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### Mechanism Design without Money

Lecture 9

Debt – Israel psychology market

- I was asked to give more examples for markets.
- Clinical psychology
- Offers are made in three rounds
- If you agreed in round 2, you are not allowed to back off in round 3.
- What are the problems?

Debt - A linear fraction of couples

- The number of married people is proportional to the number of people
- Simulations show decent success rates for a constant fraction of couples
- Is there a way to insert the couples, to get a stable matching?

Thm : consider a random market, with n singles, n couples and more than 20n hospitals.

With constant probability, there is no stable outcome

Proof idea – isolated markets

- Find a small structure, which prevents a stable outcome
- A few hospitals and doctors, which (if left alone) can not form a stable outcome
- Show that this small structure exists with constant probability
- Show that no one outside the structure ever enters a hospital in the isolated market

Instable structure

Hospital Preferences Capacity is 1

- For a single s and a couple c, with probability O(1/n2) we have the structure
- If the structure occurs – no “local” stable outcome
- There are n singles and n couples, so with constant probability this structure will occur

Doctor Preferences

Isolated market

- The only solution is to insert someone else to h1,h2 thus avoiding the problem
- There is an excess of positions, so if a doctor goes to h1,h2 there are hospitals which are left free. We need to show that the doctor prefers them
- A quantitative version of the Rural Hospital Theorem
- Define a probabilistic process, show it’s a martingale, use Azuma’s inequality

l Pessimistic deferred acceptance

- l-Pessimistic DA: At each step t=1; 2 …
- either a single doctor s or a couple c that has less than l temporary assignments are chosen at random and applies to the most preferred hospital or pair of hospitals on their list respectively that they haven't applied so far.
- Each hospital h assigns a doctor d if and only if no other doctor is currently assigned to h and no other doctor applied at this step to h. If some doctor d applies to h and some doctor d’ is temporarily assigned to h, h rejects both d and d’ and no doctor is never allowed to apply to h anymore.

Halting condition

- With constant probability no more than n hospitals are visited in the process 3- pessimistic DA for some < 20.
- Let Ai(t) be the number of doctors who have i positions
- Vt number of hospitals visited until time t
- Follow the variableXt = t + Vt + 15A0(t) + 10A1(t) + 5A2(t)
- Show it is a super martingale, E(Xt+1|Xt) < Xt

3-pessimistic DA bounds the number of hospitals visited

- For each entity d (single or couple) , let p1(d), p2(d) and p3(d) denote the hospitals they get
- A doctor is poor, if in the real DA it gets something which is outside of pi(d). Let k denote the number of poor entities
- But that means that the hospitals there are filled with doctors.
- And only poor doctors can fill them
- But we need to fill 3k hospitals with at most 2k doctors

What’s in today’s class?

- School choice
- Case studies: NYC and Boston
- Algorithms: the Boston mechanism, Top Trading Cycles
- More on tie-breaking
- The signaling mechanism

What is different about school choice?

Schools and students as strategic players

True indifferences

The concept of justified envy

Stability vs. optimality

The NYC High School Match

- Until 2002:
- Decentralized applications and admissions
- Only five choices allowed
- Three rounds of waiting lists, waiting lists run by mail
- Congestion (out of over 90,000 kids every year, 30,000 administratively assigned, and 17,000 receiving multiple offers)
- 30,000 students assigned to schools not on their choice list
- Schools take students’ ranking into account
- Withholding of capacity

The NYC High School Match

- Are NYC schools really two-sided matching problem?
- Schools conceal capacities
- EdOpt schools have different preferences (high scores, attendance records, etc.)
- Solution: Deferred Acceptance
- Only 12 options allowed (breaks truthful revelation, Haeringer and Klijn, JET, 2009)
- Due to historical rules about specialized schools – matching is done in 3 rounds (round 3 for unmatched kids)

The NYC High School Match

- Single tie-breaking vs. multiple tie-breaking (for schools’ indifferences)?
- Multiple tie-breaking increases number of instabilities, and it therefore constrains the efficiency
- NYC DOE saw simulations, and tried both tie-breaking rules, and decided on single tie-breaking rule

