[math-fun] Marriages, and schooling the children
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)
I was carrying over the randomization I had described earlier: each voter flips a coin to determine if they will turn out for the election. For any of the voters in the block the chance that they will improve their situation by voting red is higher than the chance that they will improve their situation by voting green, whether they assume that the others of their block vote honestly or strategically. With reasonable payoffs (say, 2 utils for green and 1 util for red) the unique best strategy for any of these voters (who are actually voting!) will be red, under the assumptions I described for other voters. Again, to emphasize: this is not a minor point or a corner case, rather the foundation of how many democracies around the world (mine included) function on a practical basis. Voters who prefer third-party candidates tend quite strongly to vote for the better (according to their preferences) of the two major-party candidates, at least in close elections.
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.
I think that analysis will probably be more useful than simulation, at least until the issues are better understood. It's far too easy to code simulations to match one's own preconceptions, and especially hard to catch oneself in the act. Charles Greathouse Analyst/Programmer Case Western Reserve University On Wed, Dec 10, 2014 at 12:43 AM, Warren D Smith <warren.wds@gmail.com> wrote:
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)
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Charles Greathouse -
Warren D Smith