Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies Benjam´ın Tello Banco de M´ exico
March 16, 2017
1 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Source: http://www.nrmp.org
2 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Roth (2002), Table 1
3 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Centralized matching via DA In a matching market that employs DA
each student submits a preference list over individual hospitals each hospital submits a preference list over individual students DA produces a stable matching with respect to submitted preferences
4 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Centralized matching via DA In a matching market that employs DA
each student submits a preference list over individual hospitals each hospital submits a preference list over individual students DA produces a stable matching with respect to submitted preferences Advantage: No student can’t benefit from misrepresenting his preferences (Dubins and Freedman, 1981).
4 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Centralized matching via DA In a matching market that employs DA
each student submits a preference list over individual hospitals each hospital submits a preference list over individual students DA produces a stable matching with respect to submitted preferences Advantage: No student can’t benefit from misrepresenting his preferences (Dubins and Freedman, 1981). Problem: Hospitals do not always have incentives to report their true preferences (Roth and Sotomayor, 1990).
4 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Centralized matching via DA In a matching market that employs DA
each student submits a preference list over individual hospitals each hospital submits a preference list over individual students DA produces a stable matching with respect to submitted preferences Advantage: No student can’t benefit from misrepresenting his preferences (Dubins and Freedman, 1981). Problem: Hospitals do not always have incentives to report their true preferences (Roth and Sotomayor, 1990).Therefore, there is no guarantee that the matching produced by DA is stable under the true preferences. 4 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
This paper
Consider the preference revelation game (for hospitals) induced by DA Roth and Sotomayor (1990): In general, there are unstable Nash equilibrium outcomes.
5 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
This paper
Consider the preference revelation game (for hospitals) induced by DA Roth and Sotomayor (1990): In general, there are unstable Nash equilibrium outcomes. This paper: Acyclicity of the hospitals’ preference profile (Romero-Medina and Triossi, 2013) is necessary and sufficient to ensure that the outcome of any dropping equilibrium is stable.
5 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Model S
H
s1
h1
s2
h2
s3
h3
s4
h4
s5
h5
∅
∅
Results
Conclusion
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Model
Ps1 —– h2
S
H
s1
h1
s2
h2
{s1 , s5 }
h1 h5
s3
h3
s4
h4
h3
s5
h5
{s3 , s5 } {s1 }
h4 ∅
Ph2 , qh2 = 2 —– {s1 , s3 }
∅
∅
{s3 } {s5 } ∅ .. . 6 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Responsiveness
Assumption: Hospitals have responsive preferences over sets of students. A hospital’s preferences are responsive if faced with two sets of students that differ only in one student, the hospital prefers the set of students containing the more preferred student as long as the hospital has unfilled positions, it prefers to fill a position with an “acceptable” student rather than leaving it unfilled
7 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Responsiveness
Assumption: Hospitals have responsive preferences over sets of students. A hospital’s preferences are responsive if faced with two sets of students that differ only in one student, the hospital prefers the set of students containing the more preferred student as long as the hospital has unfilled positions, it prefers to fill a position with an “acceptable” student rather than leaving it unfilled Remark. Under responsiveness, only rankings over individual students matter.
7 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Model
Ps1 —– h2
S
H
s1
h1
s2
h2
s3
h1 h5
s3
h3
s4
h4
s5
h5
∅
∅
h4 ∅ .. .
Ph2 —– s1 s5 ∅ .. . s4
8 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Deferred acceptance
Step 1: Each student s proposes to the hospital that is ranked first in Ps . Each hospital h considers its proposers and tentatively assigns its qh positions to these students following the preferences Ph . All other proposers are rejected. Step k, k ≥ 2: Each student s that is rejected in Step k − 1 proposes to the next hospital in his list Ps . Each hospital h considers the students that were tentatively assigned a position at h in Step k − 1 together with its new proposers. Hospital h tentatively assigns its qh positions to these students following the preferences Ph . All other proposers are rejected.
