Based on replies to other answers, the criteria are:
stays simple
one-shot
single winner
encourages “honest” answers (relative to Closest Number) aka has “good incentives”
no sealed answers, no simultaneous play
Not specified: fairness and symmetry. As far as I can tell, it’s okay if the first player has an advantage or disadvantage, or plays differently.
For two players, the Closest Number game is already a good solution. Alice honestly guesses the median, and Bob honestly guesses “higher” or “lower” by guessing either one higher or one lower. This is the pointer to my solution.
Concrete example
First let’s get a concrete example of Closest Number for three players, to show how it encourages “dishonest” answers. The game will be to guess the answer to 2d6. We’ll give ties to the first player. The optimal line is:
So not really more honest. I like this game for other reasons (non-constant-sum, meaningful choices, aesthetics of specifying a range) but it doesn’t answer the question as posed.
The Under-In-Over Game
Basically a different way of generalizing the Closest Number game for three players:
The first player specifies their 33.3% confidence range
The second player picks one of the three ranges: Under, In, or Over
The third player picks one of the remaining ranges
The first player gets whatever is left.
The first player mostly wants to be honest, unless they know that the other players are misinformed (a dangerous strategy if she’s wrong). Then the second and third player fully honestly pick the slice they think is largest. It’s … kinda dull. But it matches the criteria given. And you have to expect that a game that encourages non-strategic answers is strategically dull.
Extensions to 4+ players are an exercise for the reader, there’s a naive extension where the first player splits the range into N-1 pieces, but I think there are better options where the splitting job is divided between multiple players.
Criteria
Based on replies to other answers, the criteria are:
stays simple
one-shot
single winner
encourages “honest” answers (relative to Closest Number) aka has “good incentives”
no sealed answers, no simultaneous play
Not specified: fairness and symmetry. As far as I can tell, it’s okay if the first player has an advantage or disadvantage, or plays differently.
For two players, the Closest Number game is already a good solution. Alice honestly guesses the median, and Bob honestly guesses “higher” or “lower” by guessing either one higher or one lower. This is the pointer to my solution.
Concrete example
First let’s get a concrete example of Closest Number for three players, to show how it encourages “dishonest” answers. The game will be to guess the answer to 2d6. We’ll give ties to the first player. The optimal line is:
Aria: 5 (claims 15⁄36) Bridget: 8 (claims 11⁄36) Claude: 9 (claims 10⁄36)
This can be a fun game, but it fails the “honest” test, whatever “5″ means, it isn’t an honest guess at the median.
Failed attempt: Smallest Range
Everyone guesses a range
The winner is the person with the smallest range that contains the answer.
If nobody’s range contains the answer, everyone loses.
Ties go to whoever played first.
Minimum three players.
Optimal line in 2d6:
Aria: 4-6 (claims 12⁄36) Bridget: 8-10 (claims 12⁄36) Claude: 2-12 (claims 12⁄36)
So not really more honest. I like this game for other reasons (non-constant-sum, meaningful choices, aesthetics of specifying a range) but it doesn’t answer the question as posed.
The Under-In-Over Game
Basically a different way of generalizing the Closest Number game for three players:
The first player specifies their 33.3% confidence range
The second player picks one of the three ranges: Under, In, or Over
The third player picks one of the remaining ranges
The first player gets whatever is left.
The first player mostly wants to be honest, unless they know that the other players are misinformed (a dangerous strategy if she’s wrong). Then the second and third player fully honestly pick the slice they think is largest. It’s … kinda dull. But it matches the criteria given. And you have to expect that a game that encourages non-strategic answers is strategically dull.
Extensions to 4+ players are an exercise for the reader, there’s a naive extension where the first player splits the range into N-1 pieces, but I think there are better options where the splitting job is divided between multiple players.