A Model of Matching and Multiple Criteria Ramsey Et 04.09

A MODEL OF MATCHING WITH FRICTION AND MULTIPLECRITERIA

DAVID M. RAMSEY AND STEPHEN KINSELLA

Abstract. We present a model of matching based on two character measures. Thereare two classes of individual. Each individual observes a sequence of potential partnersfrom the opposite class. One measure describes the ”attractiveness” of an individual.Preferences are common according to this measure: i.e. each individual prefers highlyattractive partners and all individuals of a given class agree as to how attractive indi-viduals of the opposite class are. Preferences are homotypic with respect to the secondmeasure, referred to as ”character” i.e. all individuals prefer partners of a similar char-acter. Such a problem may be interpreted as e.g. a job search problem in which theclasses are employer and employee, or a mate choice problem in which the classes aremale and female. It is assumed that attractiveness is easy to measure and observablewith 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 part-ner 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 thendecide whether they wish to form a pair. It is assumed that mutual acceptance is re-quired 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 isconsidered in which character ”forms a ring” and has a uniform distribution. A set ofcriteria based on the concept of a subgame perfect Nash equilibrium is used to define thesolution of this particular game. It is shown that such a solution is unique. The generalform of the solution is derived and a procedure for finding the solution of such a gameis given.

1. Introduction

This paper presents a model of matching based on two measures. There are two classesof individual, and each individual wishes to form a partnership with an individual ofthe opposite class. Each individual observes a sequence of potential partners. Mutualacceptance is required for a partnership to form. On finding a partner, an individual ceasessearching. One measure describes ”attractiveness”. Preferences are common according tothis measure: i.e. each individual prefers highly attractive partners and all individuals ofa given class agree as to how attractive individuals of the opposite class are. Preferences

Date: April 25, 2009.Key words and phrases. game theory, job search, mate choice, common preferences, homotypic pref-

erences, subgame perfection.1

2 D. M. Ramsey & S. Kinsella

are homotypic with respect to the second measure, referred to as ”character” i.e. allindividuals prefer partners of a similar character.

For convenience, it is assumed that individuals know their own attractiveness and char-acter with certainty. Also, the distribution of character and attractiveness are assumedto be discrete with a finite support, and constant over time. These distributions maydepend on the class of the individual.

Together, the attractiveness, character and the class of an individual determine their”type”. Individuals can observe the attractiveness and character of prospective partnersperfectly. However, in order to measure character, an ”interview” is required. It isassumed that each party incurs costs when they both agree on an interview. In addition,individuals incur search costs at each stage of the search process.

At each stage of the process an individual is paired with a prospective partner. First,both individuals must decide whether they wish to proceed to the interview stage on thebasis of the attractiveness of the prospective partner. A pair is formed after the interviewonly by mutual consent. These final decisions are based on both the attractiveness andcharacter of the prospective partners. At equilibrium each individual uses a strategyappropriate to their type. The set of strategies corresponding to such an equilibrium iscalled an equilibrium strategy profile.

Such a problem may be interpreted as e.g. a mate choice problem in which the classesare male and female or a job search problem in which the classes are employer and em-ployee. In such a model of mate choice, the individuals first decide whether to court.Courtship only occurs by mutual consent. It may be the case that one of the sexes firstproposes courtship. The assumption that attractiveness can be observed very quickly, butcourtship is required to observe someone’s character is obviously a simplification. How-ever, in the case of human mate choice many traits that can be thought of as definingattractiveness (physical attractiveness, employment, economic status) are usually mea-sured quickly, whilst observation of traits defining character (political and religious views,tastes and emotions) are generally more difficult to measure. In this regard, our modelbears a strong resemblance to ‘speed dating’ models recently developed and tested byFisman et al. (2006).

In the case of such a model job search, it is assumed that a job’s attractiveness (inter-preted as pay, conditions, status) can be observed from an advert. The attractiveness ofa job seeker (interpreted as qualifications and experience) are readily seen from his/herapplication. Similarity of character may be interpreted as the ability of the employer (ordepartment) and employee to work together as a team. In certain cases, the characterof an employee might be interpreted as his/her speciality. For example, suppose two dif-ferent mathematics departments are looking for a professor. Both want someone with alarge number of publications in prestigious journals (a measure of attractiveness). How-ever, department A wants a fluid dynamicist and department B wants a game theorist,since such a professor would suit their profile/fulfil their requirements (i.e. suitability ofcharacters). This interpretation is somewhat problematic within the framework presentedhere, since it is likely that an individual’s speciality would be visible from the application.

Matching with Multiple Criteria 3

In the literature to date, job search models typically adopt a matching function ap-proach, where the employer and employee search for the perfect ‘fit’ using a set of costlycriteria. Equilibrium conditions are derived and tested for robustness once the model isbuilt, and policy recommendations follow (Burdett, 1978; Pissarides, 1994; Drewlanka,2006; Shimer and Smith, 2000). Jovanovic (1979), Hey (1982), and MacMinn (1980) arethe classic studies. Devine and Piore (1991) and Shimer and Smith (2000) survey themore recent developments.

A partnership can be viewed as an ‘experience good’, where the only way to determinethe quality of a particular match is to form the match and experience it. There have alsobeen experimental studies done on matching models, for example Roth and Sotomayor(1990) and many of his subsequent papers1.

Another strand in the literature is the search-theoretic literature developed by McCall(1970) and extended by Diamond and Maskin (1979) and others, where the job searchproblem is conceived of as a dynamic program which has to be solved in finite time, solabour markets are best described by optimal control problems and solution methods.The literature here is vast and well studied.

Job search has also been modelled as a mating or network game, with representative con-tributions being (Albelda, 1981; Beller, 1982; Peterson et al., 2000; Coles and Francesconi,2007; Fisman et al., 2008; Pissarides, 1994). In the biological literature, mate choice ismodelled as sequential observation of prospective mates. The model presented here is adevelopment of this strand.

The general model is presented in Section 2. Section 3 gives the set of criteria that wewish an equilibrium to satisfy. These conditions are based on the concept of a subgameperfect equilibrium (a refinement of the concept of Nash equilibrium).

Section 4 describes a general method for calculating the expected rewards of eachindividual under a given strategy profile. Section 5 considers the interview subgame(when individuals decide whether to form a partnership) and the application/invitationsubgame (when individuals decide whether to go on to the interview stage). Results onthe form and uniqueness of the equilibrium are derived in Section 6. An algorithm forderiving this equilibrium is also presented, together with an example. Section 7 gives abrief conclusion and suggests directions for further research.

2. The General Model

For convenience, the language of a job search problem will be used. We consider asteady state model in which the distributions of the attractiveness (qualifications) andcharacter of a job seeker, as well as of the attractiveness and character of an employer(denoted X1,js, X2,js, X1,em and X2,em, respectively), do not change over time.

1Basic models start like this: derive a utility function for the worker, U , and have them commit a costlysearch to the maximisation of U in finite time. The matching function is the mechanism which equatesworkers to firms in this process. Asymmetric information, or educational or skill-based heterogeneity isthen introduced to study an aspect of the labour market process.


