## The Median Game

Update: Apparently this game was invented already by Douglas Hofstadter who called it  “Mediocrity” and it is published in Hofstadter’s book Metamagical Themas: Questing for the Essence of Mind and Pattern. It is also called “Hruska.” (See here and here.)

Ehud Friedgut reminded me of the game MEDIAN which I proposed many years ago.

There are three players and they play the game for eight rounds. In every round all players simultaneously say a number between 1 and 8. A player whose number is (strictly) between the other two get a point. At the end of the game the winner is the player whose number of points is strictly between those of the others.

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### 4 Responses to The Median Game

1. Austin says:

The same game is mentioned in one of Doug Hofstadter’s books, under the name Mediocrity. You can take the game to higher levels recursively — the object of the level n game is to win the median number of level n–1 games.

2. When I was an undergrad, I played a level-3 game of Mediocrity (also known as Hruska) with two friends. Each level-k game consisted of 6 level k-1 games, where a level-0 game required us each to name numbers between 1 and 6. (Why 6? Each turn used a different permutation to break ties.) The last turn (#216) took us half an hour. I named 1, which gave me the low score for the 36th level-1 game, which gave me the high score for the 6th level-2 game, which gave me the middle score for the level-3 game. I won!

3. Gil Kalai says:

Dear Austin and Jeff, many thanks! I got into this game from some project about whether some insights of economic theory extends if we replace the rationality assumption by strange “rationality” rules like one based on median. Here is a link to a somewhat related (to the motivation, not the game) 2002 paper on NAJ economics, see especially section 4.2 http://www.najecon.org/v3.htm .