Chapter 16.4, Problem 12WE

To determine

To find: the matrix that shows the tournament winner using both direct and second stage dominances and to write the name of the winner.

Answer to Problem 12WE

The matrix that shows the tournament winner is $\left[\begin{array}{cccc}1& 0& 2& 1\\ 1& 0& 1& 1\\ 1& 2& 2& 0\\ 1& 0& 1& 1\end{array}\right]$.

Explanation of Solution

Given information:

In a tennis tournament J beats C, but losses to Y and A. C beats Y and A. Y beats J and A. A beats only J.

Calculation:

The matrix that represents the tennis tournament is.

  $X=\begin{array}{c}J\\ C\\ Y\\ A\end{array}\left[\begin{array}{cccc}0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 1& 1& 0\\ 1& 0& 0& 0\end{array}\right]$

  $X$ Represents the number of winners. $X^2$ represents the number of second stage dominances.

So, $X+X^2$ represents the number of winners no more than 2 steps.

  $\begin{array}{c}{X}^2=\left[\begin{array}{cccc}0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 1& 1& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{cccc}0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 1& 1& 0\\ 1& 0& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 1& 1\\ 1& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right]\end{array}$

Now,

  $\begin{array}{c}X+X^2=\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 1& 1\\ 1& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right]+\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 1& 1\\ 1& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 2& 1\\ 1& 0& 1& 1\\ 1& 2& 2& 0\\ 1& 0& 1& 1\end{array}\right]\end{array}$

Therefore the matrix that shows the tournament winner is $\left[\begin{array}{cccc}1& 0& 2& 1\\ 1& 0& 1& 1\\ 1& 2& 2& 0\\ 1& 0& 1& 1\end{array}\right]$ .

The row sums of $X+X^2$ show that the Y, with a total of 5 direct and second stage dominances, are the tournament.