Predicting the Performance of Teams of Bounded Rational Decision-makers Using a Markov Chain Model

Transcription

1 The InTITuTe for ytem reearch Ir TechnIcal report Predicting the Performance of Team of Bounded Rational Deciion-maer Uing a Marov Chain Model Jeffrey Herrmann Ir develop, applie and teache advanced methodologie of deign and analyi to olve complex, hierarchical, heterogeneou and dynamic problem of engineering technology and ytem for indutry and government. Ir i a permanent intitute of the univerity of maryland, within the a. Jame clar chool of engineering. It i a graduated national cience foundation engineering reearch center.

2 Abtract Predicting the Performance of Team of Bounded Rational Deciion-maer Uing a Marov Chain Model Jeffrey W. Herrmann A. Jame Clar School of Engineering Univerity of Maryland College Par, MD jwh2@umd.edu In practice, when faced with a complex optimization problem, team of human deciionmaer often eparate it into ubproblem and then olve each ubproblem intead of tacling the complete problem. It would be ueful to now the condition in which eparating the problem i the uperior approach and how the ubproblem hould be aigned to member of the team. Thi paper decribe a mathematical model of a earch that repreent a bounded rational deciion-maer ( agent ) olving a generic optimization problem. The agent earch can be modeled a a dicrete-time Marov chain, which allow one to calculate the probability ditribution of the value of the olution that the agent will find. We compared the ditribution generated by the model to the ditribution of reult from earche of olution to traveling aleman problem. Uing thi model, we evaluated the performance of two- and three-agent team who ued different olution approache to olve generic optimization problem. In the all-at-once approach, the agent collaborate to earch the entire et of olution in a equential manner: the next agent begin where the previou agent topped. In the eparation approach, the agent eparate the problem into two ubproblem: (1) find the bet et of olution, and (2) find the bet olution in that et. The reult how that team found better olution uing eparation when high-value olution are le liely. Uing the all-at-once approach yielded better reult when the value were uniformly ditributed. The optimal aignment of ubproblem to team alo depended upon the ditribution of value in the olution pace. 1

3 Introduction In practice, human deciion-maer often eparate a complex optimization problem into ubproblem and then olve each ubproblem intead of tacling the complete problem. Thi approach i a natural trategy given the contraint that human deciion-maer have. It i epecially convenient in a team etting: different ubproblem can be aigned to different member of the team. If human deciion-maer were able to optimize, then eparating a problem into ubproblem would uually lead to olution that are inferior to thoe found by olving the problem all-at-once (only in certain condition will the optimal olution to the ubproblem form an optimal olution to the complete problem). It i well-nown, however, that real-world deciion-maer cannot optimize becaue of limit on their problem-olving capacity. Thi concept i nown a bounded rationality (Simon, 1997a). Bounded rationality reflect the obervation that, in mot real-world cae, deciion-maer have limited information and limited computational capabilitie for finding and evaluating alternative and chooing among them (March and Simon, 1993; Simon, 1997b; Gigerenzer et al., 1999). A deciion-maer cannot perfectly evaluate the conequence of the available choice. Thi prevent complete and perfect optimization. The tudy decribed in thi paper wa motivated by the following quetion: (1) when i eparating a better approach than olving the problem all-at-once? (2) what i the bet way to aign ubproblem to the member of a team? A part of an ongoing tudy of the effectivene of eparation by human deciion-maer, thi paper preent the reult of a tudy that conidered a cla of generic optimization problem, modeled team of bounded rational deciion-maer, and evaluated the quality of the olution 2

4 that thee team found uing different olution approache. The tudy wa intended to provide inight into the relationhip between the ditribution of the olution and the relative performance of the olution approache. A novel feature of thi tudy i our Marov chain model of the earch of a bounded rational deciion-maer. Thi model allow one to calculate, without reorting to computational imulation, the probability ditribution of the value of the olution that the deciion-maer will find. The remainder of the paper proceed by reviewing the concept of bounded rationality, defining eparation, and decribing the cla of optimization problem that we tudied. The paper then introduce the Marov chain model, decribe the intance analyzed, and preent the reult of the tudy before concluding with ome inight gained from thi wor. Bounded Rationality Satificing and fat and frugal heuritic are two model of bounded rationality (Gigerenzer et al., 1999). Bounded rational deciion-maer may earch until they find omething that meet their requirement (atificing), or they may ue fat and frugal heuritic that earch a limited et of object and information and mae choice uing rule that are eay to compute (and therefore quic). Thi tudy conidered only thi econd type of bounded rationality. When tacling a complex optimization problem, the deciion-maer need to determine value for variou apect of the olution. That i, he mut mae a deciion. In practice, he doe not have complete information about the available alternative and their impact on overall performance. Thu, he mut employ ome proce for generating and evaluating alternative. 3

