Question 4 (i)(b)
Nash equilibrium of the advertising game
3 marks
- Time to write
- About 6 minutes
Question
Question 4 (i)(b) · 3 marks- Firms A and B each have to decide whether or not to advertise. Their payoffs (profits in million dollars) are given below:
| Firm B: Advertise | Firm B: Does Not Advertise | |
|---|---|---|
| Firm A: Advertise | (4, 4) | (12, 2) |
| Firm A: Does Not Advertise | (2, 12) | (7, 7) |
- Determine the Nash Equilibrium of this game.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- DetermineFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
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- 1 Best responses of each firm 1 mark 2 min
- 2 Nash equilibrium identified with payoffs 1½ marks 3 min
- 3 Comment: a prisoners' dilemma ½ mark 1 min
Short on time? Step 2 carries 1½ of the 3 marks: guard its minutes.
Try it first. About 6 minutes for 3 marks. Write it out, then open the solution.
Solution
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Meaning. A Nash equilibrium is a pair of strategies in which each player's strategy is his best response to the other's, so that neither has any incentive to change his choice on his own.
| B: Advertise | B: Does Not Advertise | |
|---|---|---|
| A: Advertise | (4, 4) | (12, 2) |
| A: Does Not Advertise | (2, 12) | (7, 7) |
- If B advertises, A's best response is to advertise (4 > 2); if B does not advertise, A's best response is again to advertise (12 > 7).
- If A advertises, B's best response is to advertise (4 > 2); if A does not advertise, B's best response is again to advertise (12 > 7).
The Nash equilibrium. The only cell in which both firms are playing a best response to each other is (Advertise, Advertise), with payoffs (4, 4) million dollars. From that cell, a firm that stopped advertising on its own would drop from 4 to 2, so neither moves. In every other cell at least one firm gains by switching. Since each firm's dominant strategy is to advertise, the equilibrium is also an equilibrium in dominant strategies.
Why the firms do not reach (7, 7). Both firms would earn more, 7 each, if neither advertised. But (Does Not Advertise, Does Not Advertise) is not stable: each firm, expecting the other to stay quiet, could raise its profit from 7 to 12 by advertising. This is a prisoners' dilemma: each firm's rational choice leads to an outcome worse for both than a joint one, so a Nash equilibrium need not be the best outcome.
Answer. The Nash equilibrium is both firms advertise, with profits of $4 million each.
Where the 3 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Best responses of each firm | 1 |
| 2 | Nash equilibrium identified with payoffs | 1.5 |
| 3 | Comment: a prisoners' dilemma | 0.5 |
| Total | 3 |
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