Decision Making Problem in Division of Cognitive Labor with Parameter Inaccuracy: Case Studies

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1 Americn Journl of Informtion Systems, 04, Vol., No., 6-0 Avilble online t Science nd Eduction Publishing DOI:0.69/jis--- Decision Mking Problem in Division of Cognitive Lbor with Prmeter Inccurcy: Cse Studies Jin Hun Zhng, Khin Wr Wr Htike, Ammr Od, Ho Zhng * School of Informtion Science nd Engineering, Centrl South University, Chngsh, Hunn, P.R. Chin *Corresponding uthor: ho@csu.edu.cn Received December, 03; Revised December 7, 03; Accepted December 9, 03 Abstrct Scientific communities will be more effective for society if scientists effectively divide their cognitive lbor. So one wy to study how scientists divide their cognitive lbor hs become n importnt re of reserch in science. This problem ws firstly discovered nd studied by Kitcher. Lter on, Kleinberg nd Oren pointed out tht the model proposed by Kitcher might not be relistic. We investigte the imct of the imprecise rmeter in project selection results. In this per, we further our study on this issue. We study the policy of decision mking problem bsed on the modified division of cognitive lbor model with the ssumption tht scientist is wre of the existence of the imprecise rmeters nd provide the detiled nlyticl results. And we provide decision rule to minimize the possible loss bsed on error probbility estimtion. Keywords: cognitive lbor, imprecise rmeters Cite This Article: Jin Hun Zhng, Khin Wr Wr Htike, Ammr Od, nd Ho Zhng, Decision Mking Problem in Division of Cognitive Lbor with Prmeter Inccurcy: Cse Studies. Americn Journl of Informtion Systems, no. (04: 6-0. doi: 0.69/jis---.. Introduction The best known pproch for modeling congnitive lbor hs been creted by Philip Kitcher [,] nd Michel Strvens [3,4]. Kitcher proposed tht the progress of science will be optimized when there is n optiml distribution of cognitive lbor within the scientific community. However, the min rgument of the model proposed by Kitcher hs mde mny ssumptions tht might not be relistic. The division of cognitive lbor model is procedure by which scientists clculte their expected rewrds. Scientists clculte their mrginl contribution to the probbility of the success of project nd then uses this informtion to estimte the expected rewrd nd yoff. They ssumed tht scientists re utility-mximizes, the division of cognitive lbor mong number of pre-defined projects, (distribution ssumption every scientist knows the distribution of cognitive lbor before they choose wht project to work on nd finlly (success function ssumption ech project hs success function, nd tkes the units of cognitive lbor s input nd the minly objective is probbilities of success of project. Kitcher found tht, when scientists mde their decisions out of their personl interests for wrds, the result might be even better thn the pure one. Kitcher employed mthemticl model to support his rgument. We got lot of lessons from st experiences becuse the seemed mostly unlikely projects (or theory might be proved to be the correct one in the end. Kitcher mentions in his work, tht when those high-minded gols re replced by bser motives such s thirst for fme, some scientists will utomticlly choose the second project towrds the improvement of the totl probbility of success. The min rgument of division of cognitive lbor model is tht the ssumptions of the model re not relistic. In division of cognitive lbor model, Kitcher nd Strevens used representtive-gent pproch tht mens they ssumed tht every individul gent who is exctly the sme sitution s ll the others. In this reserch, we keep only the representtive gent pproches like Kitcher becuse we emphsize only bout success function ssumption with rmeter inccurcy. Like Gilbert sid, we overrte or underrte the odds tht it will occur. And hence we