player $i$. A player is “strong/weak” if their ability is positive/negative. In the above tournament, competitive games are won by the player that is more able. Strong players win solitaire games. Likewise, competitive games are won if the combined abilities of the two players equals that of a strong player.

By construction, we have the following.

Proposition 4.9**.**

Let $\phi\in S_{n}^{\pm}$ be a signed permutation with signed permutation matrix $A_{\phi}$. Then the mean score sequence of $T_{\phi}$ is $A_{\phi}\rho_{C_{n}}$.

Proof.

By construction, the mean score sequence of $T_{\phi}$ is

[TABLE]

Indeed, if $\phi(i)=j>0$, then player $i$ wins its solitaire game, worth 1 point, and all competitive and collaborative games against players $i^{\prime}$ such that $|\phi(i^{\prime})|<j$, for a total of $j-1$ additional points. The wins and loses from competitive and collaborative games against all other players cancel out. Indeed, if some $\phi(i^{\prime})>j$ (resp. $\phi(i^{\prime})<-j$) then player $i$ loses/wins its competitive/collaborative (resp. collaborative/competitive) game with player $i^{\prime}$. Hence player $i$ wins $j=\phi(i)$ points in total. The case that $\phi(i)=j<0$ is symmetric, and follows by similar reasoning. ∎

We are now ready to prove the main result of this section, which implies Theorem 1.4 in the case that $\Phi=C_{n}$.

Theorem 4.10**.**

A vector $x\in{\mathbb{R}}^{n}$ is a mean score sequence of a $C_{n}$-tournament on complete graph $K_{C_{n}}$ of type $C_{n}$ if and only if $|x|\preceq_{w}\rho_{C_{n}}$.

Proof.

It is clear that these conditions are necessary. On the other hand, suppose that $|x|\preceq_{w}\rho_{C_{n}}$. Then, by Theorem 4.6, there exists an absolutely doubly stochastic matrix $A$ such that $x=A\rho_{C_{n}}$. By Theorem 4.5, there exists numbers $\lambda_{\phi}\in[0,1]$ for all $\phi\in S_{n}^{\pm}$ summing $\sum_{\phi\in S_{n}^{\pm}}\lambda_{\phi}=1$ so that and $A=\sum_{\phi\in S_{n}^{\pm}}\lambda_{\phi}A_{\phi}$. Therefore by Proposition 4.9,

[TABLE]

where $x_{\phi}$ is the mean score sequence of the tournament $T_{\phi}$ corresponding to $\phi$. Hence, to conclude, consider the random tournament $T_{x}$ with probabilities

[TABLE]

and

[TABLE]

By construction, $T_{x}$ has mean score sequence $x$. ∎

4.3. Probabilistic proof

For $x,y\in{\mathbb{R}}^{n}$, we have that $x\preceq_{w}y$ (see Section 1.7) if and only if for all continuous increasing convex functions $\varphi$ we have that

[TABLE]

See [44, Sections 3.C.1.b and 4.B.2] for a proof. As such, Theorem 4.6 above can be viewed as a special case of Strassen’s coupling theorem [61], in the specific case of uniform probability distributions on discrete multisets. Indeed, let $\mu_{x}$ be uniform on $\{x_{1},\ldots,x_{n}\}$ and $\mu_{y}$ uniform on $\{y_{1},\ldots,y_{n}\}$. Then $\mu_{x}$ is bounded by $\mu_{y}$ in the increasing stochastic order, written as $\mu_{x}\preceq_{\rm inc}\mu_{y}$, if and only if (4.1). In this case, by [61], there is a coupling, that is, a joint distribution of random variables $(X,Y)$ with marginals $\mu_{x}$ and $\mu_{y}$, for which

[TABLE]

Using this, we give a probabilistic proof of Theorem 4.10 above, similar in spirit to the “football” proof of Moon’s classical theorem in [4]. See the discussion after (4.3) below for an informal sports interpretation of the probabilistic construction given by the following proof.

