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.

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