A payoff matrix must be read player by player
Never compare one firm’s payoff with the rival’s payoff.
In a one-shot prisoner’s dilemma, the Nash equilibrium may be inefficient because
- players cannot observe their own possible payoffs
- unilateral defection is rational although mutual cooperation pays more
- each player lacks a best response to the other’s action
- the rules require both players to choose the cooperative action
- no combination of actions can be a Nash equilibrium
unilateral defection is rational although mutual cooperation pays more Each player protects its own payoff by choosing the dominant action even though mutual cooperation would create a larger joint payoff.
A Nash equilibrium is an outcome in which
- both firms earn the highest joint payoff
- each firm chooses the cooperative action
- no player gains by changing strategy alone
- every player uses a dominant strategy
- government enforces the chosen strategies
no player gains by changing strategy alone At a Nash equilibrium, each strategy is a best response to the other. No player benefits from a unilateral deviation.
In a prisoner’s-dilemma setting, a dominant strategy is one that
- maximizes total industry profit
- works only if rivals cooperate
- produces equal payoffs for both players
- requires a binding agreement
- is optimal regardless of the rival’s choice
is optimal regardless of the rival’s choice A dominant strategy is best for a player under every action available to the rival.
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To find Firm A’s best response, hold Firm B’s strategy fixed and compare A’s payoffs across A’s choices. Repeat for every B choice. A dominant strategy is best for A regardless of B’s action. Then perform the same analysis for B.
The two numbers in a cell belong to different players, usually row player first and column player second. Never compare A’s payoff with B’s payoff to select A’s strategy. Strategy asks what gives one player the larger own payoff conditional on the rival’s column or row.
For A, hold B at High price and compare A’s 60 with 80. Low is better. Hold B at Low and compare A’s 20 with 35. Low is better again. Thus Low is dominant for A. Repeat horizontally for B’s second payoff. Both comparisons must be completed before naming equilibrium.
A Nash equilibrium is a pair of mutual best responses: neither player benefits by changing alone. It need not be fair, efficient, or the highest combined payoff. A player can have no dominant strategy while the game still has a Nash equilibrium.
Dominance is a property of one strategy across all rival actions. Nash equilibrium is a property of a strategy pair. A game can have an equilibrium even if neither player has a dominant strategy, and it can have multiple equilibria. Do not use the terms interchangeably.
| B: High price | B: Low price | |
|---|---|---|
| A: High price | 60, 60 | 20, 80 |
| A: Low price | 80, 20 | 35, 35 |
In this example low price is dominant for both firms: 80 exceeds 60 when the rival chooses high, and 35 exceeds 20 when the rival chooses low. Low-low is the Nash equilibrium even though high-high gives both more.
The outcome is a prisoner’s dilemma: individual incentives lead to low-low, while high-high has larger joint and individual payoffs. Choosing high unilaterally is unstable because the rival can gain by switching low. A socially or jointly better cell is not an equilibrium if a player wants to deviate.
| Concept | Test | What it does not require |
|---|---|---|
| Best response | Highest own payoff for one fixed rival action | Best payoff in whole matrix |
| Dominant strategy | Best response to every rival action | Rival having dominance |
| Nash equilibrium | Mutual best responses in one cell | Highest joint payoff |
| Joint-profit maximum | Largest sum of payoffs | Stability against unilateral deviation |
No dominant strategy, still an equilibrium
A restaurant prefers a downtown location if its rival chooses suburbs, but suburbs if the rival chooses downtown. Neither location is always best. If the rival has opposite incentives, downtown-suburbs and suburbs-downtown can both be mutual best responses. Nash equilibrium does not require dominance.
If payoffs change, erase old best-response marks and recompute. A fine, subsidy, or repeated-game value can alter only one cell yet change dominance or equilibrium. Labels such as “cooperate” carry no mathematical privilege. Displayed payoffs decide.
In a sequential game, timing and credible responses matter, often represented by a game tree rather than a simultaneous matrix. The economics exam’s standard matrix questions are simultaneous unless timing is stated. Use the structure shown.
Reza is an experienced Math instructor and a test-prep expert who has been tutoring students since 2008. He has helped many students raise their standardized test scores--and attend the colleges of their dreams. He works with students individually and in group settings, he tutors both live and online Math courses and the Math portion of standardized tests. He provides an individualized custom learning plan and the personalized attention that makes a difference in how students view math.
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