Centering an asymmetric tabular

Question

I'm trying to center a payoff table asymmetrically but I give up and I ask to you.
The problem is that this type of table appears different than how LaTeX sees it. In fact, the first column should not be considered part of the table when centering. How can I tell this to LaTeX?

Here is the table:

As you see the table appears right shifted.
And here is the code found on tex.stackexchange.com and modified:

\documentclass{article}
\usepackage{caption}
\usepackage{adjustbox}
\newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}}

\begin{document}

\begin{center}
\begin{tabular}{rcccc}
\label{tab:inspectionGame}
&&$a$&$b$\\
\cline{3-4}
&\multicolumn{1}{p{0.3cm}}{\vfill\centering$A$}&
\multicolumn{1}{|p{1.5cm}}{\hfill $4$\newline $3$\hfill}&
\multicolumn{1}{|p{1.5cm}|}{\hfill $1$\newline $-3$\hfill}\\
\cline{3-4}
&\multicolumn{1}{p{0.3cm}}{\vfill\centering$B$}&
\multicolumn{1}{|p{1.5cm}}{\hfill $4$\newline $-3$\hfill}&
\multicolumn{1}{|p{1.5cm}|}{\hfill $-1$\newline $-4$\hfill}\\\cline{3-4}
\end{tabular}
\captionof{figure}{ Ash nazg durbatulûk, ash nazg gimbatul.}
\end{center}

\end{document}

EDIT: curiously someone had exactly my same problem (Center table by end of cell) just few hours ago. I try to apply that solution and if it works I'll delete my question.

EDIT2: Maybe that solution is not exactly what I need or maybe I'm still too noob to generalize :-(

Answer 1

Make the left labels zero wide, and also remove the intercolumn space around the first column. I also straightened the input.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{array}
\usepackage{caption}

\newcommand{\leftlabel}[1]{%
  \makebox[0pt][r]{#1\hspace{\tabcolsep}}%
}
\newcommand{\toplabels}[2]{%
  \multicolumn{1}{@{}c@{}}{}&\multicolumn{1}{c}{#1}&\multicolumn{1}{c}{#2}
}
\newcommand{\payoff}[2]{%
  \hfill#1\newline#2\hfill
}

\begin{document}

\begin{center}
%% just to show the center
\hspace*{\fill}\hspace{-.2pt}\rule{0.4pt}{1cm}\hspace*{\fill}

\begin{tabular}{@{}r@{}|m{1.5cm}|m{1.5cm}|}
\toplabels{$a$}{$b$}\\
\cline{2-3}
\leftlabel{$A$} & \payoff{$4$}{$3$} & \payoff{$1$}{$-3$} \\
\cline{2-3}
\leftlabel{$B$}& \payoff{$4$}{$-3$} & \payoff{$-1$}{$-4$} \\
\cline{2-3}
\end{tabular}
\captionof{figure}{Ash nazg durbatulûk, ash nazg gimbatul.}
\label{tab:inspectionGame}
\end{center}

\end{document}

Answer 2

\documentclass{article}
\usepackage{array,ragged2e}
\newcolumntype{M}[1]{>{\Centering}m{#1}}
\newcolumntype{R}{>{\RaggedLeft}p{1em}}

\newcommand{\MC}[1]{\multicolumn{1}{c}{#1}}
\newcommand\Cell[2]{$\array{@{}R @{\hspace{7mm}} R @{}} & #1\\#2 & \endarray$}

\begin{document}

\begin{table}\centering
$\begin{array}{r|M{1.6cm}|M{1.6cm}|}
 \MC{} & \MC{a}    & \MC{b}          \\ \cline{2-3}
     A & \Cell{4}{3}  & \Cell{1}{-3} \\ \cline{2-3}
     B & \Cell{4}{-3} & \Cell{-1}{-4}\\ \cline{2-3}
\end{array}$
\caption{ Ash nazg durbatulûk, ash nazg gimbatul.}\label{tab:inspectionGame}
\end{table}

\end{document}

Answer 3

A solution with multirow:

        \documentclass{article}

        \usepackage{caption}
        \usepackage{array, multirow}

        \begin{document}

        \begin{center}
        $ \begin{array}{@{}r *{2}{|r@{\hspace{2.4em}}r}|}
         \multicolumn{1}{c}{}& \multicolumn{2}{c}{a}& \multicolumn{2}{c}{b}\\
        \cline{2-5}
         \multirow{2}{*}{$A$}& & 4 &  & 1 \\
           & 3 &  &  -3  & \\
        \cline{2-5}
         \multirow{2}{*}{$B$}& & 4 &  & -1 \\
           & -3 &  & -4  & \\
        \cline{2-5}
        \end{array} $
        \captionsetup{font = footnotesize}
        \captionof{figure}{Ash nazg durbatulûk, ash nazg gimbatul.}
        \label{tab:inspectionGame}
        \end{center}

        \end{document}