An Introduction to Decision Theory

Game theory studies decisions in which the outcome depends partly on what other people do. Chess is a paradigmatic example. Before I make a move, I always carefully consider what my opponent's best response will be, and if the opponent can respond by doing something that will force a checkmate, she can be fairly certain that I will do my best to avoid that move. Both I and my opponent know all this, and this assumption of common knowledge of rationality (CKR) determines which move I will eventually choose, as well as how my opponent will respond. I do not consider the move to be made by my opponent to be a state of nature that occurs with a fixed probability independently of what I do. On the contrary, the move I make effectively decides my opponent's next move.

Chess is, however, not the best game to study for newcomers to game theory. This is because it is such a complex game with many possible moves. Like other parlor games, such as bridge, monopoly and poker, chess is also of limited practical significance. In this chapter we will focus on other games, which are easier to analyze but nevertheless of significant practical importance. Consider, for example, two hypothetical supermarket chains, Row and Col. Both have to decide whether to set prices high and thereby try to make a good profit from every item sold, or go for low prices and make their profits from selling much larger quantities. Naturally, each company's profit depends on whether the other company decides to set its prices high or low.

If both companies sell their goods at high prices, they will both make a healthy profit of $100,000. However, if one company goes for low prices and the other for high prices, the company retailing for high prices will sell just enough to cover their expenses and thus make no profit at all ($0), whereas the other company will sell much larger quantities and make $120,000. Furthermore, if both companies set their prices low each of them will sell equally much but make a profit of only $10,000. Given that both companies wish to maximize their profit, what would you advise them to do? Consider the game matrix in Table 11.1.