Question 3: Bayesian Nash equilibrium Consider the two games/payoff matrices in Figures 1 and 2 ("Game A" and "Game B", respec- tively). The overall extensive-form game G, which is characterized by incomplete information, is defined as follows. ⚫ Nature first draws either Game A or Game B, with Pr[Game A] = 1. Player 1 observes Nature's choice perfectly, while Player 2 does not observe it. Thereafter, the two players simultaneously choose their actions a₁ € (Up, Down) and a₂ € (Left, Right), respectively. Finally, the players receive their payoffs, which are given by the payoffs in Game A or B, depending on Nature's choice. (a) Draw a game tree that shows the extensive form of the incomplete-information game G described above. (b) Solve for the Bayesian Nash equilibria of the game G. 2 Player 2 Left Right Up 1,1 4,0 Player 1 Down 0,4 3,3 Figure 1: Game A (see Question 3) Player 2 Left Right Up 0,0 0,1 Player 1 Down 1,0 3,3 Figure 2: Game B (see Question 3)
Question 3: Bayesian Nash equilibrium Consider the two games/payoff matrices in Figures 1 and 2 ("Game A" and "Game B", respec- tively). The overall extensive-form game G, which is characterized by incomplete information, is defined as follows. ⚫ Nature first draws either Game A or Game B, with Pr[Game A] = 1. Player 1 observes Nature's choice perfectly, while Player 2 does not observe it. Thereafter, the two players simultaneously choose their actions a₁ € (Up, Down) and a₂ € (Left, Right), respectively. Finally, the players receive their payoffs, which are given by the payoffs in Game A or B, depending on Nature's choice. (a) Draw a game tree that shows the extensive form of the incomplete-information game G described above. (b) Solve for the Bayesian Nash equilibria of the game G. 2 Player 2 Left Right Up 1,1 4,0 Player 1 Down 0,4 3,3 Figure 1: Game A (see Question 3) Player 2 Left Right Up 0,0 0,1 Player 1 Down 1,0 3,3 Figure 2: Game B (see Question 3)
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.7P
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![Question 3: Bayesian Nash equilibrium
Consider the two games/payoff matrices in Figures 1 and 2 ("Game A" and "Game B", respec-
tively). The overall extensive-form game G, which is characterized by incomplete information, is
defined as follows.
⚫ Nature first draws either Game A or Game B, with Pr[Game A] = 1. Player 1 observes
Nature's choice perfectly, while Player 2 does not observe it. Thereafter, the two players
simultaneously choose their actions a₁ € (Up, Down) and a₂ € (Left, Right), respectively.
Finally, the players receive their payoffs, which are given by the payoffs in Game A or B,
depending on Nature's choice.
(a) Draw a game tree that shows the extensive form of the incomplete-information game G
described above.
(b) Solve for the Bayesian Nash equilibria of the game G.
2
Player 2
Left
Right
Up
1,1
4,0
Player 1
Down
0,4
3,3
Figure 1: Game A (see Question 3)
Player 2
Left
Right
Up
0,0
0,1
Player 1
Down
1,0
3,3
Figure 2: Game B (see Question 3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4aa8b1da-55a3-479a-aa80-dfddd30b4e07%2Fb8745453-3254-4232-9149-57335afbd40e%2Fglzhlfq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3: Bayesian Nash equilibrium
Consider the two games/payoff matrices in Figures 1 and 2 ("Game A" and "Game B", respec-
tively). The overall extensive-form game G, which is characterized by incomplete information, is
defined as follows.
⚫ Nature first draws either Game A or Game B, with Pr[Game A] = 1. Player 1 observes
Nature's choice perfectly, while Player 2 does not observe it. Thereafter, the two players
simultaneously choose their actions a₁ € (Up, Down) and a₂ € (Left, Right), respectively.
Finally, the players receive their payoffs, which are given by the payoffs in Game A or B,
depending on Nature's choice.
(a) Draw a game tree that shows the extensive form of the incomplete-information game G
described above.
(b) Solve for the Bayesian Nash equilibria of the game G.
2
Player 2
Left
Right
Up
1,1
4,0
Player 1
Down
0,4
3,3
Figure 1: Game A (see Question 3)
Player 2
Left
Right
Up
0,0
0,1
Player 1
Down
1,0
3,3
Figure 2: Game B (see Question 3)
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