CS234r: Markets for Networks and Crowds
Instructors: Nicole Immorlica and Brendan Lucier
Problem Set #2
Due: Apr 1, 2016
Answer 4 of the following 5 questions.
1. (30 points) This question tests your basic comprehension of the lectures.
(a) (10 points) Market equilibria. For each of the following combinatorial markets
give a market equilibrium or prove that none exists. (Note: V (∅) = 0 in all valuation
functions.)
• Items: {a,
pb, c, d}, agents: {1, 2}.
V1 (S) = |S| for all S ⊆ {a, b, c, d}
V2 (S) = 12 |S| for all S ⊆ {a, b, c, d}
• Items: {a, b, c, d}, agents: {1, 2}.
V1 (S) = 78 for all S 6= ∅
V2 (S) = max {|S|, 2} for all S 6= ∅
(b) (5 points) Double auctions. Prove that the VCG double auction is strategyproof
for both buyers and sellers.
(c) (15 points) Single item revenue maximization. Suppose a seller with a single
item faces two buyers.
• Your opponent has a value v2 distributed uniformly in the interval [0, 2] whereas
you have a value v1 distributed uniformly in the interval [0, 1]. The seller
is running a first-price auction. Your opponent used revenue equivalence to
derive strategies s1 (v1 ) = v1 /2 and s2 (v2 ) = min(1/2, v2 /2) and has decided to
bid according to strategy s2 (v2 ). Show that bidding s1 (v1 ) is not your bestresponse, and so your opponent’s calculation was incorrect. Find the error
in your opponent’s application of revenue equivalence.1 Conclude there is no
efficient equilibrium.
• Use revenue equivalence to compute an equilibrium of the first-price auction
with reserve r ∈ [0, 1] for two bidders with values distributed uniformly in
the interval [0, 1]. Make sure you prove the computed solution is indeed an
equilibrium.
• We now return to the original value distribution, i.e., v1 is distributed uniformly
on [0, 1] and v2 is distributed uniformly on [0, 2]. Design a revenue-optimal
auction for this setting.
2. (30 points) In this problem we will consider a market consisting of buyers, sellers, and
traders. There is one type of good in the market, and each seller has one unit of the
good to sell. Sellers have no value for the good, and no costs. Each buyer i is interested
in buying one copy of the good, and has some value vi for it. Traders have no value
1
If you are unable to find the error, please email us for the solution; you need it for the next bullet.
for the good, but they can purchase goods from the sellers and then resell them to the
buyers. Each seller (buyer) can sell to (buy from) some subset of the traders. Buyers
cannot buy directly from sellers.
Suppose each trader offers a (possibly different) price to each of its sellers and each of
its buyers. These prices, along with trade decisions by the buyers and sellers, form an
equilibrium if
• each seller accepts a highest offered price,
• each buyer accepts a lowest offered price no greater than their value,
• each trader sells at most the number of goods that they buy, and
• no one trader is able to change their prices and increase their profit. (Noting that
the buyers and sellers might change their choices given a change in prices).
For each of the following markets, draw the seller-trader-buyer interactions as a tripartite
graph, and describe the set of pricing equilibria.
(a) (5 points) There is one seller, one buyer, and one trader. The buyer has value 1.
The trader can trade with the buyer and the seller.
(b) (5 points) There is one seller, one buyer, and two traders. The buyer has value 1.
Each of the two traders can trade with both the buyer and the seller.
(c) (5 points) There is one seller, three buyers, and three traders. Call the buyers
{B1, B2, B3} and the traders {T 1, T 2, T 3}. For each x ∈ {1, 2, 3}, trader T x can
trade with the seller and buyer Bx, and buyer Bx has value x.
(d) (5 points) There are three sellers {S1, S2, S3}, three buyers {B1, B2, B3}, and two
traders {T 1, T 2}. Each buyer has value 1. Trader T 1 can trade with S1, S2, B1,
and B2. Trader T 2 can trade with S2, S3, B2, and B3.
(e) (5 points) There are three sellers {S1, S2, S3}, four buyers {B1, B2, B3, B4}, and
two traders {T 1, T 2}. For each x ∈ {1, 2, 3, 4}, buyer Bx has value x. Trader T 1
can trade with S1, S2, B1, and B2. Trader T 2 can trade with S2, S3, B2, B3, and
B4.
(f) (5 points) How does the equilibrium in the last part change if trader T 2 can no
longer trade with seller S2?
3. (30 points) In this problem we explore the design of search ad auctions. There are
m advertisement slots and n advertisers. Advertiser i has value vi for a click. Slot s
has click-through rate ws , meaning an ad in slot s receives a click with probability ws .
Assume slots are labeled in decreasing order of click-through rate, i.e., ws ≥ ws+1 for all
s. The outcome of an auction is a matching of ads to ad slots.