For convenience, we suppose that X1,es, X1,js, X2,es and X2,js are discrete randomvariables which take integer values and have a finite support, and that the attractivenessof a job seeker or employer is independent of his/her character2. It will also be assumedthat the set of possible characters of job seekers coincides with the set of possible charactersof employers.

The type of an individual can be defined by their attractiveness and character, togetherwith 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].

Preferences are common with respect to attractiveness, i.e. all job seekers prefer jobswith a high attractiveness measure3. Also, preferences are homotypic with respect tocharacter, i.e. an employer prefers employees with a similar measure of character.

To be more precise, suppose the reward obtained by a type xjs job seeker from takinga job with a type xem employer is g(x2,js,xem), where g is strictly increasing with respectto x1,em and strictly decreasing with respect to |x2,js − x2,em|.

Similarly, the reward obtained by a type xem employer from employing a type xjs jobseeker is h(x2,em,xjs), where h is strictly increasing with respect to x1,js, and strictlydecreasing with respect to |x2,js − x2,em|.

The population is assumed to be large.At each moment n, (n = 1, 2, . . .) each unemployed individual is presented with a

prospective job picked at random. Thus it is implicitly assumed that the operationalratio of unfilled jobs to job seekers is one, and that individuals can observe as manyprospective jobs as they wish. Suppose the ratio of the number of searching job seekersto the number of searching employers is r, where r > 1. This situation can be modelledby assuming that a proportion r−1

rof employers have attractiveness −∞ (in reality job

seekers paired with such an employer at moment i meet no employer then). An individualcannot return to a prospective job (or job seeker) found earlier, nor would they wantto, given their preferences and past negative experiences with that prospective employer(employee).

Suppose that employees can observe the ‘type’ of a prospective job perfectly. Theattractiveness of an employer can be observed on the basis of an advert, while an interviewis required to observe the character of an employer.

Similarly, an employee’s qualifications can be measured almost instantaneously by em-ployers on the basis of an application, but in order to observe character an interview isrequired.

The (non-negative) search costs incurred at each stage by a job seeker and by anemployer are c1,js and c1,em, respectively. These reflect the costs of not having a job andnot having a position filled at each stage, respectively. The costs of interviewing are c2,js

and c2,em to job seekers and employers, respectively. The costs of applying for a job arec3,js.

2In further work we hope to relax this rather restrictive assumption.3A proxy for which would be the starting wage.


It is assumed that the costs of advertising a vacant position are incorporated into thecosts of not having a position filled. Adverts often give quite explicit information regardingthe qualifications required. It is assumed that this information is implicitly given by themeasure of attractiveness of a job.

Empirical evidence on search costs abounds in the literature, for instance Petersonet al. (2000) considers job search costs to be the primary cause of ‘sticky’ wages andlow labour market mobility, when contrasting the European and US labour markets.Devine and Piore (1991) also presents empirical evidence on search costs, and in thelarge Lucas/Prescott literature incorporating job search costs into macroeconomic models,Andolfatto (1996) is representative.

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

If the job seeker applies, the employer must then decide whether to proceed with aninterview or not, based on the qualifications of the job seeker. If either the job searcherdoes not apply or the employer does not wish to interview, the two individuals carryon searching. In some scenarios—for example day by day manual labour—it might payindividual employers to immediately accept a prospective employee for a job withoutinterview. However, to keep the strategy space as simple as possible, it is assumed thatindividuals must always interview before forming a pair.

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 theother as a partner or not. If acceptance is mutual, then a job pair is formed. Otherwise,both individuals continue searching. Since at this stage both individuals have completeinformation regarding the type of the other, it may be assumed that these decisions aremade simultaneously.

A job seeker’s total reward from search is assumed to be the reward gained from thejob taken minus the total search costs incurred. Hence, the total reward from search of ajob seeker of type xjs from taking a job with an employer of type xem after searching forn1 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 from employing a jobseeker of type xjs after searching for k1 moments and interviewing k2 job seekers is givenby

h(x2,em,xjs)− k1c1,em − k2c2,em.

Let π define the strategy profile used in the job search game, referred to as the su-pergame Γ. This is the game in which each individual plays a sequence of games of theform described above with job seekers being paired at each stage with a random job untila partner is found. This supergame depends on the distributions of attractiveness andcharacter for both employers and employees, together with the search, application and in-terview costs. π describes the strategy used by each member of the population accordingto their type (assumed to be a pure strategy).


The game played by a job seeker and employer on meeting can be split into two sub-games. The first will be referred to as the application/invitation subgame, in which thepair decide whether to proceed to an interview or not. The second subgame is called theinterview game and at this stage both parties must decide whether to accept the other ornot.

These subgames will be considered in Section 5. Since this game is solved by recursionin the manner developed by Spear (1989, 1991), we first consider the interview subgame.

These two subgames together define the game played when a job seeker of type xjs meetsan employer of type xem and the strategy profile is π. This game is denoted G(xjs,xem; π).

Suppose that in a mate choice game males initially solicit courtship with females. In thedescription given above, males and females play the roles of job searchers and employers,respectively.

3. Conditions for a Solution to the Game

We look for a Nash equilibrium profile π∗ of Γ which satisfies some desirable criteria(outlined below). When the population play according to the strategy profile π∗, then noindividual can gain by using a different strategy to the one defined by π∗.

We look for a Nash equilibrium strategy profile πN of Γ that satisfies the followingadditional conditions:

Condition 1: In the interview game, a job seeker accepts a prospective job (respec-tively, an employer offers a position to a job seeker) if and only if the reward fromsuch 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 fromthe resulting interview subgame minus the costs of interviewing is as least as greatas her expected reward from future search.

Condition 3: A job seeker only applies for a job if his expected reward from ap-plying minus the costs of applying for the job is at least as great as his futureexpected reward from search.

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

In mathematical terms, Conditions 1, 2, 3 are necessary and sufficient conditions toensure that when the population use the profile πN , each individual plays according toa subgame perfect equilibrium in the appropriately defined application/invitation andinterview subgames.

Since the sets of character measures for job seekers and employers coincide, the mostpreferred employees of a type [x1,em, x2,em] employer are job seekers of maximum qualifi-cations who have character x2,em. The most preferred jobs of a type [x1,js, x2,js] employeeare jobs of maximum attractiveness which have character x2,js. Condition 1 states thatin the interview game a job seeker will always accept his/her most preferred job. This isclear as the job seeker’s future expected reward from search must be less than the reward

4Again, this is a restrictive assumption we hope to relax in further work.


from obtaining his most preferred job. Moreover, if a job seeker accepts a job that wouldgive him a reward of k in the interview game, then he must accept any job that wouldgive him a reward of at least k. It follows that the acceptable difference in character isnon-decreasing in the attractiveness of the prospective partner. Several labour marketstudies have found empirical evidence that employers and employees are happiest withlabour market choices they view as similar to themselves in some respects (eg. labourmarket type, class, educational level), (Peterson et al., 2000; Beller, 1982; Albelda, 1981).