5 March and Simon (1993) decribe the general characteritic of human problem-olving in organizational deciion-maing. The firt characteritic i that maing a complex deciion involve maing a large number of mall deciion. The econd characteritic i that problemolving ha a hierarchical tructure in which olving any problem goe through phae that, in turn, require olving more detailed ubproblem. The general concept of eparation i related to thee two characteritic. The third characteritic i that problem-olving conit of earching for poible olution (cf. Nutt, 2005). The fourth characteritic i that problem-olving include creening procee that evaluate the olution that are found. The fifth characteritic i that problem-olving ha not only random component (uch a finding and evaluating olution) but alo a procedural tructure that allow it to yield good olution. The propoed model of a bounded rational deigner i motivated by thee lat three characteritic. Wang and Chiew (2008) decribed human problem olving a a higher-layer cognitive proce that can be conidered a a earch proce, though it require other cognitive procee uch a abtraction, analyi, ynthei, and deciion-maing. Their model of a generic problem olving proce i a earch that iteratively generate and evaluate potential olution. Thu, we conclude that earch i a valid model of bounded rational deciion-maing. Herrmann (2010) too a imilar approach but conidered eparation of a pecific engineering deign optimization problem a model of progreive deign procee. Hi reult indicated that well-deigned progreive deign procee are the bet way to generate profitable product deign. Thi paper conider a very different problem but addree the ame iue: how doe the eparation affect the quality of the olution? It alo conider the bet way to allocate the reource of a team of deciion-maer to the ubproblem in the eparation. 4

6 An important apect of bounded rationality i that the reource and time available for problem-olving are limited. Conequently, the propoed model of a bounded rational deciionmaer incorporate limit that will contrain the amount of time available for the earch (and the quality of the olution found). A great variety of model, including agent-baed imulation model, have been ued to tudy team in general (cf. Crowder et al., 2012). Team of bounded rational deciion-maer who perform the ame ta repeatedly were tudied by Boettcher and Levi (1982, 1983) and Levi and Boettcher (1983), who modeled the deciion-maing proce of each peron a a twotage proce that incorporated ituation aement and repone election. The condition of bounded rationality i modeled a a contraint on the rate at which each peron can proce information. The deciion-maing proce wa later extended to include information fuion, ta proceing, and command interpretation tage and repreented uing Petri net (Levi, 2005). Gurnani and Lewi (2008) tudied collaborative, decentralized deign procee in which the model of the individual deciion-maer (the deigner) were baed on the idea of bounded rationality. In their model, the value choen by each deigner wa determined by randomly ampling from a ditribution around the (locally) optimal olution. Thi model wa meant to repreent the mitae that deigner mae due to bounded rationality. Their reult howed that incorporating bounded rationality led to more deirable olution in a collaborative, decentralized deign proce in which the deigner had different objective and no way to coordinate their activitie. Herrmann (2010) preented a method for aeing the quality of a product deign proce by meauring the profitability of the product that the proce generate. Becaue deign deciion maing i a type of earch, the method imulated the choice of a bounded rational 5

