21 Important Games
Several games appear repeatedly throughout game theory because they capture fundamental strategic situations. These games highlight the core challenges of cooperation, competition, and coordination that arise across economics, politics, biology, and social interactions. Before examining these games, we first define Pareto optimality, a concept essential for analyzing their outcomes.
21.1 Pareto Optimality
A strategy profile Pareto dominates another if every player strictly prefers the first outcome to the second. For example, if there are two player \(1\) and \(2\), the profile \((x, y)\) Pareto dominates \((x', y')\) when:
\[ u_1(x, y) > u_1(x', y') \mbox{ and } u_2(x, y) > u_2(x', y'). \]
A strategy profile is Pareto optimal (or Pareto efficient) if no other profile Pareto dominates it. This represents an outcome where no player can be made better off without making someone else worse off.
21.2 Coordination Game
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 1, 1 | 0, 0 |
| \(b\) | 0, 0 | 1, 1 | |
Players want to coordinate on the same type of action. Both \((a, a)\) and \((b, b)\) are Nash equilibria and both are Pareto optimal. This captures situations like choosing which side of the road to drive on—it doesn’t matter which we choose as long as everyone chooses the same.
21.3 Anti-Coordination Game
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 0, 0 | 1, 1 |
| \(b\) | 1, 1 | 0, 0 | |
Players want to choose different types of actions from each other. Both \((a, b)\) and \((b, a)\) are Nash equilibria and both are Pareto optimal. This models situations where players benefit from differentiation, such as two businesses choosing different market niches to avoid direct competition, or drivers choosing different routes to reduce traffic congestion.
21.4 Coordination and Competition
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 0, 0 | 2, 1 |
| \(b\) | 1, 2 | 0, 0 | |
Players want to miscoordinate, but each prefers a different outcome. Row prefers \((a, b)\) while Column prefers \((b, a)\). Both equilibria are Pareto optimal. This captures conflicts where coordination is needed but the players’ preferences diverge.
21.5 Chicken (Hawk-Dove Game)
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 2, 2 | 1, 3 |
| \(b\) | 3, 1 | 0, 0 | |
Both \((a, b)\) and \((b, a)\) are Nash equilibria. Playing \(b\) (the aggressive strategy) yields the best outcome if the opponent backs down by playing \(a\) (the passive strategy), but mutual aggression \((b, b)\) is disastrous. This represents conflicts where backing down is costly but mutual aggression is catastrophic, such as the game of “chicken” where two drivers speed toward each other and each must choose whether to swerve or stay the course.
21.6 Stag Hunt
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 3, 3 | 0, 2 |
| \(b\) | 2, 0 | 1, 1 | |
Both \((a, a)\) and \((b, b)\) are Nash equilibria, but \((a, a)\) Pareto dominates \((b, b)\). Playing \(a\) (the cooperative strategy) yields the best outcome if both cooperate, but playing \(b\) (the safe strategy) is safer. This game captures the tension between risk and mutual benefit in cooperation.
21.7 Prisoner’s Dilemma
| Column | |||
| \(a\) | \(b\) | ||
| Row | \(a\) | 3, 3 | 0, 4 |
| \(b\) | 4, 0 | 1, 1 | |
The Prisoner’s Dilemma represents a fundamental conflict between individual rationality and collective welfare. Typically, strategy \(a\) represents cooperation while strategy \(b\) represents defection. Each player has a dominant strategy to defect, leading to the unique Nash equilibrium \((b, b)\) with payoffs \((1, 1)\). Yet both players would be better off cooperating, achieving the outcome \((a, a)\) with payoffs \((3, 3)\). The puzzle is that individual rationality leads both players to defect, producing an outcome that is Pareto dominated by mutual cooperation. The game models situations from arms races to environmental regulation, where rational individual choices produce collectively inferior outcomes. See Kuhn (2025) for an overview of this important game.