Condition 4 states that the Nash equilibrium strategy should be stationary. This reflectsthe following facts:

1: An individual starting to search at moment i faces the same problem as onestarting at moment 1.

2: Since the search costs are linear, after searching for i moments and not findinga job, an individual maximises his/her expected reward from search simply bymaximising the expected reward from future search (i.e. by ignoring previouslyincurred costs).

Many strategy profiles may lead to the same pattern of applications, interviews andemployment. For example, suppose that there are three levels of attractiveness and char-acter, which are the same for both employers and employees. Consider π1, the strategyprofile according to which:

a: job seekers apply to employers of at least the same attractiveness;b: employers are willing to interview job seekers of at least the same attractiveness;c: in the interview game, individuals of the two extreme characters accept prospec-

tive job matches of either the same character or of the central character;d: in the interview game individuals of the central character only accept prospective

partners of the same character.

Under π1 only job seekers and employers of the same attractiveness will proceed tointerview. Pairs are only formed between individuals of the same type.

Suppose π2 differs from π1 in that employers only invite job seekers of the same at-tractiveness for interview and in the interview game individuals only accept prospectivepartners of the same character. The pattern of applications, interviews and pair formationunder these two strategy profiles would be the same.

We might also be interested in Nash equilibrium profiles that satisfy the followingconditions.

Condition 5: In the application/invitation subgame, an employer of type xem iswilling to interview any job seeker of qualifications not lower than required levelof qualifications, denoted t(xem).

Condition 6: Suppose two employers have the same character, then the most at-tractive one will be at least as choosy as the other when inviting candidates forinterview.


One might consider analogous conditions for the rules used by job seekers. However,it seems reasonable that lowly qualified job seekers will not apply for a highly attractivejob, as by doing so they will incur the costs of applying while the employer is expected toreject them. Since employers only have to decide whether to invite candidates that haveapplied to them (i.e. it seems quite possible that the job seeker will ultimately find thejob acceptable), it would seem desirable that they use a threshold rule in this game.

For example, consider an employer of low attractiveness. Under a profile satisfyingCondition 5, she would be willing to interview a job seeker with high qualifications,although such a job seeker would never accept her in the interviewing game. So, althoughPizza Hut might like a Harvard PhD graduate making pizzas, this is not a realistic match.However, we expect that under the forces of labour market competition, individuals donot apply for prospective jobs they are sure to reject in the interview game, as this wouldbe costly to them. It is simply not worth the Harvard graduate’s time and energy toapply to Pizza Hut. Hence, in practice Pizza Hut would not have to decide whether toinvite such a candidate for interview for such a post. That is to say, the employer wouldbe indifferent between the two actions that are purely hypothetically available to her. Inthis case a threshold rule is reasonable.

Condition 6 reflects the intuition that highly attractive employers will be more choosythan less attractive employers.

4. Deriving the Expected Payoffs Under a Given Strategy Profile

Given the strategy profile used by a population, we can define what applications aremade, which pairs of types of individuals proceed to the interview stage, and which pairsof types of individuals form a job pairing.

From this it is relatively simple to calculate the expected length of search and theexpected number of interview rounds and/or applications of an individual of a given type.Let p(xjs) be the probability that a job seeker is of type xjs and q(xem) be the probabilitythat an employer is of type xem.

Let M1(xem; π) be the set of types of employee that an employer of type xem willinterview (i.e. the employee applies to the employer and the employer invites for interview)under the strategy profile π.

Similarly, let F1(xjs; π) be the set of types of employer that an employee of type xjs

will be interviewed by.Define M2(xem; π) to be the set of types of employee that eventually pair with a employer

of type xem.Similarly, define F2(xjs; π) to be the set of types of employer that eventually pair with

an employee of type xjs.Finally, define F0(xjs; π) to be the set of types of employers that a job seeker of type

xjs will apply to.By definition M2(xem; π) ⊆ M1(xem; π), F2(xjs; π) ⊆ F1(xjs; π) ⊆ F0(xjs; π).


The expected length of search of a job seeker of type xjs, Ljs(xjs; π), is the reciprocalof the probability of finding an employer with which he will pair at a given stage. Theexpected number of interviews of such a job seeker, Djs(xjs; π), is the expected length ofsearch, times the probability of being interviewed at a given stage. The expected numberof applications, Ajs(xjs; π), can be calculated in an analogous manner.

Hence,

Ljs(xjs; π) =1∑

xem∈F2(xjs;π) q(xem); Djs(xjs; π) =

∑xem∈F1(xjs;π) q(xem)∑xem∈F2(xjs;π) q(xem)

Ajs(xjs; π) =

∑xem∈F0(xjs;π) q(xem)∑xem∈F2(xjs;π) q(xem)

.(1)

Similarly, the expected length of search of an employer of type xem, Lem(xem; π), and theexpected number of interviews held by such an employer, Dem(xem; π), are given by

(2) Lem(xem; π) =1∑

xjs∈M2(xem;π) p(xjs); Dem(xem; π) =

∑xjs∈M1(xem;π) p(xjs)∑xjs∈M2(xem;π) p(xjs)

.

The expected reward of a type xjs job seeker from accepting a position under thestrategy profile π is the expected reward from taking a job with an employer given thather type is in the set F2(xjs; π). Hence, the job seeker’s expected total reward from search,Rjs(xjs; π), is given by

(3) Rjs(xjs; π) =

∑xem∈F2(xjs;π) q(xem)g(x2,js,xem)∑

xem∈F2(xjs;π) q(xem)− Cjs(xjs; π),

where Cjs(xjs; π) = c1,jsLjs(xjs; π) + c2,jsDjs(xjs; π) + c3,jsAjs(xjs; π) are the expectedsearch costs of such a job seeker under the strategy profile π. Similarly,(4)

Rem(xem; π) =

∑xjs∈M2(xem;π) p(xjs)h(x2,em,xjs)∑

xjs∈M2(xem;π) p(xjs)− c1,emLem(xem; π)− c2,emDem(xem; π).

5. The Interview and Offer/Acceptance Subgames

5.1. The Interview Subgame. Assume that the population are following a strategyprofile π.

The job seeker and employer both have two possible actions: accept the prospectivepartner, denoted a, or reject, denoted r. As argued above, we may assume that thesedecisions are taken simultaneously.

Also, we ignore the costs already incurred by either individual, including the costs ofthe present interview, as they are subtracted from all the payoffs in the matrix, and hencedo not affect the equilibria in this subgame.


Suppose the job seeker is of type xjs and the employer is of type xem. The payoff matrixis 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)]

).

The appropriate Nash equilibrium of this subgame is for the job seeker to accept the jobif and only if g(x2,js,xem) ≥ Rjs(xjs; π) and the employer to accept the job seeker if andonly if h(x2,em,xjs) ≥ Rem(xem; π).

For convenience, we assume that when h(x2,em,xjs) = Rem(xem; π), an employer alwaysaccepts the job seeker (in this case she is indifferent between rejecting and accepting himfor the job). Similarly, if g(x2,js,xem) = Rjs(xjs; π), it is assumed that a job seeker alwaysaccepts an employer.

