Faculty Working Paper 91-0143 330 B385 1991:143 COPY 2 STX The Distributed Learning System: A Group Problem-Solving Approach to Rule Learning JUL / 2 wi Riyaz Sikora Michael J. Shaw Department of Business Administration Department of Business Administration Bureau of Economic and Business Research College of Commerce and Business Administration University of Illinois at Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 91-0143 College of Commerce and Business Administration University of Illinois at (Jrbana-Champaign June 1991 The Distributed Learning System: A Group Problem-Solving Approach to Rule Learning Riyaz Sikora* e-mail: sikora@uxh.cso.uiuc.edu and Michael J. Shaw** e-mail: mshaw@uxl.cso.uiuc.edu Department of Business Administration *Doctoral candidate in Information Systems **Associate Professor of Information Systems The Distributed Learning System: a Group Problem-Solving Approach to Rule Learning Abstract The disciplines of Distributed Artificial Intelligence (DAI) and Group Decision Support Systems (GDSS) are both concerned with building systems to aid the problem-solving by a group of agents, artificial in former and humans in latter. Although they have the same goal, these two fields have for the most part remained separate. The aim of this paper is to bridge this gap between DAI and GDSS research by building a computational model capturing the group problem-solving behavior of agents, be they artificial or humans. We choose the problem of rule learning as the domain of our study. We present the design, implementation, and evaluation of the Distributed Learning System (DLS), a system modelling the distributed or group problem-solving approach to rule learning. We do so by first extracting a set of core group parameters and group processes involved in group problem-solving, which have been identified by the researchers in GDSS and DAI fields, and then designing a computational model incorporating these. Our results show that, with the same amount of resources allocated,the group problem-solving approach with multiple agents gives better results than a centralized single-agent approach. Our model confirms some of the findings from the GDSS domain and shows in a formal manner the effect of various situational variables on group performance. Some of the implications of this approach to the design of DAI systems, to the development of GDSSs, to the machine learning methods, and to problem-solving strategies are also discussed. Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/distributedlearn91143siko 1. Introduction Technical developments in electronic communication, computing, and decision support, coupled with new interest on the part of organizations to improve meeting effectiveness, are spurring research in the area of group decision support systems (GDSS) (DeSanctis and Gallupe [1987]). At the same time there is also a growing need to develop systems which can integrate and coordinate the working of intelligent, autonomous information systems. For example, in an organization using different expert systems in their production, purchasing, and marketing departments, there is a definite need for interaction and co-ordination among these expert systems for solving problems that need more than one area of expertise. The recent advances in processor fabrication and communication technologies coupled with the above factors have led to the increased interest in Distributed Artificial Intelligence (DAI) systems (Decker [1987]), which concern with coordinating the working of artificial intelligent agents. Although the fields of GDSS and DAI have the same common goal of building systems to aid the problem solving by a group of agents (human experts in former and artificial intelligent agents in latter), they have remained as two separate fields. One of the main objectives of this paper is to bridge the gap between these fields by extracting the relevant features or situational variables which can be common to both the systems, and presenting a generic framework for group problem-solving involving a hybrid network of agents be they human or artificial. The growing interest in group decision support systems is in most part the result of the need to improve group decision making in organizations. The belief in the efficiency of group work, or group problem-solving (GPS), was reinforced by research in the 1930's that showed groups could solve problems in large numbers and with greater speed than could isolated individuals (Shaw [1932]). In other words, the group problem-solving activity gives rise to the phenomenon of emergent intelligence (Shaw&Whinston [1989]) wherein a group can perform problem-solving tasks better than what the sum of the individuals' abilities can do. Moreover, research has shown that groups sometimes do not exceed or even equal the performance of the best individual in the group - referred to as group losses or process losses (Shaw[1978], Brown[1988]) - because the process of working together as a group sometimes gets in the way of and diminishes group performance (Steiner[1966]). This productivity loss in group activity results mainly from information loss, information distortion and/or making decisions without sufficient alternatives to consider (Kraemer & King [1988]). One of the main objectives of the research related to GDSS has been to design systems which can best maximize the synergistic effect of group problem solving and minimize group losses by removing common communication barriers, providing techniques for structured decision analysis, and systematically directing the pattern, timing, or content of discussion (DeSanctis and Gallupe[1987]). Some examples of group problem solving may be organizational decision making where group of managers deliberate upon strategic planning, a creative group of people participating in a brainstorming session, a group of experts from different domains jointly design a product etc. One of the aims of this research is to address the issue: Under what conditions does a group problem-solving strategy, using multiple agents, perform better than a centralized, single-agent approach in a problem-solving situation? We tackle the above issue by extracting a set of core group parameters and group processes which have been identified by the GDSS and DAI research communities, and developing a computational model which can simulate such a group problem solving activity. Several researchers in GDSS community (DeSanctis and Gallupe[1987]; Nunamaker et al.