Question 1) Repeated Cournot

Consider a repeated version of the Cournot model discussed in class, with two firms i = 1, 2, demand function

P (Q) = (A - Q)⁺,

marginal production costs c_i > 0 (satisfying c₁, c₂ < A) and discount factor α ∈ (0, 1).

a) Find out a strategy profile that induced these firms to cooperate in this market (i.e., to collude by producing in total the monopoly quantity (A - c/2)) for sufficiently patient firms. How large does the discount factor δ needs to be to sustain cooperation?

b) Now suppose firms want to sustain cooperation in an asymmetric way. Describe a strategy profile that induces these firms to cooperate, for large enough discount factor δ, in an asymmetric way, i.e., to produce the monopoly quantity in total (which maximizes total profits in the stage game) with firm 1 producing share α₁ ∈ (0, 1) of this quantity and firm 2 producing share α₂ ∈ (0, 1) of this quantity, with α₁ + α₂ = 1. Suppose that α₁ < α₂. Find how large the discount factor δ needs to be for cooperation to be sustained (Hint: you can use modified grim-trigger strategies for this question).

Question 2) A simple bayesian game

Suppose players 1 and 2 have to jointly decide whether to go to a BC Lions (L) or Canucks (C) game.

a) What are the Nash equilibria of this game (pure and mixed)?

b) Let's turn this game into a Bayesian game. Now let's assume that player 2 might be a friend or a foe to player 1. If player 2 is a friend he has exactly the same payo? as in item (a). If he is a foe, then he does not like to be together with player 1!.  So player 2 gets a payo? of 4 if he goes to a Canucks game alone, a payo? of 1 if he goes to a BC Lions game alone, and a payo? of zero if he ends up in the same place as player 1. Player 2 is either a friend or foe with probability ½.

Player 1 is always a friend, and so has payo?s identical to the one presented in item (a). Formally describe the Bayesian game described (present all the elements that constitute a Bayesian Game).

c) What are the pure Bayes-Nash equilibria of this game?

Question 3) Prisoner dilemma with alternating actions

Consider the following version of the prisoner's dilemma:

Assume that the game represented is played for infinite periods, with discount factor δ ∈ [0, 1).

a) When can the cooperative outcome (C, C) be sustained via grim-trigger strategy profile?

b) Draw the set of feasible and individually rational payo?s.

c) What is the highest symmetric individually rational payo? profile (i.e., the highest payo? profile (v, v) consistent with feasibility and individual rationality)?

d) Consider the following modified grim-trigger strategy: both players alternate between the action profile (C, D) and (D, C) as long as everyone has followed this plan so far. If someone played di?erently, then play D forever.

Formally, let the set of alternating histories be (which includes the initial node)

H_A = {Φ, (C, D), ((C, D), (D, C)), ((C, D), (D, C), (C, D)),. . .}.

The strategy profile proposed is (s₁^MGT, s₂^MGT) such that:

Player 1 plays as follows:

• s₁^MGT(h_t) = C if h_t ∈ H_A and t is an odd number.
• s₁^MGT(h_t) = D otherwise (i.e., if h_t ∈ H_A and t is even or h_t ∉ H_A).

Player 2 plays as follows:

• s₂^MGT(h_t) = C if h_t ∈ H_A and t is an even number (including zero).
• s₁^MGT(h_t) = D otherwise (i.e., if h_t ∈ H_A and t is odd or h_t ∉ H_A).

What is the average discounted payo? generated by this strategy profile?

e) Under what conditions is (s₁^MGT, s₂^MGT) a SPE? (Hint (i): due to symmetry you only need to check for potential deviations of player 1. Hint (ii): there are three types of histories - cooperative odd period, cooperative even period and non-cooperative. You need to check for one-shot deviations in each of these).

Question 4) First price auction

Consider a first price auction in which two bidders are competing for one good. Each bidder has valuation v_i distributed uniformly on [2, 3].

a) What are the elements that describe a Bayesian game? Describe all these elements for the auction described.

b) Find numbers α, β ≥ 0 such that both bidders using bidding strategy b(v)= α + βv is a Bayes-Nash equilibrium of this model. (Hint 1: Assume that bidder -i uses the strategy described above and solve for the optimal bid for bidder i, assuming he has value v. In a Bayes-Nash equilibrium, this optimal bid needs to be identical to α + βv, for any v ∈ [2, 3].  Hint 2: If bidder -i follows bidding rule b(v) = α + βv and bidder i bids b ∈ [α + 2β, α + 3β], the probability with which bidder i wins the auction is ((b-α/ β) - 2)).