Although it appears late in this book, this is a core chapter about non-cooperative games. It studies extensive games, which are game trees with imperfect information. Typically, players do not always have full access to all the information which is relevant to their choices.

Congestion means that a shared resource, such as a road, becomes more costly when more people use it. In a congestion game, multiple players decide on which resource to use, with the aim to minimize their cost. This interaction defines a game because the cost depends on what the other players do.

In this chapter we start with the systematic development of non-cooperative game theory. Its most basic model is the game in strategic form, the topic of this chapter. The available actions of each player, called strategies, are assumed as given. The players choose their strategies simultaneously and independently, and receive individual payoffs that represent their preferences for strategy profiles (combinations of strategies).

In this final chapter, we explain a new equilibrium concept called correlated equilibrium that is more general than Nash equilibrium. It allows for randomized actions of the players that depend on an external signal (like a traffic light) that is observed by the players (typically in different ways) so that their actions can be correlated.

Zero-sum games are games of two players where the interests of the players are directly opposed: One player’s loss is the other player’s gain. Competitions between two players in sports or in parlor games can be thought of as zero-sum games.

In a non-cooperative game, every outcome is associated with a payoff to each player that the player wants to maximize. The payoff represents the player’s preference for the outcome. The games considered so far often have a pure-strategy equilibrium, where the preference applies to deterministic outcomes and is usually straightforward.

As we showed in Section 6.5, Nash (1951) proved the existence of a mixed equilibrium in a finite game with the help of Brouwer’s fixed-point Theorem 6.4. Fixed-point theorems are powerful tools for proving the existence of many equilibrium concepts in economics. Brouwer’s theorem is the first and most important of these.

This chapter is about the geometric structure of equilibria in two-player games in strategic form. It shows how to quickly identify equilibria with qualitative “best-response diagrams”.

This chapter considers game trees, the second main way for defining a non-cooperative game in addition to the strategic form. In a game tree, players move sequentially and (in the case of perfect information studied in this chapter) are aware of the previous moves of the other players. In contrast, in a strategic-form game players move simultaneously. In this “dynamic” setting, a play means a specific run of the game given by a sequence of actions of the players.

Combinatorial game theory is about perfect-information two-player games, such as Checkers, Go, Chess, or Nim, which are analyzed using their rules. It tries to answer who will win in a game position (assuming optimal play on both sides), and to quantify who is ahead and by how much. The topic has a rich mathematical theory that relates to discrete mathematics, algebra, and (not touched here) computational complexity, and highly original ideas specific to these games.