k/2\rfloor-1},i_{\lfloor k/2\rfloor},i_{\lfloor k/2\rfloor+1},\ldots,i_{k}$
and the value $M[n][k]$ as a solution to ${\mathcal{P}}(w,k)$.

The correctness of the algorithm is based on the following simple remark: if
$w[1:n]$ can be transformed into a $k$-universal word by a sequence of $m$
insertions, such that in the respective sequence of insertions transforms
$w[1:i_{\lfloor k/2\rfloor}]$ into a ${\lfloor k/2\rfloor}$-universal word,
then the following hold:

  * •

$w[1:i_{\lfloor k/2\rfloor}]$ can be transformed into a ${\lfloor
k/2\rfloor}$-universal word by $p$ insertions;

  * •

$w[i_{\lfloor k/2\rfloor}+1:n]$ can be transformed into a ${\lceil
k/2\rceil}$-universal word by $m-p$ insertions.

Thus, solving ${\mathcal{P}}(w,k)$ can be reduced to computing $i_{\lfloor
k/2\rfloor}$ and solving recursively ${\mathcal{P}}(w[1:i_{\lfloor
k/2\rfloor}],{\lfloor k/2\rfloor})$ and ${\mathcal{P}}(w[i_{\lfloor
k/2\rfloor}+1:n],{\lceil k/2\rceil})$.

The time complexity $T(n,k)$ of solving ${\mathcal{P}}(w,k)$ is then
$T(n,k)=O(nk)+T(i_{\lfloor k/2\rfloor},\lfloor k/2\rfloor)+T(n-i_{\lfloor
k/2\rfloor},{\lceil k/2\rceil})$. It is easy to show that $T(n,k)\in O(nk)$.

Assume that the algorithm computing $H[n][k]$ and $M[n][k]$ needs $cn$ space
for some constant $c$.

One can show by induction on $n+k$ that the space complexity $S(n,k)$ of
solving ${\mathcal{P}}(w,k)$ is upper-bounded by $dn$, where $d\geq c$ is a
constant, $n=|w|$, and $k\leq n$. The case $n+k=2$ is trivial. Assume that
this is true for $n+k\leq r$. We show that it is true for $n+k=r+1$. To solve
${\mathcal{P}}(w,k)$ we first compute the values $M[n][k]$ and $H[n][k]$ in
$cn\leq dn$ space, for some constant $c$. The main observation we make here is
that the space we used in this computation can then be reused. Then we solve
${\mathcal{P}}(w[1:i_{\lfloor k/2\rfloor}],\lfloor k/2\rfloor)$ using
$di_{\lfloor k/2\rfloor}\leq dn$ space (which is actually reused space).
Finally, we solve ${\mathcal{P}}(w[i_{\lfloor k/2\rfloor}+1:n],\lfloor
k/2\rfloor)$ using $d(n-i_{\lfloor k/2\rfloor})\leq dn$ space (which is, once
more, reused space). Thus, $S(n,k)$ is upped bounded by $dn$.

This concludes our proof, and shows the statement of the theorem for the case
of increasing the universality index of a word $w$ by insertions to a value
$k$ with $k\leq|w|$.

We immediately get that in the case of increasing the universality index of a
word $w$ by insertions to a value $k$ with $k>|w|$ we need $O(n)$ space to
reach $n$-universality, and the rest of the construction can be done trivially
in $O(k\sigma)$ space (we just need to write down the output).

The case of decreasing the universality index of a word $w$ by deletions to a
value $k<\iota(w)$ can be treated in an identical way (using the same divide-
and-conquer trick). ∎

######  Theorem 7.

Let $w$ be a word, with $|w|=n$ and $alph(w)=\\{\mathtt{a},\mathtt{b}\\}$. Let
$k\geq\iota(w)$ be a positive integer. We can compute in $O(|w|)$ time the
minimal number of insertions needed to obtain a $k$-universal word from a
given binary word $w$.

###### Proof.

We show that, if $\ell$ is the universality index of $w$, we have that the
number $i$ of insertions, that we want to compute, is $i=0$ if $k\leq\ell$,
$i=k-\ell$ if $\ell<k\leq n-2\ell$, and $i=2k-n$ if $n-2\ell<k$.

Define the mapping $\overline{\cdot}:\Sigma\rightarrow\Sigma$ by
$\overline{\mathtt{a}}=\mathtt{b}$ and $\overline{\mathtt{b}}=\mathtt{a}$. The
claim holds immediately for $k\leq\ell$ by Theorem 1. Assume $k>\ell$. Since
$\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}^{\ell}\subseteq\operatorname{Subseq}_{2\ell}(w)$
there exists
$w_{1},\dots,w_{\ell}\in\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}$ and
$r_{1},\dots,r_{\ell}\in[n]$ with $w[r_{j},r_{j}+1]=w_{j}$ for all
$j\in[\ell-1]$. Notice that by the choice of $\ell$ we have $r_{j}+1<r_{j+1}$.
Set $u_{1}=w[1:r_{1}-1]$, $u_{\ell+1}=w[r_{\ell}+2:n]$, and
$u_{j}=w[r_{j-1}+2:r_{j}-1]$ with $u_{s}=\varepsilon$ if the first index is
strictly greater than the second, for $2\leq j\leq\ell$ and $s\in[\ell]$. This
implies $w=u_{1}w_{1}u_{2}\dots u_{\ell}w_{\ell}u_{\ell+1}$ and $u=u_{1}\dots
u_{\ell+1}$ is of length $n-2\ell$. If $n-2\ell\geq k>\ell$ set

$u^{\prime}=u[1]\overline{u[1]}u[2]\overline{u[2]}\dots
u_{k-\ell}\overline{u[k-\ell]}u[k-\ell+1]\dots u[n-2\ell]$

and $u_{s}^{\prime}$ accordingly for $s\in[\ell+1]$. Then
$w^{\prime}=u_{1}^{\prime}w_{1}u_{2}^{\prime}\dots
u_{\ell}^{\prime}w_{\ell}u_{\ell+1}^{\prime}$ is obtained from $w$ by $k-\ell$
insertions and by the definition of $u^{\prime}$ we have
$\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}^{k}\subseteq\operatorname{Subseq}_{2k}(w^{\prime})$.
This implies that $w^{\prime}$ is $k$-universal. If $k>n-2\ell$ set

$u^{\prime}=u[1]\overline{u[1]}u[2]\overline{u[2]}\dots
u_{k-\ell}\overline{u[k-\ell]}u[k-\ell+1]\overline{u[k-\ell+1]}\dots
u[n-2\ell]\overline{u[n-2\ell]}$

and $u_{s}^{\prime}$ accordingly for $s\in[\ell+1]$. Then
$w^{\prime}=u_{1}^{\prime}w_{1}u_{2}^{\prime}\dots
u_{\ell}^{\prime}w_{\ell}u_{\ell}^{\prime}(\mathtt{a}\mathtt{b})^{k-n+\ell}$
is obtained from $w$ by $n-2\ell+2(k-n+\ell)=2k-n$ insertions. By definition
of $u^{\prime}$ and the appended $(\mathtt{a}\mathtt{b})^{k-n+\ell}$ we get
$\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}^{\ell+n-2\ell+k-n+\ell}=\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}^{k}$
is a subset of $\operatorname{Subseq}_{k}(w^{\prime})$ and by Theorem 1,
$w^{\prime}$ is $k$-universal. This proves that with $i$ insertions a
$k$-universal word can be obtained from $w$.

We prove now that $i$ is minimal. Suppose that $i$ is not minimal and let
$i^{\prime}<i$ be the minimal number of insertions such that the obtained word
$w^{\prime}$ is $k$-universal. By Theorem 1 we have
$\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}^{k}\subseteq\operatorname{Subseq}_{2k}(w^{\prime})$
and there exists
$w_{1},\dots,w_{k}\in\\{\mathtt{a}\mathtt{b},\mathtt{b}\mathtt{a}\\}$ such
that $w_{1}\dots w_{k}$ is a subsequence of $w^{\prime}$. Let
$j^{\prime\prime}$ be the number of $w_{s}$ which were inserted completely and
$j^{\prime}$ be the number of $w_{s}$ in which one letter was already in $w$
and one is inserted, i.e. $i^{\prime}=j^{\prime}+2j^{\prime\prime}$. This
implies $k\leq\ell+j^{\prime}+j^{\prime\prime}$. By the first part of the
proof we have $k=i+\ell$. If $\ell<k\leq n-2\ell$ we get
$i^{\prime}+\ell<i+\ell=k\leq\ell+j^{\prime}+j^{\prime\prime}$ and thus
$i^{\prime}<j^{\prime}+j^{\prime\prime}$ which contradicts
$i^{\prime}=j^{\prime}+2j^{\prime\prime}$. If $k>n-2\ell$ we get with
$n+i^{\prime}\geq 2\ell+2j^{\prime\prime}+2j^{\prime}$ (for $w^{\prime}$’s
length)

$2(\ell+j^{\prime}+j^{\prime\prime})\leq
n+i^{\prime}<n+i=2k=2(\ell+j^{\prime}+j^{\prime\prime}).$

Hence, $i$ is minimal.

By Theorem 2 the decomposition into $w_{1},\dots,w_{\ell}$ can be found in
run-time $\mathcal{O}(n)$. The word $u_{s}^{\prime}$ for $s\in[\ell+1]$ can be
constructed by going once one left to right while inserting after each letter
$x$ the opposite letter $\overline{x}$ until $k-\ell$ insertions are reached.
If $k>n-2\ell$ for each letter a new one is inserted and the remaining
$(\mathtt{a}\mathtt{b})^{k-n+\ell}$ letter are simply appended. ∎

##  Appendix C Appendix: The computational model

General algorithmic framework: Our results are of algorithmic nature. The
computational model we use is the standard unit-cost RAM with logarithmic word
size: for an input of size $n$, each memory word can hold $\log n$ bits.
Arithmetic and bitwise operations with numbers in $[1:n]$ are, thus, assumed
to take $O(1)$ time. Numbers larger than $n$, with $\ell$ bits, are
represented in $O(\ell/\log n)$ memory words, and working with them takes time
proportional to the number of memory words on which they are represented. In
all the problems, we assume that we are given a word $w$, with $|w|=n$, over
an alphabet $\Sigma=\\{1,2,\ldots,\sigma\\}$, with $|\Sigma|=\sigma\leq n$.
That is, we assume that the processed words are sequences of integers (called
letters or symbols, each fitting in $O(1)$ memory words). This is a common
assumption in string algorithms: the input alphabet is said to be an integer
alphabet. For a more detailed general discussion on this model see, e.g.,
[11], and for its use in the particular case of algorithms related to
subsequences see [23].

