Consider the same setting as in Problem 2 with two potential buyers and known distribution functions F1(1) 1 and F2(2) 22 2 on 0, 1 (that is, the maximum possible value 1) (a) Write the virtual surplus functions 1(1), 2(2) and show that they are weakly increas ing (that is, show that F1 and F2 are ...

a The virtual surplus functions for bidders 1 and 2 are calculated as follows 11 111 22 222 23 To sh ...

Consider the same setting as in Problem 2 with two potential buyers and known distribution functions: F1(1)

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Question:

Consider the same setting as in Problem 2 with two potential buyers and known distribution functions: F1(θ1) = θ1 and F2(θ2) = 2θ2 − θ2 on [0, 1] (that is, the maximum possible value θ ̄ = 1).
(a) Write the virtual surplus functions ψ1(θ1), ψ2(θ2) and show that they are weakly increas- ing (that is, show that F1 and F2 are regular).
(b) In the revenue maximizing mechanism (given by equations (4), (5), and (6)), find the conditions on θ1 and θ2 under which the good is sold to neither bidder.
(c) In the revenue maximizing mechanism, find the conditions on θ1 and θ2 under which the good is sold to bidder 1 and under which the good is sold to bidder 2.
(d) In the (θ1, θ2)-coordinates represent three regions in which: 1. there is no sale, 2. buyer 1 gets the good, 3. buyer 2 gets the good; and calculate the probability that nobody gets the object. Hint: Note that this probability is evaluated according to the distribution of bidders’ valuations.