(a) (15 points) Let xi be the probability agent i is served by an auction, and pi be his
expected payment. Given a valuation profile, label agents in decreasing order of
value, i.e., vi ≥ vi+1 for all i. An outcome consisting of allocation (x1 , . . . , xn ) and
Page 2
payments (p1 , . . . , pn ) is envy-free for valuation profile (v1 , . . . , vn ) if no agent wants
to swap allocation and payment with another agent, i.e., if
vi xi − pi ≥ vi xj − pj
for all agents i and j. Show the following characterizes the minimally-priced2 envyfree allocations:
P (i) monotonicity, i.e., xi ≥ xi+1 , and (ii) payment identity, i.e.,
pn = 0, pi = n−1
j≥i vj+1 (xj − xj+1 ).
(b) (5 points) Find the envy-free outcome for search ads with the maximum welfare.
(c) (10 points) Design an auction for search ads that maximizes welfare in a truthful
dominant-strategy equilibrium. Give a formula for the payment rule. Compare this
to the envy-free payment rule derived in part (b).
4. (30 points) You are a customer trying to buy computing resources from a cloud provider.
Each day, the available resources are sold as CPU units. For simplicity, assume that no
customer (including you) can use more than one CPU unit in a day. Each day, you can
purchase a CPU unit in one of two ways: you can pay a fixed price set in advance, or
you can bid in a spot market where the provider sells any leftover units not sold by fixed
prices. If there are k CPU units for sale in the spot market, the top k highest-bidding
customers each win one unit, and pay the (k + 1)st highest bid.
There are 9 other customers, besides you, trying to buy CPU units. The total number
of resources available each day is a uniformly random number between 2 and 8. Assume
that every other customer has a value drawn uniformly between 0 and 10 for getting a
computing resource, each day.
In each of the following scenarios, describe a strategy for purchasing CPU units (take
the fixed price or bid in the spot market, and what to bid in the latter case). Unless
otherwise specified, your goal is to maximize expected utility (value minus costs). You
will also provide a written description (1-2 paragraphs each) explaining why you
chose the strategy you did. It is more important to explain your reasoning than to try
to find a precisely optimal ranking – in many cases, there is not a single “best” strategy.
(a) (6 points) You need a single CPU unit, and it must be on a specific day. The fixed
price that day is 6. You have value 7 for obtaining the unit.
(b) (6 points) You need a single CPU unit, and it must be on a specific day. But now
you have a choice of 2 different datacenters from which to buy the CPU unit. You
don’t care which datacenter you get the CPU unit from, but you only get value for
one, and each of their spot markets resolve simultaneously. Each datacenter has 9
other customers wanting to buy CPU units, as described above. You have value 10
for obtaining the unit, and the fixed price that day (for each datacenter) is 7.
(c) (6 points) We are now back in the case of a single datacenter, but now you need 3
CPU units over the course of a week. You can use at most 1 CPU unit each day,
so you need to win a unit on at least 3 days that week. Your value is 20 if you get
at least 3 units, otherwise your value is 0. The fixed price each day is 6.
2
We disallow negative payments.
Page 3
(d) (6 points) This part is the same as the previous part, but now instead of having a
value for getting at least 3 units, you are tasked with getting at least 3 units while
paying at most your budget, which is 14. If you succeed, you get a payoff of 1;
otherwise your payoff is 0. The fixed price each day is 5.
(e) (6 points) You are now taking the role of the cloud provider. Suppose that all
buyers have payoffs similar to the one described in the previous part, where the
budget for each buyer is uniformly distributed between 10 and 20. You need to
choose a single fixed price, which will be offered each day. Your goal is to maximize
revenue – what price do you choose?
5. (30 points) Reading comprehension / mechanism design proposal. For this question, do
one of the following:
(a) Prepare a discussion of one of the following papers from the course reading list.
Summarize the model and contribution, discuss the key new ideas and techniques,
and/or suggest some next steps for further research that builds upon the paper. Do
you like the paper? Does the model capture the important features of the market
being studied? Are the results insightful? What are the limitations of the paper?
• Cole and Fleischer, “Fast-Converging Tatonnement Algorithms for One-Time
and Ongoing Market Problems”
• Ben-Zwi, Henzinger, and Loitzenbauer, “Ad Exchange: Envy-Free Auctions
with Mediators”
• Immorlica, Stoddard, and Syrgkanis, “Social Status and Badge Design”
OR
(b) Choose a real-world market not discussed in class, and propose a mechanism for
resolving that market. You should limit yourself to a market with transferable
utility. Why is your mechanism appropriate? What are the special features of your
market that deserve consideration, and how does your mechanism handle them? If
you wish to do this question, you should send a proposal to the course email
account briefly describing the market you want to analyze; we will respond within
24 hours letting you know if it is appropriate.
Page 4