<p>Let <span class="math-container">$I = \{1, 2, 3\}$</span> denote the set of individuals and <span class="math-container">$X = [a, b]\subset \mathbb{R}^+$</span> with <span class="math-container">$a, b \in \mathbb{R}^+$</span> and <span class="math-container">$a &lt; b$</span>, be the set of alternatives. For each <span class="math-container">$i\in I$</span> and some <span class="math-container">$p_i\in X$</span> the following utility function captures individual i's preferences over each alternative <span class="math-container">$x \in X$</span>:</p>
<p><span class="math-container">$$u(x; p_i) = -(x - p_i)^2 \tag{1}$$</span></p>
<p>Let <span class="math-container">$R_m$</span> denote the set of preference relations represented by a member of the class of utility functions defined above. Let <span class="math-container">$R$</span> denote the set of complete and transitive preference relations defined over <span class="math-container">$X$</span>.</p>
<p><span class="math-container">$\textbf{Question 1:}$</span> In this framework and for these assumption on the domain of preferences, does there exist a Social Welfare Function, <span class="math-container">$F : R^3_m \to R$</span> that satisfies Independence of Irrelevant Alternatives and Weak Pareto?</p>
<p><span class="math-container">$\textbf{Question 1:}$</span> Suppose now the Planner knows the functional form of the utilities, but does not observe the parameter <span class="math-container">$p_i$</span>. Consider the following rule <span class="math-container">$\sigma$</span> that assigns a unique element of <span class="math-container">$X$</span> to each profile <span class="math-container">$(p_1, p_2, p_3) \in X^3$</span>:
<span class="math-container">$$\sigma(p_1, p_2, p_3) = \text{med}\{p_1, p_2, p_3\} \tag{2}$$</span></p>
<p>where for each triplet of numbers <span class="math-container">$(\alpha, \beta, \gamma) \in X^3$</span>, <span class="math-container">$\text{med}\{\alpha, \beta, \gamma\}$</span> is the median among them. Does <span class="math-container">$\sigma$</span> satisfy Strategy-Proofness?</p>

CC BY-SA 4.0
2024-01-16T01:06:28.710Z
Is this from some problem set?
Michael Greinecker
CC BY-SA 4.0
2024-01-16T07:51:34.540Z
Yes, this is the last one...
Oliver Queen
CC BY-SA 4.0
2024-01-16T07:56:31.770Z
In that case you ought to [document your own effort](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions).
Michael Greinecker
<microeconomics><mathematical-economics><social-welfare><social-choice>
A question in social choice and preference aggregation
Oliver Queen
# A question in social choice and preference aggregation

(Asked by: username_0 on 2024-01-15)

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?

**Tags:** microeconomics, mathematical-economics, social-welfare, social-choice
> **Comments:**
> **username_1**: Is this from some problem set?  
> **username_0**: Yes, this is the last one...  
> **username_1**: In that case you ought to [document your own effort](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions).  

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## 0 answers

