How to Read a Payoff Matrix: Best Responses and Nash Equilibrium
Read a payoff matrix one player at a time. Fix the rival’s action, compare your player’s own alternatives, and then check both players for equilibrium. Work through the price and location examples, distinguish ties from strict gains, and try three original interactive questions.
For a best response, compare one player’s own payoffs.
Check Three Game-Theory Definitions
These exercises specify simultaneous, one-period choices and higher-is-better payoffs. In the firm examples, the numbers are comparable profit units, with A’s payoff first and B’s second. We check the displayed pure-strategy choices; mixed-strategy equilibria are a separate topic. A model’s best responses do not guarantee how people will behave in a real market.
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 In the standard one-shot prisoner’s dilemma, defection is strictly better for each player against either rival action. Mutual cooperation nevertheless gives each player more than mutual defection. The conflict concerns the supplied payoffs, not an assumption that players cannot read the matrix.
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 a best response to every rival action. In the strict prisoner’s-dilemma comparisons here, it gives a strictly higher own payoff each time. If one comparison is a tie, distinguish weak from strict dominance rather than silently treating the tie as a gain.
Video Lesson: Game Theory and Oligopoly
A focused video lesson from CrashCourse.
Hold One Rival Action Fixed
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.
Identify the Player and Payoff Order
Check the stated payoff order before reading any cell; row player first is a convention, not a universal rule. Here, A chooses rows and receives the first entry, while B chooses columns and receives the second. For A’s best response, move between rows within one fixed column. For B’s, move between columns within one fixed row. Comparing A with B inside one cell answers neither best-response question.
Complete Both Firms’ Comparisons
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.
Test Profitable Unilateral Changes
A Nash equilibrium combines best responses: no player can increase its own payoff by changing strategy alone while the rival’s strategy stays fixed. A tie does not create a profitable deviation, so more than one action can be a best response. Equilibrium need not maximize joint profit, and a player does not need a dominant strategy for an equilibrium to exist.
Distinguish Strict and Weak Dominance
Keep the tests separate. Strict dominance requires a strictly higher payoff than an alternative against every rival action. Weak dominance allows equality in some comparisons, with a strict improvement in at least one. Nash equilibrium concerns a strategy pair and excludes profitable unilateral changes, including when best responses tie. A game may have several pure equilibria or none; that last result does not rule out mixed-strategy equilibrium.
Read the Original Price Matrix
| A’s price / B’s price | 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.
At High–High, either firm can improve its own payoff from 60 to 80 by switching to Low while the other stays High. At Low–Low, switching alone reduces the deviator’s payoff from 35 to 20. Thus Low–Low is the equilibrium of this matrix, while High–High gives both firms more profit. Those two producer payoffs do not measure consumer welfare or total social surplus.
Keep the Four Concepts Separate
| 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 the firms’ profit payoffs | Stability against unilateral deviation |
Find Equilibria without a Dominant Location
Suppose both restaurants prefer a different location from the other, as in the original matrix below. A prefers Suburbs when B chooses Downtown, but Downtown when B chooses Suburbs. B has the same preference for separation. Neither player has a dominant location, yet (Downtown, Suburbs) and (Suburbs, Downtown) are both pure-strategy Nash equilibria. The stated payoffs establish this result; the restaurant names alone would not.
| A’s location / B’s location | B: Downtown | B: Suburbs |
|---|---|---|
| A: Downtown | (3, 3) | (9, 7) |
| A: Suburbs | (7, 9) | (4, 4) |
Recheck a Changed Payoff
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.
Read the Timing Assumptions
First check timing. In a simultaneous-choice exercise, neither player observes the rival’s current choice before selecting its own action. Sequential games instead specify an order and information available at later decisions, often shown in a game tree. A matrix alone does not universally establish timing; follow the problem’s stated structure.
Check the Game-Theory Definitions
OpenStax’s oligopoly lesson introduces interdependent choices and prisoner’s-dilemma reasoning. The Stanford Encyclopedia of Philosophy’s game-theory entry distinguishes strategy, equilibrium and weak-dominance issues. The restaurant payoffs and the changed-payoff exercise below are original classroom examples, rather than copied source exercises.
Trace Best Responses and a Payoff Tie
Use the restaurant matrix to check both asymmetric location choices. Then return to the original price matrix and change only A’s payoff at (Low, Low) from 35 to 20. B’s payoff there stays 35, and every other entry is unchanged. Does A still strictly prefer Low in both columns?
Check your reasoning
In the restaurant game, if B chooses Downtown, A compares 3 with 7 and chooses Suburbs. If B chooses Suburbs, A compares 9 with 4 and chooses Downtown. For B, when A chooses Downtown, Suburbs gives 7 rather than 3; when A chooses Suburbs, Downtown gives 9 rather than 4. The two off-diagonal cells are mutual best responses. Neither player has a dominant location. These are the two pure-strategy equilibria; no claim is made that they exhaust mixed-strategy possibilities.
In the changed price game, A still prefers Low when B chooses High because 80 exceeds 60. When B chooses Low, A now gets 20 with either High or Low. Low weakly dominates High for A, but it no longer strictly dominates it. Mark both tied best responses in that column rather than choosing one arbitrarily.
B’s own payoffs have not changed, so Low remains strictly dominant for B. Both (A High, B Low) and (A Low, B Low) are now pure-strategy Nash equilibria: B chooses its best response, and A cannot improve by moving between tied payoffs of 20. Changing one player’s entry can change the equilibrium set without changing the other entry in that same cell.
Try Three Payoff-Matrix Questions
Choose one answer, then check your reasoning. These are short learning checks, not an exam score. If the controls are unavailable, the explanations remain readable below each question.
Answer D. Keep B’s High column fixed and compare the first entries in A’s two rows: Low gives 80 and High gives 60. Comparing the two players within one cell, or changing B’s action during the comparison, does not test A’s best response.
Answer A. Against Downtown, each player prefers Suburbs; against Suburbs, each prefers Downtown. Thus each off-diagonal cell combines best responses, while the preferred location changes with the rival’s choice. Mixed strategies are outside this question.
Answer C. A’s comparison when B chooses High remains 80 versus 60 in favor of Low. When B chooses Low, A’s two payoffs are now 20 and 20. Low is never worse and is strictly better in one comparison, which gives weak rather than strict dominance.
Connect Payoffs and Market Incentives
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