Specialists and Generalists: Equilibrium Skill Acquisition Decisions in Problem-solving Populations
SSpecialists and Generalists: Equilibrium Skill AcquisitionDecisions in Problem-solving Populations
Katharine A. Anderson b, ∗ a Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Abstract
Many organizations rely on the skills of innovative individuals to create value,including academic and government institutions, think tanks, and knowledge-based firms. Roughly speaking, workers in these fields can be divided into twocategories: specialists, who have a deep knowledge of a single area, and gener-alists, who have knowledge in a wide variety of areas. In this paper, I examinean individual’s choice to be a specialist or generalist. My model addresses twoquestions: first, under what conditions does it make sense for an individual toacquire skills in multiple areas, and second, are the decisions made by individu-als optimal from an organizational perspective? I find that when problems aresingle-dimensional, and disciplinary boundaries are open, all workers will spe-cialize. However, when there are barriers to working on problems in other fields,then there is a tradeoff between the depth of the specialist and the wider scope ofproblems the generalist has available. When problems are simple, having a widevariety of problems makes it is rational to be a generalist. As these problemsbecome more difficult, though, depth wins out over scope, and workers againtend to specialize. However, that decision is not necessarily socially optimal–ona societal level, we would prefer that some workers remain generalists.
Keywords:
Skill acquisition, specialization, jack-of-all-trades, problem solving,knowledge based production, human capital
JEL Codes: J24, O31, D00, M53, I23 ∗ Corresponding author: +1-(412) 427-1904
Email address: [email protected] (Katharine A. Anderson)
Preprint submitted to Journal of Economic Behavior and Organization September 10, 2018 a r X i v : . [ phy s i c s . s o c - ph ] J u l hanks to Scott Page and Ross O’Connell. This work was supported by theNSF and the Rackham Graduate School, University of Michigan. Computingresources supplied by the Center for the Study of Complex Systems, Universityof Michigan. pecialists and Generalists: Equilibrium Skill AcquisitionDecisions in Problem-solving Populations Katharine A. Anderson b, ∗ b Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA
1. Introduction
Many organizations rely on the skills of innovative individuals to create value.Examples include academic institutions, government organizations, think tanks,and knowledge-based firms. Workers in these organizations apply a variety ofskills to in order to solve difficult problems: architects design buildings, bio-chemists develop new drugs, aeronautical engineers create bigger and betterrockets, software developers create new applications, and industrial designerscreate better packaging materials. Their success–and thus the success of the or-ganizations they work for–is dependent on the particular set of skills that theyhave at their disposal, but in most cases, the decision of which skills to acquireis made by individuals, rather than organizations. The perception is that theseworkers choose to become more specialized as the problems they face becomemore complex (Strober (2006)) . This perception has generated a countervailingtide of money and institutional attention focused on promoting interdisciplinaryefforts. However, we have very little real understanding of what drives an indi-vidual’s decision to specialize.Roughly speaking, workers in knowledge-based fields can be divided into twocategories: specialists, who have a deep knowledge of a single area, and general-ists, who have knowledge in a wide variety of areas. In this paper, I consider an ∗ Corresponding author: +1-(412) 427-1904
Email address: [email protected] (Katharine A. Anderson) This dichotomy is often summed up in the literature via a metaphor used by Isaiah Berlin
Preprint submitted to Journal of Economic Behavior and Organization September 10, 2018 ndividual’s decision to be a specialist or a generalist, looking specifically at twopreviously unaddressed questions. First, under what conditions does it makesense for an individual to acquire skills in multiple areas? And second, are thedecisions made by individuals optimal from an organizational perspective?Most of the work done on specialists and generalists is focused on the rolesthe two play in the economy. Collins (2001) suggests that specialists are morelikely to found successful companies. Lazear (2004 and 2005), on the otherhand, suggests that the successful entrepreneurs should be generalists–a theorysupported by Astebro and Thompson (2011), who show that entrepreneurs tendto have a wider range of experiences than wage workers. Tetlock (1998) findsthat generalists tend to be better forecasters than specialists. In contrast, a widevariety of medical studies (see, for example, Hillner et al (2000) and Nallamothuet al (2006)), show that outcomes tend to be better when patients are seenby specialists, rather than general practitioners. However, none of this workconsiders the decision that individuals make with respect to being a specialistor generalist. While some people will always become generalists due to personaltaste, the question remains: is it ever rational to do so in the absence of apreference for interdisciplinarity? And is the decision that the individual makesoptimal from a societal perspective?There is evidence that being a generalist is costly. Adamic et al (2010) showthat in a wide variety of contexts, including academic research, patents, andcontributions to wikipedia, the contributions of individuals with greater focustend to have greater impact, indicating that there is a tradeoff between thenumber of fields an individual can master, and her depth of knowledge in each.This should not be surprising. Each of us has a limited capacity for learning newthings–by focusing on a narrow field of study, specialists are able to concentrate in an essay on Leo Tolstoy: “The fox knows many things, but the hedgehog knows one bigthing” (Berlin (1953)) In other words, foxes are generalists with a wide variety of tools toapply to problems (albeit sometimes inexpertly) and hedgehogs are specialists who have asingle tool that they can apply very well.
