Creating a Payoff Matrix Using LaTeX Tabular Environment

Please post complete minimal examples rather than code fragments. They make it much easier to help.

Maybe something like this?

I don't know why you would want 'Player Y' centred over any box, so I've assumed you don't really mean it.

\documentclass{article}
\usepackage{multirow,array}
\begin{document}
  \begin{table}
    \setlength{\extrarowheight}{2pt}
    \begin{tabular}{cc|c|c|}
      & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $Y$}\\
      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$A$}  & \multicolumn{1}{c}{$B$} \\\cline{3-4}
      \multirow{2}*{Player $X$}  & $A$ & $(x,y)$ & $(x,y)$ \\\cline{3-4}
      & $B$ & $(x,y)$ & $(x,y)$ \\\cline{3-4}
    \end{tabular}
  \end{table}
\end{document}

Or possibly

  \begin{table}
    \setlength{\extrarowheight}{2pt}
    \begin{tabular}{*{4}{c|}}
      \multicolumn{2}{c}{} & \multicolumn{2}{c}{Player $Y$}\\\cline{3-4}
      \multicolumn{1}{c}{} &  & $A$  & $B$ \\\cline{2-4}
      \multirow{2}*{Player $X$}  & $A$ & $(x,y)$ & $(x,y)$ \\\cline{2-4}
      & $B$ & $(x,y)$ & $(x,y)$ \\\cline{2-4}
    \end{tabular}
  \end{table}

Check out the sgame package, which nicely packages this all up for you: https://www.economics.utoronto.ca/osborne/latex/sgame.pdf

Your example game could be written as:

\begin{game}{2}{2}[Player~X][Player~Y]
& $A$ & $B$ \\
$A$ &(x,y) &(x,y)\\
$B$ &(x,y) &(x,y)
\end{game}