7 deigner for each ubproblem uing earch algorithm. The earche, which were limited to a fixed number of iteration, had random component (either randomly electing a olution or randomly moving to a point near the exiting olution) and a procedural tructure to eep trac of the bet olution found. The reult howed that decompoing a problem into ubproblem yield a better olution than olving the entire problem at once when bounded rational earch i employed. Herrmann (2012) decribed a tudy in which different approache for eparating the Inventory Slac Routing Problem (a complex vehicle routing problem) were imulated. Again, a random earch wa ued to imulate how a bounded rational human deciion-maer would olve each ubproblem. The reult howed that the tructure of the eparation and the objective ued in each ubproblem ignificantly affected the quality of the olution that are generated. Additional detail about the Inventory Slac Routing Problem can be found in the report by Montjoy and Herrmann (2012). Hong and Page (2004) modeled problem-olver of limited ability a earche that they called heuritic, and each heuritic earched a finite et of olution until it could not find a better olution. Thu, the problem-olver i conducting a type of hill-climbing earch. Hong and Page tudied team of uch problem-olver and identified condition under which a divere et of problem-olver will liely perform better than a team of high-performing individual. In their wor, the problem-olver earched a finite et of olution (the ize ranged from 200 to 10,000). A problem-olver wa characterized by how many and which point near the current olution it would conider. Eentially, different problem-olver had different neighborhood definition. The reearch decribed in the current paper tarted by modeling deciion-maer lie the problem-olver that Hong and Page conidered, but the earch pace ued are omewhat 6

8 different. Thi reearch did not meaure the value of diverity; intead it tudied how eparating the problem and allocating the team reource affect the quality of the olution that team of problem-olver generated. Separation A eparation i a proce that olve a equence of ubproblem. A large problem i divided into ubproblem, and the olution to one ubproblem provide the input to one or more ubequent ubproblem. Note that the eparation doe not have to be a imple equence of ubproblem; it may have ubproblem that are olved in parallel at place. A given eparation pecifie a partial order in which the ubproblem are olved. A different order of ubproblem would be a different eparation and would lead to a different olution. The ubproblem objective function are urrogate for the original problem objective function. Thee urrogate come from ubtituting impler performance meaure that are correlated with the original one, eliminating component that are not relevant to that ubproblem, or from removing variable that will be determined in another ubproblem. Problem Formulation Thi tudy conidered team of bounded rational deciion-maer (problem-olver, or agent) and different approache for olving a generic cla of optimization problem. An optimization problem require finding an optimal olution in a pace of many olution. In thi tudy, every point in the olution pace ha a value in the et {1,, N}, where the value N i the optimal value, and a point with value i i better than a point with value j if and only if i > j. The number of point in the olution pace i not relevant. The point in the olution pace are grouped into et. The value of a et i the average value of the point in the et. In engineering deign, a et correpond to a conceptual deign, 7

9 and the point in the et correpond to the detailed deign that are aociated with that conceptual deign. For intance, a et may be the baic path of a highway that i to be built between two citie, while the point in that et are the pecific roadway that follow that general path. A team of two agent can olve thi type of problem uing two approache. In the all-atonce approach, the firt agent earche the entire pace of olution until he top at a locally optimal olution, and then the econd agent begin earching from that point until he top at a locally optimal olution (becaue the econd agent begin a new earch, a point that i locally optimal for the firt agent may not be locally optimal for the econd agent; thi will dicued in more detail in the next ection). The agent can alo eparate the problem into two ubproblem that are olved equentially. The firt ubproblem i to find the bet et, and the econd ubproblem i to find the bet point within that et. The firt agent earche the et until he top at a locally optimal et, and then the econd agent earche that et of point until he top at a locally optimal olution. If the team conit of three agent, then they can collaborate in the all-at-once approach; the third agent can continue the earch by tarting where the econd agent topped. If they eparate the problem, the third agent can help the firt agent by continuing the earch of the et or help the econd agent by continuing the earch of the point in that et. Modeling Agent Searching In the generic optimization problem, a problem-olver ( agent ) mut earch a pace of olution (or et); the agent want to find the bet point (et), that i, the one that ha the larget value, but he i limited, however, by the limitation of the earch trategy. In thi tudy we 8

10 aumed that the agent earch had the following characteritic: The agent begin by randomly electing a point in the olution pace. From the current point, the agent chec another randomly elected point and accept the new point if it i better (ha a greater value) than the current point. The agent earch i governed by a parameter K. If the agent chec K conecutive point that are not better than the current point, then the agent top hi earch. (Increaing K will increae the effort pent earching and will increae the expected value of the olution found.) The earch for a et wor in the ame way. There are W type of et and an ordering over the value of the et uch that a et of type a i better than a et of type b if a > b. The quality of the olution found by the agent i a random variable that depend upon K and the ditribution of value in the earch pace. Let px ( ) be the probability that a randomly elected point will have the value x, for all x { 1,, N}. Let vt () be the value of the agent current point after checing t point (where t = 0 correpond to the initial point). The tochatic proce vt (), t = 1, 2,, i generated by a Marov chain. The tate of the Marov chain i the value of the agent current point and the number of point checed ince accepting that point. Let ( vt (), nt ()) be the tate at tep t, where vt () i the value of the agent current point and nt () equal the number of point checed ince accepting that point. The number of tate i finite. The tate are the element of { 1,, N} { 0,, K} have nt (). All tate that = K are aborbing tate becaue the earch top when thi occur. All of the other tate are tranient. Thu, the ditribution of vt () converge to the ditribution over the aborbing tate (1, K),, (N, K). The probability tranition matrix for thi Marov chain can be decribed a follow: At t = 0, (0) 0 Pv(0) = x = px ( ). n =, and { } 9

