score interval: score with additional protection against the chicken dilemma
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@casimir said in score interval: score with additional protection against the chicken dilemma:
depends on how candidates compare on each ballot
Recall that preferences can be coded in a matrix, and matrices add.
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@isocratia said in score interval: score with additional protection against the chicken dilemma:
Isn't the chicken dilemma kind of a myth? It contradicts the theorem that approval and score voting elect the Condorcet winner under 100% tactical voting.
Perfectly-informed, perfectly-rational tactical voting generate a Condorcet winner. This is true of all (relevant) voting systems (including plurality), and is the inherent nature of a Condorcet winner.
The problem is that a very wide variety of factors stand in the way of "perfect." There is decent hindsight evidence suggesting that Gary Johnson would have been the Condorcet winner of the 2016 US Presidential election. Consider the informational, institutional, and political barriers preventing the Democrats from nominating Gary Johnson, advancing a Johnson strategy instead of a Clinton strategy. We'd sooner have held the election on Jupiter.
In prisoner's dilemma, it is just as rational for two criminals who have telepathy to have each other's backs as it is for two criminals who are isolated to sell each other out. Neither outcome should be surprising.
Chicken dilemma is similar. If Sanders voters have perfectly single-peaked preferences and only care about maximizing their preferences in this one election, then they will line their perfectly-rational and well-behaved butts up behind Clinton to beat Trump.
But man, political hostage-taking is so in right now. "No, screw that, you elect me if you don't want the other side to win!" This approach might even be mathematically rational if your previous assumptions about this person's utilities were wrong. Maybe Sanders and his voters care more about influencing future elections than winning just this one. Maybe Matt Gaetz just want to run for Florida Governor. Maybe someone is a full-on accelerationist, who believes the best way to address problems is to first make them worse.
I can't think of a point in American history where intra-group picking of "BETRAY" has ever been so prolific as it is right now.
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It's not exactly true of plurality voting. In plurality, there can be multiple Myerson-Weber equilibria (fixed points where the voters vote tactically based on beliefs about the likely winners that are later matched by the actual result). The Condorcet winner wins in some but not all.
To use a contrived example:
~50% of voters: A > B > C
~50% of voters: A > C > BIf the voters believe that A is one of the two likely frontrunners, then A will win with 100% of the vote. That is one Myerson-Weber equilibrium.
But if the voters believe that B and C are the likely frontrunners, then A will lose with 0% of the vote. That is the other Myerson-Weber equilibrium. And even though they would all be better off if they all voted for A, any individual voter unilaterally switching their vote to A will only make the outcome worse from their own perspective.
Approval voting eliminates this kind of absurd equilibrium simply by allowing voters to vote for multiple candidates instead of "switching" their vote from one candidate to another. In the example above, if voters believe that B and C are the frontrunners, then they all approve A anyway, and A wins with 100% approval. So the second, suboptimal equilibrium is eliminated.
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@chocopi said in score interval: score with additional protection against the chicken dilemma:
Perfectly-informed, perfectly-rational tactical voting generate a Condorcet winner. This is true of all (relevant) voting systems (including plurality), and is the inherent nature of a Condorcet winner.
I think this is missing the most important caveat, which is mentioned above. The problem isn't information, it's that any such move has to be perfectly coordinated. Voters need to take actions that are individually irrational, despite there being no way to enforce these strategies. The reason Johnson lost is because it would be individually irrational for a voter to switch from backing Trump to backing Johnson, unless they expected everyone else to do the same. (Whereas anyone who preferred Johnson to Clinton and Trump should've approved Johnson.)
IIRC Vox did a poll with approval in 2016, and found Johnson neck-and-neck with Clinton (which lines up with polls they'd be competitive in a one-on-one).
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@chocopi said in score interval: score with additional protection against the chicken dilemma:
In prisoner's dilemma, it is just as rational for two criminals who have telepathy to have each other's backs as it is for two criminals who are isolated to sell each other out. Neither outcome should be surprising.
Chicken dilemma is similar. If Sanders voters have perfectly single-peaked preferences and only care about maximizing their preferences in this one election, then they will line their perfectly-rational and well-behaved butts up behind Clinton to beat Trump.Wait, but it's not similar. The prisoner's dilemma is taught in game theory classes as a way to teach students a simple game with a dominant (always-best) strategy. The chicken dilemma is a way to teach students about equilibrium-refinement.
Basically, there are some games with multiple Nash equilibria, i.e. there are many "potentially rational" strategies you could use. To predict which one someone will actually use, we need to put reasonable constraints on what kinds of beliefs we'd consider rational. There's two pure Nash equilibria for chicken: If you're 100% certain your opponent will swerve, you should plow ahead. If you're 100% certain they won't, you should swerve.