2. Model
I construct a two period model. In period 1, the workers face a distributionof problems and each worker chooses a set of skills. In period 2, a problem isdrawn from the distribution, and the workers attempt to solve it using the skillsthey acquired in period 1. I will solve for the equilibrium choice of skills inperiod 1.Let S be the set of all possible skills. The skills are arranged into 2 dis-ciplines, d and d , each with K skills, s d ...s Kd . An example with six skillsarranged into two disciplines is shown in Figure 2.1. A specialist is a person whochooses skills within a single discipline. A generalist is a person who choosessome skills from both disciplines.A problem, y , is a task faced by the workers in the model. A skill is a pieceof knowledge that can be applied to the problem in an attempt to solve it. Eachskill s kd ∈ S has either a high probability ( H ) or a low probability ( L ) of solvingthe problem. I will define a problem by the matrix of probabilities that eachskill will solve the problem. That is, y = y y ... ... y K y K where y kd = H ifskill k in discipline d has a high probability of solving the problem and L if ithas a low probability of solving the problem. So, for example, if there are two Skills are defined as bits of knowledge, tools, and techniques useful for solving problemsand not easily acquired in the short run. See Anderson (2010) for a model with a similartreatment of skills. iscipline 1 (d )s s s Discipline 2 (d )s s s Figure 2.1: Two disciplines, each with three skills. disciplines, each with three skills, a problem might be y = L HH HH L meaning that two of the skills in each discipline have a high probability of solvingthe problem, and one skill in each discipline has a low probability of solving theproblem. Define h ≡ − H and l ≡ − L .The mechanics of the model are as follows. In period 1, the workers, i ...i N ,each choose a set of skills A i ⊂ S . In period 2, the workers attempt to solvea problem using those skills. I will assume that workers have a capacity forlearning skills, which limits the number of skills they can obtain. In the currentcontext, I will assume that all workers all have the same capacity for learningnew skills, and that all skills are equally costly to obtain. Let M ∈ Z + representan individual’s capacity for new skills and let q = 1 be the cost of acquiring a The case where workers have different capacities would obviously be an interesting exten-sion, as would the case where different skills had different costs. c , for learning skills in a newdiscipline. That is, a worker pays c to obtain the first skill in a discipline,and q = 1 for every additional skill in that discipline. For simplicity, I willassume that M = K + c . This assumption means that a specialist can obtainall K skills in one discipline, and a generalist can obtain a total of K − c skillsspread over the two disciplines.Although workers in period 1 do not know the particular problem they willface in period 2, they do know the distribution, ∆ , from which those problemswill be drawn. In particular, they know the probability that each skill will bean H skill or an L skill. For simplicity, I will make two assumptions about thedistribution of problems: 1) skills are independent, meaning that the probabilitythat skill s kd is an H skill is independent of the probability that skill s k (cid:48) d (cid:48) is anH skill and 2) skills are symmetric within disciplines , meaning that every skillin a discipline has an equal probability of being an H skill. This knowledge of the distribution of problems can be translated into knowl-edge about individual skills. Let δ d be the probability that a skill in discipline d is an H skill–that is, δ d = E [ P rob ( y kd = H )] where the expectation is takenover the distribution of problems, ∆ . The vector of probabilities in the twodisciplines, δ = [ δ , δ ] , is known ex ante. Workers choose their skills in period 1 to maximize their expected probabilityof solving the problem in period 2. A Nash equilibrium of this game is a choice ofskill set for each worker in the population, A = { A ...A N } , such that no workerhas an incentive to unilaterally change her skill set, given the distribution ofproblems. This assumption means that skills must be applied more or less independently. That is,it cannot be the case that skills are used in combination to solve problems, or that skills buildon one another. This assumption simplifies the decision making process for generalists. When skills aresymmetric within a discipline, a generalist’s skill acquisition decision is simply a division ofher skills across the two disciplines–within a discipline, she can choose her skills at random. . Results: Specialization and Barriers Between Disciplines In this section, I consider two questions. The first question concerns indi-vidual decision-making–what is the equilibrium skill acquisition decision of theworkers? Under what conditions do individuals decide to generalize? The sec-ond question concerns the optimality of that population from an organizationalperspective. Is the equilibrium population optimal?Note that in order to simplify the exposition, I will consider a special casewhere all disciplines are equally useful in expectation–that is, where δ = δ = δ .It is straightforward to generalize the results to a case where δ (cid:54) = δ (seeAppendix for the details). Given that generalists pay a significant penalty for diversifying their skills,it is difficult to explain the existence of generalists in the population. Theorem1 states that if workers can work on any available problem, then there will beno generalists in equilibrium.
