Chapter 5 Zero-Sum Games
We begin the first of several chapters on applications of linear optimization beyond the more straight forward examples we’ve seen so far, further highlighting the power and versatility of this theory. This particular chapter is about competitive zero-sum games. Two players playing against each other in some game with finite choices may find that the efficacy of a choice depends on their opponents choice. A good choice in one scenario may be a poor choice in another. Moreover, if one always makes the same choices, your opponent may be able to anticipate this move, so it makes sense to vary your choices. How should one vary them, and how might your opponent vary theirs? Questions like this are part of an area of mathematics called game theory.
In Section 5.1 we describe these games and propose a scheme to solve for the optimal strategy for both players. In Section 5.2 we prove that this strategy is valid and that optimal strategies exist. Then in Section 5.3 we see how to apply these principles to games with random components.
