# What is the points valuation for this problem (assuming that $\!5\!-\!$pointer worth $1\!$ point is standard)

Given 4 teams in a tournament, they versus against all the other teams once for each to reach best-possible place. Three points are awarded to the team winning a match, with no points awarded to the losing team. If the game is drawn, each team receives one point. I wonder about the real relative points value is not similar to what we used to know.
For example: If a team gains
+12 points, they definitely win the tournament. Valuatively they have ∞ points.
+11 points, since that's unreal and impossible. Mathematically they have 0 (even minus-X) points.
+5 points, they have completed all their matches. I think that 5-pointer worth 1 point is standard.
+0 points, maybe they still haven't have a match, they should have 0.Y points instead of 0 points.
Question. What is the points valuation for this problem (assuming that 5-pointer worth 1 point is standard)? Actual points value is always linearly compared with each other and I'm not interested in this big picture.

• What's the point of giving different points for points? Jun 20, 2022 at 9:53
• I believe I might get it now. Can you confirm whether my following thinking is correct? The link to the chess pieces was initially not clear. But possibly you are thinking of the utility of the loss, draw and win. While the these outc9mes score you 0, 1, 3 points the actual value of those outcomes might be different. Jun 20, 2022 at 10:27

Here is an initial investigation.

In a system where l = 0 points, d = 1 points, w = 3 points, then the potential outcomes are (playing 3 games against the 3 opponents)

0: lll
1: dll
2: ddl
3: ddd wll
4: wdl
5: wdd
6: wwl
7: wwd
9: www


When we give slightly different points for the win, e.g. 2.5, then this does not change the order lot

0: lll
1: dll
2: ddl
2.5: wll
3: ddd
3.5: wdl
4.5: wdd
5: wwl
6: wwd
7.5: www


The only difference is that the win-loss-loss and draw-draw-draw are not anymore the same (but this doesn't win you the competition).

With slightly more difference, 2 points for a win, we would get a bigger difference

0: lll
1: dll
2: ddl wll
3: ddd wdl
4: wdd wwl
5: wwd
6: www


Now there is a bigger change and win-loss-loss, win-draw-loss and win-win-loss are suddenly equal to other alternatives (with draws). This increases the value of a draw.

So with points for a win between 2 and 3 there is not much change. This indicates that the utility of a win versus a draw is indeed not the same as the actual scores, since we have the different scoring systems where the utility is the same. It doesn't matter whether we would give a win a score of 2.01 or 2.99, the end result will be the same (however 2 points or 3 points for a win, do have different implications, and especially the 2 points makes a big difference).

How to continue from here is, I imagine to make simulations with some sort of gambling system where teams can have some ways to change towards more or less offensive (with some associated changes in the probabilities of the game outcome) and find out which strategy would be the most successful in winning the competition.

A problem with such simulation is that it is not realistic. Probably one could also use real data to figure out what sort of outcomes are realistic (and include more complex mechanisms like goals scored which are involved when two teams and up with a tie).

In addition, a problem is that there are many different situations and the problem is broad. If it is a world championship then teams will advance when they are 1st or 2nd (and draws might be important for this second place). But in some other competition it might be all about being 1st (and draws are having little influence on who is gonna be 1st). Also the utility is variable. Depending on the current position in the competition and where the other teams are standing it might be different (this is a problem with the utility of the chess pieces that you linked to, rooks are not useful in the opening, but in the end game they are much more important, it is variable). And it also depend on the current standing in the game and the strength of the opponent (if you are 6-0 behind against a strong opponent, then it might be better to safe your energy for the next game).