ABSTRACT
Throughout our discussion of strategic behavior and static games, players were confronted with a choice of discrete pure strategies. In the prisoner’s dilemma, for example, the suspects had to decide whether to confess or remain silent. While this was perfectly reasonable, there are many other games in which the players must choose from among an infi nite number of possible strategies. In the pricing game depicted in Figure 1.2, for example, fi rms A and B had to choose from just two strategies: high price or low price. But what does it mean to charge a “high price” or a “low price”? Should we defi ne a high price strategy as any price that is higher than the price charged by a rival? If so, there are still an infi nite number of possible prices to choose from, which means that there are an infi nite number of strategy profi les and payoffs. In most business and economic applications, such as pricing and output decisions, expenditures on alternative advertising strategies, product differentiation, capital investment, auction bids, bargaining positions, and so on, it makes more sense to allow players to choose from a continuum of strategies rather than from a discrete few. In this chapter we will expand our search for a Nash equilibrium to include games involving continuous pure strategies.