If a job seeker rejects an employer, then the employer is indifferent between accepting orrejecting the job seeker. By using the rule given above, an employer will take the optimalaction whenever a job seeker “mistakenly” accepts a job offer.

Under these assumptions, this subgame has a unique Nash equilibrium and value. Letv(xjs,xem; π) = [vjs(xjs,xem; π), vem(xjs,xem; π)] denote the value of this game, wherevjs(xjs,xem; π) and vem(xjs,xem; π) are the values of the game to the job seeker andemployer, respectively.

We now consider the application/invitation game.

5.2. The Application/Invitation Subgame. Once the interview subgame has beensolved, we may solve the application/invitation subgame and hence the game G(xjs,xem; π).

As before, we assume that the population is following a strategy profile π.The possible actions of a job seeker are n — do not apply for a job and a — apply for

the job. The possible actions of an employer are i - invite for interview and r - reject anapplication.

These actions are based on the attractiveness of the prospective partner. As before, wemay ignore the costs that have been previously incurred. Since the order in which theactions are taken is important, we must consider the extensive form of this game, whichis given below.


��

��

��

@@

@@

@@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.

Here, v(xjs,xem; π) = [vjs(xjs, x1,em; π), vem(x1,js,xjs; π)] denotes the expected valueof the interview game given the strategy profile used by the population, the measuresof attractiveness of the pair, and the fact that an interview followed. In defining thesepayoffs, it is assumed that the players are following the appropriate strategy from thestrategy profile π.

The application/invitation game must be solved by recursion. The employer only hasto make a decision in the case where the job seeker has applied. She should invite forinterview if and only if her expected reward from interviewing is at least as great as theexpected reward from future search, i.e.

vem(x1,js,xem; π)− c2,em ≥ Rem(xem; π).

Suppose the application costs are strictly positive. If no employer of the attractivenessobserved wishes to invite the job seeker for interview, then the job seeker should not apply.

Otherwise, the job seeker should apply if

r(x1,js, x1,em; π)[vjs(xjs, x1,em; π)−c2,js]+[1−r(x1,js, x1,em; π)]Rjs(xjs; π)−c3,js ≥ Rjs(xjs; π),

where r(x1,js, x1,em; π) is the probability that a randomly chosen employer of attractivenessx1,em wishes to invite a job seeker of qualifications x1,js for interview.

It should be noted that when a job seeker makes his decision he has no informationregarding the character of an employer. However, the fact that a job seeker applies fora job may implicitly give the employer some information regarding his character. In thenext section we consider an adaptation of the general model in which a job applicationdoes not transfer any information about the character of a job searcher, either explicitlyor implicitly. We hope to investigate the question of such implicit transfer of informationin a future paper.

In the next section we present the specific adaptation of the model and propose analgorithm to solve such a game, as an algorithmic game in the tradition of Velupillai(1997); Nisam et al. (2007).


6. A Modification of the General Model

Realisations of the general form of the problem described above may well have complexsolutions. This partly results from edge effects, i.e. the fact that some individuals haveextreme characters. Suppose the distribution of character is uniform. Consider job seek-ers of maximum attractiveness. Those of central character are expected to have a higherexpected reward than those of extreme character (see ?). It may well be the case thatat equilibrium job seekers of maximum attractiveness and extreme character should bewilling to apply to employers of lower attractiveness than job seekers of maximum attrac-tiveness and central character would be willing to apply to. This means that employersof medium attractiveness who receive applications from highly attractive job seekers maywell implicitly have the information that the applicant has an extreme character. Thatis to say, the distribution of the character of the job seeker given such an application willnot be uniform but will be concentrated on extreme characters. This effect is likely to bemore pronounced when the distribution of character is concentrated on central values.

We make the following modification of the general model, in order to make the typessymmetric with respect to character and class:

1: The distributions of character and attractiveness are independent of class. Fur-thermore, 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 both1 and r − 1. The difference between characters i and j is defined to be thedifference 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.eg(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.

Such a problem will be referred to as ”quasi-symmetric”. It is not symmetric, since inthe application/invitation game, job seekers move first. However, suppose the strategyprofile used is symmetric with respect to class. That is to say that for all i and j, if atype [i, j] job seeker is willing to apply to an employer of attractiveness k1 and is willingto accept a job offer from a type [k2, l] employer, then a type [i, j] employer is willing toinvite a job seeker of attractiveness k1 for interview and is willing to offer a job offer to atype [k2, l] job seeker. Under such a strategy profile, the expected payoff of an individualis independent of class, i.e. for a symmetric strategy profile Rjs([i, j]; π) = Rem([i, j]; π) =R([i, j]; π).

We wish to find a symmetric equilibrium for such a problem. Such an equilibriumsatisfies the following conditions:

1: The strategy profile is symmetric with respect to class.2: The strategy profile is symmetric with respect to character. That is to say that if

a type [i, j] job seeker is willing to apply to an employer of attractiveness k1 andis willing to accept a job offer from a type [k2, l] employer, then a type [i, j + m]


job seeker is willing to apply to an employer of attractiveness k1 and is willing toaccept a job offer from a type [k2, l + m] employer.

At such an equilibrium the expected payoff of an individual is independent of his/hercharacter. Before presenting an algorithm for deriving such an equilibrium, we will proveseveral results regarding the form of such an equilibrium. Denote the set of attractivenessand character levels by A and C, respectively.

Theorem 1. At a symmetric equilibrium π∗ of a quasi-symmetric game satisfying condi-tions 1-4 the reward of an individual is non-decreasing in attractiveness.

Proof. Assume otherwise. From the symmetry of the equilibrium, it suffices toassume that there exists some j ∈ C and i, k ∈ A such that i > k and R([i, j]; π∗) <R([k, j]; π∗). First consider the interview game. From the assumptions made, it followsthat the characters of the two players are independent and come from the marginal dis-tribution of character. Since R([i, j]; π∗) < R([k, j]; π∗), the set of job seekers acceptableto a type [k, j] employer is contained within the set of job seekers acceptable to a type[i, j] employer. Suppose a type [k, j] job seeker pairs with employers of type in T1. Type[i, j] employers are always preferred by job seekers to type [k, j] employers. Hence, byaccepting job seekers of type in T1, an employer of type [i, j] is able to ensure at least thesame expected reward in any of the possible interview subgames as a type [k, j] employer.

Suppose now that under π∗ job seekers of attractiveness in the set A1 apply to and areinvited to interview by any employer of attractiveness k. Since in the interview game atype [i, j] employer is no more choosy than a type [k, j] employer and is always preferredas a partner, a job seeker cannot have a lower expected reward from applying to a type[i, j] employer than from applying to a type [k, j] employer. From the assumption that theequilibrium is symmetric, it follows that a job seeker of attractiveness a ∈ A1 should applyto any employer of attractiveness i. From the arguments made regarding the interviewsubgame, it follows that a ”mutant” employer of type [i, j] can ensure at least the samereward in the game G(x, [i, j]; π∗) as a type [k, j] employer in the game G(x, [k, j]; π∗) byusing the same strategy as an employer of type [k, j]. This is a contradiction and thusR([i, j]; π∗) ≥ R([k, j]; π∗).

