What is the payment for each winning bidder

Reference no: EM131205665

Q1. There are two firms 1 and 2, both producing the same kind of product. Firm i has two possible unit cost c_i^H and c_i^L, where c_i^H ≥ c_i^L> 0. Assuming that firm i produces q_i units of the product and firm i's unit cost is c_i ∈ {c_i^H, c_i^L}, i =1, 2, then the payoff of firm i is

u_i(q₁,c₁,q₂,c₂) = (120 - q₁ - q₂ - c_i) x q_i.

Assume that c₁^H = 40, c₁^L = 40, c₂^H = 60, c₂^L = 20. Answer the following questions.

Assume that firm 1 knows that firm 2 will choose c₂ = c₂^H. If both firms will simultaneously decide on the number of units to produce where firm i produces q_i units of the product, what are the values of q₁ and q₂ that will form a Nash equilibrium? You need to show the derivations and the result.
Assume that firm 1 knows that firm 2 will choose c₂ = c₂^Hwith probability 1/3, and choose c₂ = c₂^L with probability 2/3. If both firms will simultaneously decide on the number of units to produce where firm 1 produces q₁ units, and firm 2 produces q₂^H(q₂^L, respectively) units when its unit cost c₂ is c₂^H (c₂^L, respectively), what are the values of q₁, q₂^H and q₂^L that will form a Nash equilibrium? You need to show the derivations and the result.

Q2. Four bidders 1, 2, 3, and 4 bid for three items I₁, I₂, and I₃ using VCG auction. We use w_jk to denote bidder j's value for item k. Assume that w₁₁ = 300, w₁₂ = 200, w₁₃ = 100; w₂₁ = 310, w₂₂ = 210, w₂₃ = 110; w₃₁ = 320, w₃₂ = 220, w₃₃ = 120; w₄₁ = 330, w₄₂ = 230, w₄₃ = 130. Answer the following questions.

Which bidders are the winners in this auction? For each winning bidder, which item does it win?
What is the payment for each winning bidder?

Q3. This question is concerned with the paper, "Routing in Max-min Fair Networks: A Game Theoretic Approach", discussed in class (ICNP'2010).

Did the paper prove the existence of a Nash equilibrium?
Does the game have a unique Nash equilibrium?
In your view, what is the major contribution of this paper?
In your view, what is the major weakness of this paper?

Q4. This question is concerned with the paper, "Crowdsourcing to Smartphones: Incentive Mechanism Design for Mobile Phone Sensing", discussed in class (MobiCom'2012).

Did the paper prove the existence of a Stackelberg equilibrium?
Does the game have a unique Stackelberg equilibrium?
Is the auction designed in this paper a single auction or a double auction?
Does the auction guarantee that the number of winners is greater than zero?

Reference no: EM131205665

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