Coexistence of Pure and Mixed Strategy Nash Equilibria in Strategic Games
The concept of Nash equilibrium in game theory is fundamental to understanding strategic interactions among rational decision-makers. Pure and mixed strategy Nash equilibria, while distinct in their definitions, are not mutually exclusive. This article explores the conditions under which both types of equilibria can coexist in strategic games, providing a comprehensive overview of their coexistence and the implications thereof.
Definitions and Basics
Pure Strategy Nash Equilibrium: This occurs in a scenario where each player chooses a single, deterministic action as their best response to the strategies chosen by other players. Importantly, in a pure strategy equilibrium, players do not randomize over their choices.
Mixed Strategy Nash Equilibrium: This scenario involves players randomizing over their strategies, choosing each action with a certain probability. Mixed strategies are particularly relevant in games where no pure strategy equilibrium exists, or where players prefer to keep their actions unpredictable.
Conditions for Coexistence
Existence of Both: Some games allow for the coexistence of both pure and mixed strategy Nash equilibria. For example, in a coordination game, multiple pure strategy equilibria may coexist alongside a mixed strategy equilibrium. In such games, the players might find it profitable to coordinate on different pure strategies, or to mix their strategies to avoid predictability.
Example: Consider a game with two strategies, A and B. If both players choose A, they receive a payoff of 2.2. If both choose B, they receive 1.1. In this scenario, a mixed strategy equilibrium might exist where players randomize between A and B, leading to a different expected payoff.
Role of Dominance
Dominant Strategies: In some games, a pure strategy exists that is dominant—meaning it is the best response regardless of what the other players do. When both players have dominant strategies, the game is dominated and the mixed strategy equilibrium does not exist. A well-known example is the Prisoner's Dilemma, where players have dominant strategies that result in the dominant strategy Nash equilibrium.
However, in games without dominant strategies, the presence of pure strategy equilibria does not preclude the existence of mixed strategy equilibria. For instance, in games like the , games have an odd number of Nash equilibria. Consequently, any game that has an even number of purely strategic equilibria will necessarily have at least one mixed equilibrium.
Complexity in Nash Equilibria
Many games exhibit a highly complex equilibrium structure, with a large number of Nash equilibria. In repeated games, every strategy pair that is not worse than the minimax strategy can potentially be a Nash equilibrium. The minimax strategy represents the worst-case response to the opponent’s strategies.
For example, in a repeated game, every strategy pair that isn’t dominated by another strategy can be a Nash equilibrium. This complexity underscores the importance of considering both pure and mixed strategy equilibria when analyzing strategic interactions.
Conclusion
While pure and mixed strategy Nash equilibria serve distinct roles in strategic games, they can coexist and are not mutually exclusive. The presence of one does not preclude the existence of the other. Understanding the conditions under which both types of equilibria can coexist provides valuable insights into the strategic behavior of rational players.
Therefore, in analyzing strategic games, it is crucial to consider both pure and mixed strategy Nash equilibria to fully capture the complexity and richness of strategic interactions.