9 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) Table: True preferences Hospitals
Students
Ph1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
P s1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
10 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) Table: True preferences Hospitals
Students
Ph1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
P s1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under true preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1
10 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) Table: True preferences Hospitals
Students
Ph1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
P s1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under true preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1
End of the algorithm. Output: µ(h1 ) = {s3 , s4 }, µ(h2 ) = {s2 }, µ(h3 ) = {s1 }. 10 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
11 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under submitted preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1 , h1 rejects s3
11 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under submitted preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1 , h1 rejects s3 Step 2: s3 proposes to h3 , h3 accepts s3 and rejects s1
11 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under submitted preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1 , h1 rejects s3 Step 2: s3 proposes to h3 , h3 accepts s3 and rejects s1 Step 3: s1 proposes to h1 , h1 accepts s1
11 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
DA under submitted preferences Step 1: s1 proposes to h3 , s2 proposes to h2 , s3 proposes to h1 , s4 proposes to h1 , h1 rejects s3 Step 2: s3 proposes to h3 , h3 accepts s3 and rejects s1 Step 3: s1 proposes to h1 , h1 accepts s1
End of the algorithm. Output: µ0 (h1 ) = {s1 , s4 }, µ0 (h2 ) = {s2 }, µ0 (h3 ) = {s3 }. 11 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Example (Roth and Sotomayor, 1990) cont. Table: Submitted preferences Hospitals
Students
Ph0 1 , qh1 = 2
Ph2 , qh2 = 1
Ph3 , qh3 = 1
Ps1
Ps2
Ps3
Ps4
s1
s1
s3
h3
h2
h1
h1
s2
s2
s1
h1
h1
h3
h2
s3
s3
s2
h1
h1
h3
h2
s4
s4
s4
The profile (Ph0 1 , Ph2 , Ph3 ) is an equilibrium The matching µ0 (in red) is unstable: (h1 , s3 ) is a blocking pair
12 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Acyclicity (Romero-Medina and Triossi, 2013) H = {h1 , . . . , hn } and S = {s1 , . . . , sm }.
13 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Acyclicity (Romero-Medina and Triossi, 2013) H = {h1 , . . . , hn } and S = {s1 , . . . , sm }. A cycle (of length 4) is given by (Ph1 , Ph2 , Ph3 , Ph4 , s1 , s2 , s3 , s4 ) s2 Ph1
Ph2
s1
s3
Ph4
Ph3 s4
13 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Acyclicity (Romero-Medina and Triossi, 2013) H = {h1 , . . . , hn } and S = {s1 , . . . , sm }. A cycle (of length 4) is given by (Ph1 , Ph2 , Ph3 , Ph4 , s1 , s2 , s3 , s4 ) s2 Ph1
Ph2
s1
s3
Ph4
Ph3 s4
A preference profile is acyclic if it does not contain cycles of any length.
13 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Acyclicity (Romero-Medina and Triossi, 2013) H = {h1 , . . . , hn } and S = {s1 , . . . , sm }
Table: A cycle of length 4 Ph1 .. .
Ph2 .. .
Ph3 .. .
Ph4 .. .
s1 .. .
s2 .. .
s3 .. .
s4 .. .
s2 ...
s3 ...
s4 ...
s1 ...
∅ .. .
∅ .. .
∅ .. .
∅ .. .
A preference profile is acyclic if it does not contain cycles of any length. 14 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Dropping strategies (Kojima and Pathak, 2010)
Table: Dropping strategies for h1 Ph1
Ph0 1
Ph001
Ph0001
Ph0000 1
Ph00000 1
Ph00000 1
s1
s1
s1
s1
s1
s1
s1
s2
s2
s2
s2
s2
s2
s2
s3
s3
s3
s3
s3
s3
s3
s4
s4
s4
s4
s4
s4
s4
15 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Dropping strategies
Dropping strategies are exhaustive (Kojima and Pathak; 2010, Lemma 1), i.e., fixing the other hospitals’ strategies, the match obtained from any strategy can be replicated or improved upon by a dropping strategy.
16 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Dropping equilibria
A dropping equilibria is a Nash equilibrium where each hospital plays a dropping strategy. Remarks Any stable matching can be obtained as the outcome of a dropping equilibrium (Jaramillo, Kayı and Klijn; 2013, Proposition 1). There are Nash equilibrium outcomes that cannot be obtained as the outcome of a dropping equilibrium (Jaramillo, Kayı and Klijn; 2013, Example 1).
17 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Results
Theorem 1 Assume that the hospitals’ preference profile is acyclic. Let Q be a dropping equilibrium. Then, DA(Q) is stable at P.
18 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Results
Theorem 1 Assume that the hospitals’ preference profile is acyclic. Let Q be a dropping equilibrium. Then, DA(Q) is stable at P. Proposition 1 Assume that the hospitals’ preference profile has a cycle. Then, there are preferences Ps for each s ∈ S, and a capacity qh and a dropping strategy Qh for each h ∈ H such that the profile Q is a Nash equilibrium and DA(Q) is not stable.
18 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Remark
The restriction to dropping equilibria is crucial for the result. Example 1 in Jaramillo, Kayı and Klijn (2013) exhibits a market where (i) the hospitals’ preference profile is acyclic (ii) there is an equilibrium in which one hospital does not play a dropping strategy (iii) the equilibrium outcome is unstable
19 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Conclusion
We provide further theoretical evidence as to why markets that employ DA perform well in real-life applications.
20 / 20
Introduction
Model
Example
Acyclicity
Dropping strategies
Results
Conclusion
Conclusion
We provide further theoretical evidence as to why markets that employ DA perform well in real-life applications. Challenge: Find a restriction that ensures the stability of all Nash equilibria outcomes.
20 / 20
