If all action profiles in a payoff matrix are Pareto optimal does this automatically mean that the game is a constant-sum?

Question

My guess is no, because constant sum means that the payoff is constant in every action profile, but I'm not quite sure.

Answer 1

Having all the action profiles in a payoff matrix Pareto optimal does not imply that the game is constant sum. A counter example is given by

\begin{array}{|c|c|c|} \hline & L & M & R \\ \hline H & (3, 4) & (4, 3) & (5, 2) \\ \hline T & (6, 0) & (2, 5) & (1, 6) \\ \hline \end{array}

As we can see they are all Pareto optimal. This game is not constant sum though because of $(T, L)$.