# A question in social choice and preference aggregation

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?

• Is this from some problem set? Jan 16 at 1:06
• Yes, this is the last one... Jan 16 at 7:51
• In that case you ought to document your own effort. Jan 16 at 7:56