It is interesting to see whether we can extend our results for more general
input alphabets, so when we drop the assumption that if the input word is $w$,
then $w$ is over the alphabet $\Sigma=\\{1,\ldots,\sigma\\}$ with
$\sigma\leq|w|$. In the next paragraph we will follow the similar discussion
made in [23].

A computational model used in string algorithms assumes that the input is over
general ordered alphabets (see [37, 38, 22] and the references therein). More
precisely, the input is a sequence of elements from a totally ordered set
${\mathcal{U}}$ (i.e., string over ${\mathcal{U}}$). The operations allowed in
this model are those of the standard Word RAM model, with one important
restriction: the elements of the input cannot be directly accessed nor stored
in the memory used by the algorithms; instead, we are only allowed to compare
(w.r.t. the order in ${\mathcal{U}}$) any two elements of the input, and the
answer to such a comparison-query is retrieved in $O(1)$ time. In this model,
it holds that sorting the elements of an input sequence requires at least
$\Omega(n\log n)$ comparisons. An implementation of our algorithms, where we
first sort the letters of the input word, map them to words over
$\\{1,\ldots,n\\}$, and then use the same strategies as the ones described for
the case of integer alphabets, would require some additional $O(n\log n)$
computational time, due to the sorting.

In fact, one cannot hope to go under $\Omega(n\log n)$ comparisons in the
respective model of computation. Indeed, in this framework, it also holds that
testing the equality of two sets of size $O(n)$ requires $\Omega(n\log n)$
comparisons [16]. We can show the following lower bounds.

######  Theorem 8.

Let $w$ be a word, with $|w|=n$, $\text{alph}(w)=\Sigma$, and universality
index $\iota(w)$. Let $k$ be an integer with $n\geq k$. Computing the minimal
number of insertions (respectively, deletions, or substitutions) needed to
transform $w$ into a $k$-universal word requires $\Omega(n\log n)$ comparisons
(so $\Omega(n\log n)$ time as well).

###### Proof.

Let $S=\\{s_{1},\ldots,s_{n}\\}$ and $T=\\{t_{1},\ldots,t_{m}\\}$ two sets,
with $m\leq n$. We define the alphabet $\Sigma=S\cup T\cup\\{\$,\\#\\}$, where
the letters $\$$ and $\\#$ do not occur in $S$. We define the word

$w=\$s_{1}\cdots s_{n}\\#\$t_{1}\cdots t_{m}\\#.$

We want to show that $S=T$ if and only if the number of insertions needed to
transform $w$ into a $2$-universal word is $0$.

The left to right implication is trivial. The right to left implication is
also easy to show. If $w$ is $2$-universal, then it has two arches. These
arches must be $\$s_{1}\cdots s_{n}\\#$ and $\$t_{1}\cdots t_{m}\\#$
(otherwise one would need to insert one of the separators). This means that
the letters $s_{1},\ldots,s_{n}$ are the same as $t_{1},\ldots,t_{m}$. So
$S=T$.

Thus, to check the equality $S=T$ we can compute the minimal number of
insertions needed to make $w$ $2$-universal. Thus, this requires at least
$\Omega(|w|\log|w|)$ comparisons. As $|w|=n+m+4\in O(n)$, the statement
follows.

We can similarly show that $S=T$ if and only if the minimal number of
substitutions needed to transform $w$ into a $2$-universal word is $0$.
Similarly to the case of insertions, the lower bound is easily obtained now.

Finally, we can show that $S\neq T$ if and only if the minimal number of
deletions needed to transform $w^{\prime}=s_{1}\cdots s_{n}t_{1}\cdots t_{m}$
into a $0$-universal word (w.r.t. $\text{alph}(w^{\prime})$) is exactly $1$.
Indeed, each element $s_{i}$ occurs exactly once in $S$ and each element
$t_{j}$ occurs exactly once in $T$. So, each letter $s_{i}$ (respectively,
$t_{j}$) may occur at most twice in $w^{\prime}$ (if it is contained in both
$S$ and $T$). Clearly, if there is a letter that occurs exactly once, then the
minimal number of deletions needed to transform $w^{\prime}=s_{1}\cdots
s_{n}t_{1}\cdots t_{n}$ into a $0$-universal word is $1$, but also this letter
occurs only in $S$ or only in $T$. So $T\neq S$. If all letters occur twice,
then the minimal number of deletions needed to transform
$w^{\prime}=s_{1}\cdots s_{n}t_{1}\cdots t_{n}$ into a $0$-universal word is
$2$, and $T=S$.

Thus, to check the equality $S=T$ we can compute the minimal number of
deletions needed to make $w^{\prime}$ $0$-universal. Thus, this requires at
least $\Omega(|w|\log|w|)$ comparisons as well. As $|w|=n+m\in O(n)$, the
statement follows. ∎

So, having faster algorithms in this model of computation requires finding
better methods than the dynamic programming approach we used.

In an intermediate model, we can assume that the input is a sequence of
elements from a totally ordered set ${\mathcal{U}}$ (i.e., string over
${\mathcal{U}}$) whose elements can be stored in a constant number of memory
words. The operations allowed in this model are those of the standard Word RAM
model. So, in other words, the letters are from $[1:n^{d}]$ for some constant
$d$, if the input word has length $n$. An implementation of our algorithms,
where we first sort the letters of the input word, map them to words over
$\\{1,\ldots,n\\}$, and then use the same strategies as the ones described for
the case of integer alphabets, runs in exactly the same complexity as stated
in the main part of the paper, as a set of $n$ numbers from $[1:n^{d}]$, where
$d$ is a constant, can be sorted in $O(n)$ time using Radix-sort.


# Retrodictive Modelling of Modern Rugby Union:
Extension of Bradley-Terry to Multiple Outcomes

Ian Hamilton, University of Warwick
David Firth, University of Warwick

## Abstract

Frequently in sporting competitions it is desirable to compare teams based on
records of varying schedule strength. Methods have been developed for sports
where the result outcomes are win, draw, or loss. In this paper those ideas
are extended to account for any finite multiple outcome result set. A
principle-based motivation is supplied and an implementation presented for
modern rugby union, where bonus points are awarded for losing within a certain
score margin and for scoring a certain number of tries. A number of variants
are discussed including the constraining assumptions that are implied by each.
The model is applied to assess the current rules of the Daily Mail Trophy, a
national schools tournament in England and Wales.

Keywords: Bradley-Terry, entropy, networks, pairwise comparison, ranking,
sport.

##  1 Introduction

There is a deep literature on ranking based on pairwise binary comparisons.
Prominent amongst proposed methods is the Bradley-Terry model, which
represents the probability that team $i$ beats team $j$ as

$P(i\succ j)=\frac{\pi_{i}}{\pi_{i}+\pi_{j}}\quad,$

where $\pi_{i}$ may be thought of as representing the positive-valued strength
of team $i$. The model was originally proposed by Zermelo, (1929) before being
rediscovered by Bradley and Terry, (1952). It was further developed by
Davidson, (1970) to allow for ties (draws); by Davidson and Beaver, (1977) to
allow for order effects, or, in this context, home advantage; and by Firth
(https://alt-3.uk/) to allow for standard association football scoring rules
(three points for a win, one for a draw). Bühlmann and Huber, (1963) showed
that the Bradley-Terry model is the unique model that comes from taking the
number of wins as a sufficient statistic. Later, Joe, (1988) showed that it is
both the maximum entropy and maximum likelihood model under the retrodictive
criterion that the expected number of wins is equal to the actual number of
wins, and derived maximum entropy models for home advantage and matches with
draws. These characterisations of the Bradley-Terry model may be seen as
natural expressions of a wider truth about exponential families, that if one
starts with a sufficient statistic then the corresponding affine submodel, if
it exists, will be uniquely determined and it will be the maximum entropy and
maximum likelihood model subject to the ‘observed equals expected’ constraint
(Geyer and Thompson,, 1992). In this paper, the maximum entropy framing is
used as it helps to clarify the nature of the assumptions being made in the
specification of the model.

Situations of varying schedule strength occur frequently in rugby union. They
are apparent in at least five particular scenarios. First, in two of the top
club leagues in the world — Pro14 and Super Rugby — the league stage of the
tournament is not a round robin, but a conference system operated with an
over-representation of matches against teams from the same conference and
country. Second, in professional rugby such situations occur at intermediate
points of the season, whether the tournament is of a round robin nature or
not. Third, in Europe, a significant proportion of teams in the top domestic
leagues — Pro14, English Premiership, Top14 — also compete in one of the two
major European rugby tournaments, namely the European Rugby Champions Cup and
the European Rugby Challenge Cup. The preliminary stage of both these
tournaments is also a league-based format. If the results from the European
tournaments are taken along with the results of the domestic tournaments then
a pan-European system of varying schedule strengths may be considered. Fourth,
fixture schedules may be disrupted by unforeseen circumstances causing the
cancellation of some matches in a round robin tournament. This has been
experienced recently due to COVID19. Fifth, schools rugby fixtures often exist
based on factors such as geographical location and historical links and so do
not fit a round robin format. The Daily Mail Trophy is an annual schools
tournament of some of the top teams in England and one team from Wales that
ranks schools based on such fixtures.

