Game Theory in Tic Tac Toe
I initially thought about writing my blog post on game theory in chess, but then I realized that chess is extremely complicated and would require multiple essays to explain. So, I went down a few notches in complexity and picked tic tac toe! The beauty of game theory is that while it can be used to model extremely simple situations, like the prisoner’s dilemma, it can also be used for extremely complex situations, like chess, though the latter requires a lot more computing power.
Because tic tac toe is a game with a finite amount of possible grid positions and finite amount of ways to fill up the grid with different combinations of 5 Xs and 4 Os, this makes tic tac toe game that can be completely analyzed for a perfect strategy in which neither player wins, concluding in a draw. There are ways to figure it out purely mathematically but that goes beyond the scope of our class, so instead, with (relatively) simple game theory we can still figure out the best strategies. While reading the article I saw some really interesting applications of what we learned in class, especially Nash equilibria. The author of the article links to a picture that shows the best strategy depending on what option the opponent picks. Since X goes first, the best opening strategy is one of the corners, since, if the O player doesn’t pick the center, then the X player wins. Also, if the X player picks the center as the first move, then the optimal move for the second player is to pick the corner. This makes (Corner, Center) and (Center, Corner) Nash equilibria. As the players progress through the game, the number of choices dwindles down until the first player plays their last move (assuming both players played optimally). Throughout the game, each time a player goes to make a choice, they are faced with a set of payoffs for each move that they could possibly make, and since it’s possible to play optimally, then the difference in payoff between an optimal move and a sub-optimal move is the difference in winning (or drawing in the event of the other person playing optimally) and losing.
Source: http://ibmathsresources.com/2013/11/24/game-theory-and-tic-tac-toe/