<p>Let $\Omega$ be the set of all feasible allocations with an element $\omega \in \Omega$.  Consider $I$ agents such that the utility of agent $i$ is described by $u_i(\omega)$.</p>

<p>Definition 1: $\omega \in \Omega$ is weakly Pareto-optimal if $\nexists \omega' \in \Omega$ such that $\forall i$ $u_i(\omega') &gt; u_i(\omega)$.  Weak Pareto-optimality is basically just saying that for any other allocation, $\omega'$, in the set of feasible allocations that not everyone can strictly prefer the alternative.</p>

<p>Definition 2: $\omega \in \Omega$ is strongly Pareto-optimal if $\nexists \omega' \in \Omega$ such that $\forall i$ $u_i(\omega') \geq u_i(\omega)$ with strict inequality for at least one $i$.  Strong Pareto-optimality is a little stronger in the sense that your allocation must be strictly preferred by some to all other allocations and everyone else can be indifferent.</p>

<p>I hope that is helpful.</p>

CC BY-SA 3.0
2015-01-21T22:21:37.747Z
Awesome. That was perfect.
Stan Shunpike
CC BY-SA 3.0
2015-01-21T22:16:29.150Z
2015-01-21T22:16:29.150Z
cc7768
<p>I'd like to have a mathematical version of the following two definitions just because I remember symbols better than words. But I lack the math prowess to convert them from words to symbols. Can someone assist me? </p>

<p>Def 1:</p>

<p>A feasible allocation is <strong>weakly Pareto-optimal</strong> if there is no alternative (feasible) allocation such that <em>everyone</em> prefers the alternative to the original. </p>

<p>Def 2: </p>

<p>A feasible allocation is <strong>strongly Pareto-optimal</strong> if there is no alternative (feasible) allocation such that at least one person prefersthe alternative, and everyone else is indifferent. </p>

<microeconomics><pareto-efficiency>
Converting word definitions of Pareto-Optimal into math symbols
Stan Shunpike
# Converting word definitions of Pareto-Optimal into math symbols

(Asked by: username_0 on 2015-01-21)

I'd like to have a mathematical version of the following two definitions just because I remember symbols better than words. But I lack the math prowess to convert them from words to symbols. Can someone assist me?

Def 1:

A feasible allocation is **weakly Pareto-optimal** if there is no alternative (feasible) allocation such that *everyone* prefers the alternative to the original.

Def 2:

A feasible allocation is **strongly Pareto-optimal** if there is no alternative (feasible) allocation such that at least one person prefersthe alternative, and everyone else is indifferent.

**Tags:** microeconomics, pareto-efficiency

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## 1 answers

### Answer by username_1 on 2015-01-21. Score: 6 | Accepted answer

Let $\Omega$ be the set of all feasible allocations with an element $\omega \in \Omega$. Consider $I$ agents such that the utility of agent $i$ is described by $u\_i(\omega)$.

Definition 1: $\omega \in \Omega$ is weakly Pareto-optimal if $\nexists \omega' \in \Omega$ such that $\forall i$ $u\_i(\omega') > u\_i(\omega)$. Weak Pareto-optimality is basically just saying that for any other allocation, $\omega'$, in the set of feasible allocations that not everyone can strictly prefer the alternative.

Definition 2: $\omega \in \Omega$ is strongly Pareto-optimal if $\nexists \omega' \in \Omega$ such that $\forall i$ $u\_i(\omega') \geq u\_i(\omega)$ with strict inequality for at least one $i$. Strong Pareto-optimality is a little stronger in the sense that your allocation must be strictly preferred by some to all other allocations and everyone else can be indifferent.

I hope that is helpful.

> **Comments:**
> **username_0**: Awesome. That was perfect.  

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