�

Theorem 2. At a symmetric equilibrium π∗ of a quasi-symmetric game satisfying con-ditions 1-4 job seekers of maximum attractiveness apply to employers of attractivenessabove a certain threshold.

Note: From the symmetry of such an equilibrium, it follows that employers invite jobseekers of attractiveness above the same threshold for interview.

Proof. Let the maximum level of attractiveness be imax. From Theorem 1, indi-viduals of attractiveness imax have the highest expected reward. Since g(l, [imax, j]) ≥g(j, [i, l]), it follows that if a type [imax, j] job seeker wishes to accept an offer from a type[i, l] employer, then acceptance is mutual. Denote the set of character levels of employersof attractiveness k who are mutually acceptable to a type [i, j] job seeker in the interview


subgame by B(k; [i, j]). Suppose k0 ≤ i]. If l /∈ B(k0; [i, j]), then l /∈ B(k; [i, j]) for k < k0,since a type [k0, l] employer is always preferred to a type [k, l] employer.

Suppose that a type [imax, j] job seeker is interviewed by an employer of attractivenessk. The job seeker’s expected reward from this subgame is

vjs(k, [imax, j]; π∗) = Rjs([imax, j]; π

∗)

1−∑

l∈B(k;[imax,j])

p•,l

+∑

l∈B(k;[imax,j])

g(j, [k, l])p•,l,

where p•,l denotes the marginal distribution of the character level. Since g(j, [k, l]) ≥Rjs([imax, j]; π

∗) when l ∈ B(k; [imax, j]) and g(j, [k, l]) is increasing in k, vjs(k, [imax, j]; π∗)

is non-decreasing in k. Hence, job seekers of maximum attractiveness should apply to ofemployers of attractiveness above a certain threshold (they must apply to employers ofmaximum attractiveness, since there are no better offers). �

Theorem 3. At a symmetric equilibrium π∗ of a quasi-symmetric game satisfying con-ditions 1-4, employers of attractiveness i are prepared to interview job seekers of attrac-tiveness in [k1(i), k2(i)], where k2(i) is the maximum attractiveness of an job seeker whoapplies to an employer of attractiveness i for interview. In addition, k1(i) and k2(i) arenon-decreasing in i and k1(i) ≤ i ≤ k2(i). By symmetry, analogous results hold for jobseekers of attractiveness i.

Proof. Suppose that individuals of attractiveness imax are only prepared to pairwith other individuals of attractiveness imax. In this case, there will be no interviews inwhich only one individual is of attractiveness imax. The problem faced by job seekers ofattractiveness imax − 1 then reduces to a problem in which they are the most attractivepartners (it may be assumed that employers that will not invite them for interview areeffectively of attractiveness −∞). It follows directly from Theorem 2 that job seekers ofattractiveness imax − 1 apply to employers of attractiveness [k1(imax − 1), k2(imax − 1)],where k2(imax−1) = imax−1 < k2(imax) = imax and k1(imax−1) ≤ imax−1 < k1(imax) =imax.

Now suppose k1(imax) ≤ imax − 1. Thus employers of attractiveness imax are the mostattractive to invite job seekers of attractiveness imax − 1 for interview. First we showthat a job seeker of attractiveness imax − 1 should apply to an employer of attractivenessk1(imax). Since an employer of attractiveness imax invites a job seeker of attractivenessk1(imax) for interview, we have ∀j ∈ C

R([imax, j], π∗) ≤ vem(k1(imax), [imax, j]; π

∗)− c2,em

R([imax, j], π∗) ≤ R([imax, j]; π

∗)

1−∑

l∈B(k1(imax);[imax,j])

p•,l

+

+

∑l∈B(k1(imax);[imax,j])

p•,lg(j, [k1(imax), l])

− c2,em.


Rearranging this inequality leads to

(5)

∑l∈B(k1(imax);[imax,j])

p•,l[g(j, [k1(imax), l])−R([imax, j]; π∗)]

− c2,em ≥ 0.

An employer of type [imax−1, j] invites job seekers of attractiveness k1(imax) for interview,if

R([imax − 1, j], π∗) ≤ vem(k1(imax), [imax − 1, j]; π∗)− c2,em.

Rearranging this equation as above, we obtain ∑l∈B(k1(imax);[imax−1,j])

p•,l[g(j, [k1(imax), l])−R([imax − 1, j]; π∗)]

− c2,em ≥ 0.

From Theorem 1, the set of job seekers of attractiveness k1(imax) an employer of type[imax, j] accepts in the interview game must be contained in the set of such job seekers anemployer of type [imax − 1, j] accepts. Also, from the form of the equilibrium payoffs andsymmetry with respect to character, if an individual is willing to pair with another of notgreater attractiveness then acceptance is mutual. It follows that B(k1(imax); [imax, j]) ⊆B(k1(imax); [imax − 1, j]). Set C = B(k1(imax); [imax − 1, j]) \ B(k1(imax); [imax, j]). Wecan write the above inequality in the form∑

l∈B(k1(imax);[imax,j])

p•,l[g(j, [k1(imax), l])−R([imax − 1, j]; π∗)]+

+

{∑l∈C

p•,l[g(j, [k1(imax), l])−R([imax − 1, j]; π∗)]

}− c2,em ≥ 0.

From Equation (5), the first sum is greater than c2,em. The second sum is non-negativesince g(j, [k1(imax), l]) ≥ R([imax − 1, j]; π∗) for l ∈ B(k1(imax); [imax − 1, j]). Hence,employers of attractiveness imax − 1 should invite job seekers of attractiveness k1(imax)for interview.

We now show that if an employer of attractiveness imax − 1 should interview a jobseeker of attractiveness k0, then she should interview a job seeker of attractiveness k,where k0 < k ≤ k2(imax − 1). In order to show this, we consider the following two cases:1. k ≤ imax − 1 and 2. k > imax − 1.

Suppose k ≤ imax− 1. From Theorem 1, if a job seeker of type [k, l] is acceptable to anemployer of type [i, j] in the interview game, then acceptance is mutual. By assumptionwe have ∑

l∈B(k0;[imax−1,j])

p•,l[g(j, [k0, l])−R([imax − 1, j]; π∗)]

− c2,em ≥ 0.


If a job seeker of type [k0, l] is acceptable to an employer of type [imax − 1, j] in theinterview game, then a job seeker of type [k, l] is mutually acceptable to an employerof type [imax − 1, j]. Now g(j, [k0, l]) < g(j, [k, l]), B(k0; [i, j]) ⊆ B(k; [i, j]) and for anyl ∈ B(k; [i, j]), we have g(j, [k, l]) ≥ R([i, j]; π∗). It follows that ∑

l∈B(k;[imax−1,j])

p•,l[g(j, [k, l])−R([imax − 1, j]; π∗)]

− c2,em ≥ 0.

