Go
OK. I have completed my analysis on Boolean tic-tac-toe, and I have found
that my original hypothesis was flawed by an incorrect understanding of
the rules. However, I am not retracting this proposal because I have
found that the _first player_ can force a win -- for ANY level of
repetitive antiplay.
Consider standard tic-tac-toe, and regard it as boolean tic-tac-toe with
antiplay allowed 0 times per square. The first player will always win,
if he acts rationally, because two rational tic-tac-toe players will
always force it to a cat's game, upon which the first player wins (with 5
symbols to second's 4).
The first player, then, can always win boolean tic-tac-toe if he can
reduce it to standard tic-tac-toe; that is, he can win by eliminating
anti-play. He can do this with a twofold strategy.
1) Never give the second player the opportunity to anti-play -- do this
by never placing two of your symbols in a row, column or diagonal unless
the third space is already occupied by your opponenent's symbol; never
give yourself the winning move, but force your opponent to give it to you.
2) Create a situation where if ever the second player gives the first
player the ability to anti-play, the first player can immediately win the
game.
For the analysis below, 0 is an unmarked square
True, to force the win, opens center. The two distinct responses are corner
and side. True then plays diametrically opposite the response. Distinct
grids at this point are:
f00 0f0 False is now faced with an interesting dilemma. If he places
0t0 0t0 an f anywhere that lines up the two moves, true can antiplay his
00t 0t0 (false's) first move and win. Avoiding this, he must play
elsewhere. True responds with the same tactic he did the
first time -- diametric play. Distinct grids now are:
f00 0f0 (the other response to the second case above yields
ttf ftt the same grid as the only response to the first)
00t 0t0
False is faced with the same dilemma. True responds as before.
Distinct grids:
f0t ft0 (response to the second case above yields one of the
ttf ttf grids produced from the left case.
f0t 0ft
No further winning moves are possible. Anti-play is eliminated from the
game, which goes to the cat, and true wins, 5 spaces to 4.
As you can see, changing the number of times anti-play is allowed is
irrelevant to the game; true can always win by using anti-play as a
blocking tool until it is impossible to use, stalemating the game for his
victory.