<p>Such an example does not exist.</p>

<p>Suppose non-dictatorship is violated. WLOG, let individual <span class="math-container">$1$</span> be the dictator among <span class="math-container">$n$</span> individuals. That is, the social choice function always selects an alternative that ranks highest in <span class="math-container">$1$</span>'s preference, i.e. <span class="math-container">$$f(\succ_1,\dots,\succ_n)\in\arg\max\{\succ_1 | x\in X\}.$$</span></p>

<p>For weak Pareto efficiency to also be violated, we must have two alternatives <span class="math-container">$y,z\in X$</span> such that <span class="math-container">$y\succ_iz$</span> for every <span class="math-container">$i=1,\dots,n$</span>, but the social choice function chooses <span class="math-container">$z$</span>, i.e. <span class="math-container">$f(\succ_1,\dots,\succ_n)=z$</span>. Yet this clearly contradicts the above condition of dictatorship, in that <span class="math-container">$z\notin\arg\max\{\succ_d | x\in X\}$</span>. </p>

CC BY-SA 4.0
2019-11-18T21:22:41.307Z
2019-11-18T21:22:41.307Z
Herr K.
<p>Can you give an example of a social choice rule and a social welfare function that violates the conditions of Weak Pareto criterian and non dictatorship simultaneously?</p>

CC BY-SA 4.0
2024-12-06T16:16:52.867Z
Can you given example of a Social Choice Rule that violates condition Weak Pareto Criterion as well as Non-dictatorship ? Fully prove your claim.
Shiv Kumar
<microeconomics>
Example which violates weak pareto and non dictatorship
Eco
# Example which violates weak pareto and non dictatorship

(Asked by: username_0 on 2019-11-18)

Can you give an example of a social choice rule and a social welfare function that violates the conditions of Weak Pareto criterian and non dictatorship simultaneously?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Can you given example of a Social Choice Rule that violates condition Weak Pareto Criterion as well as Non-dictatorship ? Fully prove your claim.  

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## 1 answers

### Answer by username_2 on 2019-11-18. Score: 4 | Accepted answer

Such an example does not exist.

Suppose non-dictatorship is violated. WLOG, let individual $1$ be the dictator among $n$ individuals. That is, the social choice function always selects an alternative that ranks highest in $1$'s preference, i.e. $$f(\succ\_1,\dots,\succ\_n)\in\arg\max\{\succ\_1 | x\in X\}.$$

For weak Pareto efficiency to also be violated, we must have two alternatives $y,z\in X$ such that $y\succ\_iz$ for every $i=1,\dots,n$, but the social choice function chooses $z$, i.e. $f(\succ\_1,\dots,\succ\_n)=z$. Yet this clearly contradicts the above condition of dictatorship, in that $z\notin\arg\max\{\succ\_d | x\in X\}$.


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