Proof of Theorem 4.10.

Suppose that $x\in{\mathbb{R}}^{n}$ satisfies

[TABLE]

Then, applying (4.2) in the case that $X$ is uniform on $\{|x_{1}|,\ldots,|x_{n}|\}$ and $Y$ is uniform on $\{1,\ldots,n\}$, we obtain sub-probability measures $\mu_{i}$ on $[n]$ for which

1. (1)

$\sum_{j=1}^{n}j\mu_{i}(j)=|x_{i}|$, for all $i\in[n]$ 2. (2)

$\sum_{i=1}^{n}\mu_{i}(j)\leq 1$, for all $j\in[n]$.

The matrix $S$ with entries $s_{ij}=\mu_{i}(j)$ is doubly sub-stochastic. First, we extend $S$ to a doubly stochastic matrix (see, e.g., von Neumann [67]) by adding some $\varepsilon_{ij}\in[0,1]$ to each entry, that is, so that all rows $\sum_{j=1}^{n}(s_{ij}+\varepsilon_{ij})=1$ and columns $\sum_{i=1}^{n}(s_{ij}+\varepsilon_{ij})=1$. Then, we define probability measures $\nu_{i}$ on $[\pm n]$ by

[TABLE]

In other words, for $j\in[n]$, if $x_{i}$ is positive/negative then we put the extra weight $s_{ij}$ on positive/negative $j$. Note that if $x_{i}=0$ then all entries $s_{ij}=0$ in the $i$th row of $S$. By construction,

• (3)

$\nu_{i}$ has mean $x_{i}$, for all $i\in[n]$

• (4)

$\sum_{i=1}^{n}[\nu_{i}(j)+\nu_{i}(-j)]=1$, for all $j\in[n]$.

For probability measures $\nu,\hat{\nu}$ let

[TABLE]

and

[TABLE]

where $X,\hat{X}$ are independent random variables distributed as $\nu,\hat{\nu}$.

We claim that

[TABLE]

Note that, given this, the proof is complete, taking

[TABLE]

and

[TABLE]

Informally speaking, each time that player $i$ is involved in a game, they score an independent number (possibly negative) number of points, distributed as $\nu_{i}$. Competitive games (worth $1/2$ point) are won by the player with higher score and lost (worth $-1/2$ point) by the other player. In the case of a tie, no points are awarded. Likewise, cooperative games are won (worth $1/2$ point each) by both players if their combined score is postive, and lost (worth $-1/2$ point each) if their combined score is negative. If their combined score is 0, no points are awarded. Finally, in solitaire games, a player wins (worth $1$ point) if their score is positive, loses (worth $-1$ point) if their score is negative, and if their score is 0 then no points are awarded.

To verify (4.3), we proceed as follows. Let $\lambda$ be uniform on $\{-n,\ldots,n\}$. Since, by symmetry, all

[TABLE]

it follows, by linearity and (3), that

[TABLE]

Note that

[TABLE]

where $\hat{\lambda}$ is uniform on $[\pm n]$. Therefore

[TABLE]

By (4), we have that

[TABLE]

and so, by the law of total probability, it follows that

[TABLE]

Combining (4.4) and (4.5), we obtain (4.3), as required. ∎

4.4. Algorithmic proof

Finally, in this section, we present a constructive proof of Theorem 1.4, via a recursive procedure which can be viewed as a continuous Coxeter analogue of the Havel–Hakimi [31, 28] algorithm.

Theorem 4.11**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$ and $x\in{\mathbb{R}}^{n}$. If $|x|\preceq_{w}\rho_{\Phi}=v_{n}+\delta_{\Phi}1_{n}$ then we can construct a random $\Phi$-tournament on the complete $\Phi$-graph $K_{\Phi}$ with mean score sequence $x$, that is, probabilities such that

[TABLE]

A key ingredient is the following result from majorization theory (see [44, Section 4], and the discussion therein about the results of Hardy, Littlewood and Pólya [30], Karamata [38] and Tomíc [66]).