[1988]; George et al.[1990]) have identified a set of situational variables to be the most important in the design of GDSS: for the independent variables, group size, type of information exchange, member proximity, leadership, anonymity, and task confronting the group have been shown important for the dependent variables, decision outcomes (decision quality, number of unique alternatives generated, consensus), process outcomes (time to decision, participation, satisfaction with the process), and group coherence. Since we are concerned primarily in extracting features that might be relevant for both artificial as well as human agents— that is, our aim is to study GPS as a computational process among problem-solving agents— we choose to ignore the behavioral factors such as proximity and satisfaction. In addition, research related to DAI (Bond and Gasser[1988]) has identified several important factors in designing DAI systems: Description, decomposition, distribution, and allocation of tasks; interaction, language, and communication; coherence and co-ordination; inter-agent disparities; synthesis of results. Based on these results, this research is aimed at studying the impacts of group size (number of agents), problem decomposition and task allocation, and diversity among the agents on the performance of the group, and developing a scheme for synthesizing the individual solutions of the agents. Rule induction was chosen as the problem domain of our study for its importance in automating the construction of knowledge-base systems from examples. The problem of rule induction can be simply stated as follows: based on a set of positive and negative examples of a concept infer a description or hypothesis of the concept which correctly explains all the positive examples without covering any of the negative examples. From the standpoint of McGrath's (1984) taxonomy of task types, rule induction may be characterized as a combination of intellective, creativity, and preference. As a result of its involvement of these multiple task types, it can potentially benefit from the use of group problem solving. The problem of rule learning or induction from examples is a very widely studied problem in the area of machine learning. Algorithms like Version-spaces (Mitchell[1977]), AQ(Michalski[1983]), ID3(Quinlan[1986]), and PLSl(Rendell[1986]) are a few of the successful algorithms for learning from examples. It has been well established that inductive learning can be viewed as a problem-solving process, and all these algorithms operate on a training data set to find the concept (or rules) for explaining the data. A GPS method for rule learning should parallel quite well with the "group induction" process studied in the psychology literature. For example, Laughlin&Shippy(1983) describe an experiment in group induction to address the issue of whether "a cooperative group is able to induce a general principle that none of the group members could have induced alone, or the group merely adopts an induction proposed by one or more of the group members." Applying the GPS approach as a new learning strategy, we consider the multi-agent approach to learning from examples in which the data set is divided into different sub-sets and given to different agents (inductive learning programs). The learning results from the agents are then synthesized into the final solution. The resulting system modelling the group problem-solving activity for the task of rule learning is called Distributed Learning System (DLS). The empirical results of DLS show that (1) group problem-solving approach, using multiple agents, gives better results than the single-agent approach when appropriate parameter values and group processes are enforced; (2) the group performance does depend on the problem decomposition; (3) the group performance depends on the group size, and peaks at a specific size; (4) the group performance depends on the diversity among the agents and is best when there is maximum diversity; and (5) the group synthesis step is crucial for the success of group problem solving, and can be modelled by a genetic algorithm to capture the group processes and information exchange involved. More importantly, based on the above results, we present a generic process model for group problem solving and present some of its implications to the areas of DAI, GDSS, and machine learning. The rest of the paper is organized as follows: in §2 we review the research done in GDSS and DAI fields and discuss the research issues considered in this paper, in §3 we present the computational model of group problem-solving approach to learning and discuss the Distributed Learning System (DLS), an implementation of the proposed model, in §4 we present an example which shows in detail the advantage of group problem-solving approach over single-agent approach, in §5 we look into each of the issues mentioned in §2 and present the empirical results which suggest possible solutions, in §6 we present the summary of findings together with the implications of the Distributed Learning System to developing DAI systems, GDSS, to machine learning methods, and to problem solving strategies and finally in §7 we conclude. 