Theorem 1.
If skills are independent and symmetric within discipline, andworkers can work on any available problem, then no worker will ever want to bea generalist and the equilibrium population will contain only specialists.Proof.
The ex ante probability that a specialist in discipline i will be able tosolve a problem from a given distribution, ∆ , is E [ P ( S i )] = (cid:88) y P rob ( one of skills solves y ) ∗ ∆ ( y )= (cid:88) y (1 − P rob ( none do )) ∗ ∆ ( y )= 1 − K (cid:88) n i =0 h n i l K − n i (cid:18) Kn i (cid:19) δ n i (1 − δ ) K − n i = 1 − ( δh + (1 − δ ) l ) K where n i is the number of H skills in discipline i in a particular problem, y .9ow, consider a generalist who is spreading his skills across both disciplines.The ex ante probability that a generalist with x skills in discipline and K − c − x skills in discipline 2 will solve a problem from a given distribution, ∆ , is E [ P ( G )] = 1 − (cid:88) y P rob ( none of skills solve y ) ∗ ∆ ( y )= 1 − ( δh + (1 − δ ) l ) K − c − ( δh + (1 − δ ) l ) K > − ( δh + (1 − δ ) l ) K − c , and thus no individual willever be a generalist in two disciplines. (See Appendix for the same result with δ (cid:54) = δ )Note that this result generalizes to a case with more than two disciplines.Generalists do worse as they add skills in additional disciplines, so this resultholds regardless of the number of disciplines a generalist spreads himself across. Theorem 1 clearly indicates that when workers can solve problems in otherfields, there is no advantage to being a generalist. However, in practice, theremay be many barriers between disciplines that prevent a worker in one disci-pline from solving problems in another. Cultural or institutional barriers mayprevent her from working on questions in other disciplines, either because re-sources are not forthcoming or because it is difficult to get compensated forwork in other areas. Communication barriers are also a significant impedimentto interdisciplinary work–although a software engineers may have skills usefulin solving user interface problems, field-specific jargon may make it difficult forher to communicate her insights. If communication barriers are severe enough,she may even have difficulty understanding what open questions exist. Finally,a person in one field may simply be unaware of problems that exist in otherfields, even if her skills would be useful in solving them.Barriers to working on problems outside ones discipline give us the abilityto talk about the “scope” of a worker’s inquiry. Generalists are able to work ona broader set of problems, and thus their scope is larger than that of specialists.10here is therefore a tradeoff between the depth of skill gained through special-ization and the scope gained through generalization. A specialist has a depthof skill that gives her a good chance of solving the limited set of problems inthe area she specializes in. Generalists have a limited number of skills, but areable to apply those skills to a much broader set of problems. Thus, the choicebetween being a specialist and a generalist can be framed in terms of a tradeoffbetween the depth of one’s skill set and scope of one’s problem set.More formally, choice of whether to specialize will depend on two parameters.First, let π ( δ, h, l ) ≡ ( δh + (1 − δ ) l ) be the expected probability that a skillwon’t be able to solve a problem drawn from ∆ . When π is large, the probabilitythat any one skill will solve the problem is very low. Thus, we can think ofproblems becoming more difficult as π increases. Second, let φ be the fractionof all problems that occur in discipline 1. When φ is very large or very small,most of the problems fall in one field or another, limiting the value of increasingthe scope of the problem set.These two parameters– φ and π –define a range in which workers will chooseto generalize in equilibrium. This range is illustrated in Figure 3.1. This di-agram illustrates the tradeoff between depth and scope. When problems areeasy to solve, scope is more valuable than depth. However, as π increases andproblems become more difficult, depth wins out over scope, and the range inwhich individuals choose to generalize shrinks.Theorem 2 summarizes these results. Theorem 2.
