is there a difference between counting those better than you vs counting those worse than you in relative scoring games with ties allowed

Question

This is my first question so let me know if I am doing something wrong.

Imaging a relative scoring game. What I mean by this is a game with a set number of players... lets just say 100 where the score a player gets for a round is based upon the ranking they achieve. The game consists of multiple rounds. So for instance first place in any round would get 100 points and second place would get 99 then 3rd place 98...and so on and so forth until the last player gets 1 point. At the end of the game the person with the most overall points wins.

If there is a tie both players get the higher value. So for example a tie for second place would result in two people getting a score of 99 and then 4th place would get 97 points.

My friend is incorrectly trying to suggest that counting the amount of players you have beaten and counting the amount of players that have beaten you refers to the exact same thing, however I think that because it is possible to tie, these are different.

In a very mathematical sort of way, how can I prove to him that they are different. My thinking is that if you beat 80 people it does not necessarily mean that 19 people have beaten you...(or vice versa) Perhaps only 10 people beat you because you are in the middle of a 9 way tie.

Answer 1

You are correct. Your points for the round is $(100 -$ How many have beat you$)$, or $($Number of players you've beat $+$ number of players you're tied with$)$, and these two notions are equivalent. If you leave out "number of players you're tied with" from the last one, what you get will not be directly linked with how many points you get.

Answer 2

Imagine you have this game scoring format as you have described it. In general, there are 3 ways of handling a tie. I think most people would agree that the best way is to do as track/swimming do: the two people who tied for 3rd are each awarded the average of the points for 3rd and 4th place (as an example), as to keep the allotment of points per event standardized. In the event that this is not the case, there are two ways to handle ties. One can determine to award points based upon the number of people who you beat, or the number of people who beat you. In the event of a tie, you have neither defeated nor been defeated by the person who tied you. Just as one can determine to award points on how many people you defeat, another can similarly look at it from the opposite end- by how many people have defeated you. Each situation will handle the allotment of points to ties differently. You are describing a scenario in which the people who tie are both awarded the higher score- the score corresponding to the place/rank in which they finished. It would be wrong in such a scenario to say “the scoring is essentially determined by how many people you beat” because if that were the case, both people would be awarded the points which correspond with the lower place (ie 4th place points in a tie for 3rd). Rather, the points awarded in a format such as this correspond to the number of people who beat YOU. If your friend was saying it was generally equally right to look at it from the other perspective, and was then insisting that in the scenario where the people who tie are both award the higher points it is actually the correct way to think about the scoring system, he would have been right.