It follows that an employer of attractiveness imax − 1 should invite a job seeker of attrac-tiveness k, where k0 ≤ k ≤ imax − 1, for interview.

We now consider imax − 1 < k ≤ k2(imax − 1), i.e. k = imax and job seekers ofattractiveness imax apply to employers of attractiveness imax. If a job seeker of type[imax, l] accepts an employer of type [imax−1, j] in the interview game, then acceptance ismutual. From the symmetry of the strategy profile, a job seeker of type [imax, j] acceptsan employer of type [imax − 1, l] in the interview game. By assumption we have ∑

l∈B(imax−1;[imax,j])

p•,l[g(j, [imax − 1, l])−R([imax, j]; π∗)]

− c2,em ≥ 0.

We have B(imax−1; [imax, j]) = B(imax; [imax−1, j]), g(j, [imax, l]) > g(j, [imax−1, l]) andR([imax, j]; π

∗) ≥ R([imax − 1, j]; π∗). It follows that∑l∈B(imax;[imax−1,j])

p•,l[g(j, [imax, l])−R([imax − 1, j]; π∗)]− c2,em ≥ 0.

Thus when job seekers of attractiveness imax apply to employers of attractiveness imax−1,such employers should invite job seekers of maximum attractiveness for interview.

Hence, the theorem has been proved in the case i = imax − 1. The rest of the proof isby recursion. It is necessary to prove that the theorem holds for attractiveness level i− 1given that it holds for attractiveness level i. This recursion step is analogous to the proofof the statement for i = imax − 1 and is thus omitted. �

Hence, the general form of the equilibrium is intuitive. Job seekers should apply toemployers (and employers should invite job seekers) who are of a similar level of attrac-tiveness. It should also be noted that at such an equilibrium a type [i, j] employer willpair with a type [i, j] job seeker.

Due to the assumption of no application costs, at such an equilibrium a job seeker ofattractiveness i is indifferent between applying to an employer who would not invite himfor interview and not applying. In this case we should check the condition based on theconcept of subgame perfectness. This states that a job seeker should apply given that hisexpected reward in the interview resulting from a ”mistaken” invitation is greater thanthe expected reward from future search. Suppose the employer would not offer a positionto any job seeker of attractiveness i. The job seeker should not apply, in order to avoidthe interview costs when there is no prospect of employment.


It is possible that an employer would offer a job seeker of lower attractiveness in theinterview game but, due the costs of interview and the risks of obtaining an intervieweeof inappropriate character, would not invite such a job seeker for interview. In this casewe would have to check the condition given above. This will be considered in the examplegiven later.

We now propose an algorithm for finding a symmetric equilibrium profile. One coulduse a policy iteration algorithm (see Velupillai (2000)). Such an algorithm would startfrom some strategy profile π0. πn+1 is defined to be the best response of an individualto πn. If such an algorithm converged, it would converge to a Nash equilibrium strategyprofile. However, such an algorithm is not guaranteed to converge and initial resultsindicated that when there are a large number of types convergence is slow. We thusdecided to develop an algorithm that is adapted to the specific form of the game.

6.1. The Algorithm. Define m = b r+12c, i.e. the median character level if the number

of character levels is odd and a character level neighbouring the median otherwise. Sincethe equilibrium is assumed to be symmetric with respect to character and class, it sufficesto consider employers of character m. Let i ≥ k. Suppose an employer of attractiveness iinvites job seekers of attractiveness k for interview and an employer of type [i, j] acceptsa job seeker of type [k, l]. From the form of a symmetric equilibrium, it follows that ajob seeker of type [k, l] will apply to a pair with an employer of type [i, j]. Hence, theproblem faced by an employer of maximum attractiveness is essentially an optimisationproblem. This problem can be solved as follows

1: Assume that employers of maximum attractiveness only invite job seekers of max-imum attractiveness for interview. These job seekers can be ordered accordingto the preferences of type [imax, m] employers. We consider strategy profiles πt,t = 0, 1, 2, . . . m. Under strategy πt employers of maximum attractiveness offerpositions to job seekers whose characters do not differ by more than t. We cal-culate R([imax, m]; π0), R([imax, m]; π1), . . . in turn and each expected reward givesus a lower bound for the expected reward of an individual of attractiveness imax

at equilibrium. It should be noted that if R([imax, m]; πt) ≤ g(m, [imax, m+ t+1]),then a type [imax, m] employer should pair with a type [imax, m+ t+1] job seeker.Otherwise, a type [imax, m] employer should accept such a partner (see ?). Hence,we can derive the optimal strategy of type [imax, m] employers when they onlyinterview job seekers of attractiveness imax.

2: If the lower bound on R([imax, m]; π∗) calculated from part 1 is less than thereward obained by a type [imax, m] employer from pairing with a type [imax−1, m]job seeker, it may be optimal for employers of maximum attractiveness to interviewjob seekers of the second highest level of attractiveness. We can order job seekersof the top two levels of attractiveness with regard to the preferences of a type[imax, m] individual. By considering strategies in which type [imax, m] employersare prepared to pair with successively less preferred job seekers as in 1, we can


derive the optimal strategy of such employers given they interview job seekers ofthe two highest levels of attractiveness.

3: If required, we can derive the optimal strategies of type [imax, m] individuals giventhat they interview job seekers of the u highest levels of attractiveness, where u isat least three and not more than the number of attractiveness levels. In this waywe can derive the equilibrium strategy of a type [imax, m] employer. Note that itis assumed that if such an employer is indifferent between two strategies, then sheuses the strategy which maximises the number of attractiveness levels interviewed,together with the number of types of job seeker that she will eventually pair with.Due the form of the equilibrium and the finite number of types, this solution iswell defined and unique. The strategies used by other employers of maximumattractiveness can be found using the symmetry of the profile with respect tocharacter.

Hence, the problem faced by an employer of maximum attractiveness can be solved bysolving a sequence of optimal stopping problems. Once this has been done, the problemfaced by an employer of type [imax − 1, m] reduces to an optimisation problem in whichthe set of job seekers of maximum attractiveness who apply to and pair with her havebeen derived as above. The optimal strategy of such an employer can be calculated in thesame way as above. In such a way, we can derive the equilibrium strategies of employersof successively lower attractiveness levels.

The theorem below follows directly from the construction of the symmetric equilibrium.

Theorem 4. Assume that if according to the algorithm given above an employer is in-different between two strategies, then she uses the strategy which maximises the numberof attractiveness levels interviewed, together with the number of types of job seeker thatshe will eventually pair with. There exists a unique symmetric equilibrium of the ”quasi-symmetric” game described above.

It should be noted that the algorithm presented here is similar in its approach to thesolution of a two-sided job search game considered by Collins and McNamara (?)

6.2. 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}. It is assumed thatthe distributions of both attractiveness and character are uniform. 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) betweenthe characters of the pair.

First we consider employers of maximum attractiveness. Suppose they only invite jobseekers of attractiveness 7 for interview. The ordered preferences of a [7, 4] individualare 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 4partners give a reward from pairing of 7, 6, 5 and 4 respectively. We successively includejob seekers starting from the most preferred until no increase in the expected payoff from


search starts decreasing. Let πi denote any strategy profile in which [7, 4] employers pairwith job seekers from the first i groups described above, i = 1, 2, 3, 4.

