How do I create a test to see if my A.I. is perfect?

Question 1

You should start by observing that move 9 is always forced: there is only one empty square on the board. Move can be considered 8 forced as well, because after seven moves there could be exactly three situations:

O can win on the next move, in which case it takes the win
Placing an X in either one of the two remaining squares wins the game for X, in which case O has lost regardless of its next move
X has zero or one path to victory, in which case O blocks to force a draw

This means that the game is over after at most seven moves.

Also observe that there are only three opening moves: the center, a corner, or a side. It does not matter which of the four corners or sides you take, because the board can be rotated to match a "canonical" opening (the upper-left corner or the middle of the top side).

You can now build your state analysis code. Starting with each of the three possible openings, search with backtracking up to six additional moves using all squares that are open by the time you make the move. After each move, analyze the position to see if X or O has already won; mark wins by X as Wx, and wins by O as Wo. The remaining positions are undecided.

Do not explore positions after Wx or Wo: simply return to the prior step, reporting the win by the corresponding side.

When you reach the seventh move, statically analyze the position to decide if it is one of the three situations described above is applicable, marking the position a Wx, Wo, or a Draw.

Now to the most important step: when you backtrack to the move N-1 by the player p,

If one of the moves that you try is such that all position at the next level becomes Wp, declare the current position a Wp as well.
If all of the moves that you try lead to the win of the opponent, declare the current position a win for the opponent
Otherwise, declare the current position a Draw, and return to the prior level.

If you do this right, all three opening positions will be classified as a Draw. You should see some forcible wins after three moves.

Running this procedure classifies each position as a Wx, Wo, or a Draw. If your AI gets you a win for the player p in a position classified as Wp, or gets you a draw in a position classified as a Draw, then your AI is perfect. If, on the other hand, there are positions that are statically classified as Wp in which the AI gets p only a draw, then your AI engine needs an improvement.

Additional reading: you can find additional insights into the game in this article describing methods of counting possible games of Tic-Tac-Toe.

Question 2

My feeling is that the stats you're quoting are already pretty good. Two expert Tic-Tac-Toe players will always end in a tie, and there is no way to force a win if your opponent knows how to play the game.

Update

There's probably a more elegant wayt o prove the correctness of your A.I., but the most straightforward approach would be the brute force one. Just enumerate all possible board positions as a game tree, and prune the branches that lead directly to a loss. Then for each branch in the tree you can work out the probability of win resulting from following that branch. Then you just need to test your A.I. on each board position and make sure it's picking the branch with the highest probability of a win.

Question 3

What you're doing is more linear optimisation than A.I... I'll not describe all the linear algebra of the Tic-Tac-Toe here, there's plenty of examples on the net.

So using linear algebra, you don't have to prove anything about your results (searching for magic statistics, etc), because your results can be validated by a simple solution-injection in the original equation.

In conclusion, there is two cases :

You're using simple "deduction" logic (which is in reality non-formal linear algebra formulation) : we can't found a ready-to-use method for checking your results without look at your code. EDIT : as Andrew Cooper suggests, brute force can be a ready to use method without seeing at your code.
You're using formal linear algebra formulation : your results can be validated by a simple solution-injection in the original equation.

Question 4

The only thing you can compare is one potential move against another. Whenever it's the computer's turn to make a move, have it play out all possible games from that point on, and choose the move that leads to the highest possible amount of wins. You can't always win, but you can give the opponent more chances to make a bad move.

Question 5

Or, you can always try the tic tac toe algorithm in the link below:

Tic Tac Toe perfect AI algorithm: deeper in "create fork" step

Question 6

given that we know

one cannot force a win
with optimal strategy one cannot lose

your AI has already proven to be optimal if

you did search the full tree when playing against it
and your AI is deterministic (if it were rolling the dice at certain stages you would have had to play against all combinations)

It did not lose, you cannot demand it to win. the wins it did do not count, as your full tree search included bad moves as well. that's all, you are done.

just for fun:
if you had no a priori knowledge about the chances to win/draw/lose a game a common strategy would be to persistently save lost positions. on the next game you would try to avoid them. if you can't avoid a move to a lost position you found another one. this way you can learn not to lose against a certain strategy (if possible) or to avoid an error in your strategy.

Question 7

In order for your tic-tac-toe AI to be proven correct, it needs to satisfy two conditions:

It must never lose.
When the opponent deviates from optimal play, it must win.

Both conditions derive from the fact that if both players play optimally, the tic-tac-toe always ends in a draw.

One automatic method of determining whether your program fulfills these two conditions is to construct what is called a "minimax tree" of every possible tic-tac-toe game. The minimax tree completely characterizes the optimal move for each player, so you can use it to see if your program always selects the optimal move. This means that my answer essentially boils down to, "Write a perfect AI, and then see if it plays the same way as your own AI." However, the minimax algorithm is useful to know, and to my knowledge, this is the only way to test if your AI actually plays optimally.

Here is how the minimax algorithm works (For a gif explanation, see Wikipedia. There's also some pseudocode in the Wikipedia article on minimax.):

Beginning with the tic-tac-toe setup under consideration, construct a tree of all possible subsequent moves. The initial position at the root node. At the lowest level in the tree, you have all of the possible final positions.
Assign a value of +1 to all final positions in which the first player wins, a value of -1 to all moves in which the second player wins, and a value of 0 to all ties.
Now we propagate these values up the tree to the root node. Assume that each player plays optimally. In the last move, Player One will select any move that has a value of +1, i.e. a move that wins the game. If no move has a value of +1, Player One will select a move with value 0, tying the game. Thus, nodes where it is player Player One's move are assigned the maximum value of any of their child nodes. Conversely, when it is Player Two's move, they prefer to select moves with a value of -1, which win them the game. If no winning moves are available, they prefer to tie the game. Thus, nodes where it is Player Two's turn are assigned a value equal to the minimum of their child nodes. Using this rule, you can propagate values from the deepest level in the tree all the way up to the root node.
If the root node has a value of +1, the first player should win with optimal play. If it has a value of -1, the second player should win. If it has a value of 0, optimal play leads to a draw.

You can now determine, in each situation, whether your algorithm selects the optimal move. Construct a tree of all possible moves in tic-tac-toe, and use the minimax algorithm to assign +1, 0 or -1 to each move. If your program is Player One, it is optimal if it always selects the move with the maximum value. If it plays as Player Two, it is optimal if it always selects the move with the minimum value.

You can then loop through every move in the tree, and ask your AI to select a move. The above tells you how to determine if the move it selects is optimal.

Question 8

I would use a decision tree to solve this problem.

Putting it in simple words, decision trees are a method to recursively calculate the expectancy (and chance) of the end result. each "branch" in the tree is a decision who's expectancy is calculated from the sum of (value * chance) possible for this decision.

in a limited options scenario (like tic-tac-toe) you can have the entire tree pre-calculated and therefore after each move of the human player (chance) you can make choose (decision) the next branch witch has the highest expectancy to win.

In a chess game the solution is similar but the tree is not pre-built: after each move the computer calculates the value for every possible move on the board for n depth forward. choosing the best, second best or n-th best expectancy depending on the difficulty of the game selected by the player.