A Matching Model with Friction and Multiple Criteria

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

Today

Idea
1

Model
2

The Interview and Oﬀer/Acceptance Subgames
3
Quasi Symmetric Game

Example
4


Idea


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.


What we do

A set of criteria based on the concept of a subgame perfect Nash
equilibrium is used to deﬁne the solution of this particular game.


What we do

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.


Assumptions

Attractiveness is easy to measure and observable with certainty.
BUT to observe the character of an individual, an interview (or
courtship) is required.


Assumptions

Hence, on observing the attractiveness of a prospective partner an
individual must decide whether he/she wishes to proceed to the
interview stage.


Assumptions

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.


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.


Story

job seeker.
If either the job searcher does not apply or the employer does not
wish to interview, the two individuals carry on searching.


Story

job seeker.
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.


Story

job seeker.
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.


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.


Model

The type of an individual can be deﬁned by their attractiveness and
character, together with their role (employer or job seeker).


Model

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 ].


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.


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.


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:


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.


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.


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)]


The Application/Invitation Subgame
d
d
d
d Seeker: a
Job
Job Seeker: n
d
©
‚
d
e
[Rjs (xjs ; π), Rem (xem ; π)] e
e
Employer: r eEmployer: i
e
©
…
e
[Rjs (xjs ; π) − c3,js , Rem (xem ; π)] v(xjs , xem ; π) − (c2,js + c3,js , c2,em )
Fig. 1: Extensive form of the application/invitation game.


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 .


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
conditions 1-4 job seekers of maximum attractiveness apply to employers
of attractiveness above a certain threshold.

Theorem
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).

Algorithm


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.


Example
{1, 2, 3, 4, 5, 6, 7}.
7
Consider employers of maximum attractiveness.


Example
{1, 2, 3, 4, 5, 6, 7}.
7
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.


Example
{1, 2, 3, 4, 5, 6, 7}.
7
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.

Payoﬀs

1 1
R([7, 4]; π1 ) = 7 − 49 × − 7 × = −1
7 7
19 49 171 11
− ×−×=
R([7, 4]; π2 ) =
3 3 737 3
29 49 171 21
− ×−×=.
R([7, 4]; π3 ) =
5 5 757 5


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


Further Work

Diﬀerent information paths within search processes
Make interviewing costs independent


Further Work

Diﬀerent information paths within search processes
Make interviewing costs independent
Non uniform distributions of character—superstars/Susan Boyle.


A Matching Model with Friction and Multiple Criteria

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