Is this for a one-shot game or are you doing this over many iterations with players getting some number of points each round?
One simple method (if you are doing multiple rounds) is to rank players each round (Closest=1st, Second Closest=2nd, etc) and assign points as follows:
Points = Number of Players—Rank
So say there are 3 players who guess as follows:
Player 1 guesses 50
Player 2 guesses 49
Player 3 guesses 51
And say the actual number is 52.
So their ranks for that round would be:
Player 1: 2nd place (Rank 2)
Player 2: 3rd place (Rank 3)
Player 3: 1st place (Rank 1)
And their scores would be:
Player 1: 3 − 2 = 1 point
Player 2: 3 − 3 = 0 points
Player 3: 3 − 1 = 2 points
I think this works better if you are calculating a winner over many rounds, so that there is a new ranking and new awarding of points on each round. The same is true of least squared error, which you mention, and most of the other methods of incentivizing players to try to guess the mean expected value.
I could also think of other ways to incentivize this, and to use confidence intervals, but they all add complexity to the points calculations.
In that case, the options are really limited and the main simple ideas for that (eg: guess before you know other player’s guesses) have been mentioned already.
One other simple method for one-shot number games I can think of is:
Automatic Interval Equalization:
When all players guesses are known, you take the two players whose guesses are closest and calculate half the difference between them. That amount is the allowable error, and each player’s interval is his or her guess, plus or minus that allowable error.
You win if and only if the answer is in your interval.
Example:
Player 1 guesses 44
Player 2 guesses 50
Player 3 guesses 60
The allowable error for this would be ((50-44)/2) = 3
So the winning intervals would be:
Player 1: 41-47
Player 2: 47-53
Player 3: 57-63
This would result in at most one winner (unless the answer is half way between the 2 closest guesses). Everyone’s winning interval would be the same size and none would overlap. And nobody would have an incentive to guess near someone else’s (stated or expected) guess, unless they thought the answer was actually close to that.
However, it has the disadvantage that a lot of such contests would end up with no winner.
Is this for a one-shot game or are you doing this over many iterations with players getting some number of points each round?
One simple method (if you are doing multiple rounds) is to rank players each round (Closest=1st, Second Closest=2nd, etc) and assign points as follows:
Points = Number of Players—Rank
So say there are 3 players who guess as follows:
And say the actual number is 52.
So their ranks for that round would be:
And their scores would be:
I think this works better if you are calculating a winner over many rounds, so that there is a new ranking and new awarding of points on each round. The same is true of least squared error, which you mention, and most of the other methods of incentivizing players to try to guess the mean expected value.
I could also think of other ways to incentivize this, and to use confidence intervals, but they all add complexity to the points calculations.
Alas, it’s usually one-shot.
In that case, the options are really limited and the main simple ideas for that (eg: guess before you know other player’s guesses) have been mentioned already.
One other simple method for one-shot number games I can think of is:
Automatic Interval Equalization:
When all players guesses are known, you take the two players whose guesses are closest and calculate half the difference between them. That amount is the allowable error, and each player’s interval is his or her guess, plus or minus that allowable error.
You win if and only if the answer is in your interval.
Example:
This would result in at most one winner (unless the answer is half way between the 2 closest guesses). Everyone’s winning interval would be the same size and none would overlap. And nobody would have an incentive to guess near someone else’s (stated or expected) guess, unless they thought the answer was actually close to that.
However, it has the disadvantage that a lot of such contests would end up with no winner.