Dominant and Dominated Strategies
Dominant and dominated strategies are key concepts in game theory, helping analyze decision-making in competitive business environments.
Dominant and Dominated Strategies refer to fundamental concepts in game theory and strategic decision-making, where players choose strategies based on expectations about the actions of others and the resulting payoffs.
A dominant strategy is one that yields a higher payoff for a player regardless of what the other players do. In other words, a strategy is dominant if it is the best response to every possible strategy combination of the opponents. Choosing a dominant strategy guarantees the player the most favorable outcome no matter how others act.
A dominated strategy, in contrast, is a strategy that results in a lower payoff than another strategy, no matter what the opponents choose. A strategy is dominated if there exists another strategy that always provides a better payoff. Rational players will never choose dominated strategies because there is always a strictly better alternative.
Definition and Identification of Dominant Strategies
Strictly Dominant Strategy
A strategy is strictly dominant if it strictly outperforms all other strategies for a player in every scenario. Formally, for player i with strategies s_i and s_i', s_i is strictly dominant if for every strategy profile of other players s_{-i},
where u_i denotes the payoff function of player i.
Weakly Dominant Strategy
A strategy is weakly dominant if it is at least as good as any other strategy in every scenario and strictly better in at least one. Formally,
How to Identify Dominant Strategies
To identify a dominant strategy, compare payoffs across all possible responses of opponents. A strategy that consistently provides a higher or equal payoff, and sometimes strictly higher, qualifies as dominant.
Definition and Identification of Dominated Strategies
Strictly Dominated Strategy
A strategy s_i of player i is strictly dominated if there exists another strategy s_i' such that for every strategy profile of the other players s_{-i},
This means s_i is never the best choice and is always worse than s_i'.
Weakly Dominated Strategy
A strategy s_i is weakly dominated if there exists s_i' such that
Elimination of Dominated Strategies
Dominated strategies can be systematically eliminated from consideration, simplifying the analysis of strategic interaction. This process is known as iterative elimination of dominated strategies and is a key technique to predict rational outcomes.
Practical Implications of Dominant and Dominated Strategies
Strategic Decision-Making
Players prefer dominant strategies because they simplify decision-making by removing uncertainty about opponents’ actions. When a dominant strategy exists, rational players select it to maximize payoffs.
Existence in Games
Not all games possess dominant strategies. Many games require players to anticipate opponents’ moves and respond optimally, often leading to equilibrium concepts such as Nash equilibrium where no player can improve unilaterally.
Example: Prisoner’s Dilemma
In the Prisoner’s Dilemma, each prisoner has two strategies: Cooperate or Defect. Defecting strictly dominates cooperating because it yields a better payoff regardless of the other’s choice. Both players defecting is the dominant strategy equilibrium, even though mutual cooperation would result in a better collective outcome.
Summary Table of Key Concepts
| Concept | Definition | Payoff Comparison | Player Preference |
|---|---|---|---|
| Strictly Dominant | Strategy always better than alternatives | Strictly greater payoff in all cases | Always chosen if exists |
| Weakly Dominant | Strategy at least as good as others, sometimes strictly better | Greater or equal in all cases, strictly greater in some | Preferred if no strictly dominant exists |
| Strictly Dominated | Always worse than another strategy | Strictly lower payoff in all cases | Never chosen |
| Weakly Dominated | Never better, sometimes worse | Lower or equal payoff in all cases, strictly lower in some | Avoided in rational play |
Role in Game Theory Analysis
Dominant and dominated strategies serve as foundational tools for simplifying complex strategic interactions. By identifying and eliminating dominated strategies, analysts reduce the strategy space, focusing on plausible and rational outcomes. This leads to clearer predictions and understanding of strategic behavior in competitive and cooperative environments.
The concept also underpins more advanced solution concepts and equilibrium refinements, guiding strategic reasoning in economics, business, political science, and beyond.