2. Group Problem-Solving In order to build a computational model for group problem-solving, it is essential to first understand the underlying dynamics and to address issues such as what the parameters on which the group performance depends are and what the group processes involved are. In this section we review the research done related to GDSS and DAI, both of which are concerned with the problem- solving processes among a group of agents. After identifying a set of group parameters and processes which might be important in the design of a GPS system, we present five propositions which hypothesize the effects of the parameters on the group performance. 2.1 GDSS: A Review DeSanctis and Gallupe (1985) define GDSS as "an interactive computer-based system that facilitates the solution of unstructured problems by a set of decision makers working together as a group." The main objective of a GDSS is to improve productivity from group meetings of managers or experts by removing the communication barriers, providing a means for structured decision analysis, and directing the pattern, timing , or content of the discussion. Typically, a GDSS involves a group of decision makers with access to a computer, viewing screen, database, decision model(s), and, in many cases, a "facilitator" who supports the group in use of the technology, instructs them on the use of decision model, coordinates the group's activity, and documents the group's work. Kraemer and King (1988) give a comprehensive review of the GDSS systems in use. Several researchers have studied the effects of different situational variables on the design of a GDSS. For example, DeSanctis and Gallupe (1987) suggest "... that the design of support systems be driven by three factors: the size of the group, the presence or absence of face-to-face interaction (i.e., proximity), and the task confronting the group." They further view the group decision process as a result of interpersonal communication - the exchange of information among members. In order to help design a GDSS, they identify three levels of GDSSs. Each level alters the group communication process to a different degree. At the simplest level, GDSSs provide features aimed at removing common barriers to group work and communication, such as unequal consideration of ideas, dominance by individuals, peer pressure, and loss of autonomy. At the second level, GDSSs provide specific group techniques aimed at structuring the group's work and decision processes. At the third level, GDSSs structure group communication patterns and select and arrange the rules to be applied during a meeting. They also point out the importance of structured approaches to decision making. In general research suggests that adding structure to decision process positively impacts decision outcomes (Smith[1973]). The key effect of structured approaches appears to be increasing member participation (White et al.[1980]), focussing the problem, avoiding conformity pressures, and keeping the group on track (Gallupe [1985], DeSanctis and Gallupe[1987]). Similarly George et al. (1990) identify a set of situational variables to be the most important in the design of GDSS. For the independent variables: communication media, leadership, anonymity. For the dependent variables: decision outcomes measured by decision quality, number of unique alternatives generated, and consensus and process outcomes measured by time to decision, participation, and satisfaction with the process. 2.2 Distributed Problem-Solving: A Review Distributed Problem-solving (DPS) 1 , a sub-field of DAI, concerns with how the solving of a particular problem can be divided among different modules (or 'agents' in a multi-agent systems) that cooperate at the level of dividing and sharing knowledge about the problem and about the developing solution (Lesser & Corkill[1987]; Smith & Davis[1981]). There are four phases of the DPS approach: (1) problem decomposition, (2) sub-problem distribution, (3) sub-problem solution, and (4) answer synthesis. Figure 2.1 below shows the phases of distributed problem solving (Smith & Davis [1981]). Insert figure 2.1 here Bond & Gasser (1988) identify several issues related with the design of any Distributed AI system, which can help provide a framework for understanding the basic components of our DPS method. (1) Description, Decomposition, Distribution, and Allocation of Tasks: Problem description refers to the formulation of the problem or the representation used for the problem. Decomposition refers to the question of breaking up the problem into sub-problems which can be solved by the agents. Decomposition choices are usually dependent on how the problem is described. Since the distributed individual solutions have to be finally integrated, the choices of decomposition which maintain mutual consistency and provide as much of the global view as possible are preferred. In some circumstances, the issue of redundancy among the sub-problems enters into the decomposition problem. Choices about redundancy are related to the tradeoffs between efficiency and reliability; redundancy should be eliminated to improve efficiency, but may be necessary for reliability. Allocation of tasks refers to the problem of deciding which sub-problems to assign to which agents. (2) Interaction, Language, and Communication: The problem of interaction, language, and *In this paper we use the terms group problem solving and distributed problem solving interchangably. In general, GPS stresses the use of a group of agents and DPS emphasizes that the tasks to be solved are distributed. communication is considered important because it makes it possible for