I thought about an example game to use to illustrate what "Pareto-optimal" means, and I can't think of an outcome of Shotgun game (rock, paper, scissors) played by three players that would be Pareto-optiomal.
I was just wondering whether I'm not mistaking?
PS. I saw this question: Nash Equilibria for zero-sum games (Rock Paper Scissors) but it's about the overall optimal strategy, not Pareto-optimal.
The scoring is like this:
When you draw against another player, you get 0 points.
When you win against another player, you get 1 point.
When you loose against another player, you get -1 point.
The score is a sum of two outcomes of your match against two other players. Example: player A has scissors, player B has scissors and player C has rock, then players A and B have each 0 (from draw) - 1 (defeated by the rock), i.e. both have -1. Player C has 2 (since that player defeated both A and B and both victories give this player a +1).
Oh... but now that I've described it, I think that every such game would be Pareto-optimal, because improving one's scoring will necessary worsen the scoring of someone else. Ouch, sorry :(