Groups of diverse problem solvers can outperform groups of high-ability problem solvers

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1 Groups of diverse problem solvers ca outperform groups of high-ability problem solvers Lu Hog ad Scott E. Page Michiga Busiess School ad Complex Systems, Uiversity of Michiga, A Arbor, MI ; ad Departmet of Fiace, Loyola Uiversity, Chicago, IL Edited by William J. Baumol, New York Uiversity, New York, NY, ad approved September 17, 2004 (received for review May 25, 2004) We itroduce a geeral framework for modelig fuctioally diverse problem-solvig agets. I this framework, problem-solvig agets possess represetatios of problems ad algorithms that they use to locate solutios. We use this framework to establish a result relevat to group compositio. We fid that whe selectig a problem-solvig team from a diverse populatio of itelliget agets, a team of radomly selected agets outperforms a team comprised of the best-performig agets. This result relies o the ituitio that, as the iitial pool of problem solvers becomes large, the best-performig agets ecessarily become similar i the space of problem solvers. Their relatively greater ability is more tha offset by their lack of problem-solvig diversity. A diverse society creates problems ad opportuities. I the past, much of the public iterest i diversity has focused o issues of fairess ad represetatio. More recetly, however, there has bee a risig iterest i the beefits of diversity. I the legal cases surroudig the Uiversity of Michiga s admissios policies ad i efforts to curtail affirmative actio i Califoria, Texas, ad elsewhere, there have bee claims that diverse perspectives improve collective uderstadig ad collective problem solvig. Coicidet with this political ad legal wraglig has bee a effort o the part of scholars to idetify how to exploit this diversity both i solvig hard computatioal problems (1, 2) ad i huma orgaizatios (3). I the commo uderstadig, diversity i a group of people refers to differeces i their demographic characteristics, cultural idetities ad ethicity, ad traiig ad expertise. Advocates of diversity i problem-solvig groups claim a likage amog these sorts of diversity (which we will refer to as idetity diversity) ad what we might call fuctioal diversity, differeces i how people represet problems ad how they go about solvig them. Give that likage, they coclude that, because of their greater fuctioal diversity, idetity-diverse groups ca outperform homogeeous groups (4 6). Buildig o earlier ideas from the psychology ad artificial itelligece literatures (7), we describe a mathematical framework for modelig problem solvers that captures the fuctioal diversity that cogitive psychologists ad orgaizatioal theorists claim is correlated with idetity diversity. I our framework, agets possess iteral represetatios of problems, which we call perspectives, ad algorithms that they use to locate solutios, which we call heuristics. Together, a perspective-heuristic pair creates a mappig from the space of possible solutios to itself. A diverse group is oe whose agets mappigs are diverse. Our perspective-heuristic framework is ot miimal, because we show i a earlier paper (8) that two problem solvers with distict perspectives ad heuristics ca act idetically i the space of solutios. However, the advatage of the full framework is that it geeralizes models i the computer sciece literature that focus o diverse heuristics (1, 2), ad models i the orgaizatioal behavior ad psychology literature, which ofte emphasize diverse perspectives (3, 4, 6). The coclusio that idetity-diverse groups ca outperform homogeeous groups due to their greater fuctioal diversity rests upo a well accepted claim that if agets across groups have equal ability, fuctioally diverse groups outperform homogeeous groups. It has also bee show that fuctioally diverse groups ted to outperform the best idividual agets, provided that agets i the group are early as good (1). These results still leave ope a importat questio: Ca a fuctioally diverse group whose members have less ability outperform a group of people with high ability who may themselves be diverse? The mai result of our paper addresses exactly this questio. Cosider the followig sceario: A orgaizatio wats to hire people to solve a hard problem. To make a more iformed decisio, the