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.

The expected reward from searching of a type [7, 4] employer under π3 is greater than 4.It follows that [7, 1] and [7, 7] individuals should not be accepted in the interview game.

Since our lower bound (4.2) on the expected return from search of a type [7, 4] is lessthan the reward obtained from pairing with a job seeker of the same character and thesecond highest attractiveness (6), we now consider strategy profiles in which type [7, 4]employers interview job seekers of attractiveness 6 and 7. We only have to consider:

1: Strategy profiles in which job seekers of attractiveness [6, 4] are acceptable in theinterview game. If this were not the case, then a type [7, 4] employer would beincurring unnecessary costs.

2: Prospective partners who give a reward higher than the lower bound on theexpected reward from search.

The ordered preferences of the employer among the set of job seekers interviewed andsatisfying criterion 2 above is: group 1 is {[7, 4]}, group 2 is {[7, 3], [7, 4], [7, 5]} and group3 {[7, 2], [7, 6], [6, 3], [6, 5]}. We only need to consider strategy profiles of the following twotypes: a) type [7, 4] employers pair with those from groups 1 and 2 above, denoted π5, b)type [7, 4] employers pair with those from all three groups, denoted π6. We have

R([7, 4]; π5) =25

4− 49

4× 1

7− 14

4× 1

7= 4

R([7, 4]; π6) =45

8− 49

8× 1

7− 14

8× 1

7=

9

2.

We now consider strategy profiles in which type [7, 4] employers interview job seekers ofattractiveness at least 5. Since the present lower bound on R([7, 4]; π∗) is 4.5, we onlyneed consider strategy profiles in which [7, 4] employers pair with the same types of jobseekers as in π6, but with the addition of type [5, 4] job seekers. Denote such a strategyprofile by π7. We have

R([7, 4]; π7) =50

9− 49

9× 1

7− 21

9× 1

7=

40

9<

9

2.

It is clear that type [7, 4] employers should not interview job seekers of attractiveness4. It follows that at a symmetric equilibrium, type [7, 4] employers invite job seek-ers of attractiveness 6 and 7 for interview and offer a job to those of type in M7 ={[7, 2], [7, 3], [7, 4], [7, 5], [7, 6], [6, 3], [6, 4], [6, 5]}. It should be noted that a type [7, 4] em-ployer should offer a job to a type [5, 4] job seeker given that an interview occurs. However,


due to the costs of interviewing, the low probability of finding an acceptable partner ofattractiveness 5 and the relatively small gain obtained from such a partnership comparedto the expected reward from search, an employer of type [7, 4] should not invite such ajob seeker for interview. Define Mi + [s, t] to be the set of [k + s, j + t] where [k, j] ∈ Mi.From the symmetry of the equilibrium with respect to character, an employer of type[7, m + t] invites all job seekers of attractiveness 6 and 7 for interview and offers a job tojob seekers in M7 + [0, t].

Now we consider employers of type [6, 4]. We first consider strategy profiles under whichindividuals of maximum attractiveness follow the strategy derived above and employersof attractiveness 6 invite job seekers of attractiveness 6 and 7 for interview. From thesymmetry of the game with respect to class and character, since employers of type [7, 4]pair with job seekers of type [6, 3], [6, 4] and [6, 5], it follows that employers of type [6, 4]will pair with job seekers of type [7, 3], [7, 4] and [7, 5]. They must also pair with those oftype [6, 4]. The expected reward of a type [6, 4] employer is

R([6, 4]; π8) =25

4− 49

4× 1

7− 14

4× 1

7= 4.

The next most preferable partners of those interviewed are job seekers of type [6, 3] and[6, 5]. The expected reward obtained by extended the set of acceptable partners to includethese types is given by

R([6, 4]; π9) =35

6− 49

6× 1

7− 14

6× 1

7=

13

3.

This is greater than the expected reward from accepting the next most preferred types([6, 2] and [6, 6]). Hence, we can now consider strategy profiles in which employers of type[6, 4] invite job seekers of attractiveness at least 5 for interview. The only case we needto consider is extending the set of acceptable job seekers to include those of type [5, 4].We have

R([6, 4]; π10) =40

7− 49

7× 1

7− 21

7× 1

7=

30

7<

13

3.

It follows that employers of type [6, 4] should invite job seekers of attractiveness 6 and 7 forinterview and should offer jobs to job seekers of a type in {[7, 3], [7, 4], [7, 5], [6, 3], [6, 4], [6, 5]}.In these cases acceptance is mutual. It should also be noted that employers of type [6, 4]should accept job seekers of type [7, 2] or [7, 6] in the interview game. However, in thesecases acceptance is not mutual. Job seekers of type [5, 4] would be accepted in the inter-view game, but are not interviewed.

It can be seen that employers and job seekers of the top two levels of attractivenesspair together, but not with any other individuals. The problem faced by employers ofattractiveness 5 is analogous to the problem faced by those of attractiveness 7 (they arethe most attractive of the remaining employers. It follows that M5 = M7− [2, 0]. Arguingiteratively, Mi = Mi+2 − [2, 0] for i = 2, 3, 4. Employers of attractiveness 3 or 5 invitejob seekers of the same attractiveness or of attractiveness one level lower for interview.Employers of attractiveness 2 or 4 invite job seekers of the same attractiveness or of


attractiveness one level higher for interview. We have R([5, 4]; π∗) = 2.5 and R([4, 4]; π∗) =73, so in the interview game employers of attractiveness 4 and 5 accept any job seekers

giving them a reward of at least 3 (the reward from a pairing is by definition an integerin this game). Also, R([3, 4]; π∗) = 0.5 and R([2, 4]; π∗) = 1

3, so in the interview game

employers of attractiveness 2 and 3 accept any job seekers giving them a reward of atleast 1.

Since job seekers of attractiveness 7 and 6 do not apply to employers of attractiveness5, such employers are indifferent to inviting such job seekers for interview. We shouldcheck the relevant equilibrium condition based on the concept of subgame perfectness, i.e.if a job seeker of attractiveness 7 did ”by mistake” apply to such an employer should theemployer invite the job seeker for interview. In the interview game, a job seeker of type[7, 4] would only accept an employer of type [5, 4]. Hence, the expected reward of a type[5, 4] employer from inviting a job seeker of attractiveness 7 for interview would be

vem(7, [5, 4]; π∗) =1

7× 7 +

6

7× 2.5− 1

7= 3 > R([5, 4]; π∗).

It follows that an employer of attractiveness 5 should invite a job seeker of attractiveness7 for interview. Arguing similarly, an employer of attractiveness 3 should invite a jobseeker of attractiveness 5 for interview.

No job seekers of attractiveness 6 or 7 would accept an offer from an employer ofattractiveness 4 in the interview game. It follows that employers of attractiveness 4should not invite job seekers of attractiveness 6 or 7 for interview. Arguing similarly, anemployer of attractiveness 2 should not invite any job seeker of attractiveness above 3 forinterview.