overrte or underrte the ctul vlue of the gin [5]. He gve lot of exmples to prove tht there re errors t the odds nd vlue of gin when we decide wht the right thing is to do. In this per, we introduce perturbtion into the bsic model rmeters nd studied the policy decisions with its imct on the originl distributions. The theoreticl nd experimentl results demonstrte tht, under the provided conditions, the distributions re different from those obtined in the idel cse. The rest of the per is orgnized s follows. In section II, the relted works re introduced. We study nlyticlly the policy decisions under the division model with imprecise rmeters nd derive the close-form conditions in section III. Section IV bestows some experimentl results for number of cses. Finlly, section V concludes the per.. Relted Works In this section, we presents others work which is relted to this reserch. We exmine nd review some of the

2 Americn Journl of Informtion Systems 7 existing solutions on different spects of division of cognitive lbor model in science. The ide of division of cognitive lbor model, due to Kitcher, is one of the most striking fetures of modern socil community. However, in recent time there re mny reserches tht re pointing out the wekness of Kitchers model. There re lrge number of pproches for lerning the structure nd community of socil network. Nowdys, most of young scientists nd gents re fcing with decision problem. They choose method or mke decision ccording to their own st experience, nd lso following their peer or neighbors. Bl nd Goyl presented very generl model in economic [6]. Lnghe nd Grieff studied the division of cognitive lbor in science [7]. They generlized tht Kitchers conclusion bout the division of cognitive lbor in science is not robust ginst chnges in his single stndrd view to multiple stndrds. In recent times, Jon Kleinberg nd Sigl Oren proposed n ide to improve the socil optimum in scientific credit lloction [8]. They dpted Kitcher model nd showed tht the misllocting scientific credit mechnisms might be good wy to obtin the socil optimlity. However, t [8] they built the credit lloction model to choose one mong projects of vrying levels of importnce nd difficulty of projects. The division of cognitive lbor is one of the most conspicuous model in modern socil community. There re lrge number of pproches for lerning the structure nd community of socil network. The division of cognitive lbor model hs lso been rgued by Weisberg nd Muldoon s epistemic lndscpe pproch [9] nd Zollmn s epistemic networks pproch [0,]. In [9], they considered the originl model in [] is too idel, e.g., ll the gents know the distribution of cognitive lbor t ll times nd division of cognitive lbor model is not robust to chnge in the distribution ssumption. They built robustness model, where gents ctully clculte their mrginl contributions to their project success using specific function. In recent time, most of young scientists nd gents re fcing with decision problem. They choose method or mke decision ccording to their own st experience, nd lso following their peer or neighbors. Socil network community hve been studied in number of domins; in economic, Bl nd Goyl presented very generl model [] nd lso nother clss of pproches to simple model of herd behvior, see work by [3,4]. They lso study socil network community by following the bsic model of DeGroot [5] with the setting of individul gents re connected in socil network nd updte their beliefs repetedly tking into ccount the verged weight of their neighbors opinions cn rrive shred opinion [6,7,8]. We reviews some of the existing solutions nd on different spects of nlyzing theories developed by prior studies in the re of socil science, hhowever, they do not provide ny detiled theoreticl nlysis or remedy on the imct of the vrition on model rmeters [9]. We recently modeled the imct of imprecise rmeters theoreticlly nd we provide detiled nlyticl results for this issue [0]. 