Question

Consider a two-player, sequential-move game where each player can choose to play right or left. Player 1 moves first. Player 2 observes player 1’s actual move and then decides to move right or left. If player 1 moves right, player 1 receives \(0 and player 2 receives \)25. If both players move left, player 1 receives \(5 and player 2 receives \)10. If player 1 moves left and player 2 moves right, player 1 receives \(20 and player 2 receives \)20.

a. Write the above game in extensive form.

b. Find the Nash equilibrium outcomes to this game.

c. Which of the equilibrium outcomes is most reasonable? Explain

Short answer

a. The extensive form is as follows

b. The Nash equilibria in this simultaneous game will be (0,25) and (20,20).

c. The most reasonable Nash equilibrium will be the second equilibrium obtained in question (b).

Step-by-step solution

Finding the extensive form

a.

The sequential-move game extensive form for Player 1 (PI) and Player 2 (P2) in which each one can choose the left or right option and according to the following indications:

Finding Nash equilibrium

b.

In this situation, Player 1 is the one who plays first and Player 2's decisions are based on the decision made by Player 1. Therefore, it is a simultaneous-move games. In this case, there are two Nash equilibria. The first equilibrium occurs when Player 1 decides to play to the right, in which he considers it the best option since if he chooses the left, player 2 will be able to choose the left and obtain a loss of 5 while player 2 wins 10.Therefore, he prefers to play on the right and get 0 and player 2 gets 20. However, this is in a hypothetical situation in which Player 1 thinks that he can be tricked by Player 2 into making losses.

The seond desired Nash equilibrium in this game is for Player 1 to move to the left and Player 2 to move to the right. This is more feasible since Player 2 has incentives to play to the right and obtain twice the benefit than if he plays to the left, while Player 1 will also obtain a benefit of 20 that is greater than 0 if he plays only on the right.

Therefore, the Nash equilibria in this simultaneous game will be (0,25) and (20,20).

Finding the more reasonable equilibrium

c.

The most reasonable Nash equilibrium will be the second equilibrium obtained in question (b) in which Player 1 plays first to the left and Player 2 plays to the right. Here, both players will obtain a profit of 20(20,20). It is the most reasonable since Player 2 has no incentive to play left if he can play right and get double the profit Likewise, if Player 1 knows this, it will not be convenient for him to play to the right and obtain 0 profits.