Stuff economists like

by Colleen

I was in a meeting the other day and a situation came up where the organizer asked some of the attendees to pick a number between 1 and 20. The first lady picked 17. The next lady picking was clearly distracted and didn't realize she was supposed to pick a number. She thought the numbers were page numbers and was rifling through her binder, until the organizer clarified. Then she looked flustered and blurted out, "16".

My first thought was 16? what? Okayyyy...but even before I could mentally finish extending the "ayyyyyy", another thought flashed through my mind and I was like, THAT'S BRILLIANT!* See what she did there? (albeit, probably inadvertently) And, ok, maybe everyone else in the civilized world has already realized this, but it took until the third week of my 28th year for me to, so excuuuuuuussse me. This is what I realized:

In a 2 person game where each person guesses a number between a certain range to try and be closest to a pre-selected number, the optimal strategy for the second guesser is to pick the next consecutive higher number if the first number chosen falls in the bottom half of the range, and to pick the next consecutive lower number if the first number chosen falls in the top half of the range. (Obviously, the optimal strategy for the first person is to pick the midpoint, if possible :)

So taking the real life example that occurred in my meeting, by choosing 17, the first person has a 4/20, or 20% chance, of being the closest guess. By choosing 16, the second lady effectively locked in all the remaining numbers for herself, and has a 16/20, or 80% chance of being the closest guess. (Happy little aside: This is totally and most completely game theory at its purest. Love it :)

If the second lady had tried to balance out the first lady's guess by choosing 3, she would be automatically closer to the lower numbers, but she would concede 11 - 16 to the first chooser (1st's chance would become 50%, 2nd's would decrease to 45%), and they would both lose, should the actual number turn out to be 9 or 10.

Multiple person game:

Now, this can be extended further, and in the case of the meeting last night, my mind probably would have been completely blown had the third chooser then said 15! (Which I will totally do, should this exact opportunity ever arise again) However, players have to be aware of how many total players there are in the game and adjust their strategies appropriately. Choosing 15 is genius if you're in a 3 person game, but stupid if you're in a 4 person game. The general strategy in a multi-person game is to try and choose last, and then select the number that "locks in" the greatest range of numbers for you. If you're not choosing last, but you're aware of how many people are playing, you will want to pick a number that decreases the range available for the next people choosing, but still gives you some room to work with..so picking a consecutive number to the first as the second player is a very bad idea in theory, because you could get locked in to just that one number, BUT...if not everyone you're playing with realizes the optimal strategy is to lock you in (like in my meeting), then it's not that bad an option - just riskier.

The ideal situation is to pick first and mess it up for everyone else by choosing the midpoint, or last and then work off the ranges available to you by picking a consecutive number of an earlier guess which locks in the biggest range left.





*if you were playing a two-person game.