Lemma 4.12** ([44, Section 4.B]).**

Let $x,y\in{\mathbb{R}}^{n}$. We have that $x\preceq_{w}y$ if and only if

[TABLE]

where $z^{+}=\max\{z,0\}$.

We will use the following special case.

Lemma 4.13**.**

Let $\Phi$ be a root system of type $B_{n}$, $C_{n}$ or $D_{n}$ and $x\in{\mathbb{R}}^{n}$. Then $|x|\preceq_{w}\rho_{\Phi}$ if and only if

[TABLE]

where $\phi_{\ell}(z)=(z-\ell)^{+}$.

Although the details of the following proof are somewhat technical, the overall idea is rather intuitive, and boils down to a natural greedy algorithm. After all relevant quantities have been defined, a will give a detailed informal description of the construction, after (4.9) below.

Proof of Theorem 4.11.

The proof is by induction on $n$. Let $x\in{\mathbb{R}}^{n}$ with $|x|\preceq_{w}(v_{n}+\delta_{\Phi}1_{n})$ be given. For ease of exposition, and without loss of generality, we assume that $|x_{1}|\leq|x_{2}|\leq\cdots\leq|x_{n}|$.

In the base case $n=1$ we have $|x_{1}|\leq\delta_{\Phi}$. If $\Phi=D_{1}$, we are done, since $x_{1}=0$ and there are no probabilities to be defined. On the other hand, if $\Phi=B_{1}$ (resp. $\Phi=C_{n}$) then $|x_{1}|\leq 1/2$ (resp. $|x_{1}|\leq 1$). In these cases, we put $p_{1}^{h}=x_{1}+1/2$ (resp. $p_{1}^{\ell}=(x_{1}+1)/2$).

For the inductive step, we describe a recursive algorithm that, in each step, assigns probabilities to all games involving the most extreme (either the weakest or strongest, whichever is more extreme) remaining player.

Note that if $x_{n}=0$, then in fact all $x_{j}=0$. In this case, we can simply put all $p_{ij}^{\pm}=1/2$ and $p_{i}^{h}=1/2$ (resp.  $p_{i}^{\ell}=1/2$) if in type $B_{n}$ (resp. $C_{n}$). Hence, suppose that $x_{n}\neq 0$. We first consider the case that $x_{n}<0$. The case that $x_{n}>0$ follows by a symmetric argument (as explained in Case 2 below).

Case 1 ($x_{n}<0$). In this case, we find $p^{\pm}_{nj}$ such that

[TABLE]

and

[TABLE]

where

[TABLE]

Note that

[TABLE]

so (informally speaking) $x_{j}^{\prime}$ is the average number of points yet to be earned by player $j<n$, after winning on average $p_{nj}^{+}-p_{nj}^{-}$ points from games with player $n$. We also note that the $p_{nj}^{\pm}$ will be chosen in such a way that order is preserved, that is, so that $|x_{1}^{\prime}|\leq\cdots\leq|x_{n-1}^{\prime}|$.

Since $|x|\preceq_{w}\rho_{\Phi}$, we have $|x_{n}|-\delta_{\Phi}\leq n-1$. Therefore, since $x_{n}<0$, it follows that $n-1+\delta_{\Phi}+x_{n}\geq 0$.

For $j<n$, let $I_{j}=[|x_{j}|-1,|x_{j}|]$ be the unit interval with right endpoint $|x_{j}|$. For $\gamma\geq-1$, let

[TABLE]

In other words, $\ell_{j}(\gamma)$ is the length of the interval to the right of $\gamma$, plus any such length to the left of the origin counted twice.