The NYC High School Match

- Outcome:
- Only 3,000 did not receive any school they chose (compare to 30,000 the previous year)
- The reasons: relieving congestion (many offers and acceptances, instead of only three rounds), giving each student a single offer (instead of people getting multiple offers), allowing ranking of 12 instead of 5 schools, but also…
- The results continued to be better and better each year (comparing rankings), even though there were no changes to the algorithm… hmmm…
- The answer: schools have learned to stop withholding capacity!
- Open question: how to do appeals? (TTC? but that’s later)

Boston Public Schools (BPS)

About 4000 kids in each cohort. Four cohorts are making choices: K, 1, 6, and 9.

Priorities (= schools’ preferences) come from walking zones, siblings, and random tie-breaking

Boston Public Schools (BPS)

- Until 2006 the mechanism used is “The Boston Mechanism” (but also used in many other places):
- Step : Each student that is still unmatched applies to her most preferred school
- Step : Each school fills its quota as much as possible with those applicants that it prefers the most, and rejects the rest
- Problems with the “Boston Mechanism”:
- Does not produce stable matchings
- Truth-telling is not dominant (far from it)
- Not immediately clear that something is wrong…
- Those who do not play strategically get hurt

Boston Public Schools (BPS)

- Unlike NYC – unclear that the market is two-sided:
- No gaming by schools
- Lots of people in each priority class, and looks like priorities are meant to help parents select schools
- If the market is actually one-sided, then stable matchings are not Pareto optimal (it is better for people to trade priorities)

Top Trading Cycles

Introduced in Shapley and Scarf (1974), but attributed to David Gale.

Draw a graph where each agent is a node, with each agent pointing to his/her/its most preferred match.

Remove a cycle, and redraw the edges, now each agent points to most preferred match among those remaining.

Repeat until all nodes are removed.

Top Trading Cycles

Theorem (Shapley and Scarf, 1974): the outcome of TTC is in the core.

Theorem(Roth, 1982): TTC is strategyproof.

Boston Public Schools (BPS)

- So there were two options for Boston:
- DA – Strategyproof, stable, selects student-optimal matching (except for tie-breaking issues)
- TTC – Strategyproof, Pareto efficient for the students
- The most important thing: that the algorithm will be strategyproof. This levels the playing field and allows gathering data about actual preferences over schools.
- The DA algorithm was chosen because it is more transparent and easier to explain to parents.

A bit more on tie-breaking

Student-optimal stable matchings

DA-STB

DA-MTB

Proposition: For any set of strict preferences for students and weak preferences for schools, any matching that can be produced by deferred acceptance with multiple tie-breaking, but not by deferred acceptance with single tie-breaking is not a student-optimal stable matching.

A bit more on tie-breaking

Example:

There are three schools and three students .

Three stable matchings from student-proposing DA with different tie-breaking rules:

Note that while all are stable, is not student-optimal, because dominates .

Stable improvement cycles

- Based on Erdil and Ergin(AER, 2008)
- Given a stable matching , strict preferences for students and priorities for the schools, a stable improvement cycleconsists of students such that:
- (every student is matched to a school)
- (every student prefers the school the next student is currently allocated)
- , where the argmax is taken with respect to school ’s priorities.
- Given a stable improvement cycle create a new matching:

Proposition: is stable and it (weakly) Pareto dominates .

Stable improvement cycles

Theorem: Fix the preferences and priorities, and let be a stable matching. If is (weakly) Pareto dominated by another stable matching, then admits a stable improvement cycle.

Corollary: In order to find a student-optimal stable matching, we can run deferred acceptance, and then find and implement stable improvement cycles until none are left.

Stable improvement cycles

Theorem (Abdulkadiroglu, Pathakand Roth, 2008): For any tie breaking rule , there is no mechanism that is strategy-proof and dominates .

Furthermore, when considering stable improvement cycles, it is kind of clear what kind of manipulations might be profitable. It is worthwhile to list schools that are over-demanded and in which you might have priority in order to replace them with people who have priority in other schools that you actually want.

