代写微观经济学作业-代写Exercises-Microeconomics代写

Practice Exercises 3

Spring 2021

代写微观经济学作业 The payoffs depend on player’s private information types; θ ∈ {0, 2} represents player 1’s type, while γ ∈ {1, 3} represents player 2’s type.

1.Consider the following Bayesian game.  代写微观经济学作业

There are two players, four possible states of nature, Θ = {1, 2, 3, 4}, and two actions for each player, A1= {U, D} and A2 = {L, R}. Each player observes her own signal (or type): Player 1’s types are T1 = {t1, t1} and player 2’s types are T2 = {t2, t2}. The probability of each state is (and this is common knowledge). Furthermore, the mapping from states to signals is common knowledge, and is given by the function µ : Θ T1 × T2 with µ(1) = (t1, t2) µ(2) = (t’1, t2) µ(3) = (t1, t2) µ(4) = (t1, t2).

The payoffs depend on the realized state of nature, and are captured by the four payoff matrices shown below:

代写微观经济学作业
代写微观经济学作业

Find the unique Bayesian Nash equilibrium of the game.

代写微观经济学作业
代写微观经济学作业

2.Two people are involved in a dispute. Person 1 does not know whether person 2 isstrong or weak; however, she assigns probability α to person 2 being strong.  代写微观经济学作业

Person 2 is fully informed. Each person can either fight (F) or yield (Y). Each person’s preferences are represented by the expected payoffs of 0 if she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields; if both people fight, then their payoffs are (1, 1) if person 2 is strong and (1, 1) if person 2 is weak. Formulate this situation as a Bayesian game and find the Bayesian Nash equilibria (note: your answer may differ depending on the value of α).

3.Supposetwo players play the Bayesian game, tt, where player 1 can choose actions from S1 = {X, Y } and player 2 can choose actions from S2 = {L, R}.

The payoffs depend on player’s private information types; θ  {0, 2} represents player 1’s type, while γ  {1, 3} represents player 2’s type. Each player learns her own type before play begins, but does not know the other player’s type. Nevertheless, the players hold a common prior belief p with p(θ, γ) = 1 for all (θ, γ). The payoffs are summarized in the matrix below:

代写微观经济学作业
代写微观经济学作业

Find the Bayesian Nash equilibria for this game.

4.Considera public goods provision game, with n  Each individual must choose whether or not to contribute to the public good, and the public good is provided if and only if at least one individual contributes.

The value  of the good is vi  to individual i: where each vi is independently and identically distributed across individuals, and is uniformly distributed on [0, 1]. The total payoff to an individual is the value of the good (if provided) minus the cost of provision (which is c if the individual provides the good, and zero otherwise). Solve for a symmetric Bayesian Nash equilibrium of this game where each individual i provides the good if and only if vi exceeds a critical threshold v. How does the probability that the good is provided at all vary with n? 代写微观经济学作业

5.Two neighboring kingdoms (Kingdom 1 and Kingdom 2) face an invading horde of Bar- barians. The Barbarians (who are not active players in this game) may be attackingfrom the North or from the South. The two kingdoms must each decide how to set up their defenses, but do not know ex ante whether the Barbarians will attack from the North or the South. In particular, each kingdom must choose between strategy A and strategy B.The two kingdoms share a common prior belief that the Barbarians will attack from the North with probability 1 and from the South with probability 代写微观经济学作业

In addition, when the attack comes from the North,  Kingdom 1 will be able to see a dust cloud created by the Barbarians before it needs to choose a strategy.

Kingdom 2 is unable to observe the dust cloud from its location. When the attack comes from the South, Kingdom 2’s trained scouts will be able to smell the horses if the winds are favorable, which occurs with probability 1 . Assume that Kingdom 1 does not have scouts who can detect the smell of the horses when the attack comes from the South, regardless of whether or not the winds are favorable. (Note: You can also assume that neither kingdom is able to directly observe whether or not the winds are favorable.)

The payoffs to the two kingdoms depend on their chosen strategies and on whether the attack comes from the North or the South. The payoff matrices for each combination are given below.

代写微观经济学作业
代写微观经济学作业

(a)Set this up as a Bayesian game by carefully specifying the players, types, strategies, and payoffs for the game. You should think carefully about the different “types” for each player, and about the information that each type possesses when making their decision.

(b)Solvefor the unique Bayesian Nash equilibrium of the game.

(c)Now suppose that Kingdom 1 also has scouts who can detect the smell of thehorses when the attack comes from the South and the winds are favorable. Note that this assumption does not give Kingdom 1 any more information about where the attack is coming from. Nevertheless, show that the assumption allows for a different Bayesian Nash equilibrium and briefly explain why it matters.

 

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代写微观经济学作业
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