3. Theoreticl Anlysis The fundmentl eqution of division of cognitive lbor model built by Kitcher is s follows []: n pi( n ρi( e ( In this eqution, ρ i nd k re ll rmeters nd k 0 is clled the responsiveness. And ssume tht there re N scientists (denoted by Si, i 0,,..., N working on M projects (denoted by PJi, i 0,,..., M. pi ( n represents the probbility of success when n scientists re working on PJ i. Ech scientist is ssumed to choose project tht mximizes his/her probbility of success. When project is successful, ll scientists working on it would eqully shre the credit. From the spect of ech scientist, the principle of choice is ssumed to be bsed on the rewrd tht he/she might receive, scientist would choose project with the lrgest rewrd pi( ni/ n i. Kitcher provided nlyticl results in [] for the bsic model. However, in relity, it is not possible tht model rmeters re known precisely by ech scientist. In this per, we only consider the cse when the rmeter ρ i devites from its true vlue. Although we conducted theoreticl nlysis on the imct of imprecise rmeters [3], we do not give the right decision tht scientist should do in this sitution. In this per, we would focus on the decision mking issue bsed on nlyticl models for some specil cses. 3.. One Scientist nd Two Projects Cse For this cse, the scientist is denoted by S nd two projects re denoted by PJ nd PJ. The generl model in Eq. cn be rewritten s follows: p( ρ( e p( ρ( e In rel pplictions, the estimted model rmeters ˆρ, ˆρ might not be the true vlues. If we ssume the mesured ρ contins some perturbtion, ˆρ ρ, ˆ ρ ρ + x where x is the dded perturbtion, nd the bove equtions cn be rewritten s [3] p ( ρ ( e p( ( ρ + x( e Nturlly, the perturbtion x would hve gret imct on this model. When p( > p(, S will switch from PJ to PJ due to better estimted success probbility. We use P e to represent this probbility of error, which cn be denoted by the following eqution: P e P( PJ P( PJ PJ + P( PJ P( PJ PJ P( PJ P( pˆ ( > pˆ ( PJ + P( PJ P( pˆ ( > pˆ ( PJ It cn be rewritten s: P( PJ P( x > ρ ρ PJ + P( PJ P( x < ρ ρ PJ

3 8 Americn Journl of Informtion Systems It cn be further rewritten s: + ρ ρ P( PJ f( x + P( PJ f( x ρ ρ It cn be seen tht P e is function of ρ ρ. In order to mke the minimum probbility of miscrrige justice, the bove formul does derivtion on ρ ρ s: P( PJ f( ρ ρ + P( PJ f( ρ ρ. ( ρ ρ So tht the vlue of the derivtive to zero, we obtin: P( PJ f( ρ ρ P( PJ f( ρ ρ Without generlity, we ssume ρ > ρ nd f( x f( x. The scientist would choose project with much lrger rewrd. If the scientist would choose project PJ, the probbility of error is written s: + P( pˆ ( > pˆ ( P( x > ρ ρ f ( x ρ ρ Otherwise, when the scientist would choose project, the probbility of error is: ρ ρ P( pˆ ( > pˆ ( P( x < ρ ρ f ( x If rndom vrible x follows the uniform distribution, f( x ( x b We obtin: + + f ( x ρ ρ ρ ρ ρ ρ ρ ρ f ( x ( If ρ- ρ ( + b/, would hold. The scientist would choose project due to better estimted success probbility. ( If ρ- ρ > ( + b/, < would hold. The scientist would choose project to get much more rewrd or success probbility. If rndom vrible x follows the norml distribution, We obtin: e f( x e πδ ( x µ /( δ P f x e + + ( x µ /( δ ( ρ ρ ρ ρ πδ ρ ρ ρ ρ ( x µ /( δ f ( x e πδ ( If ρ-ρ µ, we hve. In this cse, the scientist would choose project to work due to better estimted success probbility. ( If ρ-ρ > µ, < would hold. The scientist would choose project to get much more rewrd or success probbility. 