the agents to combine their efforts. However, several questions such as: what kinds of interactions are possible? what is the result of this interaction? What kinds of communication are possible? etc., arise in the design of a DPS system. (3) Coherence and Coordination: Coherence refers to how well the system behaves as a unit, along some dimensions of evaluation. Typical dimensions of evaluation could be solution quality, efficiency, clarity or conciseness of the final solution etc. Coordination is a property of interaction among some set of agents performing some collective activity. Coordination and coherence are partially related - better coordination may lead to greater efficiency coherence. However, good local solutions by the agents do not necessarily add up to good global behavior, because good local solutions may have unfortunate global effects. This is especially important in the context of inductive learning where the local solutions generated by the agents can be local optima. We will see how DLS overcomes this problem of local vs. global optimum. (4) Modeling other agents and Organized Activity : This concerns with the knowledge each agent has about what other agents are doing and what other agents know, so that they can organize their activities. The main question concerning this is what knowledge and how should the knowledge of other agents be represented and organized? Typical types are knowledge of agents capabilities, resources, demands, beliefs, goals, plans etc. (5) Inter agent disparities: Uncertainty and Conflict : This concerns with the ability of the agents to cope with problems of disparity and uncertainty between their objectified representations and the affairs to which the representations refer. There is also the important question of conflict resolution between the agents. Negotiation is often proposed in DAI research as a conflict- resolution and information-exchange scheme (see for example, Davis & Smith [1983]). (6) Synthesis of Results: This concerns with combining the solutions generated by the individual agents (often partial and incomplete) to form the complete solution of the problem. As we will see this is the critical step in the DLS because the individual solutions generated by the agents are usually local optimum and have to be combined to get a globally optimum solution. Contract Net is an example of a distributed problem-solving system(Davis & Smith [1983], Smith[1980], Smith & Davis[1981]). The primary goal of this system was opportunistic, adaptive task allocation among a collection of problem solvers, using a framework called "negotiation", based on task announcements, bids, and awarded "contracts". In this, an agent needing help (called the manager) with a task can divide it into subtasks and negotiate a contract for each subtask with the other agents (called contractors). The manager for a task makes a task announcement giving a description of the task and eligibility requirements of the agents who may bid for the task. The manager - contractor relationship ceases to exist when the task is completed. The contract net protocol thus dynamically decomposes problems, allocates the tasks, organizes agents in heirarchies for the purposes of control in achieving these tasks, and disbands the heirarchies once the tasks are complete. Variations of contract net protocol have been used in wide applications. For example, Malone's Enterprise system (Malone et al.[1988]) uses a specialized version of contract net protocol, called Distributed Scheduling Protocol (DSP), for the task of scheduling jobs on machines. In it the processors send out "requests for bids" on tasks to be done, and other processors respond with bids giving estimated completion times that reflect machine speed and currently loaded files. The DSP specializes the selection criterion of the contract net protocol to assign tasks to the best machine available at run time (either local or remote). 2.3 Research Issues Based on the prior research on GDSS and DAI, we can identify a set of core group parameters and group processes which might be relevant for the design of a generic GPS. As mentioned, one of the main objectives of this research is to investigate whether and under what conditions distributed or group problem-solving approach, using multiple agents, is a better strategy than using a centralized single agent approach and to demonstrate the phenomenon of emergent intelligence. This can be described by the following proposition: Proposition 1 : With the same amount of resources allocated 2 , multiple agents, using a group problem solving approach give better performance than a single agent solving the entire problem. As previously explained, there are advantages(e.g., emergent intelligence) and disadvantages(e.g., group losses) associated with group problem solving. We are interested in finding out, behavioral factors aside, if there are any computational benefits in using group problem solving. In order to address the above issue, we have to understand the dynamics of group problem solving, i.e., what are the parameters on which the group problem solving behavior depends? and what are the group processes underlying the dynamics of GPS? This is important for two reasons. First, identifying the main parameters and group processes involved in group problem solving or decision making will help us in building computational models that can duplicate the GPS behavior, and second, having such a computational model can help us in either validating any existing theories about the effect of the identified parameters on group performance or in identifying new relationships between group performance and the parameters. Thus, by this exercise, we would have not only learned about the effect of different parameters on the group 