Question $1$

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Consider a collective choice problem where $N=\{1,2\},Z=\{x,y,z\},T_1=\{t_1^{**}\},T_2=\{t_2^{\text{'}},t_2^{\text{'}\text{'}}\},$ and where the agents' Bernoulli payoff functions $u_i$:$Z{T}_{i}R,$ for iin$\{1,2\},$ satisfy $u_1(x,t_1^{**})>u_1(y,t_1^{**})>u_1(z,t_1^{**}),u_2(z,t_2^{\text{'}})>u_2(y,t_2^{\text{'}})>u_2(x,t_2^{\text{'}}),$ and $u_2(y,t_2^{\text{'}\text{'}})>u_2(x,t_2^{\text{'}\text{'}})>u_2(z,t_2^{\text{'}\text{'}}).$

Furthermore, agent $1$ assigns probability $in(0,1)$ to type $t_2^{\text{'}}$ of agent $2,$ and probability $1-$ to type $t_2^{\text{'}\text{'}}.$

Then the state space $T=T_1{T}_2$ contains

elements, and the set of all deterministic $($i$.$e$.,$ not involving any randomisations $--$as defined in the lecture$)$ decision rules contains

elements.

The number of deterministic decision rules that are also ex post efficient is equal to

and the number of deterministic decision rules that are both ex post efficient and truthfully implementable is equal to

Now consider a stochastic $($as defined in tutorial problem $1)$ decision rule hat$(d)$ that yields outcome $y$ with probability $1$ in every state where $t_2=t_2^{\text{'}},$ and yields all three outcomes in $Z$ with equal probabilities in every state where $t_2=t_2^{\text{'}\text{'}}.$ Then, given the assumptions made previously regarding $u_1$ and $u_2,$

the decision rule hat$(d)$ can be truthfully implementable, but only if further restrictions on $u_1$ and $u_2$ are imposed.

given the information provided in the problem, it is not possible to determine whether the decision rule hat$(d)$ is$,$ or can be truthfully implementable or not.

the decision rule hat$(d)$ is not truthfully implementable.

the decision rule hat$(d)$ is truthfully implementable.