We present a model of matching based on two character measures.
There are two classes of individual. Each individual
observes a sequence of potential partners from the opposite class.
One
measure describes the "attractiveness" of an individual.
Preferences are common according to
this measure: i.e. each individual prefers highly attractive partners and all individuals
of a given class agree as to how attractive individuals of the opposite class are. Preferences are
homotypic with respect to the second measure, referred to as "character" i.e.
all individuals prefer partners of a similar character.
Such a problem may be interpreted as e.g. a job search problem in which the classes
are employer and employee, or a mate choice problem in which the classes are male and
female.
It is assumed that
attractiveness is easy to measure and observable with certainty. However,
in order to observe the character of an individual, an interview (or courtship) is required.
Hence, on observing the attractiveness of a prospective partner an individual must decide whether he/she wishes
to proceed to the interview stage. Interviews only occur by mutual consent. A pair can only be formed
after an interview. During the interview phase the prospective pair
observe each other's
character, and then decide whether they wish to form a pair.
It is assumed that mutual acceptance is required for pair formation to
occur. An individual stops searching on finding a partner.
This paper
presents a general model of such a matching process. A particular case is
considered in which character "forms a ring" and has a uniform distribution.
A set of criteria based on the concept of a subgame
perfect Nash equilibrium is used to define the solution of this particular game. It is shown that
such a solution is unique. The general form of the solution is derived and a procedure for finding
the solution of such a game is given.
A Matching Model with Friction and Multiple Criteria
1. A Model of Matching with Friction and Multiple
Criteria
David M. Ramsey Stephen Kinsella
University of Limerick
{stephen.kinsella, david.ramsey}@ul.ie
April 25, 2009
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 1 / 21
2. Today
Idea
1
Model
2
The Interview and Offer/Acceptance Subgames
3
Quasi Symmetric Game
Example
4
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 2 / 21
3. Idea
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 3 / 21
4. What we do
This paper presents a general model of matching processes (job
search, speed dating).
A particular case is considered in which character “forms a ring” and
has a uniform distribution.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21
5. What we do
This paper presents a general model of matching processes (job
search, speed dating).
A particular case is considered in which character “forms a ring” and
has a uniform distribution.
A set of criteria based on the concept of a subgame perfect Nash
equilibrium is used to define the solution of this particular game.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21
6. What we do
This paper presents a general model of matching processes (job
search, speed dating).
A particular case is considered in which character “forms a ring” and
has a uniform distribution.
A set of criteria based on the concept of a subgame perfect Nash
equilibrium is used to define the solution of this particular game.
It is shown that such a solution is unique. The general form of the
solution is derived, and a procedure for finding the solution of such a
game is given.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21
7. Assumptions
Attractiveness is easy to measure and observable with certainty.
BUT to observe the character of an individual, an interview (or
courtship) is required.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21
8. Assumptions
Attractiveness is easy to measure and observable with certainty.
BUT to observe the character of an individual, an interview (or
courtship) is required.
Hence, on observing the attractiveness of a prospective partner an
individual must decide whether he/she wishes to proceed to the
interview stage.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21
9. Assumptions
Attractiveness is easy to measure and observable with certainty.
BUT to observe the character of an individual, an interview (or
courtship) is required.
Hence, on observing the attractiveness of a prospective partner an
individual must decide whether he/she wishes to proceed to the
interview stage.
Interviews only occur by mutual consent. A pair can only be formed
after an interview. During the interview phase the prospective pair
observe each other’s character, and then decide whether they wish to
form a pair.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21
10. Story
A job seeker first must decide whether to apply for a job or not on
the basis of the job advert (the attractiveness of the job).
If the job seeker applies, the employer must then decide whether to
proceed with an interview or not, based on the qualifications of the
job seeker.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21
11. Story
A job seeker first must decide whether to apply for a job or not on
the basis of the job advert (the attractiveness of the job).
If the job seeker applies, the employer must then decide whether to
proceed with an interview or not, based on the qualifications of the
job seeker.
If either the job searcher does not apply or the employer does not
wish to interview, the two individuals carry on searching.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21
12. Story
A job seeker first must decide whether to apply for a job or not on
the basis of the job advert (the attractiveness of the job).
If the job seeker applies, the employer must then decide whether to
proceed with an interview or not, based on the qualifications of the
job seeker.
If either the job searcher does not apply or the employer does not
wish to interview, the two individuals carry on searching.
During an interview, an employee observes the character of his
prospective employer, and vice versa. After the interview finishes,
both parties must decide whether to accept the other as a partner or
not.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21
13. Story
A job seeker first must decide whether to apply for a job or not on
the basis of the job advert (the attractiveness of the job).
