The rules of game 3 were as follows: two players have nickels and quarters. At the same time, they each play one coin. If both players play the same coin, player 2 gives player 1 the average value of the coins; otherwise, player 1 gives player 2 the average value of the coins. Here is the payoff matrix for this game:
|Nickel  ||Quarter  |
|Player 1  ||Nickel||5||-15|
The lower value of this game is -15 while the upper value is 5. Can we find a pure value for the game? According to the Minimax Theorem, one of the most important results in game theory, we can. The Minimax Theorem states that every finite, two-person, zero-sum game has a value V that is the average amount that one player can expect to win if both players act sensibly.
Suppose player 2 knows which coin player 1 will play on each turn. Then it will be easy for player 2 to play a coin that makes player 2 lose money. Therefore, player 1 can't play with a pattern. Instead, he must use a mixed strategy, in which he randomly chooses to play a nickel or quarter on each turn. However, it is not necessarily true that he should play each strategy half the time. He may want to weight the strategies differently, playing one with probability p and the other with probability 1 - p. How do we figure out p?
It turns out that one property of the value of a game is that, if player 1 plays his optimal strategy, he will achieve exactly the value of the game no matter what the other player does (as long as the other player has no dominant strategies). In particular, the yield when player 1 plays agains player 2's two different pure strategies should be the same. In other words, if player 1 uses his optimal strategy, he will get the same amount of money whether player 2 always plays nickels or always plays quarters. Let's suppose that player 2 always plays nickels. Player 1 plays nickels p of the time so gains 5 cents p of the time. The other 1 - p of the time, he loses 15 cents. Overall, he wins 5p - 15(1 - p) = 20p - 15. Now, suppose player 2 always plays quarters. Player 1 plays nickels p of the time so loses 15 cents p of the time. The rest of the time, he wins 25 cents. Overall, he wins -15p + 25(1 - p) = 25 - 40p. Because he should win the same in both situations, the two winnings are the same. So, 20p - 15 = 25 - 40p. Solving for p, we find that it is 2/3. To find the amount that player 1 expects to win, we just plug this back into either of the equations and find that he should win an average of -5/3 per game. Even if player 2 figures out this strategy, he cannot do anything to change it.
Similarly, we can look at the payoff matrix from player 2's point of view and find a mixed strategy for player 2. If we do so, we find that player 2 should play nickels 2/3 of the time and quarters 1/3 of the time. If he does so, he should win an average of 5/3 cents per game.