Definition talk:Pareto Efficiency

Hi everyone,

I changed $\mathbb R^N$ for a finite set $N$ to $\mathbb R^n$ with a natural number $n$. This is the common definition of Pareto Efficiency. In the given book by Osborne, this is equivalent and what he meant (see for example Definition 11.1 where $N$ is the set of players in a game and thus leading to $N=\{1,\dots,n\}$).

Additionally I would like to add that Pareto efficiency can either be defined as "there does not exists $y\in X\setminus\{x\}$ with $x_i\leq y_i$ for all $i$", which comes from the concept of maximising a function and can thus also be called Pareto maximal, or the other way by "there does not exists $y\in X\setminus\{x\}$ with $y_i\leq x_i$ for all $i$", which comes from the concept of minimising a function and can thus also be called Pareto minimal. I have seen that in some definitions this was given as a remark, but I do not know how to do that. Mathematicians mostly use the minimal definition, economists the maximal one.

Also, a symbol for that set might be nice. Going with the maximal definition I have seen $\textrm{Max}(X)$, but then the definition should be called "minimal elements of $X$". Quite common is also $\textrm{Eff}(X)$, sadly with this we would have the confusion of minimal and maximal again. And once I have seen $\textrm{Par}(X)$, having the same problem, but: There is a generalisation of the concept, which I would like to define and which is much more often used nowadays. Instead of ordering $\mathbb R^n$ by the standard cone $\mathbb R^n_\geq=[0,\infty)^n$ (this is the order $x\leq y$ if and only iff $x_i\leq y_i$ for all $i\in\{1,\dots,n\}$ used above) it can be ordered by any cone $C$ giving $C$-efficient (or efficient or $C$-extremal) points of $X$. This set is written $\textrm{Eff}(X,C)$ or $\textrm{Ext}(X,C)$. Depending on what will be decided here I would try to make this later. The thing now is, that $\textrm{Eff}(X,C)$ also depends on minimal or maximal just even by how you look at the problem - if we have a "minimisation" problem, these minimal points will be called efficient and the other way around. The definition of extreme points is independent on that because in the name "extreme" it is reflected to be extreme with respect to the set $X$ and not to a problem. In this sense $\textrm{Par}(X)$ might be a way to get around min/max here as well.

Sources:

[title] Vector Optimization: Theory, Applications, and Extensions, [author] Johannes Jahn, [edition] 2, [DOI] 10.1007/978-3-642-17005-8 -------------- definition 4.1

[title] On generalized-convex constrained multi-objective optimization, [authors] Christian Günther and Christiane Tammer ----------------- definition 4.2

Please excuse me for writing so much here and not knowing how you handle such stuff.

Best regards Paul

Thank you for this -- I'll check it out when I have a brain cell free. (Just re-entered the world of work again, doing 10 hour days.) --prime mover (talk) 20:06, 13 September 2024 (UTC)

BTW, it would be good if you could sign your posts by using the "signature" icon at the top of the edit window, then we can tell exactly which user is talking about exactly what. --prime mover (talk) 20:09, 13 September 2024 (UTC)