Consider two discount retailer outlets in Baltimore, Walmart and Sears. These two stores simultaneously choose the price level. This simultaneous-move game is played once and then the world ends. Their payoff matrix is given below. Note that the first and second elements in the bracket are the payoffs (i.e., profits) for Walmart and Sears, respectively. What is the Nash equilibrium in the game?

Question

Consider two discount retailer outlets in Baltimore, Walmart and Sears. These two stores simultaneously choose the price level. This simultaneous-move game is played once and then the world ends. Their payoff matrix is given below. Note that the first and second elements in the bracket are the payoffs (i.e., profits) for Walmart and Sears, respectively. What is the Nash equilibrium in the game?

BlackTom AI · Accepted Answer

Nash equilibrium requires each player’s chosen action to be a best response to the other player’s action. Here both retailers simultaneously choose price high or low; payoffs (Walmart, Sears) are:
- (High, High) = (30, 35)
- (High, Low)  = (5,  65)
- (Low,  High) = (60, 5)
- (Low,  Low)  = (10, 10)

1) Relevant concepts
- Best response: the action that yields the highest payoff given the other player’s action.
- Dominant strategy: an action that is a best response regardless of the opponent’s choice.
- Nash equilibrium: a profile where each player’s action is a best response to the other’s.

2) Step-by-step reasoning (best-response comparisons)
Consider Walmart’s payoffs given Sears’ choice:
- If Sears chooses High: Walmart gets 30 if High, 60 if Low → 60>30, so Walmart prefers Low.
- If Sears chooses Low: Walmart gets 5 if High, 10 if Low → 10>5, so Walmart prefers Low.
Thus Walmart’s best response to either Sears action is Low (Walmart has a dominant strategy = Low).

Consider Sears’ payoffs given Walmart’s choice:
- If Walmart chooses High: Sears gets 35 if High, 65 if Low → 65>35, so Sears prefers Low.
- If Walmart chooses Low: Sears gets 5 if High, 10 if Low → 10>5, so Sears prefers Low.
Thus Sears’s best response to either Walmart action is Low (Sears also has a dominant strategy = Low).

3) Intermediate comparisons shown above (pairwise payoff comparisons for each possible opponent action).

4) Why the correct answer is correct
Because both firms have dominant strategies of Price = Low, the strategy profile (Low, Low) is mutual best responses. No unilateral deviation by either firm increases their payoff from (10,10): deviating from Low to High while the other stays Low would change Walmart’s payoff from 10 to 5 (worse) and Sears’ payoff from 10 to 5 (worse). Therefore (Low, Low) is a Nash equilibrium.

5) Why other options are incorrect
- (High, Low): Walmart would prefer to switch to Low (5 → 10 then 10>5), so not a Nash equilibrium.
- (Low, High): Sears would prefer to switch to Low (5 → 10), so not a Nash equilibrium.
- (High, High): Both players would prefer to deviate to Low (30→60 for Walmart; 35→65 for Sears), so not a Nash equilibrium.
- “No Nash equilibrium exists”: incorrect because (Low, Low) is a clear Nash equilibrium as shown.

6) Conclusion
The unique Nash equilibrium is (Wal-Mart, Sears) = (Price low, Price low), which matches the provided answer.

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