2 The resources include compution time, data, and communication channel, etc. 8 performance, but would have sufficient understanding for developing a computational model which effectively captures the GPS behavior. Group Parameters: Research related to the GDSS has shown that group size has an effect on the group performance. For example, DeSanctis and Gallupe (1987) state that the situational variables critical to the design of a GDSS include group size; Nunamaker, Applegate, and Konsynski (1988) report that "member satisfaction increases with the size of the group," and that 'larger groups appreciate the structuring inherent within the system . . .". This leads us to make the following proposition: Proposition 2 : The performance of group problem solving will depend on the number of the agents used. We also need to decide how we are going to break the task to give it to different members of the group, i.e., problem decomposibility. Research related to the DAI field has identified the importance of problem decomposition, however as Bond and Gasser (1988, pp. 11), report 'There has been remarkably little work in DAI that addresses problem decomposition." The choices related to decomposition are related to redundancy among the sub-problems or tasks. Redundancy can slow down the efficiency of the distributed system, on the other hand redundancy is required for reliability. Thus, we have the following proposition: Proposition 3 : The performance of group problem solving will depend on the choices related to the problem decomposition. There is also another important factor which can effect the group performance— the group diversity . In other words, we would like to know whether diversity among the group members effects the group performance. If it does, then whether it is advantageous to have diversity or homogeneity. Intuitively, it seems that diversity among the group members should be more beneficial as it provides different views of the same problem. Evidence from different fields also suggests that having such diversification has a net advantage (Brady [1985]). For example, to accelerate scientific progress, Feyerabend (1975) recommends, amongst other things, 'proliferation of theories'; Kuhn (1970) has observed periods of stagnation in scientific progress associated with reduced diversity of theories when there was one encompassing theory (such as, phlogiston). The Delphi technique for structured group problem solving (which is discussed briefly later in this section) also demonstrates the importance of diversity; (Linstone and Turoff, 1975, pp. 4) "the heterogeneity of the participants must be preserved to assure validity of the results, i.e., avoidance of domination by quantity or by strength of personality ("bandwagon effect")." All these arguments leads us to the proposition: Proposition 4 : The greater is the diversity among the agents, in a group problem solving situation, the better is the performance of the group as a whole. Group Processes: As mentioned before, in order to develop a computational model for the group problem solving activity we need to identify the processes involved in it. To begin with we need a mechanism for problem decomposition and task allocation 3. Although this is an important problem for the DAI field, most of the DAI systems developed either assume that the given problem is totally decomposible (for example, Contract net by Davis & Smith (1983)) or incorporate problem- specific decomposition mechanisms. Since the partial solutions given by each agent in the group have to be combined, mechanisms for problem decomposition which decompose the problem in such a way that each agent knows how its partial result(s) fit into the complete solution, have been proposed (Barr, Cohen, and Feigenbaum, pp. 100, 1989). In general, the decomposition mechanism should maintain mutual consistency and provide as much global view of the problem as possible should be preferred. Another important process that is part of any group problem solving activity is group interaction and co-ordination . Since the group is basically working on different parts of the same problem, hence they have to interact and coordinate their solutions so that they can be synthesized into a final complete solution. In other words there has to be an information exchange among the agents which can help improve the agents individual solutions and lead to a final solution. However, as mentioned before, in order to maximize the effect of group problem solving and minimize group losses there has to be a mechanism for structured decision analysis and a mechanism to systematically direct the pattern, timing, or content of information exchange. For example, in delphi technique 4 (Linstone and Turoff [1975]) the coordination and information exchange takes place through a central coordinator. The coordinator provides the group with a common focus, summarizes and communicates individual partial solutions to all agents, provides a new focus to the group based on problem solving in previous stage, and repeats this cycle until 3 The group process used in Electronic Meeting Systems( Nunamaker et al, 1990) can be viewed as a special case in which each participant solves the same problem and the degree of decomposition is zero. 4 which can be characterized as a method for structuring a group communication process so that the process is effective in allowing a group of individuals, as a whole, to deal with a complex problem. 