Dollar Sign Mathematics#
The actual name of the mathematical equation editor is \(\LaTeX\) which, without the graphics, is simply the word latex (pronounced La-Tek). The LaTeX equation editor is a markdown language that mathematicians use to produce publication quality graphics for mathematics expressions, equations and symbols.
The modern generation often calls it Dollar Sign Mathematics because, in a string of text, the in-line equations are set off by an opening and closing $.
Polynomials and Subscripts#
We use the carat symbol ^ for exponents and the underscore _ for subscripts.
Exponents#
The text $$7x^3-2x^2-5x+1$$ will display as below when included inside dollar signs:
Subscripts#
Subscripts are coded as $x_1 = 5$ and will displas as shown:
Gathering Multiple Equations#
We have the align environment for multiple equations that we would like to see centered and aligned. The double slash ends a line while the ampersand symbol forces the alignment at that spot. The text below shows four lines of LaTeX code, but hard returns should exist between the lines.:
\begin{align} E(Rose A) &= 5x + 2y\\ E(Rose B) &= 2x - 42y\\ \end{align}
The display looks as follows:
Text in Equations#
When we want to create the expected value of “Rose Strategy A”, the standard LaTeX displays all characters as variables. The code: $E(Rose A)$ displays as
\(E(Rose A)\)
As you can see if you look closely, the italicized font used for variables looks awkward when used for a word. However, we can use the \text{} command to print text inside the equation. The results of $E(\text{Rose A})$ looks much more pleasing to the eye:
Vectors#
The code to create the symbol for a vector is \vec{r} which displays as shown:
For the values in the matrix and brackets to enclose the matrix, we must work a bit harder. To create a column vector, we create an \(n\times 1\) matrix with a double backslash indicating the end of row in the matrix:
$\begin{array}{r} 1 \\ 2 \\ -3 \end{array}$
which displays as shown below:
We can add brackets by including \left[ to begin and \right] to close which displays as follows:
Matrices#
The basic code for creating a matrix is similar to vectors. As a matter of fact, both are created with arrays. We first create the array using \begin{array}{rr} 1 & 3 \\ 2 & -1 \end{array} where the {rr} indicate two columns of right-justified text.
If we add the two strategies for each player, \(A\) and \(B\), we need the code \begin{array}{r|rr} & A & B \\\hline A & 1 & 3 \\ B &2 & -1 \end{array} to produce this:
Adding Lines#
The game matrix above includes vertical and horizontal lines. Between the columns, we change {rrr} to {r|rr} which produces the vertical line betwee the first and second columns. The \hline command inserts a horizontal between the two lines it is placed between.
Thus, the code \begin{array}{r|rr} & A & B \\\hline A & 1 & 3 \\ B &2 & -1 \end{array} will display as shown:
Matrix Games#
We will nest our game matrix inside another matrix that provides captions for the players. The external matrix will be created with \begin{array}{cc}&\text{Column Player}\\\text{Row Player}&0\end{array}
Note the following:
The \(0\) is a place holder showing where our game matrix will reside.
We leave an empty cell in this array to align the elements properly.
Adding Player Subtitles#
We will simply insert the code for the matrix where the \(0\) displays above. The combined code looks like this:
\begin{array}{cc}&\text{Column Player}\\\text{Row Player}& \begin{array}{r|rr} & A & B \\\hline A & 1 & 3 \\ B &2 & -1 \end{array} \end{array}
Rose and Colin#
With a hat tip to Philip Straffin whose book is suggested as a supplementary text for this course, I will use the player names Rose and Colin. Why?
Rose plays the row strategies (e.g. “rows”)
Colin plays the column strategies (e.g. “columns”)
The same code as shown above work here with the text of subtitles altered appropriately: