I’m going to play this with my family.

Wow.

]]>If two people are like Dave, I guess I would go for 4. Is this one of those situations where you get in trouble by either underthinking or overthinking others?

]]>I wonder if there is some kind of a limit pattern on this as the number of rounds gets very large. It just seems that 1 and 17 could not continue as the most picked numbers. It also seems that more people (such as Dave) would begin picking 2, 3 and 4 pushing the winning numbers up. Just hypothesizing here.

]]>Holy schnikeys, Dave — you’re right! It was the wrong graph for Game 6. My apologies! That has now been updated, as well as other info in the post. (The winner for Game 6 was actually 2, not 7 as previously stated.)

So using your theory for the updated Game 6, there was no number that would won. Still, I think your choice of 3 is a good one.

]]>I would look at what number(s) I could have played that would have won:

Game 1) Not possible

Game 2) Not possible

Game 3) 4

Game 4) 2, 3

Game 5) 3, 5

Game 6) 3, 5 (Is this accidentally a graph of the Game 5 data?)

Game 7) 4

Game 8) Not possible

Game 9) 3

Game 10) Not possible

Based on this alone, I would go with “3” if I were playing in Game 11.

]]>I’ve played this game with many groups, Evan, and your theory about n/2 sounds good. But I have some data to give an even better estimate for the number to choose. In all the games here, the winning number was never greater than n/5. See the post at https://mathjokes4mathyfolks.wordpress.com/2011/04/26/analyzing-my-favorite-game/; in those games with 32 people, the winning number was never greater than n/7. And I played 5 rounds with a group of 170 people once, and the winning number with that group was never greater than n/16.

]]>Rich, see the post https://mathjokes4mathyfolks.wordpress.com/2011/04/26/analyzing-my-favorite-game/, where I show the results from this game played for 5 rounds by the same group of people. What happened there is that people chose lower numbers in consecutive rounds, until the group decided that the numbers were getting *too low*, and then the numbers popped back up again.

It seems to me that in this game, with n players, you should avoid choosing a number larger than n/2 (they *could* win, but only if there is some number smaller than n/2 that no one chose).

Also, there appears to be no equilibrium in pure strategies… the only equilibrium strategies (strategies so that everyone is happy with how they played) must involve randomisation.

However, given the counter-intuitive nature of mixed strategy equilibrium, finding the equilibrium doesn’t really help decide what you should play.

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