11 Let Pw (, jxi,, ) P{ vt ( ) xnt, ( ) i vt ( 1) wnt, ( 1) j} = = = = =. Pw (, jx,,0) = px ( ) for w= 1,, N 1, x= w+ 1,, N, j = 0,, K 1 P( w, j, w, j+ 1) = p(1) + + p( w) for w= 1,, N, j = 0,, K 1 PwKwK (,,, ) = 1 for w= 1,, N. Pw (, jxi,, ) = 0 otherwie. Conider, for example, the problem with N = 10 and px ( ) = (11 x) / 55 for x = 1,,10. The value of px ( ) are approximately (0.1818, , , , , , , , , ). Let P be the probability tranition matrix for thi Marov chain. A B 0 P= A 0 B, where 0 0 I A = and I i the 10-by-10 identity matrix B = and The probability ditribution for vt () converge to (0.0060, , , , , , , , , ). Although the mallet value are the one that are mot liely to occur in the olution pace, moderate value (lie 6, 7, and 8) are the one that 10

12 are mot liely to be found in the earch. The probability of the earch returning a olution with a value greater than or equal to 7 equal , although the probability that a randomly elected point ha a value greater than or equal to 7 equal only Comparion to Search Reult To generate ome evidence that thi model i a reaonable model of thi type of earch, we randomly generated intance of the traveling aleman problem (TSP), contructed and evaluated the correponding Marov chain model, and ued random earche to find olution to thee intance. We then evaluated the imilarity of the ditribution generated by the Marov chain model and the ditribution of the earch reult uing a chi-quared goodne of fit tet. In particular, we generated TSP intance with 12 ite located on a two-dimenional quare. The coordinate of the ite were randomly drawn according to a uniform ditribution on the interval [0, 100]. The Euclidean ditance wa ued a the ditance meaure. The 12th ite wa fixed a the lat ite in any TSP olution. The eparation involved clutering the 12 ite into three cluter of four ite (the third cluter alway included the 12th ite). Thu, each cluter include (4!)(4!)(3!) = 3,456 route. When creating the Marov chain model, every cluter wa evaluated by finding the average length of the route in that cluter. In the earche, a cluter wa evaluated approximately by finding the average length of 20 randomly-elected route in that cluter. For the Marov chain model, the route length (and cluter average) were dicretized into approximately 50 value (the exact number depended upon the ditribution of the route length). The earche include all-at-once earche with one, two, and three agent and three verion of the eparation: after one agent earche for a cluter and a econd earche the route 11

13 in that cluter (thi we call 1-1 ); after one agent earche for a cluter and two other agent earch the route in that cluter (thi we call 1-2 ); and after two agent earch for a cluter and a third agent earche the route in that cluter (thi we call 2-1 ). In general, chi-quared tet howed that the ditribution generated by the Marov chain model and the ditribution of the earch reult were imilar. Table 1 and Figure 1 how the reult for one 12-ite TSP intance. Each earch wa run 1000 time, and the route length were converted into the dicrete value ued in the Marov chain analyi. For each of the ix earche, becaue the chi-quared tatitic i le than the critical value, we do not reject the hypothei that the earch reult are ditributed according to the ditribution that the Marov chain generate. Figure 1 how the ditribution for the Separation 2-1 earch for the ame intance and the ditribution of the value for all of the route for thi intance. (Although the TSP i a minimization problem, in thee ditribution, a larger value repreent better route.) Table 1. Chi-quared tet tatitic for one 12-ite TSP intance (n = 1000). Search Tet tatitic Degree of freedom Critical value (alpha = 0.10) All-at-once one agent All-at-once two agent All-at-once three agent Separation Separation Separation