If skills are independent and symmetric within discipline, andthere are barriers to working on problems in other disciplines, then workers willgeneralize if − (cid:16) − π K − c − π K (cid:17) ≤ φ ≤ − π K − c − π K where φ is the fraction of problemsassigned to discipline 1. If φ > − π K − c − π K , then workers will all specialize indiscipline 1 and if φ < − (cid:16) − π K − c − π K (cid:17) then workers will all specialize in discipline2.Proof. In this case, the ex ante probability that a problem is solved by a special-ist is φ (cid:16) − ( δh + (1 − δ ) l ) K (cid:17) for a specialist in discipline 1 and (1 − φ ) (cid:16) − ( δh + (1 − δ ) l ) K (cid:17) .2 0.4 0.6 0.8 1.00.20.40.60.81.0 ! problems are harderproblems are easier ϕ Generalists SpecialistsSpecialists
Figure 3.1: Equilibrium skill acquisition decisions when k = 3 and c = 1 . for a specialist in discipline 2. Since generalists can work on problems in bothdisciplines, their expected probability of solving the problem is − ( δh + (1 − δ ) l ) K − c .A worker will generalize if E [ P ( S )] < E [ P ( G )] and E [ P ( S )] < E [ P ( G )] .The result follows immediately. (See Appendix for the same result with δ (cid:54) = δ ) Note that the size of the regions in which workers specialize depends on howcostly it is to diversify ones skills. As the fixed cost of learning something in anew discipline increases ( c ↑ ), the regions in which people specialize grow. In this section, I consider whether this distribution of specialists and gener-alists in the population is optimal, from a societal perspective. There is reasonto believe that it would not be. From a societal standpoint, we would like tomaximize the probability that someone manages to solve the problem. Thismeans that as a society, we would prefer to have problem solvers apply as widea range of skills as possible. But workers who diversify their skills obtain fewer12 eneralists ! Specialists (d ) Specialists (d )Society Prefers Specialists (d )Society Prefers Specialists (d ) Society Prefers GeneralistsRanges in which generalists are under-provided Figure 3.2: skills overall, which tends to make the individual want to specialize. The resultof this disconnect between individual and social welfare is a range in which gen-eralists are under provided (see Figure 3.2). As problems become more difficult,this region of suboptimality grows, as is illustrated in Figure 3.3.Theorem 3 summarizes these results.
Theorem 3.
If skills are independent and symmetric within discipline, andthere are barriers to working on problems in other disciplines, then there is arange of values for φ (the fraction of problems assigned to discipline 1) such thatgeneralists are underprovided in the equilibrium population of problem solvers.In particular, generalists are underprovided when − π K − c − π K < φ < − π N ( K − c ) − π NK or − − π N ( K − c ) − π NK < φ < − − π K − c − π K .Proof. The probability that at least one of the N problem-solvers in the popu-lation solves the problem is − P rob ( none of them do ) . If all of the individualsin the population are specialists in discipline 1, then with probability φ , eachspecialist has a probability − π K of solving the problem and π K of not solv-ing it. With probability − φ , the problem is assigned to the other discipline,and no specialist solves it. Thus, the probability of someone in a population of13 .2 0.4 0.6 0.8 1.00.20.40.60.81.0 ! problems are harderproblems are easier ϕ Workers generalize and it is optimalWorkers specialize but generalists are socially optimalWorkers specialize and it is optimal
Figure 3.3: Regions of social suboptimality for k = 3 , c = 1 , N = 10 discipline 1 specialists solving the problem is P rob ( one of N solve it ) = 1 − P rob ( none of N solve it )= 1 − [ φP rob ( none solve problem in d )+ (1 − φ ) P rob ( none solve problem in d )]= 1 − (cid:104) φP rob ( one fails ) N + (1 − φ ) ∗ (cid:105) = 1 − (cid:104) φ (cid:0) π K (cid:1) N + (1 − φ ) ∗ (cid:105) = φ (cid:0) − π KN (cid:1) On the other hand, if they are all generalists, then the probability of at leastone solving the problem is
P rob ( one of N solve it ) = 1 − P rob ( none of N solve it )= 1 − (cid:0) π K − c (cid:1) N = 1 − π N ( K − c ) Society is better off with a population of generalists when − π N ( K − c ) >φ (cid:0) − π KN (cid:1) , which is true when φ < − π N ( K − c ) − π NK . However, there is a populationof generalists when φ ≤ − π K − c − π K . It is always the case that − π K − c − π K ≤ − π N ( K − c ) − π NK .14o if − π K − c − π K < φ < − π N ( K − c ) − π NK , then society is better off with a population ofgeneralists, but has a population of specialists.We can make a similar argument for specialists in discipline 2. Society is bet-ter off with a population of generalists when − π N ( K − c ) > (1 − φ ) (cid:0) − π KN (cid:1) ,which is true when φ > − − π N ( K − c ) − π NK . However, there is a population of general-ists when φ > − − π K − c − π K . It is always the case that − − π N ( K − c ) − π NK ≤ − − π K − c − π K .So if − − π N ( K − c ) − π NK < φ < − − π K − c − π K , then society is better off with a popu-lation of generalists, but has a population of specialists. (See Appendix for thesame result with δ (cid:54) = δ )Note that the size of the regions of suboptimality will depend on the numberof individuals in the population. As N increases, the suboptimal regions becomelarger.