Since $\delta_{\Phi}\leq 1$ and $|x_{n}|\geq|x_{j}|$ for all $j<n$, it follows that

[TABLE]

Therefore, since $x_{n}<0$, we have that

[TABLE]

Note that $\sum_{j<n}\ell_{j}(\gamma)$ is decreasing continuously in $\gamma\geq-1$. Hence select (the unique) such $\gamma_{*}\in[-1,\infty)$ for which

[TABLE]

Using this quantity, we define the probabilities $p_{nj}^{\pm}$ as follows:

• •

if $x_{j}\leq 0$, put $p_{nj}^{-}={\rm length}(I_{j}\cap[\gamma_{*},\infty))$ and $p_{nj}^{+}={\rm length}(I_{j}\cap[\gamma_{*},0])$,

• •

if $x_{j}\geq 0$, put $p_{nj}^{+}={\rm length}(I_{j}\cap[\gamma_{*},\infty))$ and $p_{nj}^{-}={\rm length}(I_{j}\cap[\gamma_{*},0])$,

• •

if $\Phi=B_{n}$ (resp. $C_{n}$) put $p_{n}^{h}=0$ (resp. $p_{n}^{\ell}=0$).

Before continuing with the formal proof, let us discuss the general intuition behind our construction, and our strategy of proof going forward. The choice of $\gamma_{*}$, and the probabilities $p_{nj}^{+}$ and $p_{n}^{h}$ (or $p_{n}^{\ell}$) that it determines, has the following natural interpretation in terms of a greedy strategy for player $n$. Let us assume (Case 1 below) that player $n$ is a weak player, $x_{n}<0$. (The other case is symmetric, see Case 2 below.) In this case, they lose/forfeit their solitaire game, $p_{n}^{\ell}=0$. We must then select the remaining probabilities in such a way that

[TABLE]

To select such a $\gamma^{*}$, we think of a system of $n-1$ labelled particles on ${\mathbb{R}}$, where the $j$th particle is placed at position $|x_{j}|$. Particles $j$ farther to the right correspond to players $j$ that player $n$ would prefer to compete/cooperate with, depending on whether $x_{j}$ is negative/positive (that is, if player $j$ is weak/strong). Hence a natural greedy strategy for player $n$ is as follows. Imagine a “slider” moving at unit rate towards the origin, initially starting from the right of all particles. Once the slider touches a particle, it is “picked up” and slides along with it. Particles can travel for at most a unit distance, at which point they are “dropped off.” There are two cases to consider.

Case 1a. If the total distance travelled to the left by all particles equals $n-1+\delta_{\Phi}+x_{n}$ once the slider reaches some point $\gamma_{*}>0$ to the right of the origin, then we simply let $p_{nj}^{-}$ (resp. $p_{nj}^{+}$) be the distance travelled to the left by particle $j$ if $x_{j}<0$ (resp. $x_{j}>0$), given by ${\rm length}(I_{j}\cap[\gamma_{*},\infty))$. Note that, in this case, player $n$ has managed to win its required (average) number of points by only competing/cooperating with weak/strong players.

Case 1b. On the other hand, if some particles reach the origin before the total distance travelled reaches $n-1+\delta_{\Phi}+x_{n}$, then we modify the construction as follows. In this case, player $n$ is not be able to avoid competing/cooperating with some strong/weak players. Once the slider reaches the origin, we imagine particles at the origin simultaneously traveling to the left and right at the same rate (effectively, being held in place) until they have traveled a unit distance or else the total distance travelled (to the left and right) by all particles reaches $n-1+\delta_{\Phi}+x_{n}$ (whichever comes first). At this point, for some $\gamma_{*}\in[-1,0]$, note that the $j$th particle will have travelled ${\rm length}(I_{j}\cap[\gamma_{*},\infty))$ to the left and ${\rm length}(I_{j}\cap[\gamma_{*},0])$ to the right. If $x_{j}<0$ (resp. $x_{j}>0$) we let $p_{nj}^{-}$ and $p_{nj}^{+}$ (resp. $p_{nj}^{+}$ and $p_{nj}^{-}$) be these distances travelled to the left and right. Note that, in this case, player $n$ prioritizes competition/cooperation with weak/strong players, and cooperates/competes with such players only as necessary.