In modern rugby union the most prevalent points system is as follows:

  * 4 points for a win

  * 2 points for a draw

  * 0 points for a loss

  * 1 bonus point for losing by a match score margin of seven or fewer

  * 1 bonus point for scoring four or more tries

This is the league points system used in the English Premiership, Pro14,
European Rugby Champions Cup, European Rugby Challenge Cup, the Six
Nations111In the Six Nations there are an additional three bonus points for
any team that beats all other teams in order to ensure that a team with a 100%
winning record cannot lose the tournament because of bonus points, and the
pool stages of the most recent Rugby World Cup, which was held in Japan in
2020. In the southern hemisphere, the two largest tournaments, Super Rugby and
the Rugby Championship, follow the same points system except that a try bonus
point is awarded when a team has scored three more tries than the opposition,
so at most one team will earn a try bonus. In the French Top14 league the try
bonus point is also awarded on a three-try difference but with the additional
stipulation that it may only be awarded to a winning team. The losing bonus
point in the Top14 is also different in awarding the point at a losing margin
of five or fewer instead of seven or fewer. Together these represent the
largest club and international tournaments in the sport. For the rest of this
paper the most prevalent system, the one set out above, will be used. The
others share the same result outcomes formulation and hence a substantial
element of the model. When appropriate, methodological variations will be
mentioned that might better model the alternative try-bonus method, where the
bonus is based on the difference in the number of tries and may only be
awarded to one team. For the avoidance of confusion, for the remainder of this
paper, the points awarded due to the outcomes of matches and used to determine
a league ranking will be referred to as ‘points’ and will be distinguished
from the in-game accumulations on which match outcomes are based, which will
henceforth be referred to as ‘scores’. Likewise, ‘ranking’ will refer to the
attribution of values to teams that signify their ordinal position, while
‘rating’ will refer to the underlying measure on which a ranking is based.

It is not the intention here to make any assessment of the relative merits of
the different points systems, rather to take the points system as a given and
to construct a coherent retrodictive model consistent with that, whilst
accounting for differences in schedule strength. In doing so, a model where
points earnt represent a sufficient statistic for team strength is sought. It
is important to understand therefore that this represents a ‘retrodictive’
rather than a predictive model. This is a concept familiar in North America
where the KRACH (“Ken’s Rating for American College Hockey”) model, devised by
Ken Butler, is commonly used to rank collegiate and school teams in ice hockey
and other sports (Wobus,, 2007).

The paper proceeds in Section 2 with derivation of a family of models based on
maximum entropy, a discussion of alternatives, and the choice of a preferred
model for further analysis. Section 3 proposes estimation of the model through
a loglinear representation. In addition, the implementation of a more
intuitive measure of team strength, the use of a prior, and an appropriate
identifiability constraint, are also discussed. In Section 4 the model is used
to analyse the current Daily Mail Trophy ranking method, and in Section 5 some
concluding remarks are made.

##  2 Model

###  2.1 Entropy maximisation

The work of Jaynes, (1957) as well as the Bradley-Terry derivations of Joe,
(1988) and Henery, (1986) suggest that a model may be determined by seeking to
maximise the entropy under the retrodictive criterion that the points earnt in
the matches played are equal to the expected points earnt given the same
fixtures under the model. Taking the general case, suppose there is a
tournament where rather than a binary win/loss there are multiple possible
match outcomes. Let $p^{ij}_{a,b}$ denote the probability of a match between
$i$ and $j$ resulting in $i$ being awarded $a$ points and $j$ being awarded
$b$, with $m_{ij}$ the number of matches between $i$ and $j$. Then we may
define the entropy as

$S(p)=-\sum_{i,j}m_{ij}\sum_{a,b}p^{ij}_{a,b}\log p^{ij}_{a,b}\quad.$ (1)

This may be maximised subject to the conditions that for each pair of teams
the sum of the probabilities of all possible outcomes is 1,

$\sum_{a,b}p^{ij}_{a,b}=1\quad\text{for all $i,j$ such that $m_{ij}>0$},$ (2)

and the retrodictive criterion that for each team $i$, given the matches
played, the expected number of points earnt is equal to the actual number of
points earnt,

$\sum_{j}m_{ij}\sum_{a,b}ap^{ij}_{a,b}=\sum_{j}\sum_{a,b}am^{ij}_{a,b}\quad,$
(3)

where $m^{ij}_{a,b}$ represents the number of matches which result in $i$
being awarded $a$ points and $j$ being awarded $b$.

The entropy, $S(p)$, is strictly concave and so the Lagrangian has a unique
maximum. With $\lambda_{ij}$ being the Lagrange multiplier associated with
teams $i,j$ in condition (2), and $\lambda_{i}$ those for the retrodictive
criterion applied to team $i$ from condition (3), then the solution satisfies

$\log p^{ij}_{a,b}=-\lambda_{ij}-a\lambda_{i}-b\lambda_{j}-1\quad\text{for all
$i,j$ such that $m_{ij}>0$}\quad,$ (4)

which gives us that

$p^{ij}_{a,b}\propto\pi_{i}^{a}\pi_{j}^{b}\quad\text{for all $i,j$ such that
$m_{ij}>0$}\quad,$ (5)

where the $\pi_{i}=\exp(-\lambda_{i})$ may be used to rank the teams, and
$\exp(-\lambda_{ij}-1)$ is the constant of proportionality. This result holds
for $i,j$ such that $m_{ij}>0$ and a reasonable modelling assumption is that
it may then be applied to all pairs $(i,j)$.

This derivation presents the most general form of the maximum entropy model,
but various specific models may be motivated in this way by imposing a variety
of independence assumptions or additional conditions. Some of the main
variants are considered next.

###  2.2 Model alternatives

####  2.2.1 Independent result and try-bonus outcomes

Conceptually one might consider that points awarded for result outcomes and
the try bonus are for different and separable elements of performance within
the predominant points system that is being considered here. If that were not
the case then a stipulation similar to that imposed in Top14, which explicitly
connects the try bonus and the result outcome, could be used. While the result
outcome in rugby union is commonly presented as a standard win, draw, loss
plus a losing bonus point, it may be thought of equivalently as five possible
result outcomes — wide win, narrow win, draw, narrow loss, wide loss. This
leads to a representation of the five non-normalised result probabilities as

$\displaystyle P(\text{team $i$ beats team $j$ by wide margin})$
$\displaystyle\propto\pi_{i}^{4}$ $\displaystyle P(\text{team $i$ beats team
$j$ by narrow margin})$ $\displaystyle\propto\rho_{n}\pi_{i}^{4}\pi_{j}$
$\displaystyle P(\text{team $i$ draws with team $j$})$
$\displaystyle\propto\rho_{d}\pi_{i}^{2}\pi_{j}^{2}$ $\displaystyle
P(\text{team $j$ beats team $i$ by narrow margin})$
$\displaystyle\propto\rho_{n}\pi_{i}\pi_{j}^{4}$ $\displaystyle P(\text{team
$j$ beats team $i$ by wide margin})$ $\displaystyle\propto\pi_{j}^{4}\quad,$

where $\rho_{n}$ and $\rho_{d}$ are structural parameters related to the
propensity for narrow or drawn result outcomes respectively.

Taking the conventional standardisation of the abilities that the mean team
strength is 1, as in Ford Jr, (1957), then the probability of a narrow result
outcome (win or loss) in a match between two teams of mean strength is
$2\rho_{n}/(2+2\rho_{n}+\rho_{d})$, and that for a draw outcome is
$\rho_{d}/(2+2\rho_{n}+\rho_{d})$.

A nice feature for this particular setting is that the try bonus point
provides information on the relative strength of the teams, so that the
network is more likely to be connected. It may thus supply differentiating
information on team strength even where more than one team has a 100% winning
record.

There are four potential try bonus outcomes that may be modelled by the
probabilities:

$\displaystyle P(\text{team $i$ and team $j$ both awarded try bonus point})$
$\displaystyle\propto\tau_{b}\pi_{i}\pi_{j}$ $\displaystyle P(\text{only team
$i$ awarded try bonus point})$ $\displaystyle\propto\pi_{i}$ $\displaystyle
P(\text{only team $j$ awarded try bonus point})$ $\displaystyle\propto\pi_{j}$
$\displaystyle P(\text{neither team awarded try bonus point})$
$\displaystyle\propto\tau_{z}\quad,$

so that in a match between two teams of mean strength the probability of both
being awarded a try bonus is $\tau_{b}/(2+\tau_{b}+\tau_{z})$ and that for
neither team gaining a try bonus is $\tau_{z}/(2+\tau_{b}+\tau_{z})$.

This model would be derived through a consideration of entropy maximisation by
taking the result outcome and try bonus outcome as separable maximisations,
but then enforcing that the $\pi_{i}$ are consistent. Each of the structural
parameters may be derived by an appropriate additional condition. For example
in the case of $\rho_{d}$, the relevant condition would be that, given the
matches played, the expected number of draws is equal to the actual number of
draws.

####  2.2.2 Try bonus independent of opposition

One might choose to make an even stronger independence assumption, that the
probability of gaining a try bonus is solely dependent on a team’s own
strength and independent of that of the opposition. This has the advantage of
greater parsimony. It may be expressed as

$\displaystyle P(\text{team $i$ gains try bonus point})$
$\displaystyle\propto\tau\pi_{i}$ $\displaystyle P(\text{team $i$ does not
gain try bonus point})$ $\displaystyle\propto 1\quad,$

where $\tau/(1+\tau)$ is the probability that a team of mean strength gains a
try bonus. This model would clearly not be appropriate to the southern
hemisphere system where the try bonus was awarded for scoring three more tries
than the opposition.

####  2.2.3 Try bonus dependent on result outcome

Alternatively, the try bonus could be conditioned on the result outcome. This
of course would be necessary if modelling the points system employed in the
Top14 for example, where only the winner is eligible for a try bonus. The
conditioning could be done in a number of ways. One could consider the five
result outcomes noted already, or consider a simplifying aggregation, either
into wide win, close result (an aggregation of narrow win, draw, and narrow
loss), or wide loss; or win (an aggregation of narrow win and wide win), draw,
or loss (narrow loss and wide loss).