orgaizatio admiisters a test to 1,000 applicats that is desiged to reflect their idividual abilities i solvig such a problem. Suppose the applicats receive scores ragig from 60% to 90%, so that they are all idividually capable. Should the orgaizatio hire (i) the perso with the highest score, (ii) 20 people with the ext 20 highest scores, or (iii) 20 people radomly selected from the applicat pool? Igorig possible problems of commuicatio withi a group, the existig literature would suggest that ii is better tha i, because more people will search a larger space, but says little about ii vs. iii. The ituitio that agets with the highest scores are smarter suggests that the orgaizatio should hire ii, the idividually bestperformig agets. The ituitio that the radomly selected agets will be fuctioally diverse suggests that the orgaizatio should hire iii, the radomly selected oes. I this paper, we provide coditios uder which iii is better tha ii. Thus, the focus of our aalysis is o the tesio betwee the idividual abilities i a group ad its fuctioal diversity. Uder the set of coditios we idetify, as the iitial pool of problem solvers becomes large, the fuctioal diversity of the group of idividually best-performig agets ecessarily becomes very small. Ultimately, the gai i idividual abilities is more tha offset by the fuctioal diversity of a group of radomly selected people. It is i this sese that we might say diversity trumps ability. This tesio is established regardless of the precise ature of group cooperatio. Complemetary to our study, a computer sciece literature (2) has bee addressig the questios of how to make the diverse group as effective as possible, ad how ad whe the algorithms should share hits, iformatio, ad solutios, takig as a give that a diverse group does better tha a idividual. Orgaizatioal theorists have also focused o exploitig diversity. Their challege has bee how to ecourage people with diverse idetities ad backgrouds to work together productively (3). This paper focuses exclusively o fuctioal diversity: differeces i how people ecode problems ad attempt to solve them. The claim that perspectives ad heuristics may be iflueced by race, geography, geder, or age has much to recommed it, as does the claim that perspectives ad tools are shaped by experieces, traiig, ad prefereces. However, eve whe applyig our result to those cases whe idetity diversity has bee show This paper was submitted directly (Track II) to the PNAS office. To whom correspodece should be addressed. luhog@umich.edu by The Natioal Academy of Scieces of the USA PNAS November 16, 2004 vol. 101 o

2 to correlate with fuctioal diversity, we eed to be acutely aware that idetity-diverse groups ofte have more coflict, more problems with commuicatio, ad less mutual respect ad trust amog members (3, 9 11). The ext sectio presets the basic model of diverse problemsolvig agets. A Computatioal Experimet reports simulatio results establishig that a diverse group ca ofte outperform a group of the best. A Mathematical Theorem explores the logic behid the simulatio results ad provides coditios uder which diversity trumps ability. Some implicatios of our results are discussed i Cocludig Remarks. A Model of Diverse Problem Solvers Our model cosists of a populatio of problem solvers of limited ability who attempt to maximize a fuctio V that maps a set of solutios X ito real umbers. For example, the set of solutios could be the set of possible gasolie egie desigs, with the value fuctio geeratig the efficiecy of various desigs. Problem solvers have iteral laguages i which they ecode solutios. This iteral laguage ca be iterpreted at the eurological level, our brais perceive ad store iformatio, or metaphorically, we iterpret problems based o our experiece ad traiig. The represetatio of solutios i the problem solver s iteral laguage is called a perspective. Formally, a perspective is a mappig M from the set of solutios ito the aget s iteral laguage. A problem solver s heuristic is a mappig, deoted by A, from solutios i her iteral laguage to subsets of solutios. It captures how she searches for solutios. Give a particular solutio, the subset geerated by the mappig A is the set of other solutios the aget cosiders. I this way, the problem-solvig ability of a aget is captured by her perspective ad heuristic pair (M, A). Two agets ca