## Tuesday, April 9, 2013

### Dilemma 7: Dilemma Tic-Tac-Toe

Dilemma Tic-Tac-Toe

The grid # and the letters X and O are the same; but make these three changes in the rules:
* Player X starts first, but not in the center square;
* X and O alternate, until they fill the grid, or outcome is certain;
* Truce = both XXX and OOO rows; win/lose = only one sort;
draw = neither XXX nor OOO rows.

Here are some sample games. Numbers tell order of moves:

X5 | O6 | X3           X5 | O6 | X3           O4 | X3 | O6
O8 | O2 | X9           X7 | O2 | X9           O8 | O2 | X9
X1 | X7 | O4           X1 | O8 | O4           X1 | X7 | X5

Draw                     Truce                     X wins

X9 | X3 | O8           X5 | X9 | X7            O6 | X3 | X5
O4 | O2 | O6           O6 | O2 | O8           X7 | O2 | X9
X1 | X7 | X5            X1 | O4 | X3           O4 | X1 | O8

Truce                      Truce                     O wins

O6 | X1 | O4           X9 | X1 | O4           X5 | X7 | X3
X9 | O2 | X3           O8 | O2 | O6           O8 | O2 | X9
X5 | O8 | X7           X5 | X7 | X3            X1 | O6 | O4

Draw                       Truce                     X wins

Here is a truce where X started in the center square:
O4 | X5 | O8
O6 | X1 | X9
O2 | X7 | X3

Compare the first two games:
X5 | O6 | X3           X5 | O6 | X3
O8 | O2 | X9           X7 | O2 | X9
X1 | X7 | O4           X1 | O8 | O4
Draw                     Truce

On the second game’s sixth move, O put down O6 in the top-center square (A2), and then told X, “If you block me at C2, I’ll block you at B1, and we’ll draw. Better to grab the ABC1 file now, and let me get ABC2.” X agreed, and they truced. This is classic axial negotiation.
Dilemma tic-tac-toe is a good introduction to dilemma games; it is simple, familiar, and a truce under good axial play.
We can define dilemma tic-tac-toe on other grids, such as 4x4 (4 in a row) or 3x4 (3 in a row). In the following diagrams I leave the grid partly blank once the outcome is decided.
A 4 by 4 truce (4 in a row):
X13 | X11 | X9  | X7
X1  | O4   | X3  |
| X5   | O2  |
O12 | O6  | O10 | O8

A 3 by 4 truce (3 in a row):

O10 | O8 | O6
O4  | X1  | X7
X3  | O2  | X9
|       | X5