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Let $I = \{1, 2, 3\}$ denote the set of individuals and $X = [a, b]\subset \mathbb{R}^+$ with $a, b \in \mathbb{R}^+$ and $a < b$, be the set of alternatives. For each $i\in I$ and some $p_i\in X$ the following utility function captures individual i's preferences over each alternative $x \in X$:

$$u(x; p_i) = -(x - p_i)^2 \tag{1}$$

Let $R_m$ denote the set of preference relations represented by a member of the class of utility functions defined above. Let $R$ denote the set of complete and transitive preference relations defined over $X$.

$\textbf{Question 1:}$ In this framework and for these assumption on the domain of preferences, does there exist a Social Welfare Function, $F : R^3_m \to R$ that satisfies Independence of Irrelevant Alternatives and Weak Pareto?

$\textbf{Question 1:}$ Suppose now the Planner knows the functional form of the utilities, but does not observe the parameter $p_i$. Consider the following rule $\sigma$ that assigns a unique element of $X$ to each profile $(p_1, p_2, p_3) \in X^3$: $$\sigma(p_1, p_2, p_3) = \text{med}\{p_1, p_2, p_3\} \tag{2}$$

where for each triplet of numbers $(\alpha, \beta, \gamma) \in X^3$, $\text{med}\{\alpha, \beta, \gamma\}$ is the median among them. Does $\sigma$ satisfy Strategy-Proofness?

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