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Matching Process

The key to understanding the matching process is the concept of 'tentative acceptance'. The process is an 'applicant-proposing' one; this means that it is driven by the applicants' requests for placement. At first, all acceptances are tentative. As programs receive more requests, they will retain only those applicants they consider most desirable.

The Matching Program's role is that of an intermediary, executing binding instructions. The outcome is pre-determined by the rank lists of programs and applicants. At no point can this outcome be modified by a judgment or bias on the part of the intermediary, nor is there any element of chance.

In the following example, ten applicants (1-10) compete for eight positions offered by four programs (A-D, offering 3, 2, 2 and 1 positions respectively).

Initially, each applicant requests placement in his/her preferred program.

  • A receives requests from 2, 6 and 9; it tentatively accepts all three.
  • B receives requests from 4 and 10; it accepts 4, but did not list 10 as acceptable.
  • C receives requests from 1, 3 and 7; it accepts 1 and 7, but rejects 3, since it has only two positions.
  • D receives requests from 5 and 8; D is not interested in 5 and accepts 8, even though it is its last choice.

PROGRAMS

APPLICANTS

3

2

2

1

<<< Number of positions available

A

B

C

D

1

2

3

4

5

6

7

8

9

10

(6)

7

4

1

(C)

(A)

C

(B)

D

(A)

(C)

(D)

(A)

B

10

(4)

(1)

4

A

D

B

A

C

C

B

A

D

D

4

6

6

7

D

B

A

D

A

A

B

C

A

3

1

(7)

10

B

C

D

C

D

C

C

1

5

10

6

8

3

3

9

7

9

(8)

(9)

2

5

(2)

10

5

<<< These applicants were not listed.

Legend: (...) tentative acceptance; --- rejected request

In the next round, the rejected applicants (3, 5 and 10) request placement at their next choice. Most of these requests will create a cascade of further reactions and requests.

  • 3 requests B; B accepts.
  • 5 requests C, but C is not interested. 5 now requests A; A considers this an improvement and rejects 2. 2 now requests D, but D is not interested, nor are B and C. 2's list is now exhausted, so 2 will remain unmatched.
  • 10 requests D; this is an improvement for D, so D rejects 8. 8 now requests A; A moves up again and rejects 5. Since 5's list is now exhausted, 5 remains unmatched also.

PROGRAMS

APPLICANTS

3

2

2

1

<<< Number of positions available

A

B

C

D

1

2

3

4

5

6

7

8

9

10

[6]

7

4

1

[C]

(A)

C

[B]

D

[A]

[C]

(D)

[A]

B

10

[4]

[1]

4

A

D

[B]

A

C

C

B

[A]

D

[D]

4

6

6

7

D

B

A

D

A

A

B

C

A

3

1

[7]

[10]

B

C

D

C

D

C

C

1

5

10

6

[8]

[3]

3

9

7

9

(8)

[9]

2

(5)

(2)

2

2

10

5

5

<<< These applicants were not listed.

Legend: (...) tentative acceptance; --- rejected request

At this point the matching process ends, with 2 and 5 remaining unmatched. That at least two applicants would remain unmatched was predictable, since there were ten applicants for eight positions. Note that the matched applicants were always matched to the highest program on their list that was willing to accept them. Programs retained their preferred applicants but usually did not get only their top choices. Applicants who did not match were presented for consideration to all the programs on their list.

 

 

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