####  2.2.4 Independent offensive and defensive strengths

It seems not unreasonable to consider that a team’s ability to earn a try
bonus point might be modeled as being dependent on its own attacking strength
and the opposition’s defensive strength and independent of its own defensive
strength and the opposition’s attacking strength. This may be captured by
considering team strength to be the product of its offensive and defensive
strength

$\pi_{i}=\omega_{i}\delta_{i}\quad,$ (6)

where we consider the probability of a team $i$ scoring a try bonus in a match
as proportional to their offensive strength parameter $\omega_{i}$, and the
probability of them not conceding a try bonus as proportional to their
defensive strength parameter $\delta_{i}$.

Given $\pi_{i}$, only one further parameter per team need be defined and so
the non-normalised try bonus outcome probabilities may be expressed as

$\displaystyle P(\text{both $i$ and $j$ gain try bonuses})$
$\displaystyle\propto\frac{\pi_{i}}{\delta_{i}}\frac{\pi_{j}}{\delta_{j}}=\frac{\pi_{i}\pi_{j}}{\delta_{i}\delta_{j}}$
$\displaystyle P(\text{only $i$ gains try bonus})$
$\displaystyle\propto\frac{\pi_{i}}{\delta_{i}}\delta_{i}=\pi_{i}$
$\displaystyle P(\text{only $j$ gains try bonus})$
$\displaystyle\propto\frac{\pi_{j}}{\delta_{j}}\delta_{j}=\pi_{j}$
$\displaystyle P(\text{neither team gains try bonus})$
$\displaystyle\propto\delta_{i}\delta_{j}\quad.$

Thus the model replaces the symmetric try bonus parameters $\tau_{b}$ and
$\tau_{z}$ with team-dependent parameters. This may be derived from an entropy
maximisation by considering the try bonus outcome independently from the
result outcome. The familiar retrodictive criterion that for each team, the
expected number of try bonus points scored is equal to the actual number of
try bonus points scored, is then supplemented by a second criterion that for
each team, the expected number of matches where no try bonus is conceded is
equal to the actual number of matches where no try bonus is conceded. See
Appendix for details.

####  2.2.5 Home advantage

Home advantage could be parametrised in several ways. Following the example of
Davidson and Beaver (1977) and others, one possibility is to use a single
parameter, for example by applying a scaling parameter to the home team and
its reciprocal to the away team. This may be derived via entropy maximisation
with the inclusion of a condition that, given the matches played, the
difference between the expected points awarded to home teams and away teams is
equal to the actual difference. An alternative explored by Joe, (1988) is to
consider the home advantage of each team individually so that the rating for
each team could be viewed as an aggregation of their separate home team and
away team ratings. See Appendix for details.

###  2.3 Preferred model for the most prevalent points system

####  2.3.1 Motivating considerations

In Section 2.2, four different possible result and try-bonus models were
presented, each representing different assumptions around the independence of
these points as they related to team strengths. Additionally two possible home
advantage models were discussed.

The assumption of the independence of the result outcomes and try bonus of
Section 2.2.1 was proposed based on the conceptual separation of result
outcome and try bonus inherent in the most prevalent points system. Clearly,
for a scenario such as that faced in the Top14 where try bonus is explicitly
dependent on result outcome, a version of the dependent models presented in
Section 2.2.3 would be required. However, for modelling based on the most
prevalent points system, the introduction of so many additional structural
parameters seems unwarranted compared to the greater interpretability of the
independent model of Section 2.2.1. On the other hand, the more parsimonious
model from taking the try bonus of the two teams as independent events
requires only one less structural parameter. In work available in the
Appendix, it was found to substantially and consistently have weaker
predictive ability compared to the opposition-dependent try bonus model of
Section 2.2.1 when tested against eight seasons of English Premiership rugby
results. While predictive ability is not the primary requirement of the model,
it was in this case considered a suitable arbiter, and so the opposition-
dependent try bonus model of Section 2.2.1 is preferred.

Both the offensive-defensive model and the team specific home advantage model
require an additional parameter for every team. There may be scenarios where
this is desirable but given the sparse nature of fixtures in the Daily Mail
Trophy they do not seem to be justified here, and so the combination of a
single strength parameter for each team and a single home advantage parameter
is chosen for the model in this case.

####  2.3.2 A summary of the model

For a match where $i$ is the home team, and $j$ the away team, the model for
the result outcome may be expressed as

$\displaystyle P(\text{team $i$ beats team $j$ by wide margin})$
$\displaystyle\propto\kappa^{4}\pi_{i}^{4}$ $\displaystyle P(\text{team $i$
beats team $j$ by narrow margin})$
$\displaystyle\propto\rho_{n}\kappa^{3}\pi_{i}^{4}\pi_{j}$ $\displaystyle
P(\text{team $i$ draws with team $j$})$
$\displaystyle\propto\rho_{d}\pi_{i}^{2}\pi_{j}^{2}$ $\displaystyle
P(\text{team $j$ beats team $i$ by narrow margin})$
$\displaystyle\propto\frac{\rho_{n}\pi_{i}\pi_{j}^{4}}{\kappa^{3}}$
$\displaystyle P(\text{team $j$ beats team $i$ by wide margin})$
$\displaystyle\propto\frac{\pi_{j}^{4}}{\kappa^{4}}\quad,$

and for the try bonus point as

$\displaystyle P(\text{team $i$ and team $j$ both gain try bonus point})$
$\displaystyle\propto\tau_{b}\pi_{i}\pi_{j}$ $\displaystyle P(\text{only team
$i$ gains try bonus point})$ $\displaystyle\propto\kappa\pi_{i}$
$\displaystyle P(\text{only team $j$ gains try bonus point})$
$\displaystyle\propto\frac{\pi_{j}}{\kappa}$ $\displaystyle P(\text{neither
team gains try bonus point})$ $\displaystyle\propto\tau_{z}\quad,$

where $\kappa$ is the home advantage parameter, and $\rho_{n}$ and $\rho_{d}$
are structural parameters related to the propensity for narrow or drawn result
outcomes respectively as before.

In order to express the likelihood, additional notation is required. From now
on, the paired $ij$ notation will indicate the ordered pair where $i$ is the
home team and $j$ the away team, unless explicitly stated otherwise. Let the
frequency of each result outcome be represented as follows:

$r^{ij}_{4,0}$ home win by wide margin

$r^{ij}_{4,1}$ home win by narrow margin

$r^{ij}_{2,2}$ draw

$r^{ij}_{1,4}$ away win by narrow margin

$r^{ij}_{0,4}$ away win by wide margin

and the frequency of each try outcome as

$t^{ij}_{1,1}$ both try bonus

$t^{ij}_{1,0}$ home try bonus only

$t^{ij}_{0,1}$ away try bonus only

$t^{ij}_{0,0}$ zero try bonus

Then define the number of points gained by team $i$,
$p_{i}=\displaystyle\sum_{j}4(r^{ij}_{4,0}+r^{ij}_{4,1}+r^{ji}_{0,4}+r^{ji}_{1,4})+2(r^{ij}_{2,2}+r^{ji}_{2,2})+(r^{ji}_{4,1}+r^{ij}_{1,4})+(t^{ij}_{1,1}+t^{ij}_{1,0}+t^{ji}_{0,1}),$
and let $n=\displaystyle\sum_{i}\sum_{j}(r^{ij}_{4,1}+r^{ij}_{1,4})$ be the
total number of narrow wins,
$d=\displaystyle\sum_{i}\sum_{j}(r^{ij}_{2,2}+r^{ji}_{2,2})$ the total number
of draws, $b=\displaystyle\sum_{i}\sum_{j}t^{ij}_{1,1}+t^{ji}_{1,1}$ the total
number of matches where both teams gain try bonus points,
$z=\displaystyle\sum_{i}\sum_{j}(t^{ij}_{0,0}+t^{ji}_{0,0})$ the total number
where zero teams gain a try bonus, and
$h=\displaystyle\sum_{i}\sum_{j}4(r^{ij}_{4,0}-r^{ij}_{0,4})+3(r^{ij}_{4,1}-r^{ij}_{1,4})+(t^{ij}_{1,0}-t^{ij}_{0,1})$
be the difference between points scored by home teams and away teams. Then the
likelihood can be expressed as

$L(\boldsymbol{\pi},\rho_{n},\rho_{d},\tau_{b},\tau_{z},\kappa\ \mid
R,T)\propto\rho_{n}^{n}\rho_{d}^{d}\tau_{b}^{b}\tau_{z}^{z}\kappa^{h}\displaystyle\prod_{i=1}^{m}\pi_{i}^{p_{i}}\quad,$

where $R,T$ are the information from the result and try outcomes respectively.
It is therefore the case that the statistic $(\boldsymbol{p},n,d,b,z,h)$ is a
sufficient statistic for
$(\boldsymbol{\pi},\rho_{n},\rho_{d},\tau_{b},\tau_{z},\kappa)$.