By the assumed symmetry with respect to class, if an employer of attractiveness ishould invite a job seeker of attractiveness k for interview, a job seeker of attractivenessi should apply to an employer of attractiveness k.

It remains to determine the strategy used by a type [1, 4] employer. Since no jobseeker of greater attractiveness will apply to such an employer, we only have to considerstrategies in such an employer offers positions to successively less preferred partners ofattractiveness 1. Denote the strategy profile under which individuals of attractiveness atleast 2 use the strategies derived and employers of type [1, 4] pair with job seekers of typesin a) {[1, 4]}, b) {[1, 3], [1, 4], [1, 5]} and {[1, 2], [1, 3], [1, 4], [1, 5], [1, 6]} by π11, π12 and π13,respectively. We have

R([1, 4]; π11) = 1− 49× 1

7− 7× 1

7= −7

R([1, 4]; π12) =1

3− 49

3× 1

7− 7

3× 1

7= −7

3

R([1, 4]; π13) = −1

5− 49

5× 1

7− 7

5× 1

7= −9

5.


The expected reward of a type [1, 4] employer under π13 is greater than the reward obtainedfrom pairing with job seekers of type [1, 1] and [1, 7]. Hence, at a symmetric equilibriumemployers of type [1, 4] should not accept such job seekers.

It remains to consider the set of job seekers that an employer of attractiveness 1 shouldinvite for interview according to the conditions based on subgame perfection. No job seekerof attractiveness above 3 would ever accept a job offer from an employer of attractiveness1 at this equilibrium. Hence, employers of attractiveness 1 should never invite job seekersof attractiveness 3 for interview. Job seekers of type [3, 4] and [2, 4] would accept anoffer from a type [1, 4] employer. Arguing as in the case of employers of attractiveness 5inviting job seekers of attractiveness 6 and 7 for interview, employers of attractiveness 1should invite job seekers of attractiveness 2 and 3 for interview.

The following table gives a synopsis of the equilibrium strategy profile. Each individualshould accept a prospective partner in the interview game if the reward from such amatching is at least as great as the reward from search. For ease of presentation the setof such partners is not presented.

Attractiveness Attractiveness levels invited Expected Reward7 { 6,7 } 4.506 { 6,7 } 4.335 { 4,5,6,7 } 2.504 { 4,5 } 2.333 { 2,3,4,5 } 0.502 { 2,3 } 0.331 { 1,2,3 } -1.80

Table 1. Brief description of symmetric equilibrium for the example considered

7. Conclusion

This paper has presented a model of two-sided job search where both common andhomotypic preferences are taken into account. The preferences of employers (job seekers)are common with respect to the attractiveness of job seekers (employers) and homotypicwith respect to character. Attractiveness can be assessed immediately, but in order toassess character an interview is required.

We considered a particular type of such problems in which the distribution of attrac-tiveness and character, as well as search and interview costs, were independent of theclass of a player. Character was assumed to form a ring, such that the ”extreme” levels ofcharacter are neighbours. For convenience, the supports of attractiveness and characterwere assumed to be finite sets of integers. The distribution of character is uniform.

The form of a symmetric equilibrium in such a game which satisfies various criteriabased on the concept of subgame perfectness was derived and an algorithm to find suchan equilibrium described. It is shown that such an equilibrium exists and is unique


(assuming that if an employer is indifferent between inviting for interview or not then sheinvites, analogous assumptions are made for the interview game and the strategies of jobseekers).

The use of this combination of preferences would seem to be logical in relation tojob choice. Although there is no perfect correlation in individuals’ assessment of theattractiveness of members of the opposite labour market type, there is normally a veryhigh level of agreement, particularly among job seekers. This approach has been used,for example, by Gale and Shapley (1962). Using such an approach individuals have theirown personal ranking of the desirability of members of the opposite labour market type.The approach used here would seem to be a good compromise between the approachesused in matching and the assumptions of common preferences. These ”mixed” preferencesseem to be both reasonably tractable within the framework of searching for a job within arelatively large population and allow a general enough framework to model the preferencesof individuals reasonably well (although it would seem that modelling character as a one-dimensional variable is rather simplistic). By using a larger number of types, we couldapproximate continuous distributions of qualifications and character Ramsey (2008).

For simplicity it was assumed that individuals know their own qualifications and char-acter, whereas in practice they may have to learn about these measures over time.

Also, individuals are able to measure attractiveness and character perfectly, althoughat some cost. It would be interesting to consider different ways in which information isgained during the search process. For example, some information about the character ofa prospective labour market match may be readily available. Hence, an improved modelwould allow some information to be gained on both the attractiveness and character of aprospective partner at each stage of the decision process.

In terms of the evolution of such procedures, it is assumed that the basic frameworkis given, i.e. the model assumes that the various search, application and interview costsare given. It should be noted that labour markets may benefit from the interview processi.e. employers might find employees of different qualifications to those stated in theadvertisement, but still useful to the company.

However, this model cannot explain why such a system has evolved, only the evolutionof decisions within this framework. Individuals may lower their search costs by joiningsome internet or social group. Such methods can also lead to biasing the conditionaldistribution of the character of a prospective job in a searcher’s favour. It might bethat interviewing costs are dependent on the types of the two individuals involved. Forexample, two individuals of highly different characters might incur low interview costs, asthey realise very quickly that they are not well matched.

Also, the ability to incur interview costs may well transfer information regarding theattractiveness and/or character of a job seeker. Hence, it may be more costly to beinterviewed by highly attractive employers, since they would have strong preferences forhigh quality (i.e. attractive) job seekers.

Also, it would be useful to investigate how the payoff functions, together with therelative costs of searching and interviewing, affect the importance of attractiveness and


character in the decision process. It should be noted that using attractiveness as an initialfilter in the decision process will lead to qualifications becoming relatively more importantthan character, especially if the costs of interviewing are relatively high.

It would also be useful to adapt the algorithm to problems in which the distributionof character is not uniform and/or the set of character levels do not form a ring. In thiscase, it is expected that individuals of extreme character will usually be less choosy thanthose of a central character for a given level of attractiveness. Two major problems resultfrom this. Firstly, the form of the equilibrium will be more complex than the form of thesymmetric equilibrium given here. Any algorithm to derive an equilibrium in this casewill certainly be much slower than the algorithm outlined in this paper, which uses theform of the symmetric equilibrium and the fact that the problem can be reduced to asequence of optimal stopping problems to greatly reduce the set of strategy profiles whichmust be considered. Secondly, at such an equilibrium the fact that a highly attractive jobseeker applies to an employer of relatively low attractiveness may well give the employerinformation that the job seeker has an extreme character. This is due to the fact thathighly attractive job seekers of median character may not apply to such an employer. Wehope to develop these ideas in our forthcoming work.

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Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland.E-mail address: [email protected]

Department of Economics, Kemmy Business School, University of Limerick, Limerick,Ireland.

E-mail address: [email protected]

A Model of Matching and Multiple Criteria Ramsey Et 04.09

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