A Matching Model with Friction and Multiple Criteria

A Model of Matching with Friction and MultipleCriteria

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

1 Idea

2 Model

3 The Interview and Offer/Acceptance SubgamesQuasi Symmetric Game

4 Example

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 2 / 21

Idea

Figure: caption

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 3 / 21

What we do

This paper presents a general model of matching processes (jobsearch, speed dating).

A particular case is considered in which character “forms a ring” andhas a uniform distribution.

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

It is shown that such a solution is unique. The general form of thesolution is derived, and a procedure for finding the solution of such agame is given.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21

What we do

This paper presents a general model of matching processes (jobsearch, speed dating).

A particular case is considered in which character “forms a ring” andhas a uniform distribution.

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

It is shown that such a solution is unique. The general form of thesolution is derived, and a procedure for finding the solution of such agame is given.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21

What we do

This paper presents a general model of matching processes (jobsearch, speed dating).

A particular case is considered in which character “forms a ring” andhas a uniform distribution.

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

It is shown that such a solution is unique. The general form of thesolution is derived, and a procedure for finding the solution of such agame is given.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 4 / 21

Assumptions

Attractiveness is easy to measure and observable with certainty.

BUT to observe the character of an individual, an interview (orcourtship) is required.

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

Interviews only occur by mutual consent. A pair can only be formedafter an interview. During the interview phase the prospective pairobserve each other’s character, and then decide whether they wish toform a pair.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21

Assumptions

Attractiveness is easy to measure and observable with certainty.

BUT to observe the character of an individual, an interview (orcourtship) is required.

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

Interviews only occur by mutual consent. A pair can only be formedafter an interview. During the interview phase the prospective pairobserve each other’s character, and then decide whether they wish toform a pair.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21

Assumptions

Attractiveness is easy to measure and observable with certainty.

BUT to observe the character of an individual, an interview (orcourtship) is required.

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

Interviews only occur by mutual consent. A pair can only be formedafter an interview. During the interview phase the prospective pairobserve each other’s character, and then decide whether they wish toform a pair.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 5 / 21

Story

A job seeker first must decide whether to apply for a job or not onthe basis of the job advert (the attractiveness of the job).

If the job seeker applies, the employer must then decide whether toproceed with an interview or not, based on the qualifications of thejob seeker.

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

During an interview, an employee observes the character of hisprospective employer, and vice versa. After the interview finishes,both parties must decide whether to accept the other as a partner ornot.

If acceptance is mutual, then a job pair is formed. Otherwise, bothindividuals continue searching.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21

Story

A job seeker first must decide whether to apply for a job or not onthe basis of the job advert (the attractiveness of the job).

If the job seeker applies, the employer must then decide whether toproceed with an interview or not, based on the qualifications of thejob seeker.

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

During an interview, an employee observes the character of hisprospective employer, and vice versa. After the interview finishes,both parties must decide whether to accept the other as a partner ornot.

If acceptance is mutual, then a job pair is formed. Otherwise, bothindividuals continue searching.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21

Story

A job seeker first must decide whether to apply for a job or not onthe basis of the job advert (the attractiveness of the job).

If the job seeker applies, the employer must then decide whether toproceed with an interview or not, based on the qualifications of thejob seeker.

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

During an interview, an employee observes the character of hisprospective employer, and vice versa. After the interview finishes,both parties must decide whether to accept the other as a partner ornot.

If acceptance is mutual, then a job pair is formed. Otherwise, bothindividuals continue searching.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21

Story

A job seeker first must decide whether to apply for a job or not onthe basis of the job advert (the attractiveness of the job).

If the job seeker applies, the employer must then decide whether toproceed with an interview or not, based on the qualifications of thejob seeker.

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

During an interview, an employee observes the character of hisprospective employer, and vice versa. After the interview finishes,both parties must decide whether to accept the other as a partner ornot.

If acceptance is mutual, then a job pair is formed. Otherwise, bothindividuals continue searching.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 6 / 21

Model

We consider a steady state model in which the distributions of theattractiveness (qualifications) and character of a jobseeker, as well asof 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 andcharacter, together with their role (employer or job seeker).

The type of a job seeker will be denoted xjs = [x1,js , x2,js ]. The typeof an employer will be denoted xem = [x1,em, x2,em].

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21

Model

We consider a steady state model in which the distributions of theattractiveness (qualifications) and character of a jobseeker, as well asof 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 andcharacter, together with their role (employer or job seeker).

The type of a job seeker will be denoted xjs = [x1,js , x2,js ]. The typeof an employer will be denoted xem = [x1,em, x2,em].

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21

Model

We consider a steady state model in which the distributions of theattractiveness (qualifications) and character of a jobseeker, as well asof 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 andcharacter, together with their role (employer or job seeker).

The type of a job seeker will be denoted xjs = [x1,js , x2,js ]. The typeof an employer will be denoted xem = [x1,em, x2,em].

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 7 / 21

Model

A job seeker’s total reward from search is assumed to be the reward gainedfrom the job taken minus the total search costs incurred. Hence, the totalreward from search of a job seeker of type xjs from taking a job with anemployer of type xem after searching for n1 moments, attending n2

interviews and applying for n3 jobs is given by

g(x2,js , xem)− n1c1,js − n2c2,js − n3c3,js .