4. An Extension: Problems with Multiple Parts
In the previous section, I showed that barriers to addressing problems inother disciplines can induce problem solvers to diversify their skills. In this sec-tion, I consider an extension of the previous model, which highlights a secondscenario in which individuals can be incentivized to acquire skills in multipledisciplines: problems with multiple parts. As problems become increasinglycomplicated, they may be broken down into many different sub-problems. Al-though in some cases, these subproblems may all be best addressed within asingle discipline, in others, different subproblems will be best addressed usingdifferent skills. In this section, I show that when problems are multidisciplinary –that is, when different parts of a problem are best addressed using differentdisciplines–then a population of generalists can be sustained.
As in the previous model, skills in the set S are divided into two disciplines, d and d . Workers use their skills to address a problem, the nature of which isnot known ex ante. They will choose to be a specialist or generalist in period 115o maximize their chances of solving the problem in period 2. But now, supposeeach problem consists of two parts, y and y . In order to solve the problem,an individual must solve all parts of the problem. Each part of the problemis addressed independently by the skills in each of the disciplines. Thus, muchas before, we can define the parts of the problem by a matrix of probabilitiesthat each skill will solve the problem. That is, y i = y i y i ... ... y iK y iK where y ikd = H if skill k in discipline d has a high probability of solving part i and L if it has a low probability of solving part i .As in the previous section, I will assume that for each part of the problem,skills are independent ( P rob ( y ikd = H ) uncorrelated with P rob ( y ik (cid:48) d (cid:48) = H ) ) andskills are symmetric within disciplines ( P rob ( y ikd = H ) = P rob ( y ijd = H ) ) .As before, the probability that a given skill is an H skill is not known exante. However, the workers know the expected probability that a skill is an H skill. I will allow the expected probabilities to vary across parts of the problem–in other words, it is possible that a discipline will be more useful in solving oneof the parts of the problem than in solving the other part of the problem. Let δ id be the probability that a skill from discipline d is an H skill for part i ofthe problem. That is, δ id = E (cid:2) P rob (cid:0) y ikd = H (cid:1)(cid:3) . The matrix δ = δ δ δ δ describes a distribution of problems, ∆ , and is known ex ante. An entry in the i th column of that matrix is the vector of probabilities that a skill in each of thedisciplines will be useful for solving part i of the problem.We can categorize the problems according to the relative usefulness of thetwo disciplines in the two parts of the problem. There are two categories for theproblems:1. One discipline is as or more useful for both parts of the problem: δ i ≥ δ i ∀ i
2. One discipline is more useful for part 1 and the other discipline is more This is essentially an adaptation of Kremer’s O-ring Theory (Kremer (1993)). ategory 2: Each discipline useful for a different partCategory 1: One discipline is always as or more useful on all parts X XX X XX OOX XO O X XX O
Part 1 Part 2 Discipline 1Discipline 2
X XX XX OO X O XX O O XO X O OO OX OO O etc...
Figure 4.1: useful for part 2: δ i > δ i and δ j < δ j These categories are illustrated in Figure 4.1.If a problem falls into the first category, then the results are similar to thoseobtained in Section 3. In particular, if there are no barriers to working onproblems in other disciplines, then all workers will specialize. If a problem fallsinto the second category, then the results do not resemble any of those alreadyexplored. Problems with multiple parts, each of which is best addressed withinthe context of a different discipline, are often referred to as multidisciplinary.