But what if you're not 100% certain what your opponent will do? Then you have to look for the trembling-hand-proper equilibrium. Say I know my opponent isn't perfect: they can make mistakes or slipups in playing a strategy, or maybe some of their voters won't get the message about the strategy I'm telling them to use. Basically, I'm going to rule out any situations where 100% of my opponent's voters use a certain strategy. Maybe we can say that for each voter, there's a 99% chance they'll use the correct strategy, but a 1% chance they'll mess up their ballot.
This is a very powerful refinement, and you can use it to prove that both bullet-voting and friendly-voting are irrational in the chicken model. 100% bullet-voting risks throwing the election to the Republicans. 100% friendly-voting means you can't the election to the other subfaction with 100% probability. The rational strategy is a mixed strategy, i.e. in equilibrium some Sanders voters will support Clinton and others won't.
For example, say Clinton's faction is polling 35% and Sanders' faction is polling 25%, with Trump at 40%. Then, the optimal strategy is for >20% of Sanders' voters to approve Clinton (or equivalently, for all of them to give a 20/100 on a score ballot). This lets Clinton win with more than 1/5 * .25 + .35 = 40% of the vote, while keeping the margin between Clinton and Sanders as tight as possible in case there's a polling error.
Where exactly the scores end up depends on details like exact utilities and uncertainty. You'll get more cooperation with better polling; that lets you rule the smallest faction out of contention with 100% certainty. You also get more cooperation with more polarization (i.e. if voters care more about blocking the other party than about supporting their favorite).
The main point, though, is that bullet-voting is irrational in this chicken dilemma. Clinton is still (by far) the most-likely winner in this situation, assuming she's the Condorcet winner. We can also work out where, exactly Clinton will end up with perfect strategy, and the answer is "just above 40%, barely edging out Trump and Sanders". In approval, the optimal strategy is to randomize whether you approve Clinton. (In score, I expect a fair chunk of voters to use the intermediate scores instead because that's simpler, but it gives the same result.)
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@casimir said in score interval: score with additional protection against the chicken dilemma:
Or even simpler (but less clean):
Voters score candidates on a scale 0-5.
The two highest scoring candidates and everyone with equal or over 50% of possible score enter the runoff.
Normalize the scores for the remaining candidates.
The remaining candidate with the highest score is declared winner.This second variant behaves like STAR in the case when there is at most two candidates with 50%+ scores. This means, it's a simple Chicken Dilemma improvement to STAR.
This is getting close to practical.
It might even comply with state constitutions that require a winning candidate to receive the “the highest”, “the greatest” or “the largest” number of votes, or “a plurality of votes”. (need good lawyers.)
Could call it STIR, Score Than Instant Runoff.
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Normalize the scores for the remaining candidates.
I like normalization well enough, but are we going to be able to explain and justify it to John Q. Public?
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@casimir said in score interval: score with additional protection against the chicken dilemma:
There is a simpler variant that uses plain score and normalizes the scores in the second round.
- Voters score candidates on a scale 0-5.
- Regard any defeat as a tie when it can be reversed by subtracting all common scores between those candidates of the winner's score.
- Eliminate all candidates with at least one defeat.
- Normalize the scores for the remaining candidates.
- The remaining candidate with the highest score is declared winner.
Or even simpler (but less clean):
- Voters score candidates on a scale 0-5.
- The two highest scoring candidates and everyone with equal or over 50% of possible score enter the runoff.
- Normalize the scores for the remaining candidates.
- The remaining candidate with the highest score is declared winner.
This second variant behaves like STAR in the case when there is at most two candidates with 50%+ scores. This means, it's a simple Chicken Dilemma improvement to STAR.
Here's a particularly simple and attractive method:
- Eliminate all candidates scored below 50%.
- Use quadratic voting to pick the best remaining candidate. (Rebrand it as equal-weight voting, by framing it as taking each ballot and dividing by its "weight"—i.e. sum of squares.)
Why the first elimination step? Well in score, approval, etc. with optimal strategy and perfect information, only one candidate should get over 50% of the vote. This candidate should be the Condorcet winner. That means that for candidates scoring over 50%, voters don't have enough information to know which (if any) is the Condorcet winner.
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@lime said in score interval: score with additional protection against the chicken dilemma:
Here's a particularly simple and attractive method:
Eliminate all candidates scored below 50%.
Use quadratic voting to pick the best remaining candidate. (Rebrand it as equal-weight voting, by framing it as taking each ballot and dividing by its "weight"—i.e. sum of squares.)Unfortunately I do not understand this. A simple as possible explanation might help.
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@gregw said in score interval: score with additional protection against the chicken dilemma:
Unfortunately I do not understand this. A simple as possible explanation might help.
As in, eliminate any candidate with an average rating below 50% of the maximum.
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I do not understand this part:
Use quadratic voting to pick the best remaining candidate. (Rebrand it as equal-weight voting, by framing it as taking each ballot and dividing by its "weight"—i.e. sum of squares.)
