Total tightness implies Nash-solvability for three-person game forms

It is long known [13] that every totally tight 2-person game form is Nash-solvable, that is, it has a Nash-equilibrium for any set of player preferences. Furthermore, it is also known [2] that every totally tight 2-person game form is acyclic and dominance solvable. In this short paper we generalize the first result to 3-person game forms, leaving the general n-person case open. On the other hand, we show that the second result does not carry over to more than two players. We exhibit an example of a three-person game form which is totally tight but neither acyclic nor dominance solvable.