Essentially every election is a Nash equilibrium since no single voter can alter result.
This is why you use a randomized model in the first place!
If your basic model is that there are a thousand other voters who are going to roll dice between red, blue, or green (or x hundred of each, all flipping a coin to decide whether to stay home or vote, etc.), where your uncertainty about future events is built into the model, then you can have nontrivial equilibria.
--uh, maybe.
For a very simple example: 300 voters want green > red > blue, 400 voters want red > green > blue, 400 voters want red > blue green, and 1000 voters want blue > green > red; election method is FPTP.
--in your very simple example here: I don't see where any randomization was. In this example, no one voter can change result, therefore, this election (like almost all elections) is a Nash equilibrium, which is an utterly useless fact.
It's not hard to see that the dominant strategy for the first group is to vote red to block the plurality-winning blues.
--Nash equilibria don't have "strategies for groups." They have "strategies for individual players."
And you know as well as I do that this is very practical in voting systems around the world, e.g., in the US. You can't just throw up your hands and say it's impossible to reason about -- or I suppose you can, but that won't do any good.
--there are many cases we can try to reason, tell stories, etc. I don't think they have much if anything to do with Nash equilibria, though. I don't know what you were trying to accomplish in this message. Anyhow, back in the schools+children problem, I think you can try to quantitatively assess how well methods work, and how well they work in the face of voters trying different strategic behaviors, via computer simulation. You can (and I did) also do such sims for voting systems. In many cases it gets quite tricky and/or non-quantitative if you try to work without a computer, so, I'd say doing computer sims would probably tell you quite a lot. Telling a few stories about some contrived scenarios might also be of some use, but will probably never unconfuse the situation enough to tell us how well the marriage/matching/etc solutions work, in any quantitative way. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)