14 Population Marov chain dn. Search reult (N = 1000) Frequency Value Figure 1. The ditribution of value in a 12-ite TSP intance. Intance To compare the performance of different eparation, we generated different ditribution for the et and population according to the following cheme. Let rv (, ) be the probability of finding a point with value v when earching in a type et. Let q () be the probability of finding a type et when earching for a et. Firt, we conidered a pace with 50 type of et and 99 value. Within thi pace, there were two different ditribution over the value in a et. In the firt, which we called uniform, 1 all value were equally liely in every et. That i, r (, ) u v = for all and v

15 In the econd, which we called triangular, only ome value were poible, but the 1 poible value were equally liely in every et. In particular, for = 1,,50, r(, v ) = for v=,, + 49 and 0 otherwie. There were two ditribution over the et. In the firt ( uniform et ), all of the et 1 were equally liely: q () u = for all. In the econd ( ewed et ), the better et are le liely: q ( ) = (51 ) /1275 for all. 50 From thee ditribution, we created 11 cae (A to K). Each cae wa an interpolation between the ditribution over the value in a et or the ditribution over the et. Cae Ditribution over the value Ditribution over the et in a et rv (, ) = r(, v) + r(, v) q () = q() A u 10 t B u 10 t C u 10 t rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = qu () D rv (, ) = rt (, v) q () = qu () E 3 1 rv (, ) = rt (, v) q () = 4qu() + 4q() F 1 1 rv (, ) = rt (, v) q () = 2qu() + 2q() G 1 3 rv (, ) = rt (, v) q () = 4qu() + 4q() H rv (, ) = r(, v) q () = q() t I u 10 t J u 10 t K u 10 t rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q() u u t 50 Next, we conidered a pace with 15 type of et and 120 value. Within thi pace, there were three different ditribution over the value in a et. In the firt, which we called 1 uniform, all value were equally liely in every et. That i, r (, ) u v = for all and v. In the econd, which we called triangular, only ome value were poible, but the poible value

16 1 were equally liely in every et. In particular, for = 1,,15, r(, v ) = for v= 5( 1) + 1,,5( 1) + 50 and 0 otherwie. In the third, which we called plit, only ome value were poible, but the poible value were equally liely in every et. In particular, 1 1 r( v, ) = 1 / (16 ) for v= (32 )( 1) + 1,, (31 ) and 0 otherwie. 2 2 In thi pace, there wa only one ditribution over the et. The better et are le liely: q ( ) = (16 ) /120 for all. From thee ditribution, we created 11 more cae (L to V). Each cae wa an interpolation between the ditribution over the value in a et. Cae Ditribution over the value Ditribution over the et in a et rv (, ) = r(, v) + r(, v) q () = q() L u 10 t M u 10 t N u 10 t rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q () O rv (, ) = r(, v) q () = q() t P t 4 Q t 2 R t 4 rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q () S rv (, ) = r(, v) q () = q() T u 10 U u 10 V u 10 rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q() rv (, ) = r(, v) + r(, v) q () = q() t 50 Computational Experiment The purpoe of the computational experiment wa to compare the performance of the team olution approache. We teted K = 2 and looed at team of two and three agent. 15

17 For each cae, we conidered the all-at-once earch and the eparation. For the all-atonce earch, we contructed the probability ditribution over the value W px () = qrx ()(, ) for = 1 x = 1,, V and then contructed the probability tranition matrix and calculated the ditribution for every time tep until the probability of being in a tranient tate wa ufficiently mall (no greater than 10-6 ). Thi yielded the probability ditribution over the value that the agent find. If a another agent continue the earch, then thi ditribution wa ued a the ditribution over the initial tate (thoe with n(0) = 0). For the eparation, we firt contructed the probability tranition matrix for the earch over the type of et and uing that calculated the probability ditribution over the type that the agent find. The ditribution q() wa ued a the initial ditribution and to contruct the probability tranition matrix. Let q () be poterior ditribution over the type of et. Then, for each type of et, we contructed the probability tranition matrix for the earch over the value in that et and calculated the probability ditribution over the value that the agent find if he earche that et. Thee were combined uing q () to generate a poterior ditribution over the value. Thu, for each cae and each olution approach (all-at-once and eparation) we contructed a poterior ditribution over the value. For the all-at-once approach, we recorded thi ditribution after one agent, two agent, and three agent. For the eparation approach, we recorded thi ditribution for three verion: after one agent earche for a et and a econd earche the point in that et (thi we call 1-1 ); after one agent earche for a et and two other agent earch the point in that et (thi we call 1-2 ); and after two agent earch for a et and a third agent earche the point in that et (thi we call 2-1 ). 16

