Game Theory · Strategy
Dominant Strategy in Game Theory
The clearest solution concept in game theory — when one strategy is always best, regardless of what anyone else does.
In most strategic situations, the best thing to do depends heavily on what your opponent does. But sometimes — and this is what makes these situations especially powerful — one strategy is better than all others no matter what the other player chooses. When that happens, economists say that strategy is dominant. Finding dominant strategies is the first step in analysing any game, and in many real-world situations it gives a complete, unambiguous prediction of behaviour.
📘 Key Terms
Strictly Dominant Strategy: A strategy that gives a strictly higher payoff than all other strategies, regardless of what other players do. A rational player will always play a strictly dominant strategy.
Weakly Dominant Strategy: A strategy that gives a payoff at least as high as all other strategies, and strictly higher in at least one case.
Dominant Strategy Equilibrium: An outcome in which every player plays their dominant strategy. This is a special case of Nash Equilibrium — and a particularly robust prediction.
Strict vs Weak Dominance: The Formal Definitions
Strategy s* strictly dominates strategy s if:
u(s*, s₋ᵢ) > u(s, s₋ᵢ) for ALL strategies s₋ᵢ of other players
Strategy s* weakly dominates strategy s if:
u(s*, s₋ᵢ) ≥ u(s, s₋ᵢ) for ALL s₋ᵢ, and > for at least one s₋ᵢ
Step-by-Step Example: Advertising Game
Two firms, X and Y, simultaneously choose whether to run a marketing campaign (High Spend or Low Spend). Payoffs (X, Y) in £m profit:
| Y: High Spend | Y: Low Spend | |
|---|---|---|
| X: High Spend | (4, 4) | (9, 2) |
| X: Low Spend | (2, 9) | (6, 6) |
Does X have a dominant strategy?
• If Y plays High Spend: X gets 4 (High) vs 2 (Low) → High Spend is better
• If Y plays Low Spend: X gets 9 (High) vs 6 (Low) → High Spend is better
High Spend strictly dominates Low Spend for X. By symmetry, High Spend also dominates for Y. Dominant Strategy Equilibrium: (High Spend, High Spend) = (4, 4).
💡 Key Insight
This is another Prisoner’s Dilemma structure — both firms would earn more (6, 6) under mutual Low Spend, but each is individually pulled toward High Spend. The dominant strategy equilibrium is individually rational but collectively suboptimal — a pattern with enormous implications for competition policy, advertising law, and regulation.
Iterated Elimination of Dominated Strategies (IEDS)
When no strategy is dominant, we can still narrow down predictions using IEDS. The process: eliminate any strictly dominated strategy (one that is always worse than another), then repeat on the reduced game. If a unique outcome survives, it is the rationalizable outcome — the prediction of rational play under common knowledge of rationality.
⚠️ Common Error
Do not confuse a dominant strategy with the best response to a specific opponent strategy. A dominant strategy is best against all opponent strategies. A best response is only best against one specific opponent strategy. A player can have a unique best response without having a dominant strategy — these are different concepts.
Q1. Define a strictly dominant strategy. Using a payoff matrix of your own construction, illustrate how dominant strategy equilibrium is reached. [6 marks]
Answer: A strictly dominant strategy is one that produces a strictly higher payoff than any alternative strategy, regardless of what other players do. [Student should construct a 2×2 matrix where one strategy always gives a higher payoff row-by-row or column-by-column.] The dominant strategy equilibrium is reached because rational players will always choose their dominant strategy — the unique outcome where both players play their dominant strategies is both a Nash Equilibrium and a dominant strategy equilibrium, making it the strongest possible prediction in game theory.
References
1. Osborne, M.J. and Rubinstein, A. (1994) A Course in Game Theory. MIT Press.
2. Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Microeconomic Theory. Oxford University Press.
3. Dixit, A. and Nalebuff, B. (1991) Thinking Strategically. W.W. Norton.
4. Gibbons, R. (1992) Game Theory for Applied Economists. Princeton University Press.
5. Rasmusen, E. (2007) Games and Information. 4th edn. Blackwell.