Stable Play: Nash Equilibria in Continuous Games
148 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
integers.* For example, is a rational number, as is 1.712, which is the same as the fraction An irrational number is a number that cannot be repre- sented by a fraction, which means that it has a decimal expansion that neither terminates nor is periodic.** is an irrational number. Combining all rational and irrational numbers gives us the set of real numbers.
In this chapter, a strategy set will be either an interval of real numbers or the entire set of real numbers. We use what is referred to as a continuous strategy set, which has no gaps—no missing numbers. You might wonder why we need so many strategies. The reason is that, although the set of rational numbers is certainly large, it is a clunky set to work with because it contains lots of gaps. The greatest value of having a strategy set be an interval of real numbers, how- ever, is that it allows us to wield the 17th-century mathematical miracle of cal- culus. This we do in Section 6.3 with great efficacy, although the games in Section 6.2 do not need or use calculus.
In Section 6.2, we consider two classic competitive models, one from eco- nomics—stores competing for customers’ business through the prices they charge—and the other from politics—candidates competing for elected office through their campaign platforms. Alhough you might imagine that searching for a few Nash equilibria amidst an infinite number of strategy profiles is harder than trying to find a needle in a haystack, a bit of clever reasoning will allow us to quickly dispense with all but a few possibilities. But the pièce de ré- sistance of this chapter occurs in Section 6.3, where we show how calculus can be used to easily derive Nash equilibria. In that optional section, first the gen- eral method is described, together with the conditions that a game must satisfy in order for the method to work. Then the method is applied to explore market competition and to understand how the actions of human beings might have led to the extinction of the woolly mammoth. Finally, we investigate the logic behind matching grants and how they can increase charitable donations.