If the job seeker applies, the employer must then decide whether to
proceed with an interview or not, based on the qualifications of the
job seeker.
If either the job searcher does not apply or the employer does not
wish to interview, the two individuals carry on searching.
During an interview, an employee observes the character of his
prospective employer, and vice versa. After the interview finishes,
both parties must decide whether to accept the other as a partner or
not.
If acceptance is mutual, then a job pair is formed. Otherwise, both
individuals continue searching.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21
14. Model
We consider a steady state model in which the distributions of the
attractiveness (qualifications) and character of a jobseeker, as well as
of the attractiveness and character of an employer (X1,js , X2,js , X1,em
and X2,em , do not change over time.
Suppose X1,es , X1,js , X2,es and X2,js are discrete random variables.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21
15. Model
We consider a steady state model in which the distributions of the
attractiveness (qualifications) and character of a jobseeker, as well as
of the attractiveness and character of an employer (X1,js , X2,js , X1,em
and X2,em , do not change over time.
Suppose X1,es , X1,js , X2,es and X2,js are discrete random variables.
The type of an individual can be defined by their attractiveness and
character, together with their role (employer or job seeker).
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21
16. Model
We consider a steady state model in which the distributions of the
attractiveness (qualifications) and character of a jobseeker, as well as
of the attractiveness and character of an employer (X1,js , X2,js , X1,em
and X2,em , do not change over time.
Suppose X1,es , X1,js , X2,es and X2,js are discrete random variables.
The type of an individual can be defined by their attractiveness and
character, together with their role (employer or job seeker).
The type of a job seeker will be denoted xjs = [x1,js , x2,js ]. The type
of an employer will be denoted xem = [x1,em , x2,em ].
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21
17. Model
A job seeker’s total reward from search is assumed to be the reward gained
from the job taken minus the total search costs incurred. Hence, the total
reward from search of a job seeker of type xjs from taking a job with an
employer of type xem after searching for n1 moments, attending n2
interviews and applying for n3 jobs is given by
g (x2,js , xem ) − n1 c1,js − n2 c2,js − n3 c3,js .
Similarly, the total reward from search of a employer of type xem from
employing a job seeker of type xjs after searching for k1 moments and
interviewing k2 job seekers is given by
h(x2,em , xjs ) − k1 c1,em − k2 c2,em .
π is the strategy profile used in the job search game.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 8 / 21
18. Modeling Strategy
The game played by a job seeker and employer on meeting can be split into
two subgames. The first will be referred to as the application/invitation
subgame, in which the pair decide whether to proceed to an interview or
not. The second subgame is called the interview game and at this stage
both parties must decide whether to accept the other or not.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 9 / 21
19. Conditions for a Solution to the Game
We look for a Nash equilibrium profile π ∗ of Γ. When the population play
according to the strategy profile π ∗ , then no individual can gain by using a
different strategy to the one defined by π ∗ . We look for a Nash equilibrium
strategy profile π N of Γ that satisfies the following additional conditions:
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 10 / 21
20. Conditions
Condition 1 In the interview game, a job seeker accepts a prospective job
(respectively, an employer offers a position to a job seeker) if
and only if the reward from such a pairing is at least as great
as the expected reward from future search.
Condition 2 An employer only invites for interview if her expected reward
from the resulting interview subgame minus the costs of
interviewing is as least as great as her expected reward from
future search.
Condition 3 A job seeker only applies for a job if his expected reward
from applying minus the costs of applying for the job is at
least as great as his future expected reward from search.
Condition 4 The decisions made by an individual do not depend on the
moment at which the decision is made.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 11 / 21
21. Conditions
Condition 5 In the application/invitation subgame, an employer of type
xem is willing to interview any job seeker of qualifications not
lower than required level of qualifications, denoted t(xem ).
Condition 6 Suppose two employers have the same character, then the
most attractive one will be at least as choosy as the other
when inviting candidates for interview.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 12 / 21
22. The Interview Subgame
Suppose the job seeker is of type xjs and the employer is of type xem . The
payoff matrix is given by
Employer: a Employer: r
Job Seeker: a [g (x2,js , xem ), h(x2,em , xjs )] [Rjs (xjs ; π), Rem (xem ; π)]
Job Seeker: r [Rjs (xjs ; π), Rem (xem ; π)] [Rjs (xjs ; π), Rem (xem ; π)]
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 13 / 21
24. Quasi Symmetric Formulation of Game
1 The distributions of character and attractiveness are
independent of class. Furthermore, the distribution of
character is uniform on 0, 1, 2, . . . , r − 1.
2 The character levels are assumed to form a ring, i.e. 0 is a
neighbour of both 1 and r − 1. The difference between
characters i and j is defined to be the difference between i
and j according to mod(r ) arithmetic. Precisely, if i ≥ j,
then |i − j| = min{i − j, r + j − i}.
