Formula for Game Score Normalization

Question

My friends and I play a lot of board games together. In an effort to keep track of how we do between each game, we use a simple method to normalize the scores.

For example, let's say we have the following score for a particular game:

• Player A: 7
• Player B: 8
• Player C: 6
• Player D: 10

These would then be normalized by adding up the total and dividing each players score by that total - giving the following:

• Player A: 23
• Player B: 26
• Player C: 19
• Player D: 32

This has turned out to be very handy for ranking who's best overall, regardless of whether the scores are based on victory points, dollar amounts or any other type of unit.

However, we are stuck on how to handle games where a lower score is better. I'm sure we are missing something simple, but we cannot determine what the formula should be.

Any help would be greatly appreciated.

Answer 1

Fun question. I think your entire system should be multiplicative, not additive. In other words:

1. Name yourself the "baseline player." Your score will always be $1$.
2. Each other player's normalized score in a game is (their points)/(your points) -- or, in the case of a golf-like game, (your points)/(their points).
3. A player's net score is the product of all their normalized scores.

This will give you exact statistics such as "I am 1.17 times better at games than you." Which could be fun.