Bid-Tac-Toe is the marriage of a simple two-person game with economics. The result is a surprisingly subtle puzzle. The game of Tic-Tac-Toe (or, Noughts and Crosses across the pond) is not very interesting once it has been solved. With proper play, which is fairly easy to discover, every game ends in a tie. If, however, we alter the rules so that players must buy the right to play at each turn, the game becomes much more interesting. Here then, are the rules to Bid-Tac-Toe:

1) Each player begins with 100 dollars in their account.
2) One player is randomly chosen to make the first bid.
3) That player bids any integer amount of dollars from zero to the balance of their account.
4) The other player may either accept that bid, or outbid it by one dollar.
5) Whichever player wins the auction gets to play their mark on the 3x3 grid, and their balance is then reduced by the amount of the bid.
6) The player that lost the auction gets to make the opening bid for the next turn.
7) Play continues in this fashion until:
      a) If either player makes three in a row, they win.
      b) Else, if either player runs completely out of money, they lose.
      c) Else, if the grid fills up with no winner, the player with the most money left in their account wins.
      d) Else, if the grid fills up with no winner, and both players are left with the same amount of money (very rare, but not impossible!), then the game is truly a tie.

A complete solution to Bid-Tac-Toe should tell you the optimum bid in any situation, but, in practice, knowing what how much to bid for the first few moves will usually allow you to win the game.