3 The rewards obtained from a pairing are symmetric with
respect to class, i.e g (x2 , [y1 , y2 ]) = h(y2 , [y1 , x2 ]).
4 The cost of applying for a job, c3,js , is equal to zero,
c1,js = c1,em and c2,js = c2,em .
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 15 / 21
25. Theorems
Theorem
At a symmetric equilibrium π ∗ of a quasi-symmetric game satisfying
conditions 1-4 the reward of an individual is non-decreasing in
attractiveness.
Theorem
At a symmetric equilibrium π ∗ of a quasi-symmetric game satisfying
conditions 1-4 job seekers of maximum attractiveness apply to employers
of attractiveness above a certain threshold.
Theorem
At a symmetric equilibrium π ∗ of a quasi-symmetric game satisfying
conditions 1-4, employers of attractiveness i are prepared to interview job
seekers of attractiveness in [k1 (i), k2 (i)], where k2 (i) is the maximum
attractiveness of an job seeker who applies to an employer of
attractiveness i for interview. In addition, k1 (i) and k2 (i) are
non-decreasing in i and k1 (i) ≤ i ≤ k2 (i).
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 16 / 21
27. Example
Suppose there are seven levels of both attractiveness and character,
i.e. the support of each of X1,em , X2,em , X1,js and X2,js is
{1, 2, 3, 4, 5, 6, 7}.
Both the search costs, c1 , and the interview costs, c2 are equal to 1 .
7
The reward obtained from a partnership is defined to be the
attractiveness of the partner minus the difference (modulo 7) between
the characters of the pair.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 18 / 21
28. Example
Suppose there are seven levels of both attractiveness and character,
i.e. the support of each of X1,em , X2,em , X1,js and X2,js is
{1, 2, 3, 4, 5, 6, 7}.
Both the search costs, c1 , and the interview costs, c2 are equal to 1 .
7
The reward obtained from a partnership is defined to be the
attractiveness of the partner minus the difference (modulo 7) between
the characters of the pair.
Consider employers of maximum attractiveness.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 18 / 21
29. Example
Suppose there are seven levels of both attractiveness and character,
i.e. the support of each of X1,em , X2,em , X1,js and X2,js is
{1, 2, 3, 4, 5, 6, 7}.
Both the search costs, c1 , and the interview costs, c2 are equal to 1 .
7
The reward obtained from a partnership is defined to be the
attractiveness of the partner minus the difference (modulo 7) between
the characters of the pair.
Consider employers of maximum attractiveness.
The ordered preferences of a [7, 4] individual are as follows: first
(group one) - [7, 4], second equal (group two) - [7, 3], [7, 5], fourth
equal (group 3) [7, 2], [7, 6] and sixth equal (group 4) - [7, 1], [7, 7].
Group 1, 2, 3 and 4 partners give a reward from pairing of 7, 6, 5 and
4 respectively.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 18 / 21
30. Example
Suppose there are seven levels of both attractiveness and character,
i.e. the support of each of X1,em , X2,em , X1,js and X2,js is
{1, 2, 3, 4, 5, 6, 7}.
Both the search costs, c1 , and the interview costs, c2 are equal to 1 .
7
The reward obtained from a partnership is defined to be the
attractiveness of the partner minus the difference (modulo 7) between
the characters of the pair.
Consider employers of maximum attractiveness.
The ordered preferences of a [7, 4] individual are as follows: first
(group one) - [7, 4], second equal (group two) - [7, 3], [7, 5], fourth
equal (group 3) [7, 2], [7, 6] and sixth equal (group 4) - [7, 1], [7, 7].
Group 1, 2, 3 and 4 partners give a reward from pairing of 7, 6, 5 and
4 respectively.
Let πi denote any strategy profile in which [7, 4] employers pair with
job seekers from the first i groups described above, i = 1, 2, 3, 4.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 18 / 21
32. Equilibrium Strategy Profile
Attractiveness Attractiveness levels invited Expected Reward
{ 6,7 }
7 4.50
{ 6,7 }
6 4.33
{ 4,5,6,7 }
5 2.50
{ 4,5 }
4 2.33
{ 2,3,4,5 }
3 0.50
{ 2,3 }
2 0.33
{ 1,2,3 }
1 -1.80
Table: Brief description of symmetric equilibrium for the example considered
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 20 / 21
33. Further Work
Different information paths within search processes
Make interviewing costs independent
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 21 / 21
34. Further Work
Different information paths within search processes
Make interviewing costs independent
Non uniform distributions of character—superstars/Susan Boyle.
Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 21 / 21