10 individual solutions converge i.e., a consensus is reached. Thus, although there is no direct interaction among the agents, there is an exchange of information (between the different individual solutions) which helps the agents refine their solutions until they all converge. In short, the approach presented in this paper is to evaluate the individual complete solutions and integrate these via successive cycles of refinement The same is true in the GPS technique of Collective Induction (CI) (Laughlin [1983,85,88]). In CI, the task assigned to the human group is to induce a concept description from positive and negative examples of the concept. The experiment begins with a positive exemplar being presented to the group. Each member then comes up with a concept description which is consistent with this exemplar (individual hypothesis). This is then compared with those of the other group members and by a process of discussion, a consensus is reached on what the concept description should be (group hypothesis). This is then tested in the light of new exemplar and the cycle of problem solving is repeated for a fixed (predetermined) number of times. Thus, CI consists of several cycles wherein at the end of each cycle the individual solutions of each member are compared and a consensus is reached on what the best solution should be. Their findings showed that group induction is advantageous over single-agent induction primarily due to the agents' sharing of hypotheses and new facts generated. Similarly, in more open-ended activities which call for creativity and imagination, like 'brainstorming' which is an ideas- generating technique often used in business and advertising (Osborn [1957]), it was found that the technique was most beneficial if carried out initially in private, the interacting group then being used as a forum for combining and evaluating these individually produced ideas (Lamm and Trommsdorff [1973]). In order to develop a computational model to simulate the group problem solving activity and study the dynamics, we need a mechanism for implementing structured group interaction and co- ordination. However, there are two important sub-processes underlying the process of interaction and co-ordination which can be identified from the above examples. First is a mechanism for evaluating and identifying good individual solutions and the second is a mechanism for exploring new solutions either by modification or combination of other solutions. In delphi technique, for example, the evaluation is done implicitly by the central coordinator when it summarizes the individual solutions and provides a new focus to the group. When the coordinator summarizes the individual solutions, some of the solutions which are not "good" may get eliminated, i.e., they may not be included in the summary. Then, in the next cycle new solutions are explored by the individual members utilizing the past solutions of other members (which are presented in the form of the summary and the new focus). In other words, new solutions are produced as a result of evaluation, modification and combination of the past solutions. The same is true in collective induction, as the members compare the different solutions (evaluate) and try to reach a consensus for a common solution (i.e., combine). Thus, in order to effectively model the group interaction 11 and co-ordination of their individual solutions we need mechanisms which can simulate; {^evaluation, {^selection, {^elimination, {4)modification, and {5)combination of the individual solutions. 3. Group Problem-Solving Approach to Rule Learning 3.1 A General Framework Based on the group parameters and group processes identified in the last section, we present a general framework for group problem-solving. As mentioned before, one of the important parameters of the model would be the number of agents involved in the problem-solving i.e., the group size. The group problem-solving activity can be classified into two types: one in which all the agents work on the same problem synchronously, i.e., the problem is solved jointly by the group simultaneously and second in which the agents work on different parts of the problem asynchronously, their solutions to be synthesized finally. Examples of systems supporting the synchronous group problem-solving are the PLEXYS system (Nunamaker et al.[1988]), COLAB (Stefik et al.[1987]), and the concurrent design system described in Bond(1989). Examples of systems supporting asynchronous group problem-solving are the distributed problem-solving systems (see §2.2) exemplified by the office information system described in Woo&Lachovsky( 1986). However, irrespective of the type of problem-solving activity the group is involved in, we can identify certain generic processes involved. Since the main objective of the group is to reach a consensus on a final solution for the problem at hand, there has to be information exchange among the agents involved to facilitate consensus. Information exchange among the agents can take several forms ranging from unconstrained direct one-on-one interaction among the agents where any agent can request information from any other agent to a more structured but open ended interaction where the intermediate results of all the agents are made available to every other agent through a common channel of communication. Most of the GDSSs developed would exemplify the unconstrained end of the spectrum where as systems using Delphi technique (Linston&Turoff[1975]) are examples of systems using structured means of information exchange. Again, irrespective of the kind of information exchange involved, its most important effect is that it allows for new facts, hypothesis, and partial solutions generated by one agent in its problem- solving to be shared by other agents. These additional pieces of information shared among the agents would trigger the group of agents to new problem-solving and evidence-gathering activities which would not have been performed without the agents 1 interactions. In turn, these triggered activities would generate new facts, hypothesis, and evidence that are shared among the agents. 