3.. Two Scientists nd Two Projects Cse For this cse, two scientists re denoted by S nd S; two projects re denoted by PJ nd PJ, respectively. The distribution between scientists nd projects could be <, 0>, <0, > nd <, > three cses. Without rmeter perturbtions, S nd S would not both choose PJ, the stble distribution cnnot be < 0, > for the cse ρ > ρ. Next, we would do nlysis from the two spects of personl nd socil with rmeter perturbtions. Without generlity, we ssume the mesured ρ contins some perturbtion, ˆρ ρ, ˆ ρ ρ + x where x is the dded perturbtion. ( If the distribution is <, 0>, the success probbility or rewrd of scientists S nd S working on project PJ is written respectively s: k p( ρ( e / ( k ρ e For the cse with rmeter perturbtions, the estimted success probbility is s follows: k pˆ ( ρ( e / 0k pˆ ( ( ρ + x( e / 0 ( k ρ e Obviously, the rmeter perturbtions would not hve imct on estimted success probbility nd the ggregte probbility of success. ( If the distribution <0,> holds, the success probbility or rewrd of scientists S nd S working on project PJ is written respectively s: k p( ρ( e / ( k ρ e The estimted success probbility due to rmeter perturbtions is written s: k pˆ ( ( ρ + x( e / k ( ρ + x( e (3 If the distribution is <,>, the success probbilities of scientist S working on project PJ nd scientist S working on project PJ re written respectively s: p( ρ( e p( ρ( e

4 Americn Journl of Informtion Systems 9 ρ( e + ρ( e With rmeter perturbtions, the corresponding estimted probbility of success re s follows: pˆ ( ρ( e pˆ ( ( ρ + x( e ρ( e + ( ρ + x( e In generl, we ssume ρ > ρ nd the scientist S would choose one project to work firstly. If the scientist S chooses the project PJ, the scientist S would choose the project PJ or the project PJ. The error probbility of the scientist S choosing the project PJ or the project PJ is computed respectively by: Pp ( ( > p( Px ( > ( ρ + e / ρ + ( e k f x ρ ( + / ρ ( e ( Pp ( ( > p( Px ( < ρ + e / ρ ρ + / ρ f ( x If rndom vrible x follows the uniform distribution, f( x ( x b We obtin: ( If ρ( + e / ρ ( + b/, we obtin. For this cse, the scientist S would choose project PJ to work due to better success probbility or rewrd. And the ggregte probbility of success is: ( k ρ e ( If ρ( + e / ρ < ( + b/, > would hold. The scientist S would choose project PJ to work on. ρ( e + ( ρ + x( e Next, it is lso meningful to study from the socil spect, if the scientist S would choose one of project to work on due to his/her better personl rewrd or success probbility, is it the CO-distribution? For the distribution <, 0>, only if, it is the CO-distribution. When, x ρe ρshould hold. And the rndom vrible x follows the uniform distribution, x b. Hence, the inequlity < ρe ρ b should hold. Considering the inequlity ρ ( + e / ρ ( + b/, the scientist S would mke the choice for the benefit of both personl nd socil interests. If rndom vrible x follows the norml distribution, f( x e πδ ( x µ /( δ Similrly, we obtin: If ρ( + e / ρ µ, the inequlity P e P e would hold. The scientist S would choose project PJ due to better personl rewrd. If ρ( + e / ρ < µ, > holds. S will mke choice on PJ. If S would choose PJ due to better success probbility, while mking the distribution <,0> be the co-distribution, the inequlity x ρe ρ should hold. From the bove nlysis, we know tht when the perturbtion item x would meet certin conditions both the personl nd socil better success probbility could be stisfied One Scientist nd M Projects Cse Without generlity, let we ssume the probbilities of success for M projects stisfy p > p > p3 >... > pm. S will choose PJ with the highest probbility of success. If pi ( i > contins perturbtion item x, S will mke the choice ccording to the following computtion. P P( PJj PJ P( P( PJj > P( PJ j,, 3,..., M ej The scientist S would choose project j only if P ej is much smller thn others. 4. Simultion We study the policy decision bsed on the modified division of cognitive lbor model with the detiled simultion results. We would like to provide the probbility of error when the scientist S chooses the projects with the ssumption of rmeter inccurcy t the projects probbility of success. In this simultion experiment, the rmeter vlues re s follows: ρ 0.9, ρ 0.5, k 0.4 nd the rmeter inccurcy x is set in rndom vlue between 0 nd. Figure. The error probbility of the scientist S choosing the projects if x follows the uniform distribution Figure shows the probbility of error ccording to the scientist S choice. When the scientist S is working in