differ i either dimesio or alog both dimesios. Thus agets ca have diverse perspectives (as psychologists assume), diverse heuristics (as computer scietists assume), or both. A solutio is a local optimum for a idividual aget if ad oly if whe that aget ecodes the problem ad applies her heuristic, oe of the other solutios she cosiders has a higher value. The set of local optima for a aget together with the size of their basis of attractio determies the aget s expected performace o the problem, or what we might call the aget s ability. It follows that a group of agets ca get stuck oly at a solutio that lies i the itersectio of the idividual agets local optima. This observatio is idepedet of the procedure by which agets work together as a team. However, differet procedures for iteractig amog the agets will geerally result i differet basis of attractio for those solutios that are local optima for all of the agets. Thus, how the team works together will matter for team performace (2). A Computatioal Experimet I a series of computatioal experimets we coducted based o this framework, we fid that a collectio of diverse agets ca be highly effective collectively, locatig good ad ofte optimal solutios, cofirmig the widely accepted belief. More iterestigly, we fid that a radom collectio of agets draw from a large set of limited-ability agets typically outperforms a collectio of the very best agets from that same set. This result is because, with a large populatio of agets, the first group, although its members have more ability, is less diverse. To put it succictly, diversity trumps ability. Here, we report oe such set of computatioal experimets where the secod result is established ad highlighted. We describe i detail how the geeral framework is applied, how idividual ad collective performaces are measured, ad how diversity is defied. We cosider a radom value fuctio mappig the first itegers, {1, 2,..., }, ito real umbers. The value of each of the poits is idepedetly draw accordig to the uiform distributio o the iterval [0, 100]. Agets try to fid maximal values for this radom fuctio. I this set of experimets, we cosider oly agets who have idetical perspectives but allow their heuristics to vary. All agets ecode solutios as poits o a circle from 1 to clockwise. The heuristic that a aget uses allows her to check k positios that lie withi l poits to the right of the status quo poit o the circle. Here, 1 l ad 1 k l. For example, cosider 200, k 3, ad l 12. A problem solver with the heuristic (1, 4, 11) startig at poit 194 would first evaluate poit 195 (194 1) ad compare it with 194. If poit 194 had a higher value, she would the evaluate poit 198 (194 4). If poit 198 had a higher value, she would the check poit 9 ( ). If that poit had a higher value, she the would evaluate poit 10 (9 1). She would keep evaluatig util oe of her three checks located a higher value. Therefore, a heuristic, deoted by ( 1, 2,..., k ), where each i {1, 2,..., l} specifies the positio to check, aturally defies a stoppig poit for a search started at ay iitial poit. Deote by (i) the stoppig poit of applied to iitial poit i. We measure the performace of a aget with a heuristic by the expected value of the stoppig poits, assumig that each poit is equally likely to be the iitial poit, EV; 1 Vi. i Give k ad l, the set of heuristics is well defied. Because the order i which rules are applied matters, the total umber of uique heuristics equals l (l 1) (l k 1). They ca be raked accordig to their expected values. I our experimets, we cosidered eviromets i which a collectio of agets attempt to fid better solutios to the problem either sequetially or simultaeously. Our fidigs do ot seem to deped o which structure was assumed. I the results we report here, agets approach the problem sequetially. The first aget searches util she attais a local optimum. The secod aget begis her search at that poit. After all agets have attempted to locate higher-valued solutios, the first aget searches agai. The search stops oly whe o aget ca locate a improvemet, i.e., util the solutio lies i the itersectio of all agets local optima. The collective performace of agets is the defied as the expected value of the stoppig poits, similar to the defiitio of performace of a idividual aget. The diversity of two heuristics a ad b of the same size k, ( a, b ), is defied by a, b k k i a i, b i, k where ( a i, b i ) 1if a i b i ad 0 else. For example, for a (5, 6, 9) ad b (9, 5, 6), ( a, b ) 1, because for ay i {1, 2, 3}, a i b i. I the result we report, we set l 12 or 20, k 3, ad the umber of poits o the circle 2,000. We experimeted with l varyig betwee 6 ad 20, k varyig betwee 2 ad 7, ad varyig betwee 200 ad 10,000. Withi these parameter rages, we foud qualitatively similar pheomea. For a give class of agets defied by k ad l, we raked all of the possible agets by their expected values ad created two groups, oe cosistig of, say, the 10 best agets, the agets with the highest expected I aother set of computatioal experimets where a differet problem was beig solved, we cosider agets with the same heuristics but whose perspectives vary. Similar results were foud {Hog, L. & Page, S. E. (2002) Workig paper, Diversity ad Optimality [Loyola Uiversity (Chicago) ad Uiv. of Michiga (A Arbor)]} Hog ad Page

3 Table 1. Result of computatioal experimets Group compositio Performace Diversity, % Te agets ad l 12 Best agets (0.020) (0.798) Radom agets (0.007) (0.232) Twety agets ad l 12 Best agets (0.015) (0.425) Radom agets (0.005) (0.066) Te agets ad l 20 Best agets (0.026) (0.843) Radom agets (0.006) (0.089) Numbers i paretheses are stadard deviatios. values, ad oe cosistig of 10 agets radomly chose from the give class. For l 12, the results from a represetative sigle ru were as follows: The best aget scored 87.3; the worst aget scored 84.3; the average score of the 10 best agets was 87.1, ad the average score of the 10 radomly selected agets was The collective performace of the 10 best agets had a value of 93.2; their average diversity (averaged over all possible pairs) was The collective performace of the 10 radomly selected agets was 94.7; their average diversity was 0.92.** We preset (Table 1) the results averaged over 50 trials. The data show that, o average, the collective performace of the radomly selected agets sigificatly outperforms the group of the best agets. Moreover, the diversity measures show a strikig differece i the costituecy of the two groups. The best group does ot have early as much diversity as the radom group. Whe we elarged the group size from 10 to 20, the radom group still did better, but with a less proouced advatage. The group of the best agets became more diverse. This occurred because the set of heuristics was fiite ad fixed. Table 1 reports data with groups of 20, agai averaged over 50 trials. Next, we icrease the set of possible heuristics (or agets) by settig l 20. Agets ca ow look up to 20 spots ahead o the circle, ad the total umber of agets equals 6,840. Ituitively, we ca make the followig predictios. First, the diversity of the radom group should be greater as a result of the icrease i the umber of heuristics. Secod, this icreased diversity should improve the collective performace of the radom group. Ad third, the icrease i the umber of agets implies that the collective performace of the best group should also improve. The results of this set of experimets are preseted i Table 1. From the data, we see, i fact, that all three predictios occur. Oce agai, diversity is the key to collective performace. A Mathematical Theorem I this sectio, we develop a mathematical theorem that explais the logic behid our ew result: that a radom collectio of itelliget agets outperforms the collectio cosistig of oly the best agets. Followig is a brief summary of the theorem. I the mathematical model, agets wat to maximize a value fuctio that is assumed to have a uique maximum. Cosider a populatio of agets, deoted by, that satisfy the followig assumptios: (i) Agets are itelliget: give ay startig poit, a aget fids a weakly better solutio, ad the set of local optima ca be eumerated. (ii) The problem is difficult: o aget ca always fid the optimal solutio. (iii) Agets are diverse: for ay potetial solutio that is ot the optimum, there exists at least oe aget who ca fid a improvemet. (iv) The best aget is uique. Cosider drawig agets idepedetly from accordig to some distributio. The theorem states that with probability oe, there exist sample sizes N 1 ad N, N 1 N, such that the collective performace of N 1 draw agets exceeds the collective performace of the N 1 idividually best agets i the group of N draw agets. To formulate the theorem