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It doesn't need to have a normal distribution for this to work. And there is no incentive for parties to form even if it was a 100% polarized electorate. Howev

It doesn't need to have a normal distribution for this to work. And there is no incentive for parties to form even if it was a 100% polarized electorate. However, let's imagine a slightly more realistic case where 45% of the room prefers exactly 65 degrees, 46% prefers exactly 75 degrees, and the other 9% are evenly spread in the in-between range. That's far from a normal distribution curves, it is two big spikes and a very low area connecting them. The median preference is probably going to be around 70/71. A reasonable middle ground. The only way it won't arrive at that result is if people are consistently acting against their own interests. If people attempt to get strategic about it, and form parties and agree to vote in a particular way, it won't change the result at all. I'm not sure where you are thinking these incentives come from, but I'm saying they don't exist. What is the incentive for an individual voter of doing anything other than voting for their exact preference? If you are one of those closer to the middle, preferring, say, 72 degrees, your vote will have exactly as much upward pull if you vote for 72 compared to voting for 75. Joining the 75 party doesn't help you (but it potentially hurts you, if you were wrong about the median being lower than your preference). I'll refer again to this, where I built a demonstration app you can play with, or you can just watch the video. If you want it done with polarized distribution, let me know. (it would be easy to hack it to do so) https://pianop.ly/voting/median.html https://pianop.ly/voting/median.html

hirundo 7y ago
> 45% of the room prefers exactly 65 degrees, 46% prefers exactly 75 degrees, and the other 9% are evenly spread in the in-between range ... The median preference is probably going to be around 70/71. That does not sound right. The average would be around 70/71 but the median would be 75. So instead of a moderate compromise, 46% of the room would be happy and (at least) 45% unhappy.
- robbrown451 7y ago
  No, the median is the 50 percentile. That falls in that 9% that lie between the cold natured people (0 through 45 percentile) and the warm natured people (54 through 99 percentile). So if there are 100 people, 45 like 65 degrees, and 46 like 75 degrees. Then -- assuming they are truly spread evenly in between as I said -- the other nine would be: One person prefers 66 degrees, one person prefers 67 degrees, one person prefers 68 degrees, one person prefers 69 degrees, one person prefers 70 degrees, one person prefers 71 degrees, one person prefers 72 degrees, one person prefers 73 degrees, and one person prefers 74 degrees. The median will be 70.5, since the 50 percentile falls between 70 and 71. Now if I am the 101st person, I can move it to 70 degrees by voting for 70 or less, and move it to 71 degrees by voting for 71 or greater. If I vote for any fractional number between 70 and 71, I will get my exact choice, since I will be the 50 percentile voter. There is no reason I can get better results by lying, whether or not I am good at guessing how others will vote.

hirundo 7y ago

> 45% of the room prefers exactly 65 degrees, 46% prefers exactly 75 degrees, and the other 9% are evenly spread in the in-between range ... The median preference is probably going to be around 70/71. That does not sound right. The average would be around 70/71 but the median would be 75. So instead of a moderate compromise, 46% of the room would be happy and (at least) 45% unhappy.

robbrown451 7y ago
No, the median is the 50 percentile. That falls in that 9% that lie between the cold natured people (0 through 45 percentile) and the warm natured people (54 through 99 percentile). So if there are 100 people, 45 like 65 degrees, and 46 like 75 degrees. Then -- assuming they are truly spread evenly in between as I said -- the other nine would be: One person prefers 66 degrees, one person prefers 67 degrees, one person prefers 68 degrees, one person prefers 69 degrees, one person prefers 70 degrees, one person prefers 71 degrees, one person prefers 72 degrees, one person prefers 73 degrees, and one person prefers 74 degrees. The median will be 70.5, since the 50 percentile falls between 70 and 71. Now if I am the 101st person, I can move it to 70 degrees by voting for 70 or less, and move it to 71 degrees by voting for 71 or greater. If I vote for any fractional number between 70 and 71, I will get my exact choice, since I will be the 50 percentile voter. There is no reason I can get better results by lying, whether or not I am good at guessing how others will vote.