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?.
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Thm : consider a random market, with n singles, n couples and more than 20n hospitals.
With constant probability, there is no stable outcome
Hospital Preferences Capacity is 1
Schools and students as strategic players
The concept of justified envy
Stability vs. optimality
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
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.
Theorem (Shapley and Scarf, 1974): the outcome of TTC is in the core.
Theorem(Roth, 1982): TTC is strategyproof.
Student-optimal stable matchings
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.
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 .
Proposition: is stable and it (weakly) Pareto dominates .
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.
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.
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
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.
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)
Roth and VandeVate (1990) – Random paths to stability
Jackson and Watts (2002)
Ausubel and Milgrom (2000) on package bidding