Pythagorean Theorem to Optimize Multiple Variables?

Question

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?

Answer 1

It's not boneheaded at all! It's completely commendable that you are putting such principled thought into a problem that comes up in your day to day life.

As you say, using the normalized vs. the un-normalized price will give you different rankings on the games available to you. In fact, both of these cases fit into a general family of weighted metrics like

$$ \sqrt{w_{\text{price}} x_{\text{price}}^2 + w_{\text{rank}} x_{\text{rank}}^2 + w_{\text{versatility}} x_{\text{versatility}}^2} $$

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?

Well, I'd guess you have information that you are not using yet that you can bring to bear, your history of playing games. You've probably played games in the past that you loved, and played games in the past that you didn't, and played games in that made you feel "meh". You can play with the weights to try to approximate this history. Can you find a set of weights that seems to recover your historical preference for some games over others? After playing around for a while, can you think of more information that you could incorporate to help you influence your rankings?