Generalists have an advantage in multidisciplinary problems, because they canapply different types of skills to different parts of a problem. For example, sup-pose a scientist is look at nerve conduction in an organism. That problem mayhave elements are are best addressed using biological tools, and other elementsthat are best addressed using physics tools. An individual with both biologyand physics skills will have an advantage over someone who is forced to use (forexample) physics skills to solve both parts of the problem. The below statesthat when problems are multidisciplinary, it can be rational to be a generalist,even in the absence of barriers to working in other fields.More formally, suppose that if a worker uses the “right” discipline for a partof a problem, then there is a probability δ that a skill in that discipline is17seful ( δ = P rob (cid:0) y ikd = H (cid:1) when d is the right discipline to use for part i of the problem). If she uses the “wrong” discipline, then there is a probability δ that a skill in that discipline is useful ( δ = P rob (cid:0) y ikd = H (cid:1) when d is thewrong discipline to use for part i of the problem). This is without loss ofgenerality, because the only thing that makes a problem multidisciplinary is theordering of the usefulness of the disciplines. Further, let π = δ h + (1 − δ ) l and π = δ h + (1 − δ ) l . These represent the probability that a skill in theright discipline will not solve a part of a problem and the probability that a skillin the wrong discipline will not solve part of the problem. Note that π < π .When the efficacy of the two disciplines is very different ( π (cid:28) π ), thenusing the right skill for the job has a large effect on the probability of solvingthe problem as a whole, and it will be rational to obtain skills in multipledisciplines. Figure 4.2 illustrates the region in which individuals choose to begeneralists and specialists, and Theorem 4 summarizes the result. Theorem 4.
If skills are independent and symmetric within discipline, andproblems multidisciplinary (eg: δ = δ δ δ δ with δ > δ ) then there is aset of values of π = δ h + (1 − δ ) l and π = δ h + (1 − δ ) l such that itis individually optimal for workers to be generalists, even when there are nobarriers to solving problems in other fields. In particular, workers will becomegeneralists when (cid:16) − π K − c π K − c (cid:17) > (cid:0) − π K (cid:1) (cid:0) − π K (cid:1) Proof.
WLOG, consider the case where δ = δ δ δ δ with δ > δ . A spe-cialist in discipline i will have K skills in discipline i . The expected probabilitythat her skills will solve both parts of the problem is E [ P ( success on part 1 )] ∗ E [ P ( success on part 2 )] , which is (cid:0) − π K (cid:1) (cid:0) − π K (cid:1) where π = δ h +(1 − δ ) l and π = δ h + (1 − δ ) l .A generalist will have skills in both disciplines. In this case, it will be optimalfor a generalist to split her skills evenly between the two disciplines, and shewill obtain K − c skills in each. The expected probability that she will solve both18 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 π π S pe c i a li s t s Generalists π = π Equilibrium Skill Acquisition as a Function of π and π (Two Part Problem with K=3, c=1) Figure 4.2: Equilibrium skill acquisition decisions when problems are multidisciplinary, and k = 3 and c = 1 . parts of the problem is (cid:16) − π K − c π K − c (cid:17) .Thus, individuals choose to generalize, when (cid:16) − π K − c π K − c (cid:17) > (cid:0) − π K (cid:1) (cid:0) − π K (cid:1) . This region, as a function of π and π , is illustrated in Figure 4.2 for K = 3 and c = 1 . The boundary of this region is defined by the equation π K + π K = ( π π ) K − c − π π ) K − c + ( π π ) K .As would be expected, the region where individuals specialize shrinks asthe costs to generalizing ( c ) become smaller, relative to the individual’s total Note that when δ = δ , we have a case that fits into the first category in the taxonomy ofproblem distributions in Figure 4.1. Since δ = δ = ⇒ π = π , we can use this calculationto verify the claim made above that in the case where skills are symmetric, the results are thesame as in Section 3. ( M = K + c ) .
