An Experimental Test of House Matching Algorithms

1
An Experimental Test of House Matching Algorithms

• Onur Kesten
• Carnegie Mellon University
• Pablo Guillen
• University of Sydney

2
Mechanism Design Overview

• FCC spectrum auctions (McMillan (1994), Cramton
  (1995), McAfee McMillan (1996), Milgrom (2000)
  )
• NRMP (Roth (2002), Roth Peranson (1999))
• School choice (Abdulkadiroglu Sonmez (2003),
  Chen Sonmez (2004), Abdulkadiroglu, Sonmez,
  Pathak, Roth (2005), Kesten (2005))
• House allocation Chen Sonmez (2002)
• Kidney exchange (Roth, Sonmez, Unver (2004,
  2005), Sonmez Unver (2006))

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House allocation with existing tenants

• Problem components
• - newcomers
• - existing tenants
• - priority order
• Main application Graduate housing
• Examples Michigan, Princeton, Rochester,
  Stanford, CMU, MIT, etc.

4
Outline of the Talk

• Model
• Real-life Mechanisms
• 1. Random serial dictatorship with squatting
  rights
• 2. MIT-NH4
• A mechanism from recent theory
• 3. Top trading cycles mechanism
• Main result

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The Model

• Agents I1, 2,, n
• - Existing tenants IE
• - Newcomers IN
• Houses Hh1, h2,, hm
• - Occupied houses IO
• - Vacant houses IV
• A list of strict preferences R(Ri)iI
• A priority order f1,,n -gt I

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• A house allocation problem is a pair consisting
  of
• List of agents preferences (R)
• A priority order (f)
• An allocation is a list s.t.
• every agent is assigned at most one house
• no house is assigned to more than one agent

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What is a mechanism?

Allocations
µ1
(R, f)
Mechanism
µ2
(R, f)
µ3
(R, f)
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What is a good mechanism?

• 1. Individual rationality (existing tenants)
• 2. Fairness (priority order)
• 3. Efficiency (e.g. Pareto)
• 4. Incentive compatibility (no gaming)

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Properties of Mechanisms

• 1. Individual Rationality No existing tenant is
  assigned a house which is worse for him than his
  current house.

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Properties of Mechanisms

• 2. Fairness An agent prefers someone elses
  assignment (to his own) only if either of the
  following holds
• The other agent is an existing tenant who is
  assigned his own house
• The other agent has higher priority

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Properties of Mechanisms

• 3. Pareto Efficiency It is not possible to find
  an alternative allocation that makes
• All agents at least as well off
• At least one agent strictly better off
• However, an inefficient mechanism need not always
  select inefficient outcomes!!!

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Properties of Mechanisms

• 4. Strategy-proofness (Incentive compatibility)
• It is always a dominant strategy for each agent
  to truthfully reveal his preferences.

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Trade-offs between properties

• Proposition 1 There is no mechanism which is
  individually rational, fair, and Pareto
  efficient.

Individually rational
Fair
Strategy-proof
Pareto efficient
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Real-life Mechanisms

• 1. Random serial dictatorship with squatting
  rights
• (CMU, Duke, Harvard,
  Northwestern, Upenn, etc. )
• Each existing tenant initially decides whether to
  participate or not. If participates, gives up his
  current house
• A priority ordering f of participants is randomly
  chosen
• First agent (according to f) is assigned his
  favorite house, second agent is assigned his
  favorite house among the remaining houses, and so
  on.

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Random serial dictatorship with squatting rights

• Properties
• 1. Individual rationality
• 2. Fairness
• 3. Pareto efficiency
• 4. Incentive compatibility

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Real-life Mechanisms

• 2. MIT-NH4 Mechanism
• 1. The first agent is tentatively assigned
  his top choice among all houses, the next agent
  is tentatively assigned his top choice among the
  remaining houses, and so on, until a squatting
  conflict occurs.
• 2. A squatting conflict occurs if it is the turn
  of an existing tenant but every remaining house
  is worse than his current house. That means
  someone else, the conflicting agent, is
  tentatively assigned the existing tenant's
  current house. When this happens, solve the
  squatting conflict as follows
• Assign the existing tenant his current house and
  remove him
• Erase all tentative assignments starting after
  the conflicting agent
• 3. The process is over when there are no
  houses or agents left.

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MIT-NH4 Mechanism

• Proposition 2
• 1. Individual rationality
• 2. Fairness
• 3. Pareto efficiency
• 4. Incentive compatibility

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The best fair and individually rational mechanism

• Corollary The MIT-NH4 mechanism Pareto dominates
  any other fair and individually rational
  mechanism.

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A mechanism from recent theory

• 3. Top Trading Cycles Mechanism (Abdulkadiroglu
  Sonmez)
• Assign the first agent (according to f) his top
  choice, the second agent his top choice among the
  remaining houses, and son on, until someone
  demands the house of an existing tenant.
• If at that point the existing tenant whose house
  is demanded is already assigned a house, then do
  not disturb the procedure.
• Otherwise insert him to the top and proceed.
  Similarly, insert any existing tenant who is not
  already served at the top of the line once his or
  her house is demanded.
• If at any point, a loop forms, (it is formed by
  exclusively existing tenants and each of them
  demands the house of the tenant next in the
  loop), remove all agents in the loop by assigning
  them the houses they demand, and proceed.

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Top Trading Cycles Mechanism

• Properties
• 1. Individual rationality
• 2. Fairness
• 3. Pareto efficiency
• 4. Incentive compatibility

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SUMMARY
Individually rational
Fair
MIT-NH4
RSDwSR
TTC
Strategy-proof
Pareto efficient
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TTC vs. RSDwSR An interesting experiment

• Chen Sonmez (2002) find that
• TCC is significantly more efficient than RSDwSR
• Basically, because existing tenants decide to
  participate in TTC more often than in RSDwSR
• There is no significant difference in
  truthtelling between TTC and RSDwSR

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Our Experiment Which is better? TTC or MIT-NH4
Individually rational
Fair
MIT-NH4
TTC
Strategy-proof
Strategy-proof
Pareto efficient
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TTC vs. NH4 Experimental design

• Two treatments, 5 groups in each treatment, 12
  agents per group (8 existing tenants and 4
  newcomers)
• Existing tenants first decide whether to
  participate or not
• Then subjects report their preferences. One shot
  game
• The priority order is randomly determined,
  allocation computed and subjects paid

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TTC vs. MIT-NH4 An (even more) interesting
experiment

• We find that
• In the lab, NH4 is equally or more efficient than
  TTC
• Basically, because existing tenants decide to
  participate in NH4 more often than in TTC
• There is no significant difference in
  truthtelling between NH4 and TTC

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Our main result
Individually rational
Fair
MIT-NH4
TTC
Strategy-proof
Pareto efficient
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Thank you