Stable improvement cycles

Example (Azavedo and Leshno, 2010):

Four students, and two schools with quotas and .

Assume utility from first choice is , from staying single is , and that and .

With DA-STB with random tie-breaking the equilibrium is truthful revelation, and allocation is

Stable improvement cycles

If, however, both and report the preference then the DA-STB allocation is

and the unique Pareto efficient assignment (with respect to reported preferences) that dominates DA-STB is

And this is equilibrium.

Corollary: Consider any mechanism that is Pareto efficient with respect to reported preferences, and Pareto dominates DA-STB. In the economy described, this mechanism has a unique equilibrium assignment which is Pareto dominated by the DA-STB assignment, and is unstable with respect to the true preferences.

Signaling

- Based on Coles, Kushnir and Niederle (AEJ: Micro, forthcoming).
- Two types of signaling:
- Quality signaling
- Preference signaling
- Examples of relevant markets:
- Job market for new Ph.D. economists (Coles et al., 2010)
- Informal preference signaling (Roth and Xing, 1994)
- Internet dating markets (Lee and Niederle, 2011)
- College admissions (Avery and Levin, 2009).

Signaling

- Simple example:
- Two firms and two workers
- Preference on matching with other side are drawn at random (each order with probability ).
- Utilities: top choice , second choice , unmatched .
- One offer from each firm, then workers choose among offers.
- Without signals (eq. with anonymous strategies): each firm offers to best worker, and gets it with probability , , .
- Now suppose workers simultaneously send signals before firms offer positions: each worker sends to better firm, firm proposes to a worker who is going to accept, matching probability is , , .
- Note that there are also other equilibria.

Signaling

- In congested markets, a signaling mechanism:
- Increases the expected number of matches
- Increases the welfare of workers
- Ambiguously changes the welfare of firms
- The value of signaling mechanism:
- is high for balanced markets
- decreases when there are additional periods of interaction

More on the AEA job market

- Why is it necessary in this market? (schools not reading all applications, students submit a lot of enthusiastic applications, truncation from the top and coordination problem)
- How many signals to provide?
- Signaling mechanism is open until December
- Signals are scarce, credible and equitable
- Where should people send signals:
- Candidate 1 already has already attracted some interest from very competitive universities…
- Candidate 2 is a strong but not flashy candidate…
- Candidate 3 has an unusual background… departments don’t normally recruit from his university…

More on the AEA job market

- Where are signals effective?
- Liberal arts colleges
- Departments in towns (population < 50,000)
- “Unranked” schools (including internationals!)
- For non-current PhDs
- Departments that don’t receive many signals

Extensions

Roommate problems, multi-sided matching

Many-to-one with discrete money and substitutable preferences (Crawford and Knoer, 1981; Kelso and Crawford, 1982)

Many-to-many with responsive preferences (Roth, 1984)

Matching with contracts (Hatfield and Milgrom, 2005)

Many-to-many matching with contracts (Echenique and Oveido, 2006)

Matching in supply chains (Ostrovsky, 2008)

Matching in networks with bilateral contracts (Hatfield, Kominers, Nichifor, Ostrovsky and Westkamp, working paper)

Matching with minimum quotas, regional caps, etc. (Biro, Fleiner, Irving and Manlove, 2010, Kamada and Kojima, 2013)

Related topics

Roth and VandeVate (1990) – Random paths to stability

Jackson and Watts (2002)

Ausubel and Milgrom (2000) on package bidding

Road example

- 50 people want to get from A to B
- There are two roads, each one has two segments. One takes an hour, and the other one takes the number of people on it

1 hour

N minutes

A

B

1 hour

N minutes

Nash in road example

- In the Nash equilibrium, 25 people would take each route, for a travel time of 85 minutes

1 hour

N minutes

A

B

1 hour

N minutes

Braess’ paradox

- Now suppose someone adds an extra road which takes no time at all. Travel time goes to 100 minutes

1 hour

N minutes

A

B

Free

1 hour

N minutes

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