The class of strategic games presents examples ranging from simple decision games such as Rock-Paper-Scissors, to models for economic and political interactions, to parlor games such as Checkers and Chess. Mathematically, the focus on strategy as opposed to chance represents a paradigm shift. Whereas games of pure chance are described by numbers (probabilities and expected values), the analysis of strategic games involves various notions of “solution.” We introduce three related solution concepts in this chapter.

Given the large variety of examples of games of strategy, it is natural to introduce some groupings. Focusing on the payoffs, we obtain one useful division. A total-conflict game is a two-player game in which the sum of the payoffs of the players is always the same amount at every outcome. In a total-conflict game, one player's gain is another's loss. Examples include win-lose games and zero-sum games, as introduced in Chapter 2. A partial-conflict game is any game that is not a total-conflict game. We have seen an example of a two-player partial conflict game with Battle of the Sexes (Example 2.6).

We begin with simultaneous-move partial-conflict games. We introduce the famous Prisoner's Dilemma as a representative example, giving both the standard presentation and a version arising in economics. We also analyze an example of an auction that serves to motivate our first solution concept, a dominated strategy solution. We next consider an example of a sequential-move partial-conflict game and introduce the notion of a backward induction solution.

Turning to games of total conflict, we apply our solution concepts to the case of zero-sum games and identify the maximin strategy. Finally, we consider the class of win-lose sequential-move games called combinatorial games. We introduce the Binary Labeling Rule to solve certain examples of games in this class. Our partial results here motivate Zermelo's Theorem (Theorem 4.17), which asserts the existence of a solution to all finite combinatorial games.

Partial-Conflict Games

Games in which the players are not in total conflict with one another have proven extremely useful for modeling social and political interactions.