5. Conclusion
Being a generalist is costly. Every new area of expertise comes at consid-erable fixed cost, in the form of a new literature, new jargon, and new basicideas. However, there are clearly a large (and growing) number of individuals inresearch communities who choose to do so. This raises the question of whetherthat decision is ever individually rational? And is there a reason to believe thatfewer people choose to be generalists than is socially optimal?This paper suggests that being a generalist can be a rational decision underparticular conditions. In particular, obtaining a broad range of skills is rationalif there are significant barriers to working on questions in fields with which one isunfamiliar. Those who pay the initial price of learning the jargon and literatureof a new field reap the benefits in the form of a larger pool of problems to solve. Itcan also be rational to be a generalist if problems are multidisciplinary–that is, ifdifferent parts of a problem are best addressed using skills in different disciplines.Moreover, because individuals bear the costs of becoming generalists, we willtend to have fewer of them than is optimal from a societal standpoint. Thispotential market failure means that in some cases, it is optimal for fundingagencies and private organizations to subsidize individuals in their efforts todiversify their skills and promote interdisciplinary researchers. However, it isunclear whether our current situation is one in which such funding is required.More careful consideration of this question is a good candidate for further work.There are several elements of this model that suggest directions for futureresearch. It would be interesting to consider a case where individuals differ intheir innate capacity for learning skills. This might provide some insight intowhat types of individuals choose to become generalists. Incorporating collab-oration would be another particularly interesting extension. Collaboration hasalways been an important part of problem solving and innovation, and it hasonly become more important over time (see, among others, Laband and Tollison202000), Acedo et al (2006), and Goyal et al (2006)). There is reason to believethat in a collaborative context, the advantage to generalists would be enhanced,because generalists could connect specialists in different fields.On a more general level, there is much to be gained from a better understand-ing of specialization decisions. Research universities, government organizationssuch as NASA, and private enterprises ranging from Genentec to Google arereliant on the skills of individual problem solvers. The decisions these individ-uals make about the breadth skills they obtain have an undeniable effect onthe rate of innovation. However, we still have only a limited understanding ofthe what drives those skill acquisition decisions, and what distinguishes the roleof specialists and generalists in problem solving. Better theoretical models ofthese decisions have the potential to greatly enhance our understanding of thisimportant aspect of such organizations.
References [1] Acedo, F..J., Barroso, C., Casanueva, C., and Galan, J.L., 2006.Co-authorship in Management and Organizational Studies: AnEmpirical and Network Analysis. Journal of Management Studies43 (5) , . Journal of Political Econ-omy 144 (2) , ppendix Theorem 5 is the equivalent of Theorem 1, and states that if individuals canwork on any available problem, then there is no advantage to being a generalist.
Theorem 5.
If skills are independent and symmetric within discipline, andworkers can work on any available problem, then no worker will ever want to bea generalist and the equilibrium population will contain only specialists.Proof.
As above, the ex ante probability that a specialist in discipline i will beable to solve a problem from a given distribution, ∆ , is E [ P ( S i )] = 1 − ( δ i h + (1 − δ i ) l ) K = 1 − π Ki where π i = ( δ i h + (1 − δ i ) l ) .WLOG, suppose δ > δ . Since h < l , this means that π < π and E [ P ( S )] > E [ P ( S )] . Thus, to determine whether any individual will gener-alize, I need to compare E [ P ( S )] to E [ P ( G )] .The ex ante probability that a generalist with x skills in discipline 1, and K − c − x skills in discipline 2 solves a problem from a given distribution, ∆ , is E [ P ( G )] = 1 − ( δ h + (1 − δ ) l ) x ( δ h + (1 − δ ) l ) K − c − x = 1 − π x π K − c − x Since π < π , E [ P ( G )] is strictly increasing in x . This means that ageneralist will set x = K − c − and E [ P ( G )] = 1 − π K − c − π , which is clearlyless than E [ P ( S )] = 1 − π K .Theorem 6 is a generalized version of Theorem 2, and states the parameterrange in which individuals will choose to diversify their skills when there arebarriers to working interdisciplinarily. Theorem 6.
