224
A
s a kid, you probably played tic-tac-toe with your friends or siblings. In
this game, two players take turns claiming cells of a 3
×
3 matrix (a matrix
with three rows and three columns). Player 1 claims a cell by writing an “X” in
it; player 2 writes an “O” in each cell that he claims. A player can claim only
one cell per turn. The game ends when all of the cells have been claimed or
when one of the players has claimed three cells in a line (by claiming an entire
row, column, or diagonal). If a player has claimed three cells in a line, then he
wins and the other player loses. If all of the cells are claimed and neither player
has a line, then the game is declared a tie. Figure 17.1 depicts the playing of a
game in which player 1 wins.
Tic-tac-toe is a game of skill—nothing in the game is left to chance (no
coins are flipped, no dice rolled, etc.). It is also a game of perfect information,
because players move sequentially and observe each other’s choices. Further, it
FIGURE 17
.
1
A game of tic-tac-toe
.
Tic-tac-toe
matrix
Round 1
(Player 1
moves first)
Round 2
Round 3
X
X
O
X
X
O
Round 4
(Player 2
blocks a line)
Round 5
Round 6
(Player 2
blocks)
Round 7
(Player 1
wins with
diagonal)
X
X
O
O
X
X
X
O
X
X
O
O
X
X
O
O
X
X
O
O
X
O
17
PARLOR GAMES
225
Parlor Games
is a finite game. These facts imply that the result on page 188 (Chapter 15) holds
for tic-tac-toe: this game has a pure-strategy Nash equilibrium. What else do
we know about tic-tac-toe? It is a two-player, strictly competitive game: either
player 1 wins and player 2 loses, or player 2 wins and player 1 loses, or the play-
ers tie. Thus, the result on page 149 (Chapter 12) also applies, meaning that each
player’s Nash equilibrium strategy is a security strategy for him.
Let’s review. One of the outcomes (1 wins, 2 wins, tie) occurs in a Nash
equilibrium. Because the equilibrium strategies are security strategies, each
player can guarantee himself this equilibrium outcome. For example, suppose
“1 wins” is the equilibrium outcome. Then our mathematical analysis proves
that player 1 has a
winning strategy
,
which guarantees that he will win the game
regardless of what his opponent does
.
On the other side of the table, player 2 has
a strategy that guarantees at least a loss, regardless of what player 1 does. In this
case, player 2’s security strategy is not very helpful to her, but player 1’s secu-
rity strategy is quite helpful to him. The game is “solved” as long as player 1 is
smart enough to calculate his security strategy.
In fact, “1 wins” is
not
the equilibrium outcome of tic-tac-toe. The Nash
equilibrium actually leads to the tie outcome. Thus, our mathematical analysis
implies that each player has a strategy guaranteeing a tie. In other words, when
smart, rational players engage in tic-tac-toe, the game always ends in a tie. You
probably knew this already, because most people figure it out relatively early in
life. As a matter of fact, discovering the solution to tic-tac-toe is one of the major
stepping stones of childhood development. I am sure child psychologists put it
