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 this game in extensive form.

b.      Find the Nash equilibrium outcomes to this game.

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

Answer 1

Chapter 10, Problem 6CCQ

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1. Chapter 1

2. Chapter 2

3. Chapter 3

4. Chapter 4

5. Chapter 5

6. Chapter 6

7. Chapter 7

8. Chapter 8

9. Chapter 9

10. Chapter 10

1. 1CBE

2. 1CCQ

3. 1CE

4. 2CCQ

5. 3CCQ

6. 4CCQ

7. 5CCQ

8. 6CCQ

9. 7CCQ

10. 8CCQ

11. 9CCQ

12. 10CCQ

13. 11PA

14. 12PA

15. 13PA

16. 14PA

17. 15PA

18. 16PA

19. 17PA

20. 18PA

21. 19PA

22. 20PA

23. 21PA

24. 22PA

25. 23PA

11. Chapter 11

12. Chapter 12

13. Chapter 13

14. Chapter 14

Problem

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.

Step-by-step solution

1. Step 1 of 3

   (a)

   The extensive form of the two-player, sequential-move game where Player 1 and Player 2 can choose to play oralong with the payoffs is given as follows:

   

   Comment

2. Step 2 of 3

   (b)

   There are two Nash equilibriums in this case.

   First Nash equilibrium is {Player 1 plays right} with the outcome {$0, $25}

   Second Nash equilibrium is {Player 1 plays left, Player 2 plays right} with the outcome {$20, $20}

   The first occurs when Player 2 threatens to choose Left if Player 1 chose left and thus, Player 1 chose right. To see that these strategies indeed comprise Nash equilibrium, note the following: Given Player 2’s strategy is to choose left if Player 1 chose left, The Player1’s best response is to choose right. Thus, neither player has an incentive to change its strategy. Thus, the Player 1 earns $0 and Player 2 earns profits of $25.

   However, this Nash equilibrium involves a threat that is not credible.

   The reason Player 1 chooses right is that Player 2 threatens to choose left, if Player 1 choose left.

   Given that Player 1 choose left, Player 1 earns $20 by choosing right but only $10 by choosing left. If Player 1 chooses left, it is not in the best interest of Player 2 to choose left.

   Thus, the outcome in which Player 1 choose right because Player 2 threatens to choose left, if it chooses left is a Nash equilibrium , but it is not a sub game perfect equilibrium. It involves a threat that is not credible.

   The second Nash equilibrium for this game is for Player 1 to choose left and Player 2 to follow this move by playing Right. In particular, if Player 1 plays left , the Player 2’s best response is to play right (by playing right, Player 2 earns $20 instead of $10 it would earn by playing left).

   Given that Player 2 plays right if Player 1 plays left, Player 1’s best choice is to play left (by choosing left, Player 1 earns $20 instead of $0 it would earn by playing right). This is a sub game perfect equilibrium, because it is clearly in Player 2’s self-interest to play right whenever Player 1 chooses to play left.

   Comment

3. Step 3 of 3

   (c)

   The Nash equilibrium {Player 1 plays left, Player 2 play right} with the outcome {$20, $20} is most reasonable. This is because it is also a sub game perfect equilibrium. It is clearly in Player 2’s self-interest to play right whenever Player 1 chooses to play left.

   
   

   Comment