If skills are independent and symmetric within discipline, andthere are barriers to working on problems in other disciplines, then there is a ange of values for φ (the fraction of problems assigned to discipline 1) for whichindividuals will generalize.In particular, the ranges are as follows:If δ = δ = δ , workers will obtain K − c skills spread across the twodisciplines when − − π K − c − π K ≤ φ ≤ − π K − c − π K , K skills in discipline 1 when φ > − π K − c − π K , and K skills in discipline 2 when φ < − − π K − c − π K .If δ > δ , then workers will obtain K − c − skills in discipline 1 andone skill in discipline 2 when − (cid:16) − π K − c − π − π K (cid:17) ≤ φ ≤ − π K − c − π − π K , K skillsin discipline 1 when φ > − π K − c − π − π K , and K skills in discipline 2 when φ < − (cid:16) − π K − c − π − π K (cid:17) .If δ > δ , then workers will obtain K − c − skills in discipline 2 andone skill in discipline 1 when − (cid:16) − π K − c − π − π K (cid:17) ≤ φ ≤ − π K − c − π − π K , K skillsin discipline 1 when φ > − π K − c − π − π K , and K skills in discipline 2 when φ < − (cid:16) − π K − c − π − π K (cid:17) .Proof. In this case, the ex ante probability that a problem is solved by a spe-cialist is φ (cid:0) − π K (cid:1) for a specialist in discipline 1 and (1 − φ ) (cid:0) − π K (cid:1) for aspecialist in discipline 2. Since generalists can work on problems in both disci-plines, their expected probability of solving the problem is − π x π K − c − x where x is the number of skills the generalist chooses to acquire in discipline 1. First,suppose δ > δ . Since h < l , this means that π < π and E [ P ( G )] is strictlyincreasing in x . Thus, a generalist will choose a minimal number of skills in theless useful discipline, and E [ P ( G )] = 1 − π K − c − π .An individual will generalize if E [ P ( S )] < E [ P ( G )] and E [ P ( S )]
If skills are independent and symmetric within discipline, andthere are barriers to working on problems in other disciplines, then there is arange of values for φ (the fraction of problems assigned to discipline 1) such thatgeneralists are underprovided in the equilibrium population of problem solvers.In particular, generalists are underprovided in the following ranges:If δ = δ , then generalists are underprovided when − π K − c − π K < φ < − π N ( K − c ) − π NK or − − π N ( K − c ) − π NK < φ < − − π K − c − π K If δ > δ , then generalists are underprovided when − π K − c − π − π K < φ < − π N ( K − c − π − π NK or − − π N ( K − c − π − π NK < φ < − − π K − c − π − π K If δ > δ , then generalists are underprovided when − π π K − c − − π K < φ < − π π N ( K − c − − π NK or − − π π N ( K − c − − π NK < φ < − − π π K − c − − π K Proof.
First, suppose that δ > δ . The probability that at least one of the N problem-solvers in the population solves the problem is − P rob ( none of them do ) .If all of the individuals in the population are specialists in discipline 1, then ev-ery individual has probability φ of a problem occurring in her discipline. In thatcase, each specialist in discipline has a probability − π K of solving the problemand π K of not solving it. With probability − φ , the problem is assigned to theother discipline, and no specialist solves it. Thus, the probability of someone ina population of specialists solving the problem is P rob ( one of N solve it ) = 1 − P rob ( none of N solve it )= 1 − [ φP rob ( none solve problem in d ) + (1 − φ ) P rob ( none solve problem in d )]= 1 − (cid:104) φP rob ( one fails ) N + (1 − φ ) ∗ (cid:105) = 1 − (cid:104) φ (cid:0) π K (cid:1) N + (1 − φ ) ∗ (cid:105) = φ (cid:0) − π KN (cid:1) Through a similar argument, if everyone in the population is a specialistin discipline 2, then the probability that someone in the population solves theproblem is (1 − φ ) (cid:0) − π KN (cid:1) . 25f everyone in the population is a generalists, then the probability of at leastone person in solving the problem is P rob ( one of N solve it ) = 1 − P rob ( none of N solve it )= 1 − (cid:0) π K − c − π (cid:1) N = 1 − π N ( K − c − π N Society is better off with a population of generalists than a population ofdiscipline 1 specialists when − π N ( K − c − π N > φ (cid:0) − π KN (cid:1) , which is truewhen φ < − π N ( K − c − π N − π NK . However, there is a population of generalists when φ ≤ − π K − c − π − π K . It is always the case that − π K − c − π − π K ≤ − π N ( K − c − π − π NK . Thus,if − π K − c − π − π K < φ < − π N ( K − c − π − π NK , then society is better off with a populationof generalists, but has a population of specialists.Through a similar argument, society is better off with a population of gen-eralists than a population of discipline 2 specialists when − π N ( K − c − π N > (1 − φ ) (cid:0) − π KN (cid:1) , which is true when φ > − − π N ( K − c − π N − π NK . However, thereis a population of generalists when φ ≤ − − π K − c − π − π K . It is always the casethat − − π N ( K − c − π − π NK ≤ − − π K − c − π − π K . Thus, if − − π N ( K − c − π − π NK < φ < − − π K − c − π − π K , then society is better off with a population of generalists, buthas a population of specialists.The proof for δ > δ is similar. See the proof of Theorem 3 for the casewhere δ = δ2