Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 :(

share|cite|improve this question
What are the rules? – Arnaud Jun 2 '13 at 22:24
@Arnaud oh, I'll update the question now. – wvxvw Jun 3 '13 at 6:02
Your question and your answer are both interesting ! – Arnaud Jun 3 '13 at 7:23
@wvxvw: Why not either write up your answer and accept it (so that people can tell it has been answered), or delete it? – robjohn Jun 3 '13 at 15:43
This was flagged as belonging on Cross-Validated, but it really does not seem suited to that site. There is no game theory site, so it should stay here (besides, it has been answered). – robjohn Jun 3 '13 at 16:53
up vote 0 down vote accepted

It appears that I misapprehended the conditions I put for the problem, but after more thought it becomes apparent that, in fact any shotgun game is Pareto-optimal because whenever you improve a score of one of the players, you necessarily worsen the score of some other player.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.