Question 4 [30%]
Consider the following sponsored search auction instance I:
• 2 slots. The top slot has a known click-through rate (CTR) ctr₁ = 1 and the bottom slot has a
known click-through rate ctra=0.5.
• 2 advertisers. Advertiser 1 has a private value-per-click v₁ = 2 and advertiser 2 has a private
value-per-click U₂ = 1.
• The payoff of advertiser i, (i is either 1 or 2), who is assigned to the top slot is (v₁ - Pi), where pi
is the price charged per-click to i. The payoff of advertiser j (j is either 1 or 2 but different than
i) who is assigned at the bottom slot is 0.5-(v₁-p;) where p, is the price charged per-click to j.
P; and p; are defined by the auction rule, as follows.
Consider the following auction rule (first-price auction):
Advertisers are asked to declare their value per click (this doesn't mean that their declarations are
truthful!). Advertisers are then ranked according to their declarations and the advertiser with the
highest declaration is assigned to the slot with the highest CTR (top slot), the advertiser with the
second highest declaration is assigned to the slot with the lowest CTR (bottom slot). In case of
a tie, advertiser 1 is allocated to the top slot. The per-click payment of any advertiser is equal to
their own bid.
a. [6%] Compute the optimal/highest social welfare (sum of individual values) in I.
b. [9% ] Let bi and b2 denote the bids placed by advertiser 1 and 2, respectively, and assume b2 > b₁.
Formulate the conditions that need to be satisfied at equilibrium. The conditions should contain
only variables V₁, V2, b₁ and b₂.
c. [15% ] Write a function that takes as input the bid of advertiser 1 and calculates the best response, i.e.
the strategy/bid of advertiser 2 that results in the highest possible utility for advertiser 2.
You can (or not) follow a brute-force approach, i.e. consider all possible declarations/bids for adver-
tiser 2, calculate the associated utility and keep the bid that maximizes that utility. Copy and paste
your MATLAB code in your report.
Fig: 1
