The Condorcet Voting Game.md

The Condorcet Voting Game

This game takes place on a $N\times N$ game board. Each tile on the board can be identified by its $(x, y)$ coordinate.

The game begins with all tiles having a value of zero, except for those along the diagonal which begin with a value of one. The game ends when there is a row or column that contains only positive, non-zero values.

On your turn, you select $\frac{N(N-1)}{2}$ tiles and add one to their value. Whenever you add one to tile $(x, y)$, the value of tile's "mirror", $(y, x)$, decreases by one. The only constraints on your move are that the the tiles you select must be unique, and you may not select both a tile and its mirror.

Here's an example move on our board. The tiles colored light blue have had one added to them, the tiles colored dark blue are their mirrors and have had one subtracted from them.

You are playing with a non-zero number of players who make moves randomly. What strategy would you use if your goal was to keep the game from ending?