If you must find a single winner, allow people to change their guess after hearing the later ones, continuing until everyone is happy with their guess (and implicit range).
Thus, if player 1 was thinking “somewhere around 50” they can now pick between “exactly 50″ “lower” or “higher;” and there will be an auction to determine how far below 50 someone stops (or at what point the “51” player drops down, and maybe gets “overcut”
You could also use something like the “name that tune” algorithm:
The first guesser picks a range of values
The second guesser picks a range of values that either
A. Does not overlap with the first guesses OR
B. Covers a different number of values as the first guesser
Each subsequent guesser has the same options
A. Make a guess that doesn’t overlap anyone
B. Make a guess with a different number of values as anyone who is overlapped
Allow people to revise their ranges, as above.
Then, whoever has the smallest range that contains the true value wins.
This answer is great because it takes the problem with the initial game (one person gets to update and the other doesn’t) and returns the symmetry by allowing both players to update. The end result shows who is better at Aumann updating and should get you closer to the real answer.
If you’d rather know who has the best private beliefs to start with, you can resolve the asymmetry in the other direction and make everyone commit to their numbers before hearing anyone else’s. This adds a slight bit of complexity if you can’t trust the competitors to be honest, but it’s easily solved by either paper/pencil or everyone texting their answer to the person who is going to keep their phone in their pocket and say their answer first.
I particularly like the first suggestion as an extremely simple and lightweight modification to the standard procedure. OTOH, it does incentivise rather dishonest reports (since players still want to strategically take under-covered ranges).
The second proposal is pretty good as well, although to avoid the complexity of “do not overlap OR cover a different number of values” I might prefer to simply allow ties.
If you must find a single winner, allow people to change their guess after hearing the later ones, continuing until everyone is happy with their guess (and implicit range). Thus, if player 1 was thinking “somewhere around 50” they can now pick between “exactly 50″ “lower” or “higher;” and there will be an auction to determine how far below 50 someone stops (or at what point the “51” player drops down, and maybe gets “overcut”
You could also use something like the “name that tune” algorithm:
The first guesser picks a range of values
The second guesser picks a range of values that either A. Does not overlap with the first guesses OR B. Covers a different number of values as the first guesser
Each subsequent guesser has the same options A. Make a guess that doesn’t overlap anyone B. Make a guess with a different number of values as anyone who is overlapped
Allow people to revise their ranges, as above.
Then, whoever has the smallest range that contains the true value wins.
This answer is great because it takes the problem with the initial game (one person gets to update and the other doesn’t) and returns the symmetry by allowing both players to update. The end result shows who is better at Aumann updating and should get you closer to the real answer.
If you’d rather know who has the best private beliefs to start with, you can resolve the asymmetry in the other direction and make everyone commit to their numbers before hearing anyone else’s. This adds a slight bit of complexity if you can’t trust the competitors to be honest, but it’s easily solved by either paper/pencil or everyone texting their answer to the person who is going to keep their phone in their pocket and say their answer first.
Two good suggestions!
I particularly like the first suggestion as an extremely simple and lightweight modification to the standard procedure. OTOH, it does incentivise rather dishonest reports (since players still want to strategically take under-covered ranges).
The second proposal is pretty good as well, although to avoid the complexity of “do not overlap OR cover a different number of values” I might prefer to simply allow ties.