Consider the three-player game tree in Exercise 4.5. Recall that some or all of the strategies in a mixed equilibrium may be pure strategies.

(a) Is the following statement true or false? It is easiest to argue directly with the game tree.

For each of the players I, II, or III, the game has a mixed equilibrium in which that player plays a mixed strategy that is not a pure strategy.

(b) Is the following statement true or false? Explain.

In every mixed equilibrium of the game, at least one player I, II, or III plays a pure strategy.

Exercise 4.5

Consider the following three-player game tree. At a leaf, the topmost payoff is to player I, the middle payoff is to player II, and the bottom payoff is to player III.

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II T M B 213 1 3 0 424 2 4 1 123 3 III R 1 4 2 +2 231 350