precisely, we begi with a set of solutios X ad a give value fuctio V: X 3 [0, 1], which has a uique maximum at x*, ad V(x*) 1. The problem solvers try to locate the solutio x*, but they have limited abilities. Each problem solver uses a search rule to search for the maximum but does ot always ed up there. Suppressig the distictio betwee perspectives ad heuristics, we characterize each problem solver by a mappig : X 3 X ad a probability distributio o X. A problem solver radomly selects accordig to distributio a iitial poit where the search starts. For each x, (x) deotes the local optimum if the aget starts the search at x, that is, (x) is the stoppig poit of the search rule applied to x. I this iterpretatio, the search is determiistic: a iitial poit uiquely determies a stoppig poit. The image of the mappig, (X), is the set of local optima for problem solver. Mathematically, the mappig of a problem solver has to satisfy the followig assumptio: Assumptio 0. X, V((x)) V(x) (ii) ((x)) (x) I geeral, the set of solutios X ca be fiite, deumerable, or a cotiuum. However, to avoid measurig theoretical complicatios, we preset a simpler versio of our result where X is assumed to be fiite. This fiite versio makes the isight more straightforward, although it comes at the cost of trivializig some itricate assumptios ad argumets. For example, the group of the best-performig agets is prove below to be comprised of idetical agets. This is a artifact of the fiite versio. I the geeral versio uder reasoable coditios, the group of the best-performig agets ca be show to be similar, ot ecessarily the same. But this makes the proof more complicated. For each problem solver (, ), we measure her performace by the expected value of her search, which we deote as E(V;, ), i.e., EV;, Vxx. For the purpose of our aalysis, we assume that all agets have the same, ad that has full support. This assumptio does ot dimiish the power of our result, because all the problem-solvig ability of a aget is supposedly captured by the mappig. We ow defie the set of problem solvers we cosider. First, the problem is difficult for all agets uder our cosideratio. Assumptio 1 there exists x X, such that (x) x*. This assumptio simply ecessitates the group settig. Give that has full support, it implies that o sigle aget uder our cosideratio ca always fid the optimum. Secod, we formulate the idea of a diverse group. Assumptio 2 X{x*},? such that (x) x. This assumptio is a simple way to capture the essece of diverse problem-solvig approaches. Whe oe aget gets stuck, there is always aother aget that ca fid a improvemet due to a differet approach. Our last assumptio states that there is a uique best performer i the set of agets we cosider. **Mathematically, the expected diversity of two radomly selected agets equals Hog, L. & Page, S. E. (2002) Workig paper, Diversity ad Optimality [Loyola Uiv. (Chicago) ad Uiv. of Michiga (A Arbor)]. Hog ad Page PNAS November 16, 2004 vol. 101 o

4 Assumptio 3 (Uiqueess). argmax{e(v;, ): } is uique. Let be the uiform distributio. If the value fuctio V is oe to oe, the the uiqueess assumptio is satisfied. Therefore, i the space of all value fuctios we cosider, the uiqueess assumptio is geerically satisfied. We do ot make specific assumptios about how a group of problem solvers work together, other tha requirig that search by a group ca get stuck oly at a poit that is a local optimum for all agets i the group. A example of how this ca be achieved is that agets approach the problem sequetially: wherever a aget gets stuck, the ext perso starts the search at that poit. Let be a set of problem solvers that satisfy Assumptios 1 3 above. Let be a probability distributio over with full support. From, we draw a group of N agets; each aget is draw idepedetly from accordig to. These N agets are ordered by their idividual performaces, E(V;, ). Choose the best N 1 agets. We compare the joit performace of this group of N 1 agets with that of aother group of N 1 agets that is formed by drawig each from idepedetly accordig to. Theorem 1. Let be a set of problem solvers that satisfy Assumptios 1 3 above. Let be a probability distributio over with full support. The, with probability oe, a sample path will have the followig property: there exist positive itegers N ad N 1, N N 1, such that the joit performace of the N 1 