CS234r: Markets for Networks and Crowds
Instructors: Nicole Immorlica and Brendan Lucier
Problem Set #1
Due: Feb 26, 2016
Answer 4 of the following 5 questions.
1. (30 points) This question tests your basic comprehension of the lectures.
(a) (5 points) Exchange markets. Find an example that demonstrates the core might
not exist in an exchange market if agents have indifferences in their preferences.
(b) (15 points) Discrete allocation. Consider a discrete allocation problem. The
Random Serial Dictatorship (RSD) mechanism selects at each step a random agent
and assigns that agent his/her favorite remaining object. Prove that RSD produces
the same distribution over allocations as the Top Trading Cycles mechanism with
a uniformly random initial endowment. (Stuck? Here’s a hint.1 ).
(c) (10 points) Two-sided matching. Consider assigning workers to firms. Worker i
wants to work for at most pi hours per week. Firm j wants at most cj worker-hours
per week. A worker can be employed by at most one firm.
• Show that a stable matching might not exist.
• Let pmax = maxi pi . We now allow firms to be assigned more worker-hours than
required so long as firm j is assigned at most cj +pmax worker-hours. A matching
respecting these perturbed capacities is stable if (i) it is individually
rational,
P
(ii) there is no blocking pair (i, j) where j i µ(i) and either i0 ∈µ(j) pi0 < cj
or for some i0 ∈ µ(j), i j i0 . Prove that such a matching exists.
2. (30 points) Consider the following variant of an exchange economy with multiple goods
per agent. Each agent i brings a disjoint set Gi of goods to the market, and has a desired
set of goods Wi with Gi ∩ Wi = ∅. An assignment allocates to each agent a disjoint set of
goods. Preferences are as follows: agent i is only interested in subsets of Gi ∪ Wi of size
|Gi |, and subject to this constraint he wishes to maximize the number of goods received
from Wi . Any other set is strictly less preferable than Gi .
(a) (5 points) Show how to represent this market as a directed bipartite graph, and
prove that an IR assignment corresponds to a set of disjoint cycles in that graph.
(b) (10 points) Prove that a collection of disjoint cycles that maximizes the sum of
cycle lengths corresponds to a pareto efficient assignment.
Given a union of disjoint cycles C in directed bipartite graph G, the residual graph GC
is the following weighted directed bipartite graph. For each edge (u, v) ∈ C, GC contains
(v, u) with weight −1 (note the direction of the edge!). For each edge (u, v) ∈ G\C, GC
contains (u, v) with weight 1.
(c) (10 points) Prove that an assignment is pareto efficient if its corresponding residual
graph has no positive-weight cycles.
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Construct a one-to-one and onto mapping between endowments in TTC and orderings in RSD.
(d) (5 points) Design a polynomial-time mechanism that finds a pareto efficient assignment. You can assume, without proof, that it is possible to find a positive-weight
cycle in a weighted directed bipartite graph in polynomial time, or determine that
one does not exist.
3. (30 points) In this question we will explore school choice mechanisms when students
have strict preferences, but school preference lists can include indifferences.
(a) (5 points) One way to handle school indifference is to enforce strict preferences by
breaking ties randomly (i.e., choose a random order within each indifference class),
then run the student-proposing DA mechanism. Give an example showing that this
mechanism does not necessarily generate a Pareto efficient matching with respect
to the original preferences.
(b) (15 points) Given a stable matching µ, consider the following directed graph Gµ .
Each vertex is a school, and there is an edge from school x to school y iff there is at
least one student i such that µ(i) = x, y i x, and y weakly prefers i to any other
student j such that y j µ(j). That is, i would prefer to switch from x to y, and is
in the highest priority class (for y) of students who would like to switch to y. Prove
that if µ is not Pareto efficient, then Gµ contains a cycle.
(c) (10 points) Design a mechanism that finds a Pareto efficient matching when student
preferences are strict but school preferences might include indifferences.
4. (30 points) For this problem, you will participate in a sequence of simulated schoolchoice markets. For each market, you will choose how to rank the schools on your school
application form given your preferences, information about others’ preferences, and the
mecanism being used. You will also provide a written description (1-2 paragraphs
each) explaining why you ranked schools the way 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” ranking.
In each example, there are five schools: A, B, C, D, and E. Each school has 20
openings. There are 120 students in the market, including yourself. School preferences
are determined by a uniformly random ranking over students, drawn independently for
each school. Each student has a cardinal value for being matched to each school, and
all values are drawn independently from the uniform distributions below. Your realized
values are also listed below. The value for not getting matched to a school is 0.
School Value Distribution Your Value
A
U [10, 20]
12
B
U [6, 12]
11
C
U [4, 10]
5
D
U [3, 7]
6
E
U [1, 4]
3
∅
0
0
(a) (6 points) The Gale-Shapley mechanism is being used. You can rank all 5 schools.
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(b) (6 points) Gale-Shapley is being used, but you can rank at most 3 schools.
In the Boston mechanism, each student first proposes to their favorite school, and each
school accepts their favorite students from among those that proposed, up to the school’s
capacity. These acceptances are then fixed, and accepted students leave the market.
Each remaining student then points to their second-favorite school, and schools accept
their favorite students from among these new proposals, up to their remaining capacity.
(In particular, a school that has already reached capacity will not accept any further
students.) This repeats until all remaining students have already proposed to every
school on their list.
(c) (6 points) The Boston Mechanism is being used. You can rank all 5 schools.
(d) (6 points) The Boston Mechanism is being used, and you can rank at most 2
schools. But now every school’s preferences are are determined by a single standardized test. Scores are distributed uniformly in [0, 100]. You scored 62.
(e) (6 points) In this case you do not know your preferences for the schools; you only
know that your values will be drawn from the distributions listed above. You can
choose to interview at one or more schools, which will reveal the value you have
for them. Each interview you take costs 2 utility, and you must choose all your
interviews up front. After all interviews are complete, a Gale-Shapley mechanism
will be used. There is no limit on the number of schools listed, but students can
only list schools they interviewed at. Which schools do you choose to interview at?
(There is no standardized test — school preferences are uniformly random.)
5. (30 points) Reading comprehension / market 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?
• Akbarpour, Gharan, and Li, “Dynamic matching market design”
• Budish, “The combinatorial assignment problem”
• Ashlagi, Kanoria, and Leshno, “Unbalanced Random Matching Markets”
OR
(b) Choose a real-world market not discussed in class, and propose a mechanism for
resolving that market. You should limit yourself to an exchange market, a discrete
allocation market, or a matching market. 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.
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