Analyzing tic-tac-toe

Tic-tac-toe, or, as it is known in England, Noughts and Crosses, is an intriguing game. The rules are quite simple. The first player is ‘X’. The second ‘O’ and they take turns drawing ‘X’’s and ‘O’s on the 3 x 3 grid. The winner is the player who succeeds in completing a row, column or diagonal of his three marks, three ‘X’s or three ‘O’s.

Watch the video, to see a short analysis of some of the moves in the game. In game theory, mathematicians are interested in finding out if one of the players has an advantage on the other. We usually assume that both players are ‘clever’ and don’t make ‘silly’ mistakes, meaning, that they play the best possible available response to the other players moves.

Tic-tac-toe always ends in a draw under these condition. However, although the first player does not have a decisive advantage, meaning that if both players are clever the game cannot end with a win for him, he does has an advantage over the second player in that he has many more possible moves to achieve a win if the second player isn’t careful.

Discussion

Could you figure out how the second player can draw from all his possible responses to the first player’s moves? Why is it important that both players are considered good players? How would we analyze games where one of the players is a random player?