7 Theory of Games

Nearly every mathematician likes a good game. Let’s play! There is plenty of mathematical

analysis to be undertaken. We shall start with a few fun games and their winning strategies

before moving on to develop some of the general theory in the logic of games, ultimately

proving the fundamental theorem of finite games, due to Ernst Zermelo, which shows that

every finite game has a winning strategy for one of the players or a drawing strategy for

both players.

7.1 Twenty-One

Consider first the game of Twenty-One, in which two players cooperate to count to twenty-

one, with each player saying either the next one, two, or three numbers, starting at one.

Whoever says “twenty-one” is the winner. Perhaps our game proceeds like this.

You One, two.

Me Three, four, five.

You Six, seven, eight.

Me Nine.

You Ten, eleven.

Me Twelve, thirteen.

You Fourteen.

And so on. How shall I reply next? Who will be able to win by saying “twenty-one”?

Kindly find a partner and play the game a few times. Get a good feel for the game. Can

you find a winning strategy? This is a game that you can likely figure out.

Interlude. . .