Question

Consider the following simultaneous move game:

a. What is the maximum amount player 1 should be willing to pay for the opportunity to move first instead of moving at the same time as player 2? Explain carefully.

b. What is the maximum amount player 2 should be willing to spend to keep player 1 from getting to move first?

Short answer

Player 1 will pay $\mathrm{\$}100$ to get the first-mover advantage. Player two will pay to stop player one from getting the first move.

Step-by-step solution

Finding the sequential move production game

Given the information in the following table for both Player 1 and Player 2 and then the sequential-move production Game with the decisions is shown in the case that Player 1 can decide first.

Finding the profit

It can be seen then that if player 1 decides to play Yes, the best option for player 2 is to decide Yes as well since he will have a higher profit (400 vs 375). On the other hand, if player 1 decides to play No, player 2 will make the decision of No as well since it will generate a greater profit (525 vs 500$).

Observing this, then player 1 will only be able to choose between two options: 100$ if the decisions are Yes and Yes respectively from P1 and P2 and 300$ with the decisions of No and No respectively. The maximum amount of 600$ that is the maximum will not be able to be obtained since player 2 will not. choose Yes when player 1 chose No.

Therefore, the maximum amount player 1 should be willing to pay for the opportunity to move first instead of moving at the same time as player 2 is 100$ which is the difference between 300$ which is the minimum he could win and 400$.

Finding the maximum amount

Guided by the sequential-move production game regarding the decisions made by player 1 (Yes, Yes) and (No, No) the winning options of player 2 will be 400$ for the (Yes, Yes) decision and 525$ for the (No, No) decision. Therefore P2 will be willing to pay the difference between these two options to get the highest profit. The maximum amount he will pay will then be $\mathrm{\$}525-\mathrm{\$}400=\mathrm{\$}125.$