Econ 109 – Game Theory – Spring 2022 Problem Set 4
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Econ 109 – Game Theory – Spring 2022
Problem Set 4
Due by 11:59 pm March 15, 2023
1. In this question, your goal will be to understand whether learning in games is always valuable for players. Consider the following incomplete-information game. First, nature chooses between one of the following two A and B tables, each with probability 0]5:
Then, players 1 and 2 simultaneously choose U or D and L or R, respectively, and obtain payo住s according to the table chosen by nature. Parts I-III present variations of this game under di住erent assumptions about what players know about nature’s move.
Part I: Suppose no player observes nature’s move (and it is common knowledge).
(a) [0.5pt] Represent the game in extensive form.
(b) [0.5pt] Represent the game in Bayesian normal form.
(c) [0.5pt] Find the unique BNE.
Part II: Suppose only player 1 observes nature’s move (and it is common knowledge).
(d) [0.5pt] Represent the game in extensive form.
(e) [0.5pt] Represent the game in Bayesian normal form.
(f) [0.5pt] Find the unique BNE.
Part III: Suppose both players observe nature’s move (and it is common knowledge).
(g) [0.5pt] Represent the game in extensive form.
(h) [0.5pt] Represent the game in Bayesian normal form.
(i) [0.5pt] Find the unique BNE.
Part IV.
(j) [0.5pt] Suppose player 2 does not observe nature’s move. Does player 1 bene脯t from learning about nature’s move?
Hint: Compare player 1’s BNE expected payo扩 in (f) to that in (c).
(k) [0.5pt] Suppose player 1 observes nature’s move. Does player 2 bene脯t from learning about nature’s move?
Hint: Compare player 2’s BNE expected payo扩 in (i) to that in (f).
2. mis game is based on the book“Real Men Don’t Eat ≈iche”by Bruce Feirstein. mere are two players siting at a bar. me timing is as follows:
i. Nature determines P1’s type as“wimp”or“surly”. P1 observes it, P2 doesn’t. (and this is common knowledge)
ii. P1 chooses whether to order a beer or a quiche.
iii. P2 siting next to P1, observes P1’s choice. men P2 decides whether to start a 脯ght with P1 or not.
The extensive form is given below. Let numbers p and g denote P2’s conditional beliefs in the two information sets.
(a) [1.5pt] Is there a pooling PBE of this game? YES or NO? (Circle one).
If YES, describe one (and only one): s1 = , s2 = , p = , g = . Use space below for your argumentation.
(b) [1.5pt] Is there a separating PBE of this game? YES or NO? (Circle one).
If YES, describe one (and only one): s1 = , s2 = , p = , q = . Use space below for your argumentation.
2023-03-11