Similarly, the total reward from search of a employer of type xem fromemploying a job seeker of type xjs after searching for k1 moments andinterviewing k2 job seekers is given by

h(x2,em, xjs)− k1c1,em − k2c2,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

Modeling Strategy

The game played by a job seeker and employer on meeting can be split intotwo subgames. The first will be referred to as the application/invitationsubgame, in which the pair decide whether to proceed to an interview ornot. The second subgame is called the interview game and at this stageboth parties must decide whether to accept the other or not.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 9 / 21

Conditions for a Solution to the Game

We look for a Nash equilibrium profile π∗ of Γ. When the population playaccording to the strategy profile π∗, then no individual can gain by using adifferent strategy to the one defined by π∗. We look for a Nash equilibriumstrategy profile πN of Γ that satisfies the following additional conditions:

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 10 / 21

Conditions

Condition 1 In the interview game, a job seeker accepts a prospective job(respectively, an employer offers a position to a job seeker) ifand only if the reward from such a pairing is at least as greatas the expected reward from future search.

Condition 2 An employer only invites for interview if her expected rewardfrom the resulting interview subgame minus the costs ofinterviewing is as least as great as her expected reward fromfuture search.

Condition 3 A job seeker only applies for a job if his expected rewardfrom applying minus the costs of applying for the job is atleast as great as his future expected reward from search.

Condition 4 The decisions made by an individual do not depend on themoment at which the decision is made.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 11 / 21

Conditions

Condition 5 In the application/invitation subgame, an employer of typexem is willing to interview any job seeker of qualifications notlower than required level of qualifications, denoted t(xem).

Condition 6 Suppose two employers have the same character, then themost attractive one will be at least as choosy as the otherwhen inviting candidates for interview.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 12 / 21

The Interview Subgame

Suppose the job seeker is of type xjs and the employer is of type xem. Thepayoff matrix is given by

Employer: a Employer: rJob Seeker: aJob Seeker: r

([g(x2,js , xem), h(x2,em, xjs)] [Rjs(xjs ;π),Rem(xem;π)][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

The Application/Invitation Subgame

��

��

��

@@

@@

@@R

Job Seeker: n Job Seeker: a

[Rjs(xjs ;π),Rem(xem;π)]�

��

��

�

Employer: r

AAAAAAU

Employer: i

[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.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 14 / 21

Quasi Symmetric Formulation of Game

1 The distributions of character and attractiveness areindependent of class. Furthermore, the distribution ofcharacter is uniform on 0, 1, 2, . . . , r − 1.

2 The character levels are assumed to form a ring, i.e. 0 is aneighbour of both 1 and r − 1. The difference betweencharacters i and j is defined to be the difference between iand 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 withrespect 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

Theorems

Theorem

At a symmetric equilibrium π∗ of a quasi-symmetric game satisfyingconditions 1-4 the reward of an individual is non-decreasing inattractiveness.

Theorem

At a symmetric equilibrium π∗ of a quasi-symmetric game satisfyingconditions 1-4 job seekers of maximum attractiveness apply to employersof attractiveness above a certain threshold.

Theorem

At a symmetric equilibrium π∗ of a quasi-symmetric game satisfyingconditions 1-4, employers of attractiveness i are prepared to interview jobseekers of attractiveness in [k1(i), k2(i)], where k2(i) is the maximumattractiveness of an job seeker who applies to an employer ofattractiveness i for interview. In addition, k1(i) and k2(i) arenon-decreasing in i and k1(i) ≤ i ≤ k2(i).

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 16 / 21

Algorithm

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 17 / 21

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 theattractiveness of the partner minus the difference (modulo 7) betweenthe 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], fourthequal (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 and4 respectively.

Let πi denote any strategy profile in which [7, 4] employers pair withjob 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

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 theattractiveness of the partner minus the difference (modulo 7) betweenthe 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], fourthequal (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 and4 respectively.

Let πi denote any strategy profile in which [7, 4] employers pair withjob 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

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 theattractiveness of the partner minus the difference (modulo 7) betweenthe 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], fourthequal (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 and4 respectively.

Let πi denote any strategy profile in which [7, 4] employers pair withjob 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

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 theattractiveness of the partner minus the difference (modulo 7) betweenthe 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], fourthequal (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 and4 respectively.

Let πi denote any strategy profile in which [7, 4] employers pair withjob 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

Payoffs

R([7, 4];π1) = 7− 49× 1

7− 7× 1

7= −1

R([7, 4];π2) =19

3− 49

3× 1

7− 7

3× 1

7=

11

3

R([7, 4];π3) =29

5− 49

5× 1

7− 7

5× 1

7=

21

5.

Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 19 / 21

Equilibrium Strategy Profile

Attractiveness Attractiveness levels invited Expected Reward

7 { 6,7 } 4.50

6 { 6,7 } 4.33

5 { 4,5,6,7 } 2.50

4 { 4,5 } 2.33

3 { 2,3,4,5 } 0.50

2 { 2,3 } 0.33

1 { 1,2,3 } -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

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

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