This gives a log-likelihood, up to a constant term, of

$\displaystyle\log
L(\boldsymbol{\pi},\rho_{n},\rho_{d},\tau_{b},\tau_{z},\kappa|R,T)$
$\displaystyle=n\log\rho_{n}+d\log\rho_{d}+b\log\tau_{b}+z\log\tau_{z}+h\log\kappa+\displaystyle\sum_{i=1}^{m}p_{i}\log\pi_{i}.$

##  3 Estimation and parametrization

###  3.1 Log-linear representation

As the form of the log-likelihood suggests, and following Fienberg, (1979),
the estimation of the parameters may be simplified by using a log-linear
model. Let $\theta_{ijkl}$ denote the observed count for the number of matches
with home team $i$, away team $j$, result outcome $k$, and try bonus outcome
$l$. Furthermore let $\mu_{ijkl}$ be the expected value corresponding to
$\theta_{ijkl}$. The log-linear version of the model can then be written as

$log\>\mu_{ijkl}=\theta_{ij}+\theta_{ijk\cdot}+\theta_{ij\cdot l}\quad,$

where $\theta_{ij}$ is a normalisation parameter, and $\theta_{ijk\cdot}$ and
$\theta_{ij\cdot l}$ represent those parts due to the result outcome and try
outcome respectively. That is

  * ${\makebox[12.58705pt][r]{$\theta_{ijk\cdot}$}}=\left\\{\begin{array}[]{l l}\makebox[67.42764pt][l]{$4\alpha_{i}+4\eta$}&\makebox[154.30586pt][l]{if home win by wide margin}\\\ \makebox[67.42764pt][l]{$4\alpha_{i}+\alpha_{j}+\beta_{n}+3\eta$}&\makebox[154.30586pt][l]{if home win by narrow margin}\\\ \makebox[67.42764pt][l]{$2\alpha_{i}+2\alpha_{j}+\beta_{d}$}&\makebox[154.30586pt][l]{if draw}\\\ \makebox[67.42764pt][l]{$\alpha_{i}+4\alpha_{j}+\beta_{n}-3\eta$}&\makebox[154.30586pt][l]{if away win by narrow margin}\\\ \makebox[67.42764pt][l]{$4\alpha_{j}-4\eta$}&\makebox[154.30586pt][l]{if away win by wide margin}\\\ \end{array}\right.$

  * ${\makebox[12.58705pt][r]{$\theta_{ij\cdot l}$}}=\left\\{\begin{array}[]{l l}\makebox[67.42764pt][l]{$\alpha_{i}+\alpha_{j}+\gamma_{bb}$}&\makebox[154.30586pt][l]{if both home and away try bonuses}\\\ \makebox[67.42764pt][l]{$\alpha_{i}+\eta$}&\makebox[154.30586pt][l]{if home try bonus only}\\\ \makebox[67.42764pt][l]{$\alpha_{j}-\eta$}&\makebox[154.30586pt][l]{if away try bonus only}\\\ \makebox[67.42764pt][l]{$\gamma_{zb}$}&\makebox[154.30586pt][l]{if no try bonus for either side}\\\ \end{array}\right.$

where $\pi_{i}=\exp(\alpha_{i})$, $\rho_{n}=\exp(\beta_{n})$,
$\rho_{d}=\exp(\beta_{d})$, $\tau_{b}=\exp(\gamma_{b})$,
$\tau_{z}=\exp(\gamma_{z})$, $\kappa=\exp(\eta)$.

The gnm package in R (Turner and Firth,, 2020) is used to give maximum
likelihood estimates for
$(\boldsymbol{\alpha},\beta_{n},\beta_{d},\gamma_{b},\gamma_{z},\eta)$ and
thus for our required parameter set
$(\boldsymbol{\pi},\rho_{n},\rho_{d},\tau_{b},\tau_{z},\kappa)$. An advantage
of gnm for this purpose is that it facilitates efficient elimination of the
‘nuisance’ parameters $\theta_{ij}$ that are present in this log-linear
representation.

If modelling the try bonus dependent on the result outcome then
$\theta_{ijkl}$ would not be separated into the independent parts
$\theta_{ijk\cdot}$ and $\theta_{ij\cdot l}$, and $\theta_{ijkl}$ would need
to be specified for each result-try outcome combination. This would be the
case for example in modelling the Top14 tournament. There would be some
simplification in that case however, given that, conditional on the result
outcome there are only two try bonus outcomes, namely, winning team gains try
bonus, and winning team does not gain try bonus.

###  3.2 A more intuitive measure

Once the parameters have been estimated, they can be used to compute the
outcome probabilities. This allows for a calculation of the projected points
per match for team $i$, PPPMi, by averaging the expected points per match were
team $i$ to play each of the other teams in the tournament twice, once at home
and once away:

$\text{PPPM}_{i}=\frac{1}{2(m-1)}\displaystyle\sum_{j\neq
i}\sum_{a,b}\big{(}ap^{ij}_{a,b}+bp^{ji}_{a,b}\big{)}\quad,$

where $p^{ij}_{a,b}$ now denotes specifically the probability that $i$ as the
home team gains $a$ points and $j$ as the away team gains $b$ points.

It may readily be shown that the derivative of PPPMi with respect to $\pi_{i}$
is strictly positive so a team ranked higher based on strength $\pi_{i}$ will
also be ranked higher based on projected points per match PPPMi and vice
versa. Thus PPPMi may be used as an alternative, more intuitive, rating.

###  3.3 Adding a prior

One potential criticism of the model proposed so far is that it gives no
additional credit to a team that has achieved their results against a large
number of opponents as compared to a team that has played only a small number.
This is an intuitive idea in line with those discussed by Efron and Morris,
(1977) in the context of shrinkage with respect to strength evaluation in
sport.

An obvious way to address such a concern in the context of the model
considered in this paper is to apply a prior distribution to the team strength
parameters. According to Schlobotnik, (2018), this is an idea considered by
Butler in the development of the KRACH model. In some scenarios, one might
consider applying asymmetric priors based on, for example, previous seasons’
results. This may be appropriate if one were seeking to use the model to
predict outcomes, for example. Even then, given the large variation in team
strength that can exist from one season to the next in, for example, a schools
environment, where there is enforced turnover of players, then the use of a
strong asymmetric prior may not be advisable. In the context of computing
official rankings, it would seem more reasonable as a matter of fairness to
instead apply a symmetric prior so that rating is based solely on the current
season’s results.

This may be achieved through the consideration of a dummy ‘team $0$’, against
which each team plays two notional matches with binary outcome. From one match
they ‘win’ and gain a point and from the other they ‘lose’ and gain nothing.
Recalling that $p_{i}$ represents the total points gained by team $i$, this
adds the same value to each team’s points. The influence of this may then be
controlled by weighting this prior. As the prior weight increases, the
proportion of $p_{i}$ due to the prior increases.

Including a prior has two main advantages in this setting. One is that it
ensures that the set of teams is connected so that a ranking may be produced
after even a small number of matches. The second is that it provides one
method of ensuring that there is a finite mean for the team strength
parameters, which in turn enables the reinterpretation of the structural
parameters as the more intuitive probabilities that were originally introduced
in Section 2.2.

In the three scenarios of varying schedule strength highlighted in Section 1
that related to professional club teams, the ranking is unlikely to be
sensitive to the choice of prior weight, since at any given point in the
season teams are likely to have played a similar number of matches or to have
played sufficiently many such that the prior will not be a large factor in
discriminating between teams. Indeed when estimating rankings mid-season for a
round robin tournament there may be a preference not to include a prior so
that the estimation of PPPMi is in line with the actual end-of-season PPPMi,
without any adjustments being required. In the context of schools ranking, and
the Daily Mail Trophy in particular, this is not the case, with teams playing
between five and thirteen matches as part of the tournament in any given
season. One could consider selecting the weight of the prior based on how
accurately early-season PPPMi using different prior weights predicts end-of-
season PPPMi. However there are some practical challenges to this that are
discussed further in Section 4.3. Perhaps more fundamentally however the
determination of the weight of the prior to be used may be argued to not be a
statistical one but rather one of fairness. Its effect is to favour either
teams with limited but proportionately better records or teams with longer but
proportionately worse records, for example should a 5-0 record (five wins and
no losses) be preferred to a 9-1 record against equivalent opposition or a 6-1
record preferred to a 10-2 record? This is a matter for tournament
stakeholders and will be discussed further in the context of the Daily Mail
Trophy in Section 4.

###  3.4 Mean strength

Choosing to constrain the team strength parameters by ascribing a mean
strength of one is desirable as it allows for an intuitive meaning to be
asserted from the structural parameters in the model. This could be done in a
number of ways, two of which are discussed here.

One way would be to fit the model with no constraint and afterwards apply a
scaling factor to achieve an arithmetic mean of 1. That is let $\mu$ be the
arithmetic mean of the abilities $\pi_{i}$ derived from the model

$\mu=\frac{1}{m}\displaystyle\sum_{i}^{m}\pi_{i}\quad.$

Then by setting $\pi^{\prime}_{i}=\pi_{i}/\mu$ a mean team strength of 1 for
the $\pi^{\prime}_{i}$ is ensured.

Alternatively we might motivate an alternative mean by considering the
strength of the prior. Consider the projected points per match for a dummy
‘team $0$’ that achieves one ‘win’ and one ‘loss’ against each other team in
the tournament, as described in section 3.3. If zero points are awarded for a
‘loss’, and, without loss of generality, one point is awarded for a ‘win’, and
there is assumed to be no home advantage and bonuses, then

$\text{PPPM}_{0}=\frac{1}{m}\displaystyle\sum_{i=1}^{m}\frac{\pi_{0}}{\pi_{0}+\pi_{i}}\quad.$

The strength of team 0, $\pi_{0}$, may be selected to take any value, since it
is not a real participant in the tournament and so it may be set arbitrarily
to $\pi_{0}=1$. Intuitively since it has an equal winning and losing record
against every team one might expect it to be the mean team and therefore have
a strength of one. More formally we are setting

$\displaystyle\frac{1}{2}$
$\displaystyle=\frac{1}{m}\displaystyle\sum_{i=1}^{m}\frac{1}{1+\pi_{i}}=\sum_{i=1}^{m}\frac{\pi_{i}}{1+\pi_{i}}\quad,$

and so rearranging gives

$1=\frac{1}{m}\displaystyle\sum_{i=1}^{m}\frac{2\pi_{i}}{1+\pi_{i}}\quad,$

and by defining a generalised mean as the function on the right hand side of
this equation the required mean of one for the team strength parameters is
returned.

While the prior has been used here to give this generalised mean an intuitive
interpretation, it may be applied even without choosing to use a prior. As
such it could be particularly beneficial in the context of a tournament such
as the Daily Mail Trophy, because it is quite possible that a team will have
achieved full points and so the estimated team strength parameter $\pi_{i}$
may be infinite. If this were the case then it would not be possible to
achieve a mean of 1 using, for example, an arithmetic mean. This in turn would
mean that some of the structural parameters would be undefined also and so one
could no longer make the intuitive interpretations around propensity for draws
or narrow results based on those structural parameters. The generalised mean
defined here, on the other hand, is always finite and is therefore used in the
analysis below.

