A Unified Model of School Choice

1
A Unified Model of School Choice

• Presentation by Trevor Burnham
• University of Michigan School of Information
• 11/6/08

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School matching mechanismsA primer

• Students submit ordered preference lists over
  schools.
• Schools submit ordered preference lists over
  students.
• Matching mechanisms are magical contraptions that
  turn these lists into student-school assignments.
• For example, the Gale-Shapley Deferred
  Acceptance (DA) mechanism is widely used.
• DA has nice properties most notably, its
  envy-free, and truth-telling is a dominant
  strategy.

3
Why design a new mechanism?

• DA would be awesome if we lived in a socialist
  utopia.
• We dont.
• So some parents send their children to private
  schools.
• Problem They have to make that decision before
  they know the results of the match. This results
  in partial unraveling, thinning the match.
• Even if application deadlines were pushed back,
  the match would operate without all available
  information.

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How do we add money to the school choice model?

• We modify the Kelso-Crawford labor model (1982).
• The model is very flexible, allowing workers to
  weigh non-monetary factors in their preferences
  and allowing firms to have preferences over sets
  of workers, since workers may be complementary.
• Replace workers with students and firms with
  schools.
• One key difference Public schools arent profit
  maximizers. This complicates the model slightly.
• (Refer to your handout for notation and details.)

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Do we still get nice properties?

• The results from Kelso and Crawfords original
  paper still apply Specifically, the allocation
  produced by their salary-adjustment process
  converges to an individually rational allocation
  in finite time, and the resulting allocation is
  in the core (i.e. no coalition of students and a
  school can improve on the allocation).
• But unlike DA, truth-telling is not dominant! The
  salary-adjustment process is essentially a
  first-price auction.
• Its often advantageous to underbid for private
  schools, and to overbid for your favorite public
  school.

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Theory collides with reality

• Its probably not practical for schools to submit
  cardinal preferences for every possible
  combination of students.
• Nor can students be expected to accurately
  describe the maximum theyre willing to pay to
  attend each school.
• Can we simplify this mechanism to perform an
  equally desirable match with less information?
  ()
• And is there any way to make truth-telling a
  dominant strategy, perhaps through a Vickrey-like
  mechanism? ()
• () Possible areas for future research!

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Why let reality get in the way?

• Knowing that a core allocation for this type of
  market exists (under certain technical
  assumptions that I wont go into today) is useful
  as a basis for comparison.
• For example, Gale-Shapley DA performed after
  private school deadlines clearly produces
  allocations that are not in the core when private
  schools are considered. But what about a
  multi-phase match?
• And what about vouchers?

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Conclusion and new directions

• A larger question What should the role of public
  schools be? If we reject the radical egalitarian
  notion that education should not have a price,
  then wouldnt it be preferable to replace public
  schools with vouchers?
• Maybe, but if there a barriers to entry (e.g.
  geography), a public/private mix may be most
  efficient. And if thats the case, then we need a
  mixed matching mechanism.
• But students and schools preferences are
  complicated, so is there a practical way to
  approximate those preferences? Thats the
  million-dollar question.