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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.