Answer: The normal form of the game can be represented in a matrix format as follows:
| Take PEDs | Not take PEDs
--------------------------------------------
Take PEDs | (120(1-p), | (200(1-p), |
| 120(1-p)) | 50) |
--------------------------------------------
Not take PEDs| (50, | (80, |
| 200(1-p)) | 80) |
--------------------------------------------
b) Assuming p = 1/4, we can find the pure strategy Nash equilibrium by examining each cell in the matrix. In a Nash equilibrium, each player's strategy is optimal given the other player's strategy.
Let's analyze the matrix:
- If both players choose to take PEDs, they will each earn 120(1-1/4) = 90.
- If both players choose not to take PEDs, they will each earn 80.
- If Ulrich takes PEDs and Basso does not, Ulrich earns 200(1-1/4) = 150 and Basso earns 50.
- If Basso takes PEDs and Ulrich does not, Basso earns 200(1-1/4) = 150 and Ulrich earns 50.
Since the payoffs for taking PEDs (90) are higher than the payoffs for not taking PEDs (80) in the diagonal cells, it is not a Nash equilibrium for both players to choose not to take PEDs. However, it is a Nash equilibrium for both players to choose to take PEDs, as the payoff (90) is higher than the alternative outcomes.
c) To find the lowest level of p where both cyclists choose not to take PEDs in the equilibrium, we need to find the threshold where the payoff for not taking PEDs (80) becomes higher than the payoff for taking PEDs (90).
By comparing the payoffs, we can see that if p > 1/2, it is optimal for both players to choose to take PEDs. However, if p < 1/2, it becomes optimal for both players to choose not to take PEDs.
Therefore, the lowest level of p where both cyclists choose not to take PEDs in the equilibrium is p = 1/2.
In summary, the normal form of the game can be represented in a matrix format. The pure strategy Nash equilibrium is for both players to choose to take PEDs when p = 1/4. The lowest level of p where both cyclists choose not to take PEDs in the equilibrium is p = 1/2.