12 Thus, the amount of information shared in the agents' problem-solving efforts grows exponentially and this gives the group an advantage to reach a solution much sooner. On the other hand, the exponentially growing information would mean substantially more searching efforts in screening through all the information generated. A properly designed coordination mechanism could lead the group to evaluate all the relevant facts, hypothesis, new evidence, and partial solutions more systematically. In the next section we describe the Distributed Learning System (DLS) modelling a group problem-solving approach to the problem of rule learning. DLS uses the asynchronous type of group problem-solving approach wherein the problem is decomposed into sub-problems, allocated to different agents, and finally synthesized into a final solution. DLS incorporates a mechanism which allows for structured information exchange among the partial solutions generated by the agents. 3.2 The Distributed Learning System As mentioned before, the task of rule learning concerns with deriving a set of hypothesis or rules which can explain the set of training examples. In the traditional approach to rule learning, a single learning program is used which generates a hypothesis and successively refines it to explain all the examples. Since the process involves generating and evaluating hypotheses at each stage, it might be more efficient to use a group problem-solving approach where the examples are distributed to different learning programs and their results are synthesized to get a final hypothesis. Since each program is now working on only a fraction of the original problem and the different programs can work asynchronously, this method of distributing the amount of resources (number of examples) can make the process more efficient. At the same time since using different agents or programs provides several different alternatives or hypothesis, it potentially can give a better performance. This is the essence of Proposition 1. Insert figure 3.1 here Figure 3.1 presents the DLS, modelling group problem-solving approach to rule learning. The above figure also illustrates the different steps in the distributed problem solving approach as mentioned in §2.2. The four steps correspond to: ( 1 ) problem decomposition, (2) sub-problem allocation, (3) sub-problem solution, and (4) solution synthesis. 13 At first the problem (or data set) P is decomposed into different sub-problems Pi, P2 ... P n which are then allocated to different inductive learning programs (or agents). Each agent solves its sub-problem independently of the other agents and submits its solution to the solution synthesizer, which then synthesizes the individual solutions of the agents into a final solution or hypothesis. The synthesis step is critical in this process. Since the solutions given by the agents are local in scope but complete, the synthesis step involves more than just taking the union of the individual solutions. Each agent's solution can be thought of as an alternative hypothesis being suggested by that agent, and so the final synthesis step should correspond to the group decision-making process where in at each stage every agent proposes a solution and the group deliberates on pros and cons of each alternative. Each agent then refines its solution and the above process repeats itself until all the group members reach a consensus. In other words, for the synthesis step we need mechanisms for evaluating and refining each agent's solution. The detailed description of the four steps follow. Decomposition and Task allocation: As mentioned in §2.3, we would like to maintain mutual consistency among the sub-problems and provide as much global view of the problem in each sub-problem as possible. To this end, the problem (or data set) is decomposed into n sub-problems by using the unbiased random sampling of jackknife technique (Efron[1982]). In jackknife technique one or more data points are randomly removed from the data set to obtain a sub-sample. We use the jackknife technique of 'leave-out r ' which we define below: Definition 1 : A random sub-sample from the original data set is obtained by the jackknife technique of 'leave-out r ' (0 < r < 1) if each example in the original data set has a probability of (1- r ) of being included in the sub-sample. Since one of the issues is deciding what kind of decomposition is best, we define a decomposibility index d , which also measures the amount of overlap between subsamples. Definition 2 : The decomposibility index d , for a particular problem decomposition, is defined as the ratio of average size of each sub-problem to the size of the original problem, i.e., d = (average size of sub-problem) / (size of the original problem) In this case the size of the problem corresponds to the number of the examples. Note that < d < 1, the value of 1 corresponding to the extreme where the whole problem is allocated to all the agents. Also, for a jackknife technique of 'leave-out r ', the size of a sub- 14 problem is a binomial random variable with parameters (N, 1- r ), where N is the size of the original problem. Therefore, d =N*(l-r)/N =(l-r). As mentioned before, the decomposibility index d also defines the amount of task overlap (or redundancy as mentioned in § 2.2) among the agents. Specifically, Definition 3 : For a given decomposibility index d and the number of agents n , the total amount of task overlap, t among the agents is given by t =(d*n -1)N, where N is the size of the original problem. Note that there is no task overlap, i.e., t = 0, when d = l/n . The question of redundancy also arises here in the form of: what decomposibility index is best? As mentioned in §2.2 we would like to minimize redundancy