# Pythagorean Theorem to Optimize Multiple Variables?

New to stats, so I'm not sure if this is an already established thing or something I just made up that feels good. I have a list of board games that I'm interested in buying based on their price, their overall ranking on boardgamegeek.com and how versatile they are to accommodate different numbers of players.

Here I'm defining versatility as the rank order of the range of players the game can support. So Formula D can support 2-10 and has a versatility ranking of 1.

I want to minimize all the metrics (lowest price, highest rank, highest versatility rank) so I do:

pythagorean rank = sqrt(amazon price^2 + bgg rank^2 + versatility rank^2)


I then sort accordingly and would want to pick the one that minimizes the function. However, since the ranges are all over the place, a game with a high bgg rank could blow up the equation. So if I normalize the amazon price, bgg rank, and versatility first, then do pythagorean optimization, I get a new value. The problem is there seems to be some wildly different answers from a normalized optimization as opposed to the non-normalized.

I'm not very deep in stats, so I'm not sure if this is a completely boneheaded maneuver, but the sorted (non-normalzied) pythagorean optimization list feels more right to me than the normalized one. Is there a different way of doing this that's much better?

• Are you sure this is really a statistical question, rather than just a mathematical one? Is there some random component here? Jul 23, 2016 at 18:21

$$\sqrt{w_{\text{price}} x_{\text{price}}^2 + w_{\text{rank}} x_{\text{rank}}^2 + w_{\text{versatility}} x_{\text{versatility}}^2}$$
Chosing different values for the weights $w_{\text{price}}, w_{\text{rank}}, w_{\text{versatility}}$ will give you a diverse set of rankings, so the question becomes: how to choose one in a principled way?