idepedetly draw problem solvers exceeds the joit performace of the N 1 idividually best problem solvers amog the group of N agets idepedetly draw from accordig to. Here, there are i fact two idepedet radom evets: oe is to idepedetly draw a group of problem solvers, ad the other is to idepedetly draw a group of problem solvers ad the select a subgroup accordig to their idividual ability. The sample path we speak of i the theorem is the joit sample path of these two idepedet evets. The proof relies o two ideas. First we show (Lemma 1 below) that the idepedetly draw collectio of agets will, with probability oe, fid the optimal solutio as the group becomes large. This lemma is ituitive, give that agets draw idepedetly thus are very ulikely to have commo local optima. As the umber of agets i the group grows, the probability of their havig commo local optima coverges to zero. The secod idea relies o the uiqueess assumptio to show that, with probability oe, as the umber of agets becomes large, the best problem solvers all become similar ad therefore do ot do better tha the sigle best problem solver, who by assumptio caot always fid the optimal solutio. Cosider the first radom evet of formig a group of problem solvers: each problem solver is idepedetly draw from accordig to. Fix a sample path of this radom evet, 1. Let 1 1,..., 1 1 deote the first 1 problem solvers. The joit performace of these 1 problem solvers is the expected value of V(ỹ), where ỹ is a commo local optimum of all 1 agets. The distributio of ỹ is iduced by the probability distributio of the iitial draw,, ad a precise model of how agets work together. The proof of the lemma that follows does ot deped o what the specific model is. Without beig explicit, we assume that ỹ follows distributio 1 : X 3 [0, 1], i.e., x X, Pr ỹ x 1 x. Lemma 1. Pr{ 1 : lim 1 3 V(x) 1 (x) 1} 1. Proof: Fix ay 0 1. Defie A 1 { 1 : 1 V(x) 1 (x) }. Obviously, A 1 { 1 : 1 ( 1 ),..., 1 ( 1 ) have commo local maxima other tha x*}. Thus, PrA 1 Pr 1 : 1 1,..., 1 1 have commo local maxima other tha x*. Let m mi {(): }. Because has full support, m 0. By Assumptio 2 (diversity), for ay x X{x*}, we have ({ : (x) x}) 1 m. By idepedece, Pr 1 : 1 1,..., 1 1 x* Pr 1 : x have commo local maxima other tha x* is a commo local maximum of 1 1,..., 1 1 x* m 1 X 1 m 1. Therefore, PrA 1 X 1 m. 1 By the Borel Catelli Lemma, we have Pr 1:1 Vx 1 x ifiitely ofte 0, which implies Pr 1: lim 13 Vx 1 x 1 1. We ow prove Theorem 1. Proof of Theorem 1: Cosider the secod radom evet, where a group of agets are draw idepedetly from accordig to, ad the a subgroup of the best is formed. By Assumptio 3 (uiqueess), there is a uique problem solver i with the highest idividual performace. Call that aget *. By the law of large umbers, Pr #i,..., : i 2 * 2 : lim * 1. 3 The fractio i the above expressio is the frequecy of *ithe draw. Let be the set of sample paths ( 1, 2 ) that have both of the asymptotic properties above; i.e., defie 1, 2 : lim 1 3 Vx 1 x 1 #i,...,: i 2 * ad lim 3 By Lemma 1, Pr 1. *. Fix ay. Let 1 1 E(V; *, ), which is positive by Assumptio 1 (difficulty). From the first limit above, we kow there exists a iteger 1 0, such that for ay 1 1, Hog ad Page

5 Vx 1 x 1 1 EV; *,. From the secod limit above, there exists a iteger 0, such that for ay, #i,..., : i 2 * *. 2 Let N 1 1 ad N max{2 1 /(*), }. The N Vx 1 x EV; *,. The left-had side of the above iequality is the joit performace of the group of N 1 agets idepedetly draw accordig to. We ow prove that the right-had side term is the joit performace of the group of N 1 best agets from the group of N agets. By costructio, N. Therefore, That is, #i,...,n : i 2 * N *. 2 #i,...,n : i 2 * *N 2 1 N 1, because N 2 1 (*). This meas there are more tha N 1 umbers of agets amog the group of N agets that are the highest performig aget *. Thus, the best N 1 agets amog the N agets are all *. Therefore, their joit performace is exactly the same as the performace of *, which is E(V; *, ). To summarize, for each, there exist N 1 ad N, N N 1, such that the joit performace of the group of N 1 agets idepedetly draw accordig to is better tha the joit performace of the N 1 best agets from the group of N agets idepedetly draw accordig to. Because the set has probability 1, the theorem is prove. Cocludig Remarks The mai result of this paper provides