##  4 The Daily Mail Trophy

###  4.1 Tournament format

The Daily Mail Trophy is a league-based schools tournament that has existed
since 2013, competed for by the 1${}^{\text{st}}$ XVs of participating
schools. Participation is based on entering and playing at least five other
participating schools. Since fixtures are scheduled on a bilateral basis there
is a large amount of variability in schedule strength. In 2017/18 season, the
tournament consisted of 102 school teams, each playing between five and twelve
other participating teams, in a total of 436 matches overall. There is an
existing ranking method that has evolved over time and which seeks to adjust
for schedule strength by means of awarding additional points for playing
stronger teams as determined by their position in the previous season’s
tournament. Further details of this and of the tournament can be found at the
tournament website (https://www.schoolsrugby.co.uk/dailymailtrophy.aspx).

###  4.2 Data summary

The data for the Daily Mail Trophy have been kindly supplied by
www.schoolsrugby.co.uk, the organisation that administers the competition. The
match results are entered by the schools themselves. The score is entered, and
this is used to suggest a number of tries for each team which can then be
amended. These inputs are not subject to any formal verification. This might
suggest that data quality, especially as it relates to number of tries, may
not be reliable. However the league tables are looked at keenly by players,
coaches and parents, and corrections made where errors are found, and so data
quality, especially at the top end of the table, is thought to be good. This
analysis uses results from the three seasons 2015/16 to 2017/18. Over this
period there were 24 examples of inconsistencies or incompleteness found in
the results that required assumptions to be made. All assumptions were checked
with SOCS. Full details of these are given in the Appendix.

The results are summarised in Figure 1. In order to provide a comparison, they
are plotted above those for the English Premiership for the same season. In
comparison to the English Premiership result outcomes, there is a reduced home
advantage and a reduced prevalence of narrow results, though the overall
pattern of a higher proportion of wide than narrow results, and a low
prevalence of draws is maintained. With respect to the try bonus outcomes, the
notable difference is the higher prevalence of both teams gaining a try bonus
in the Premiership.

Daily Mail Trophy

English Premiership

(a)  (b)  (c)  (d)  (e)  (f)

Figure 1: Distribution of outcomes for Daily Mail Trophy and English
Premiership 2015/16 - 2017/18. Result outcome labels: HW - home wide win, HN -
home narrow win, DD - Draw, AN -away narrow win, AW - away wide win; Shades
indicate try bonus outcome, becoming lighter along scale: both bonus, home
bonus, away bonus, zero bonus

###  4.3 Model calibration

In the context of this model, calibration consists of two parts: a
determination of the value of the structural parameters (the model parameters
not related to a particular team); and a determination of the weight of the
prior. One approach to the structural parameters would be to allow them to be
determined each season. However, it would seem clear that, in regard to the
structural parameters, data from proximate seasons is relevant to an
assessment of their value in the current season. For example, one would not
really expect the probability of a draw between two equally matched teams to
change appreciably from season to season and so data on that should be
aggregated across seasons in order to produce a more reliable estimate for the
parameters. As can be seen from Table 1 the range for each was not large. It
was also found that varying the parameters used within that range did not
materially impact ratings under the model. The structural parameters are
therefore fixed at the mean of the three seasons’ estimated values. An
intuitive way to interpret these is by calculating, based on these parameters,
the probability of specific outcomes for a match between two teams of mean
strength. For example it can then be determined that under the model, in such
a match, the probability of a wide result is 65%, of both teams gaining a try
bonus only 1%, and perhaps most notably that the home team is 2.2 times as
likely to win as the away team.

| 2017/18 | 2016/17 | 2015/16 | Mean
---|---|---|---|---
$\rho_{n}$ | 0.384 | 0.498 | 0.463 | 0.448
$\rho_{d}$ | 0.210 | 0.184 | 0.243 | 0.212
$\tau_{b}$ | 0.042 | 0.035 | 0.050 | 0.042
$\tau_{z}$ | 2.489 | 3.075 | 2.838 | 2.801
$\kappa$ | 1.049 | 1.190 | 1.100 | 1.113

Table 1: Structural parameter values for Daily Mail Trophy 2015/16 - 2017/18

As previously mentioned there is limited scope with the Daily Mail Trophy data
to compare the prior weights based on their predictive capabilities, since it
is not a round robin format. One could look at an earlier state in the
tournament and compare to a later state where more information has become
available, but such an approach is limited both by the number of matches that
teams play (many play only five in total), by only having three seasons’ worth
of data on which to base it, and by the fact that even in the later more
informed state the estimation of the team strength will be defined by the same
model. Therefore no analysis of this kind is performed.

As discussed in section 3.3 the main aim of the use of a non-negligible prior
is to reasonably account for the greater certainty one can have on the
estimate of a team’s strength with the greater number of matches played. In
this context, details of the ranking produced using various prior weights are
presented and compared to the relevant team’s record (played, won, drawn,
lost).

(a)  (b)

Figure 2: Top10 PPPM and Rank variation with prior weight for Daily Mail
Trophy 2017/18

School | P | W | D | L | LPPM
---|---|---|---|---|---
Kingswood | 4 | 4 | 0 | 0 | 5.00
Sedbergh | 11 | 11 | 0 | 0 | 4.91
Reed’s | 10 | 10 | 0 | 0 | 4.80
Harrow | 8 | 8 | 0 | 0 | 4.50
Cranleigh | 8 | 8 | 0 | 0 | 4.63
Northampton | 7 | 7 | 0 | 0 | 4.71
St Peter’s, York | 7 | 7 | 0 | 0 | 4.43
Wellington College | 12 | 10 | 0 | 2 | 4.08
Haileybury | 7 | 6 | 0 | 1 | 4.29
Queen Elizabeth Grammar | 7 | 6 | 0 | 1 | 4.14

Table 2: Playing record for Top10, for Daily Mail Trophy 2017/18. LPPM -
league points per match - the total number of points gained, including
bonuses, divided by number of matches; P - Played, W - Win, D - draw, L - loss

Looking at Figure 2 and comparing to the information in Table 2 it can be seen
that as the prior weight is increased that, in general, teams who have played
fewer matches move lower, most notably Kingswood, and those who have played
more move higher, most notably Sedbergh. This is not uniformly true with, for
example, St Peter’s moving higher despite having played relatively few matches
and having a lower league points per match than either Kingswood or
Northampton, who they overtake when prior weight is set to 8. Of course while
the general pattern is clear and expected, the question of interest is what
absolute size for the prior should be chosen. It seems reasonable to state
that a team with a 100% winning record from four matches should not generally
be ranked higher than a team with a 100% wining record from eight matches,
assuming their schedule strength is not notably different. It certainly seems
undesirable that all of the six other teams with 100% winning records below
Kingswood should be ranked lower than them, which would imply a prior weight
of at least one and more likely 4 or higher.

Results for the other two seasons are included in the Appendix. Considerations
and comparisons in line with those above were made across the three seasons. A
reasonable case could be made for prior weights between 2 and 8, and
ultimately it is a decision that should be made by the stakeholders of the
tournament with regard to their view on the relative merit of a shorter more
perfect record as compared to a longer but less perfect record. For the
purposes of further analysis here a prior weight of 4 was chosen.

###  4.4 Results

The model may then be used to assess the current ranking method used in the
Daily Mail Trophy. As can be seen in Figure 3 there is at least broad
agreement between the two measures. However this is not a particularly helpful
way to look at the quality of the Daily Mail Trophy method, as this agreement
can be ascribed largely to the base scoring rule of league points per match,
LPPM, which both methods essentially have in common. What is of more interest
is the effectiveness of the adjustment made for schedule strength. This is
shown in Figure 4.

(a)  (b)  (c)

Figure 3: Scatterplot of Model PPPM, on x-axis, against Daily Mail Trophy
(DMT) ranking measure on y-axis. Darker colours represent higher rank in Daily
Mail Trophy. Top three teams as ranked by Daily Mail Trophy labeled.

(a)  (b)  (c)

Figure 4: Scatterplot of points adjustment to league points per match. PPPM-
LPPM, on x-axis. Adjustment due to Daily Mail Trophy (DMT) method, DMT-LPPM,
on y-axis. Darker colours represent higher rank in Daily Mail Trophy ranking.
Top three teams as ranked by Daily Mail Trophy labeled.

Here clear differences can be seen and there is a low correlation between the
measures. Not surprisingly, some of the teams who perform well in the Daily
Mail Trophy rankings seem to be those that are benefiting most from these
differences, with Wellington College in particular, winner of the Daily Mail
Trophy in two of the three seasons, being a serial outlier in this regard.
While this is concerning in its own right, the requirements on the measure are
related almost solely to the ranking that they produce, rather than the
rating. Figure 5 looks at that.

(a)  (b)  (c)

Figure 5: Scatterplot of PPPM rank on x-axis, against Daily Mail Trophy (DMT)
rank on y-axis. Darker colours represent higher rank in Daily Mail Trophy.

Here considerable differences are seen between the rankings produced by the
two different methods. In order to focus more clearly on this aspect, the
difference in ranking under the two methods is plotted against the Daily Mail
Trophy rank in Figure 6.

(a)  (b)  (c)

Figure 6: Scatterplot of Daily Mail Trophy rank on x-axis, against the gain in
rank from Daily Mail Trophy method vs PPPM. Darker colours represent higher
rank in Daily Mail Trophy.

If the top (and bottom) quintile of the Daily Mail Trophy ranking are
considered then a disproportionately positive (and negative) impact from the
Daily Mail Trophy method is seen. What is perhaps more notable is the size of
some of these rank differences, up to 28 places in a tournament of
approximately one hundred teams. Looked at across the population the mean
absolute difference in rank is approximately eight places. Looking at the
typical difference in points between two teams eight places apart then this
can be observed to be worth approximately 0.4 points per match.

It seems reasonable therefore to say that over the general population of teams
there is scope for improvement in the Daily Mail Trophy method in its approach
to adjusting for schedule strength.

