3. Suppose there are two players, 1 and 2, with type spaces $T_1 = \{t_1, t_1\}$ and $T_2 = \{t_2, t_2\}$ respectively. The set of possible outcomes is $X = \{a, b, c, d, e\}$. Let $\sim$ denote indifference and $\succ$ denote strict preference.
Player 1's preference over outcomes is: $a \sim b \succ c \succ d \succ e$ if his type is $t_1$ and $a \succ b \succ d \succ c \succ e$ if his type is $t_1$.
Player 2's preference over outcomes is: $a \sim b \succ c \succ d \succ e$ if his type is $t_2$ and $a \succ b \succ d \succ c \succ e$ if his type is $t_2$.
Consider the social choice rule:
$f(\theta) = \begin{cases} b, & \text{if } \theta = (t_1, t_2) \\ a, & \text{otherwise} \end{cases}$
Consider the direct mechanism that truthfully implements $f$ in dominant strategies. Show that truth-telling is not a player's unique weakly dominant strategy. Show that, if a player chooses a weakly dominant strategy that is not truth-telling, then $f$ is not implemented.
