An economy is an interdependent system. In the process of solving it we have deliberately pushed that interdependency into the background. The individual, both as consumer and producer, is a small part of the market and can therefore take everyone else’s behavior as given; he does not have to worry about how what he does will affect what they do. The rest of the world consists for him of a set of prices–prices at which he can sell what he produces and buy what he wants.
The analysis of strategic behavior is an extraordinarily difficult problem. John Von Neumann, arguably one of the smartest men of this century, created a whole new branch of mathematics in the process of failing to solve it. The work of his successors, while often ingenious and mathematically sophisticated, has not brought us much closer to being able to say what people will or should do in such situations. Seen from one side, what is striking about price theory is the unrealistic picture it presents of the world around us. Seen from the other, one of its most impressive accomplishments is to explain a considerable part of what is going on in real markets while avoiding, with considerable ingenuity, any situation involving strategic behavior. When it fails to do so, as in the analysis of oligopoly or bilateral monopoly, it rapidly degenerates from a coherent theory to a set of educated guesses.
“Scissors, Paper, Stone” is a simple game played by children. At the count of three, the two players simultaneously put out their hands in one of three positions: a clenched fist for stone, an open hand for paper, two fingers out for scissors. The winner is determined by a simple rule: scissors cut paper, paper covers stone, stone breaks scissors.
The game may be represented by a 3×3 payoff matrix, as shown in Figure 11-1. Rows represent strategies for player 1, columns represent strategies for Player 2. Each cell in the matrix is the intersection of a row and a column, showing what happens if the players choose those two strategies; the first number in the cell is the payoff to Player 1, the second the payoff to Player 2. It is convenient to think of all payoffs as representing sums of money, and to assume that the players are simply trying to maximize their expected return–the average amount they win–although, as you will see, game theory can be and is used to analyze games with other sorts of payoffs.
The top left cell shows what happens if both players choose scissors; neither wins, so the payoff is zero to each. The next cell down shows what happens if Player 1 chooses paper and Player 2 chooses scissors. Scissors cuts paper, so Player 2 wins and Player 1 loses, represented by a gain of one for Player 2 and a loss of one for Player 1.
I have started with this game for two reasons. The first is that, because each player makes one move and the moves are revealed simultaneously, it is easily represented by a matrix such as Figure 11-1, with one player choosing a row, the other choosing a column, and the outcome determined by their intersection. We will see later that this turns out to be a way in which any two-person game can be represented, even a complicated one such as chess.
The second reason is that although this is a simple game, it is far from clear what its solution is–or even what it means to solve it. After your paper has been cut by your friend’s scissors, it is easy enough to say that you should have chosen stone, but that provides no guide for the next move. Some quite complicated games have a winning strategy for one of the players. But there is no such strategy for Scissors, Paper, Stone. Whatever you choose is right or wrong only in relation to what the other player chooses.
While it may be hard to say what the correct strategy is, one can say with some confidence that a player who always chooses stone is making a mistake; he will soon find that his stone is always covered. One feature of a successful strategy is unpredictability. That insight suggests the possibility of a deliberately randomized strategy.
Suppose I choose my strategy by rolling a die, making sure the other player is not watching. If it comes up 1 or 2, I play scissors; 3 or 4, paper; 5 or 6, stone. Whatever strategy the other player follows (other than peeking at the die or reading my mind), I will on average win one third of the games, lose one third of the games, and draw one third of the games.
Can there be a strategy that consistently does better? Not against an intelligent opponent. The game is a symmetrical one; the randomized strategy is available to him as well as to me. If he follows it then, whatever I do, he will on average break even, and so will I.
One important feature of Scissors, Paper, Stone is that it is a zero-sum game; whatever one player wins the other player loses. While there may be strategy of a sort in figuring out what the other player is going to do, much of what we associate with strategic behavior is irrelevant. There is no point in threatening to play stone if the opponent does not agree to play scissors; the opponent will refuse, play paper, and cover your stone.