## Thursday, November 7, 2013

### Hruska

Hruska, or Mediocrity
A recursive game

The game “Hruska” was invented by Douglas Hofstadter, and named ironically after Senator Hruska, who spoke up for representation for the mediocre. The game Hruska recursively rewards mediocre play in Hruska.
Hruska is defined recursively, in levels, as follows:
Level 0 Hruska: 3 players; each picks a number from 0 to 3. The player picking the middlemost gets 1 point, the others get 0. Ties = middlemost.
Level 1 Hruska: Play 3 rounds of level-0 Hruska; sum up points won; the player with middlemost score gets 1 point, the others get 0. Ties = middlemost.
Level N+1 Hruska: Play 3 rounds of level-N Hruska; sum up points won; the player with middlemost score gets 1 point, the others get 0. Ties = middlemost.

Level N Hruska requires 3^N rounds of play. Hofstadter notes that this is a confusing game, lost in a fog of mediocrity.
Here is a faster version of Hruska, which mediocritizes itself on the fly: play Level 0 Hruska for 3 rounds; give a level-1 score to the result. Then play level 0 Hruska twice more; sum previous score with these two, middlemost gets 1 point. Then play level-0 Hruska twice more, and so on.
I see Hruska as a discrete version of the stretch-and-fold route to chaos. Hruska’s  scoring system is like the logistic map f(x)  = 4x(1-x)

#### 1 comment:

1. Sorry for the *late* comment :D

IIRC, he got around the ties by one player playing whole numbers n∈N while the other two play n+⅓ and n+⅔, respectively.