18 Reult Here we report the expected value and the probability that the agent() will find a value greater than or equal to a given threhold. The expected value of the entire population i alo given for comparion. Table 2. Expected value found in earche with W = 50 and N = 99. K = 2. Cae Population One agent Two agent Three agent Separation 1-1 Separation 1-2 Separation 2-1 A B C D E F G H I J K Table 3. Expected value found in earche with W = 15 and N = 120. K = 2. Cae Population One agent Two agent Three agent Separation 1-1 Separation 1-2 Separation 2-1 L M N O P Q R S T U V The reult how that relative performance of the all-at-once and eparation approache varie acro the cae conidered. In general, of coure, the performance of the all-at-once approach improve a more agent collaborate. The improvement (the increae in expected value) varie relatively little acro the cae conidered here. In cae C, D, E, F, G, and H, high-value olution are le liely in the entire olution pace (note the relatively low expected 17

19 value for the population in thee cae), and the all-at-once approach i inferior, a hown in Table 2 and Figure 2. The 2-1 and 1-2 eparation approache perform about the ame in thee cae and better than the 1-1 eparation approach. In cae A, B, I, and J, however, the ditribution acro the value in the entire olution pace i nearly uniform, and the all-at-once approach i uperior. The 1-2 eparation approach i uperior to the 1-1 and 2-1 eparation approache becaue the ditribution of value i nearly the ame in all et; pending more effort earching the point in the et found yield better reult than finding a better et. In cae O, P, and Q, high-value olution are le liely in the entire olution pace (note the relatively low expected value for the population in thee cae), and the all-at-once approach i inferior, a hown in Table 3 and Figure 3. Becaue high-quality et type are le liely, the 2-1 eparation approach i uperior to the 1-2 eparation approach; pending more effort finding a better et yield better reult than earching the point in a et. For cae, L, M, N, S, T, U, and V, however, the ditribution acro the value in the entire olution pace i nearly uniform, and the all-at-once approach i uperior. In cae L, M, N, U, and V, the 1-2 eparation approach i uperior to the 1-1 and 2-1 eparation approache becaue the ditribution of value i nearly the ame in all et; pending more effort earching the point in the et found yield better reult than finding a better et. Becaue the ditribution of value in the entire olution pace i uniform for cae S, T, U, and V, the performance of the all-at-once approache wa the ame in all of thee cae. The performance of the eparation approache wa inferior becaue it wa hard for the agent to find the bet type of et and changed a the ditribution of value in the et approached the uniform ditribution. 18

20 Overall, thee reult indicate that the eparation approache performed better than the all-at-once approache when the expected value of the population wa le (cae H and O); in thee cae, the better value were le liely, which decreaed the expected value and reduced the lielihood that the all-at-once approache would find olution with good value. W = 50, N = 99, K = 2: Probability that earch will return a value at leat 90 Separation 1-1 Separation 1-2 Separation One agent Two agent Three agent Population 0.4 Probability A B C D E F G H I J K Cae Figure 2. Probability that a earch will return a value of at leat 90 for cae A to K (W = 50, N = 99, K = 2). 19

21 W = 15, N = 120, K = 2: Probability that earch will return a value at leat 110 Separation 1-1 Separation 1-2 Separation 2-1 One agent Two agent Three agent 0.6 Population 0.4 Probability L M N O P Q R S T U V Cae Figure 3. Probability that a earch will return a value of at leat 110 for cae L to V (W = 15, N = 120, K = 2). Summary and Concluion Thi paper preented a Marov chain model for analyzing the earch of a bounded rational deciion-maer and dicued the reult of a tudy that analyzed the relative performance of team who are olving generic optimization problem. The reult indicate that there are ome ituation in which earching the complete pace of olution i a better approach and other ituation in which eparating the problem i a better approach. The optimal aignment of ubproblem to member of the team alo varied. The ditribution of the value in the olution pace wa a ey factor. Thi tudy, lie the one decribed by Hong and Page (2004), conidered a cla of generic optimization problem. Doing thi allowed u to vary the characteritic of the earch pace directly without the ditraction of pecific deciion variable, contraint, and objective function. 20

