A Matching Model with Friction and Multiple Criteria

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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.

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A Matching Model with Friction and Multiple Criteria

  1. 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. 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. 3. Idea Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 3 / 21
  4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  23. 23. 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. Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 14 / 21
  24. 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. 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
  26. 26. Algorithm Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 17 / 21
  27. 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. 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. 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. 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
  31. 31. 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 Ramsey & Kinsella (University of Limerick) Matching with Friction April 25, 2009 19 / 21
  32. 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. 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. 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

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