---
title: "Keeping and discarding players"
output: html_document
---

```{r setup, include=FALSE}
set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
library("hyper2")
library("magrittr")
options("digits" = 5)
```

## Functions keep() and keep_flawed()


Function `keep_flawed()` can be confusing.  Originally, the package
included a function `keep()` but this used flawed logic, hence the
name of its replacement.  Here is a tiny example that should clarify
things.  We will use a subset of `skating_table`:

```{r}
jj <- skating_table[1:3,1:6]
rownames(jj) <- letters[1:3]
jj
```


Above, rows of object `jj` correspond to competitors and columns to
judges.  Object `jj` does not function as a rank table because the
entries are not sequential (judge `J2`, for example, has no
second-ranked competitor).

We could convert explicitly:

```{r}
a <- apply(jj,2,rank)
a
```

So each column is a permutation of `123`.  So, for example judge 1
ranked `a` first, `b` third, and `c` second.  We can coerce to a
`hyper2` object with `ordertable2supp()`:

```{r}
OT <- ordertable2supp(a)
OT
```

(function `ordertable2supp()` does this automatically, so
`ordertable2supp(jj)` would have given the same result).  Now suppose
we are interested only in the first two competitors, `a` and `b`,
which are competitors 1 and 2.

```{r}
keep_flawed(OT, c("a","b"))
```

Above, function `keep_flawed()` has effectively taken `OT`, set `c=0`,
and then discarded the `c^5` term.  This process, while natural from a
computing perspective, has the effect of replacing `(a+c)^-4` with
`a^-4`, which has no natural probabilistic interpretation: we are
ignoring `c`'s victories but nevertheless interpreting the joint
strength term on the denominator [viz `a+c`] as a *loss* for `a`.

What happens if we keep only `a`?

```{r}
keep_flawed(OT,"a")
```

the above support function is meaningless (?) as, having set `b=0` we
then treat `a+b` as simply `a` which sort of makes sense until we
realise that `(a+b)^-6` reduces to `a^-6`.  It is as though `a` has
wins only 3 trials out of 8, against a competitor of known zero
strength whose (impossible) wins we ignore.  Perhaps this has a
natural interpretation in probability, but if so I don't see it.

###  Conclusions{-}

It does not seem possible to take a `hyper2` object and discard
certain players.  It seems that the only way to discard players is to
work with the original observation, discard players in the dataset
according to some criterion, and coerce to a likelihood function.