coditios uder which, i the limit, a radom group of itelliget problem solvers will outperform a group of the best problem solvers. Our result provides isights ito the trade-off betwee diversity ad ability. A ideal group would cotai high-ability problem solvers who are diverse. But, as we see i the proof of the result, as the pool of problem solvers grows larger, the very best problem solvers must become similar. I the limit, the highest-ability problem solvers caot be diverse. The result also relies o the size of the radom group becomig large. If ot, the idividual members of the radom group may still have substatial overlap i their local optima ad ot perform well. At the same time, the group size caot be so large as to prevet the group of the best problem solvers from becomig similar. This effect ca also be see by comparig Table 1. As the group size becomes larger, the group of the best problem solvers becomes more diverse ad, ot surprisigly, the group performs relatively better. A further implicatio of our result is that, i a problem-solvig cotext, a perso s value depeds o her ability to improve the collective decisio (8). A perso s expected cotributio is cotextual, depedig o the perspectives ad heuristics of others who work o the problem. The diversity of a aget s problem-solvig approach, as embedded i her perspectiveheuristic pair, relative to the other problem solvers is a importat predictor of her value ad may be more relevat tha her ability to solve the problem o her ow. Thus, eve if we were to accept the claim that IQ tests, Scholastic Aptitude Test scores, ad college grades predict idividual problem-solvig ability, they may ot be as importat i determiig a perso s potetial cotributio as a problem solver as would be measures of how differetly that perso thiks. Our result has implicatios for orgaizatioal forms ad maagemet styles, especially for problem-solvig firms ad orgaizatios. I a eviromet where competitio depeds o cotiuous iovatio ad itroductio of ew products, firms with orgaizatioal forms that take advatage of the power of fuctioal diversity should perform well. The research we cited earlier by computer scietists ad orgaizatioal theorists who explore how to best exploit fuctioal diversity becomes eve more relevat. Most importatly, though, our result suggests that diversity i perspective ad heuristic space should be ecouraged. We should do more tha just exploit our existig diversity. We may wat to ecourage eve greater fuctioal diversity, give its advatages. The curret model igores several importat features, icludig commuicatio ad learig. Our perspectiveheuristic framework could be used to provide microfoudatios for commuicatio costs. Problem solvers with early idetical perspectives but diverse heuristics should commuicate with oe aother easily. But problem solvers with diverse perspectives may have trouble uderstadig solutios idetified by other agets. Firms the may wat to hire people with similar perspectives yet maitai a diversity of heuristics. I this way, the firm ca exploit diversity while miimizig commuicatio costs. Fially, our model also does ot allow problem solvers to lear. Learig could be modeled as the acquisitio of ew perspectives ad heuristics. Clearly, i a learig model, problem solvers would have icetives to acquire diverse perspectives ad heuristics. We thak Ke Arrow, Bob Axelrod, Jea Bedar, Joatho Bedor, Steve Durlauf, Joh Ledyard, Chuck Maski, Dierdre McClosky, Jeff Polzer, Sta Reiter, ad three aoymous referees for commets o earlier versios of this paper. 1. Huberma, B. (1990) Physica D 42, Clearwater, S., Huberma, B. & Hogg, T. (1991) Sciece 254, Polzer, J. T., Milto, L. P. & Swa, W. B., Jr. (2002) Admi. Sci. Q. 47, Nisbett, R. & Ross, L. (1980) Huma Iferece: Strategies ad Shortcomigs of Social Judgmet (Pretice Hall, Eglewood Cliffs, NJ). 5. Robbis, S. (1994) Orgaizatioal Behavior (Pretice Hall, Saddle River, NJ). 6. Thomas, D. A. & Ely, R. J. (1996) Harvard Bus. Rev. September October 1996, o Newell, A. & Simo, H. (1972) Huma Problem Solvig (Pretice Hall, Eglewood Cliffs, NJ). 8. Hog, L. & Page, S. E. (2001) J. Eco. Theor. 97, MacLeod, B. (1996) Ca. J. Eco. 29, Ruderma, M., Hughes-James, M. & Jackso, S., eds. (1996) Selected Research o Work Team Diversity (Am. Psychol. Assoc., Washigto, DC). 11. 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