Here is a proposal. Not sure if I like it better or worse than my other one.
In all cases, add infinitesimal noise to everyones votes, and the final answer, to avoid ties.
If there are an odd number 2n+1≥3 of people, have everyone submit a n2n+1−n+12n+1 confidence interval. Alternate (choosing randomly which to do first) taking the unlabeled person with the lowest remaining lower bound, and labeling them “below”, and taking the unlabeled person with the highest remaining upper bound, and labeling them “above.” When one person remains unlabeled, they are given the region (r0,r1), where r0 is the greatest lower bound among people labeled “below,” and r1 is the least upper bound among people labeled “above.” Recursively distribute the remaining region less than r0 across people labeled below and the remaining region greater than r1 across people labeled above.
If there are an even number 2n≥2 of people, have everyone name their median guess, let r be the average of the two inner-most guesses, and recursively distribute the region less than r to the people who gave guesses less than r, and the region greater than r to the people who gave guesses greater than r.
Here is a proposal. Not sure if I like it better or worse than my other one.
In all cases, add infinitesimal noise to everyones votes, and the final answer, to avoid ties.
If there are an odd number 2n+1≥3 of people, have everyone submit a n2n+1−n+12n+1 confidence interval. Alternate (choosing randomly which to do first) taking the unlabeled person with the lowest remaining lower bound, and labeling them “below”, and taking the unlabeled person with the highest remaining upper bound, and labeling them “above.” When one person remains unlabeled, they are given the region (r0,r1), where r0 is the greatest lower bound among people labeled “below,” and r1 is the least upper bound among people labeled “above.” Recursively distribute the remaining region less than r0 across people labeled below and the remaining region greater than r1 across people labeled above.
If there are an even number 2n≥2 of people, have everyone name their median guess, let r be the average of the two inner-most guesses, and recursively distribute the region less than r to the people who gave guesses less than r, and the region greater than r to the people who gave guesses greater than r.
If there is 1 person, they get the entire region.