Given the nature of a tournament where there is a winner but no relegation
then there is a natural focus on the top end of the ranking. Comparisons of
the rankings for the top ten teams under the current Daily Mail Trophy method
and the model presented here are shown in Tables 3, 4, and 5.

| DMT |  | PPPM |
---|---|---|---|---
School | Rank | DMT | Rank | PPPM
Sedbergh | 1 | 7.41 | 1 | 4.65
Wellington College | 2 | 7.18 | 7 | 4.18
Cranleigh | 3 | 6.33 | 4 | 4.32
Harrow | 4 | 6.20 | 3 | 4.33
Cheltenham College | 5 | 6.16 | 8 | 4.07
St Peter’s, York | 6 | 5.83 | 6 | 4.19
Brighton College | 7 | 5.63 | 20 | 3.59
Reed’s | 8 | 5.50 | 2 | 4.38
Clifton College | 8 | 5.50 | 16 | 3.72
Haileybury | 10 | 5.49 | 10 | 4.02

| PPPM |  | DMT |
---|---|---|---|---
School | Rank | PPPM | Rank | DMT
Sedbergh | 1 | 4.65 | 1 | 7.41
Reed’s | 2 | 4.38 | 8 | 5.50
Harrow | 3 | 4.33 | 4 | 6.20
Cranleigh | 4 | 4.32 | 3 | 6.33
Kingswood | 5 | 4.31 | NR | NR
St Peter’s, York | 6 | 4.19 | 6 | 5.83
Wellington College | 7 | 4.18 | 2 | 7.18
Cheltenham College | 8 | 4.07 | 5 | 6.16
Northampton | 9 | 4.05 | 11 | 5.41
Haileybury | 10 | 4.02 | 10 | 5.49

Table 3: 2017/18: Top 10 by Daily Mail Trophy method and PPPM. NR - not ranked
due to requirement to play at least five matches

| DMT |  | PPPM |
---|---|---|---|---
School | Rank | DMT | Rank | PPPM
Wellington College | 1 | 7.22 | 3 | 4.37
Sedbergh | 2 | 6.50 | 2 | 4.43
Harrow | 3 | 6.34 | 6 | 4.22
St Peter’s, York | 4 | 6.23 | 8 | 4.06
Kirkham | 5 | 6.15 | 1 | 4.61
Canford | 6 | 6.10 | 9 | 4.02
Clifton College | 7 | 6.00 | 5 | 4.25
Rugby | 8 | 5.96 | 7 | 4.06
Brighton College | 9 | 5.90 | 4 | 4.29
Woodhouse Grove | 10 | 5.81 | 12 | 3.93

| PPPM |  | DMT |
---|---|---|---|---
School | Rank | PPPM | Rank | DMT
Kirkham Grammar | 1 | 4.61 | 5 | 6.15
Sedbergh | 2 | 4.43 | 2 | 6.50
Wellington College | 3 | 4.37 | 1 | 7.22
Brighton College | 4 | 4.29 | 9 | 5.90
Clifton College | 5 | 4.25 | 7 | 6.00
Harrow | 6 | 4.22 | 3 | 6.34
Rugby | 7 | 4.06 | 8 | 5.96
St Peter’s, York | 8 | 4.06 | 4 | 6.23
Canford | 9 | 4.02 | 6 | 6.10
St John’s, Leatherhead | 10 | 4.01 | 14 | 5.39

Table 4: 2016/17: Top 10 by Daily Mail Trophy method and PPPM.

| DMT |  | PPPM |
---|---|---|---|---
School | Rank | DMT | Rank | PPPM
Wellington College | 1 | 6.46 | 7 | 3.73
Kirkham | 2 | 6.44 | 1 | 4.41
Bedford | 3 | 6.35 | 2 | 4.37
Bromsgrove | 4 | 6.21 | 4 | 4.15
Sedbergh | 5 | 6.10 | 5 | 3.99
Woodhouse Grove | 6 | 5.65 | 19 | 3.31
Millfield | 7 | 5.21 | 13 | 3.64
Clifton College | 8 | 5.11 | 8 | 3.73
Solihull | 9 | 5.10 | 11 | 3.67
St Paul’s | 9 | 5.10 | 14 | 3.58

| PPPM |  | DMT |
---|---|---|---|---
School | Rank | PPPM | Rank | DMT
Kirkham Grammar | 1 | 4.41 | 2 | 6.44
Bedford | 2 | 4.37 | 3 | 6.35
Stockport Grammar | 3 | 4.22 | NR | NR
Bromsgrove | 4 | 4.15 | 4 | 6.21
Sedbergh | 5 | 3.99 | 5 | 6.10
Seaford College | 6 | 3.93 | 13 | 5.04
Wellington College | 7 | 3.73 | 1 | 6.46
Clifton College | 8 | 3.73 | 8 | 5.11
Queen Elizabeth Grammar | 9 | 3.69 | 17 | 4.98
Tonbridge | 10 | 3.69 | 18 | 4.94

Table 5: 2015/16: Top 10 by Daily Mail Trophy method and PPPM. NR - not ranked
due to requirement to play at least five matches

Ignoring teams that are not ranked as part of the Daily Mail Trophy having
played fewer than five matches against other participants, the top five always
appear within the top ten of the other ranking method, and the top ten always
within the top twenty. On the other hand they only agree on the first placed
team in one of the three seasons, and in the two seasons where they differ
there was no prior weight that would have resulted in the same winner as the
Daily Mail Trophy method. In particular, in 2015/16 Wellington College, who
were the winners of the tournament, are ranked seventh under the model and
were a full 0.68 projected points per match behind the leader.

##  5 Concluding Remarks

In its most general application the model presented here allows for a ready
extension of the well-known Bradley-Terry model to a system of pairwise
comparisons where each comparison may result in any finite number of scored
outcomes. For example, the model could be adapted to a situation where judges
are asked to assign pairwise preferences on the seven-category symmetric scale
made up of ‘strongly prefer’, ‘prefer’, ‘mildly prefer’, ‘neutral’ etc. if one
were prepared to assign score values to each. The maximum entropy derivation
provides a principled basis for a family of models. The application of entropy
maximisation to motivate these models also helps to clarify the various
assumptions and considerations that are essential to each. In the more
particular implementation for rugby union the family of models provided a
method for assessing teams in situations where schedule strengths vary in a
way that is consistent with the points norm of the sport. Within that family,
different models may be suitable depending on the try bonus stipulations of
the tournament, the density of matches, and the similarity of the number of
fixtures played across teams.

In the investigation of the Daily Mail Trophy the model studied here proved to
be a useful tool in highlighting concerns about the ranking method that is
currently used. It may be tempting to advocate its use directly as a superior
method for evaluating performance in that tournament. However a key element
that it lacks for a wider audience is transparency, for example as represented
in the ability of stakeholders to calculate their rating, to understand the
impact of winning or losing in a particular match, and to evaluate what rating
differences between themselves and similarly ranked teams mean in terms of how
rankings would change given particular results. The strength of the model in
accounting for all results in the rating of each team is, in this sense, also
a weakness for wider application. But even if transparency of method is seen
as a dominating requirement the model may still be useful as a means by which
alternative, more transparent methods can be assessed.

##  6 Appendix

###  6.1 Maximum entropy derivations

####  6.1.1 Offensive Defensive Strength

The offensive-defensive strength model assumes independence of the result and
try outcomes. The maximum entropy derivation is thus related solely to the try
outcome and is linked to the result outcome by the assumption that for each
team the overall strength parameter is equal to the product of the offensive
and defensive parameters. The same notation may be used as in the general
derivation, though in this case $a,b\in\\{0,1\\}$. Entropy is defined as
before as

$S(p)=-\sum_{i,j}m_{ij}\sum_{a,b}p^{ij}_{a,b}\log p^{ij}_{a,b}\quad,$ (7)

and we have the familiar condition that for each pair of teams the sum of the
probabilities of all possible outcomes is 1,

$\sum_{a,b}p^{ij}_{a,b}=1\quad\text{for all $i,j$ such that $m_{ij}>0$},$ (8)

But in this case we have two additional retrodictive criteria per team. First
that, given the matches played, for each team $i$, the expected number of
matches in which a try bonus point was gained is equal to the actual number of
matches in which a try bonus point was gained,

$\sum_{j}m_{ij}\sum_{a,b}ap^{ij}_{a,b}=\sum_{j}\sum_{a,b}am^{ij}_{a,b}\quad.$
(9)

Second that, given the matches played, for each team $i$, the expected number
of matches where a try bonus point was not conceded is equal to the actual
number of matches where a try bonus point was not conceded,

$\sum_{j}m_{ij}\sum_{a,b}(1-b)p^{ij}_{a,b}=\sum_{j}\sum_{a,b}(1-b)m^{ij}_{a,b}\quad.$
(10)

Then, for all $i,j$ such that $m_{ij}>0$, the solution satisfies

$\log
p^{ij}_{a,b}=-\lambda_{ij}-a\lambda_{i}-b\lambda_{j}-(1-b)\lambda^{\prime}_{i}-(1-a)\lambda^{\prime}_{j}-1\quad,$
(11)

which gives us that

$p^{ij}_{a,b}\propto\omega_{i}^{a}\omega_{j}^{b}\delta_{i}^{(1-b)}\delta_{j}^{(1-a)}\quad,$
(12)

where $\omega_{i}=\exp(-\lambda_{i})$,
$\delta_{i}=\exp(-\lambda^{\prime}_{i})$, and we take
$\pi_{i}=\omega_{i}\delta_{i}$.