5 0 Americn Journl of Informtion Systems project PJ, there re wys to choose for scientist S; he would follow <,> or <,0> distribution. According to the result showing in Fig., there would be only too smll error probbility if scientist chooses the project PJ. Figure. The error probbility of the scientist S choosing the projects if x follows the stndrd norml distribution Figure shows tht the error probbility for scientist S with the ssumption of rmeter inccurcy x follows the stndrd norml distribution. If scientist S chooses the project PJ to work together with scientist S, he/she could get the probbility of error over Conclusions In this per, we offer decision mking rules in the division of cognitive lbor with theoreticl nlysis on the imct of rmeter perturbtions. Due to the complexity involved in modeling the generl cse, we minly focus on severl specil cses with smll number of scientists nd projects. The nlyticl results demonstrte tht when we ssume the existence of rmeters inccurcy, we would choose the project tht minimizes the error probbility. Acknowledgement The work is supported by Ntionl Nturl Science Foundtion of Chin (No nd Specilized Reserch Fund for the Doctorl Progrm of Higher Eduction (No , the Hunn soft science project (00ZK306, Centrl Universities Fundmentl Reserch Funds for the project (0QNZT063. References [] P. Kitcher, The division of cognitive lbor, journl of philosophy, The Journl of Philosophy, vol. 87, no., pp. 5-, 990. [] P. Kitcher, The dvncement of science, New York: Oxford University Press, 993. [3] M. Strevens, The role of the priority rule in science, Journl of Philosophy, vol. 00, no., pp , 003. [4] M. Strevens, The role of the mtthew effect in science, Studies in the History nd Philosophy of Science, vol. 37, pp , 006. [5] D. Gilbert, Buried by bd decisions, Nture, pp , 0. [6] V. Bl nd S. Goyl, Lerning from neighbours, Review of Economic Studies, pp , 998. [7] DL.Rogier nd G.Mtthis, Stndrds nd distribution of cognitive lbor, Logic Journl of the IGPL, 009. [8] J.Kleinberg nd S.Oren, Mechnisms for (misllocting scientific credit, Proceedings of the 43rd nnul ACM symposium on Theory of Computing, 0. [9] M. Weisberg nd R. Muldoon, Epistemic lndscpes nd the division of cognitive lbor, Philosophy of Science, pp. 37:59-70, 009. [0] K. J. Zollmn, The communiction structure of epistemic communities, Philosophy of Science, vol. 74(5, pp , 007. [] K. J. Zollmn, The epistemic benefit of trnsient diveristy, Erkenn, vol. 7, pp. 7-35, 00. [] Bl, Venktesh nd Snjeev Goyl. "Lerning from neighbours." Review of Economic Studies, pp , 988. [3] Bnerjee, A. V. "A simple model of herd behvior." The Qurterly Journl of Economics, 07 (3, pp , 99. [4] Welch, I. "Sequentil sles, lerning, nd cscdes." The Journl of Finnce, 47(, pp , 99. [5] M. H. DeGroot. "Reching consensus." J. Americn Sttisticl Assocition, 69, 8-, 974. [6] B. Golub nd M.O. Jckson. "Nive lerning in socil networks: Convergence, influence nd the wisdon of crowds." Americn Econ. J.: Microeconomics, 000. [7] P. M. DeMrzo, D. Vynos nd J. Zweibel. "rsusion bis, socil influence, nd unidimensionl opinions." Aurterly Journl of Economics, 8(3, 003. [8] M. O. Jckson. "Socil nd Economic Networks." Princeton University Press, 008. [9] R. Muldoon nd M.Weisberg, Robustness nd ideliztion in models of cognitive lbor synthese, Synthese, vol. 83, pp. 6-74, 0. [0] K. W. W. Htike nd H. Zhng, Theoreticl study of division of cognitive lbor with imprecise model rmeters, CSSS, pp , 0.