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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.

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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.

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  • $\begingroup$ Awesome. That was perfect. $\endgroup$
    – Stan Shunpike
    Commented Jan 21, 2015 at 22:21

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