Math Problem: Win:Lose ratio

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Kueh wrote on 2012-02-11 04:15

Would it be possible for everyone to have a positive w:l for a given population?

Obviously if whatever was being won and lost was 1 on 1, then no. But what if it was 2 v 2, or n v n?

We'll just say that every loss has one win.
RebeccaBlack wrote on 2012-02-11 04:25

No... there's no way everyone could ever have a positive ratio, no matter how many people were against each other, even if the teams were uneven. The numbers have to add up to 50% and in order for this to occur, they have to all either be 50% or someone has to be under 50%.

The only exception to this is if ties somehow count as wins for both teams, but then it wouldn't be a win:loss ratio.
Yoorah wrote on 2012-02-11 07:51

To be honest, I haven't really put much thought into technically proving this, but I think that no matter how you rearrange your problem with teams, it's still going to be a zero-sum game.

The only way to avoid the zero-sum scenario is in a case where both sides benefit, an example of which would be trade. Fair trade, that is.
Kueh wrote on 2012-02-11 08:01

Quote from RebeccaBlack;770729:
No... there's no way everyone could ever have a positive ratio, no matter how many people were against each other, even if the teams were uneven. The numbers have to add up to 50% and in order for this to occur, they have to all either be 50% or someone has to be under 50%.

I'm not sure what you're trying to say here. What numbers, and who is they?

Either way, I'm pretty sure this isn't a rigorous proof.

And I think I used poor choice of words. Not every single person needs to be positive. Only the majority. Is it possible for the majority to be in the positive?
RebeccaBlack wrote on 2012-02-11 13:59

Quote from Kueh;770994:
I'm not sure what you're trying to say here. What numbers, and who is they?

Either way, I'm pretty sure this isn't a rigorous proof.

And I think I used poor choice of words. Not every single person needs to be positive. Only the majority. Is it possible for the majority to be in the positive?

Yes.

If it isn't 1v1, simply put one person on the losing team every single time and rotate the people in such a way that they're winning more than they're losing. Eventually (depending on how many people are involved), everyone but that one person will be positive.

As an example:

3v3
A, B and C vs D, E and F. F is the consistent loser.

ABC vs DEF
DEA vs BCF
etc.

Eventually, everyone but F will be on the winning team 3 times for every 2 times they're on the losing team, and thus will all be positive. F will just have a whole bunch of losses to make up for that.

No matter how many people are involved, it is always possible for all but one person to be positive by rotating them like this and forcing one person to always be on the losing team.
NomadTrooper wrote on 2012-02-11 15:11

It's a draw! Everybody wins!:skip: