Prove that the only equilibrium in the following three-player game, where Player I chooses a row (T or B), Player II chooses a column (L or R), and Player III chooses a matrix (W or E), is (T , L,W).

Guidance: First check whether there are equilibria in pure strategies. Then check whether there are equilibria in which two players play pure strategies, while the third plays a completely mixed strategy (meaning a strategy in which each one of his two pure strategies is chosen with positive probability).

After that, check whether there are equilibria in which one player plays a pure strategy, and the other two play completely mixed strategies. Finally, check whether there are equilibria in which all the players play completely mixed strategies. Note the symmetry between the players; making use of the symmetry will reduce the amount of work you need to do.