We return to the formal proof. Note that (4.4) holds by the choice of $\gamma_{*}$, and that by construction we have $|x_{1}^{\prime}|\leq\cdots\leq|x_{n-1}^{\prime}|$. Next, we verify (4.8). In doing so, we take cases with respect to whether $\gamma_{*}\in[-1,0]$ or $\gamma_{*}>0$.

Case 1a. Suppose that $\gamma_{*}>0$. Then

[TABLE]

In this case, we appeal to Lemma 4.13. Let $\ell\in\{\delta_{\Phi},1+\delta_{\Phi},\ldots,n-2+\delta_{\Phi}\}$. If $\ell>\gamma_{*}$ then

[TABLE]

Otherwise, if $\ell\leq\gamma_{*}$ then by construction we have that

[TABLE]

Therefore, since $\gamma_{*}\leq|x_{n}|$ and $x_{n}\leq 0$, it follows that

[TABLE]

and so

[TABLE]

Therefore, by Lemma 4.13, we find that $|x^{\prime}|\preceq_{w}\rho_{\Phi}$, as required.

Case 1b. On the other hand, suppose that $\gamma_{*}\in[-1,0]$. In this case, we show that $|x^{\prime}|\preceq_{w}(v_{n-1}+\delta_{\Phi}1_{n-1})$ by appealing directly to the definition of weak sub-majorization. Note that, in this case, we have that all $|x_{j}^{\prime}|=(|x_{j}|-1)^{+}$. Since $|x|\preceq_{w}\rho_{\Phi}=v_{n}+\delta_{\Phi}1_{n}$, we have that, for any $S\subseteq[n]$ of size $k$,

[TABLE]

Therefore, if $S\subset[n-1]$ is of size $k$, then

[TABLE]

where $S^{\prime}$ of size $k^{\prime}\leq k$ is the set of $j\in S$ for which $|x_{j}|>1$. Since the right-hand side is non-decreasing in $k^{\prime}\leq k$, it follows that

[TABLE]

Hence $|x^{\prime}|\preceq_{w}(v_{n-1}+\delta_{\Phi}1_{n-1})$, as required.

This concludes the proof in Case 1.

Case 2 ($x_{n}>0$). On the other hand, if $x_{n}\geq 0$, we can instead find the probabilities $q_{nj}^{\pm}=1-p_{nj}^{\pm}$ and $q_{n}^{h}=1-p_{n}^{h}$ (or $q_{n}^{\ell}=1-p_{n}^{\ell}$) by a symmetric argument. As before, fix (the unique) $\gamma_{*}$ such that

[TABLE]

Then define probabilities as follows:

• •

if $x_{j}\leq 0$, put $q_{nj}^{+}={\rm length}(I_{j}\cap[\gamma_{*},\infty))$ and $q_{nj}^{-}={\rm length}(I_{j}\cap[\gamma_{*},0])$,

• •

if $x_{j}\geq 0$, put $q_{nj}^{-}={\rm length}(I_{j}\cap[\gamma_{*},\infty))$ and $q_{nj}^{+}={\rm length}(I_{j}\cap[\gamma_{*},0])$,

• •

if $\Phi=B_{n}$ (resp. $C_{n}$) put $q_{n}^{h}=0$ (resp. $q_{n}^{\ell}=0$).

By construction, we have that

[TABLE]

Arguing as in Case 1, it can be shown that

[TABLE]

where

[TABLE]

This concludes the proof in Case 2.

To finish the proof, we note that all probabilities $p_{ij}^{\pm}$ and $p_{i}^{h}$ (or $p_{h}^{\ell}$) can be defined recursively by the above procedure, beginning with $i=n$. ∎