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}