22 In thi tudy, every agent who wa earching the ame pace wa modeled with the ame probability ditribution over the value of the next point tried; thi approach could be modified eaily, however, to model agent with different ill who ample the earch pace in different way. Becaue human deciion-maer have limitation and cannot optimize, uing welldeigned eparation can be the bet way to find quality olution and mae deciion in ituation in which trying to et everything all-at-once will lead to problem that are too large to olve well becaue the ditribution of high quality point in a large olution pace mae it difficult for the bounded rational deciion-maer to approach an optimal olution. Thee reult reinforce the concluion of Herrmann (2010) about the uefulne of eparating complex optimization problem for bounded rational deciion-maer. They alo demontrate the uefulne of thi analyi approach for comparing eparation that have different ubproblem and different allocation of team reource. Reference Boettcher, Kevin L., and A.H. Levi, Modeling the Interacting Deciionmaer with Bounded Rationality, IEEE Tranaction on Sytem, Man and Cybernetic, Volume 12, Number 3, page , Boettcher, Kevin L., and A.H. Levi, Modeling and analyi of team of interacting deciionmaer with bounded rationality, Automatica, Volume 19, Iue 6, page , Crowder, R.M., M.A. Robinon, H.P.N. Hughe, Yee-Wai Sim, The Development of an Agent- Baed Modeling Framewor for Simulating Engineering Team Wor, IEEE Tranaction on Sytem, Man and Cybernetic, Part A: Sytem and Human, Volume 42, Number 6, page ,

23 Gigerenzer, Gerd, Peter M. Todd, and the ABC Reearch Group, 1999, Simple Heuritic That Mae U Smart, Oxford Univerity Pre, New Yor. Gurnani, Ahwin, and Kemper Lewi, 2008, Collaborative, Decentralized Engineering Deign at the Edge of Rationality, Journal of Mechanical Deign, 130(12), Herrmann, Jeffrey W., Progreive Deign Procee and Bounded Rational Deigner, Journal of Mechanical Deign, Augut 2010, Volume 132, Iue 8, (8 page). doi: / Herrmann, Jeffrey W., Separating the Inventory Slac Routing Problem, Technical Report , Intitute for Sytem Reearch, Univerity of Maryland, College Par, July, Available online at Herrmann, Jeffrey W., Separating the Searche of Bounded Rational Deciion-Maer, Technical Report , Intitute for Sytem Reearch, Univerity of Maryl and, College Par, June, Available online at Hong, Lu, and Scott E. Page, Group of divere problem olver can outperform group of highability problem olver, Proceeding of the National Academy of Science of the United State of America, Volume 101, Number 46, page , Levi, Alexander H., Executable model of deciion maing organization, in Organizational Simulation, W.B. Roue and K.R. Boff, ed., John Wiley & Son, Inc., Hoboen, New Jerey, Levi, A.H., and Kevin L. Boettcher, Deciionmaing organization with acyclical information tructure, IEEE Tranaction on Sytem, Man and Cybernetic, vol.smc-13, Number 3, page ,

24 March, Jame, and Herbert Simon, 1993, Organization, econd edition, Blacwell, Cambridge, Maachuett. Montjoy, Adam, and Jeffrey W. Herrmann, Optimizing Urgent Material Delivery by Maximizing Inventory Slac, Technical Report , Intitute for Sytem Reearch, Univerity of Maryland, College Par, June 12, Available online at Nutt, Paul C., 2005, Search During Deciion-Maing, European Journal of Operational Reearch, 160, pp Simon, Herbert A., 1981, The Science of the Artificial, econd edition, MIT Pre, Cambridge, Maachuett. Simon, Herbert A., 1997a, Model of Bounded Rationality, Volume 3, The MIT Pre, Cambridge, MA. Simon, Herbert A., 1997b, Adminitrative Behavior, fourth edition, The Free Pre, New Yor. Wang, Y., and Chiew, V., 2008, On the Cognitive Proce of Human Problem Solving, Cognitive Sytem Reearch, doi: /j.cogy