####  6.1.2 Single parameter home advantage

In order to identify the home team, let the ordered pair ${ij}$ now denote $i$
as the home team and $j$ as the away team. Then under this amended notation,
define entropy as before

$S(p)=-\sum_{i,j}m_{ij}\sum_{a,b}p^{ij}_{a,b}\log p^{ij}_{a,b}\quad,$ (13)

and we have the familiar condition that for each pair of teams the sum of the
probabilities of all possible outcomes is 1,

$\sum_{a,b}p^{ij}_{a,b}=1\quad\text{for all $i,j$ such that $m_{ij}>0$},$ (14)

The retrodictive criterion is now altered to reflect the new notation,

$\sum_{j}m_{ij}\sum_{a,b}\left(ap^{ij}_{a,b}+bp^{ji}_{a,b}\right)=\sum_{j}\sum_{a,b}\left(am^{ij}_{a,b}+bm^{ji}_{a,b}\right)\quad.$
(15)

But now we also have a condition that says that the expected difference
between the number of home points and the number of away points is equal to
the actual difference,

$\sum_{i,j}m_{ij}\sum_{a,b}(a-b)p^{ij}_{a,b}=\sum_{i,j}\sum_{a,b}(a-b)m^{ij}_{a,b}\quad.$
(16)

Then, for all $i,j$ such that $m_{ij}>0$, the solution satisfies

$\log
p^{ij}_{a,b}=-\lambda_{ij}-a\lambda_{i}-b\lambda_{j}-(a-b)\lambda_{0}-1\quad,$
(17)

which gives us that

$p^{ij}_{a,b}\propto\kappa^{(a-b)}\pi_{i}^{a}\pi_{j}^{b}\quad,$ (18)

where $\kappa=\exp(-\lambda_{0})$, $\pi_{i}=\exp(-\lambda_{i})$, and the
constant of proportionality is $\exp(-\lambda_{ij}-1)$.

####  6.1.3 Team specific home advantage

Using the same notation, define entropy in the now familiar way

$S(p)=-\sum_{i,j}m_{ij}\sum_{a,b}p^{ij}_{a,b}\log p^{ij}_{a,b}\quad,$ (19)

and we have the familiar condition that for each pair of teams the sum of the
probabilities of all possible outcomes is 1,

$\sum_{a,b}p^{ij}_{a,b}=1\quad\text{for all $i,j$ such that $m_{ij}>0$},$ (20)

The retrodictive criteria are now split into home and away parts, so that we
have that, for all teams, the expected number of home points gained is equal
to the actual number of home points gained,

$\sum_{j}m_{ij}\sum_{a,b}ap^{ij}_{a,b}=\sum_{j}\sum_{a,b}am^{ij}_{a,b}\quad,$
(21)

and that, for all teams, the expected number of away points gained is equal to
the actual number of away points gained,

$\sum_{j}m_{ij}\sum_{a,b}bp^{ji}_{a,b}=\sum_{j}\sum_{a,b}bm^{ji}_{a,b}\quad.$
(22)

Then, for all $i,j$ such that $m_{ij}>0$, the solution satisfies

$\log
p^{ij}_{a,b}=-\lambda_{ij}-a.{}_{H}\lambda_{i}-b.{}_{A}\lambda_{j}-1\quad,$
(23)

where ${}_{H}\lambda_{i}$ and ${}_{A}\lambda_{j}$ are the Lagrangian
multipliers relating to the home and away criteria respectively. This gives

$p^{ij}_{a,b}\propto{}_{H}\pi_{i}^{a}{}_{A}\pi_{j}^{b}\quad,$ (24)

where the strength parameters, ${}_{H}\pi_{i}=\exp(-{}_{H}\lambda_{i})$ and
${}_{A}\pi_{j}=\exp(-{}_{A}\lambda_{j})$ denote the home and away strengths of
$i$ and $j$ respectively.

###  6.2 Data cleaning

While there were no means to validate the data independently, there were 24
occasions of identifiable self-inconsistencies or incompleteness in the data
across the three seasons of interest, 15 of which impacted the result or try
outcomes for at least one of the teams involved. The treatment of all of these
is described below. They were checked for reasonableness with SOCS, the
administrator for the tournament.

  1. 1.

Where the score could not have produced the try outcome. Since a try is worth
five points in rugby union, then the score of any team may not be less than
five times their number of tries. If swapping the number of tries recorded for
home and away teams produced consistency then this was done. If this did not
resolve the issue then the number of tries was adjusted down to the maximum
number of tries possible given the score. The number of incidences of this
were: two in 2017/18, five in 2016/17, four in 2015/16. Of those, the number
that meant a team’s try bonus status changed was just one in 2017/18 and two
in 2016/17.

  2. 2.

Where Venue had been entered as ”tbc”, the Venue was set to Neutral. Five
incidences in 2016/17, two in 2015/16.

  3. 3.

Where matches were entered as a win for one side but score and tries were both
given as 0-0. On speaking to SOCS, their speculation was that these may have
related to matches where there had been some sort of ‘gentleman’s agreement’
e.g. the teams had agreed to deselect certain players (in particular those
with representative honours), and the recording of the match was a means of
recognising that a fixture had taken place, but not giving it full status. In
our analysis, the winning team is awarded four points for a win, the losing
team one for a narrow loss, and no try bonus is awarded to either side. There
were two such results in 2017/18, and two in 2016/17.

  4. 4.

Where the try count was blank for one of the two teams, the number of tries
was taken to be the maximum number of tries possible given the score. There
was one case of this in 2016/17.

  5. 5.

Where the result outcome (Won, Draw, Loss) did not agree with the score but
did agree with the try outcome, but became consistent if the score were
reversed, then the score was reversed. One case in 2017/18. This did not
impact the analysis.

###  6.3 Daily Mail Trophy methodology

Currently the ranking is based on Merit Points, which are defined as the
average number of League Points per match plus Additional Points, awarded in
order to adjust for schedule strength.

League Points are awarded, in line with the standard scoring rule for rugby
union leagues in the UK, as:

4 points for a win

2 points for a draw

0 points for a loss

1 bonus point for losing by less than seven points

1 bonus point for scoring four or more tries

Additional Points in the Daily Mail Trophy are awarded based on the ranking of
the current season’s opponents in the previous season’s tournament:

Rank 1 to 25:  0.3

Rank 26 to 50:  0.2

Rank 51 to 75:  0.1

Otherwise:  0

So, for example, a team with eight fixtures qualifying for the Daily Mail
Trophy, with one of those against a top 25 team, three against 26-50th placed
teams, and two against 51-75th placed teams, averaging 3.2 League Points per
match, would get a Merit Points total of $3.2+1\times 0.3+3\times 0.2+2\times
0.1=4.3$.

###  6.4 Further results

(a)  (b)

Figure 7: Top10 PPPM and Rank variation with prior weight for Daily Mail
Trophy 2016/17

School | P | W | D | L | LPPM
---|---|---|---|---|---
Kirkham Grammar | 12 | 12 | 0 | 0 | 4.75
Sedbergh | 10 | 9 | 0 | 1 | 4.5
The Manchester Grammar | 4 | 4 | 0 | 0 | 4.75
Brighton College | 8 | 8 | 0 | 0 | 4.5
Wellington College | 12 | 11 | 0 | 1 | 4.42
Harrow | 9 | 8 | 0 | 1 | 4.44
Clifton College | 10 | 9 | 0 | 1 | 4.5
St John’s School, Leatherhead | 9 | 7 | 0 | 2 | 3.89
St Peter’s, York | 9 | 9 | 0 | 0 | 4.33
Canford | 10 | 9 | 0 | 1 | 4.2

Table 6: Playing record for Top10 as ranked by Model 2, for Daily Mail Trophy
2016/17. LPPM - league points per match - the total number of points gained,
including bonuses, divided by number of matches; P - Played, W - Win, D -
draw, L - loss

(a)  (b)

Figure 8: Top10 PPPM and Rank variation with prior weight for Daily Mail
Trophy 2015/16

School | P | W | D | L | LPPM
---|---|---|---|---|---
Stockport Grammar | 4 | 4 | 0 | 0 | 5
Bedford | 8 | 8 | 0 | 0 | 4.75
Kirkham Grammar | 11 | 11 | 0 | 0 | 4.64
Bromsgrove | 9 | 8 | 1 | 0 | 4.11
Sedbergh | 10 | 7 | 1 | 2 | 3.70
Seaford College | 7 | 6 | 0 | 1 | 4.14
Clifton College | 9 | 7 | 2 | 0 | 4.11
Wellington College | 13 | 9 | 0 | 4 | 3.46
Tonbridge | 9 | 7 | 0 | 2 | 3.44
Solihull | 10 | 9 | 0 | 1 | 4.10

Table 7: Playing record for Top10 as ranked by Model 2 for Daily Mail Trophy
2015/16. LPPM - league points per match - the total number of points gained,
including bonuses, divided by number of matches; P - Played, W - Win, D -
draw, L - loss

Looking at Figures 7 and 8 and comparing them to their respective leagues in
Tables 6 and 7, it may be noted, as before, that a prior weight on the larger
end of the scale is required before teams playing a smaller number of matches
are sufficiently penalised. Looking at the 2017/18 and 2015/16 seasons and the
ranking of Kingswood School and Stockport Grammar School respectively, in
particular, might suggest that of the tested priors, 4 or 8 would be most
appropriate.

An argument against this assertion might be that under current Daily Mail
Trophy rules, teams playing fewer than five matches are excluded from the
league table. In the 2017/18 and 2015/16 seasons Kingswood School and
Stockport Grammar School respectively therefore did not appear in the final
Daily Mail Trophy league table. This rule could continue to be used to deal
with cases of teams playing low numbers of matches rather than relying on the
prior to do the job entirely.

On the other hand one can credibly argue that a robust ranking model should be
able to deal with all result outcomes without an arbitrary inclusion cut off.
It is also reasonable to assert that there is still useful information from
these teams for the calibration of the model, whether they are included or not
in the final table. With this in mind it seems sensible to select a prior of 4
or 8 from the values presented here. Other than the re-ranking of Kingswood
School and Stockport Grammar School already noted the only other differences
from selecting 8 rather than 4 are that Harrow and Cranleigh swap in 2017/18,
Clifton and Brighton in 2016/17 and Solihull and Tonbridge in 2015/16. It is
not possible to say that either of these alternative rankings is definitively
right in any of these three cases. In all these cases the projected points per
match of the two teams remain very similar, and both alternatives would pass
the sensible criterion that a ranking method should be such that all other
relative rankings should not be perceivable as unreasonable by a large
proportion of the tournament stakeholders.

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