to improve efficiency but at the same time not reduce the solution quality. As we will see in the empirical results, the best performance is obtained when the redundancy (or task overlap) is minimum. There is also the factor of number of agents. In most of the DPS systems this is not considered a factor, however as we show later the performance of the DLS does depend on the number of agents used. Each generated sub-sample is allocated to an agent (inductive learning program). In the Distributed Learning System which was implemented, the split-based induction learning program PLS1 (Rendell [1986], appendix A) was used as an agent. Sub-problem solution and Solution synthesis: After the sub-problems are solved by the individual agents, their local solutions have to be combined (synthesized) to form a complete and global solution. However, as the solutions given by the agents, although complete, are local in scope (as in delphi technique and collective induction discussed before), there has to be a mechanism for structured group interaction and co-ordination which can integrate and refine the solutions over successive cycles until they all converge. As discussed in §2.3, in order to simulate this group interaction and co-ordination, we need mechanisms for evaluating and identifying good solutions and a mechanism for exploring new solutions through information exchange. We will show that a genetic algorithm can effectively model such a group behavior which can effectively lead to the synthesis of the local solutions into a global solution. We thus have the following proposition: 15 Proposition 5 : A genetic algorithm can effectively model the group processes involved in the group problem-solving activity. The reasoning behind this proposition is elaborated in the following section. 3.3 Solution Synthesis in DLS As mentioned before, most of the DPS systems address the issue of direct interaction and co- operation between the agents. In order to do this, they make many assumptions with respect to the knowledge each agent has about other agents, their beliefs, goals, type of communication etc. These assumptions limit the applicability of any particular approach. In this section we present an alternative method wherein instead of allowing the agents to interact with each other, we let their local solutions interact with each other without the control of any single agent. This is done by using a genetic algorithm (see appendix B for an introduction of genetic algorithms) which treats each agents' local solution as a single member in its initial population. The only assumption required in this approach is that there has to be a way of evaluating the partial solutions being generated by each agent. This, however, is not an unrealistic assumption because in any problem- solving situation the partial solutions can be evaluated globally by applying them to the original problem. The GA thus uses this evaluation as a fitness measure for the members of its population (i.e. local solutions of the agents) and does successive cycles of refinement to the solutions by applying the reproduction and recombination operators, until all the solutions converge. Although the mechanics of reproduction and crossover are simple, involving random number generation, string copies and some partial string exchanges, they effectively model the way humans interact when solving a problem. This is very well explained by the building block hypothesis (Goldberg [1989]) which is based on the theory of schemata (Holland [1975]). Each string in the population of n strings (which in our case corresponds to the individual solutions of the agents) represents a complete idea or prescription for performing a particular task. Substrings within each string contain various notions of what is important or relevant to the task. Viewed in this way, the population contains not just a sample of n ideas; rather, it contains a multitude of notions and rankings of those notions for task performance. Genetic algorithms ruthlessly exploit this wealth of information by (1) reproducing high-quality notions according to their performance and (2) crossing these notions with many other high performance notions from other strings (akin to information exchange). Thus, the action of crossover with previous reproduction speculates on new ideas constructed from the high performance building blocks (notions) of past trials. This is 16 similar to the group processes involved in decision making, as stated by DeSanctis and Gallupe (1987); "A group decision occurs as the result of interpersonal communication - the exchange of information among members. . .The decision process is revealed in the production and reproduction of positions regarding group action, which are directed toward the convergence of members on a final choice." The GA thus incorporates the same type of interacting mechanisms by providing a framework for implementing structured and directed information exchange. Table 3.1 summarizes the correspondences between the mechanisms involved in a group decision process, as identified in §2.3, and that in a GA. Thus, the GAs effectively model the group processes involved in group interaction and co-ordination by incorporating mechanisms for evaluating and identifying good individual solutions and for exploring new solutions by refining or recombining different solutions. This is also very similar to delphi technique, where in the central coordinator summarizes the individual solutions and provides a new focus to the group (similar to the evaluation part of the GA), which then is used by the members to refme their solutions (similar to the reproduction and recombination part of the GA) until they all reach consensus (similar to the termination criteria of convergence in the GA). Insert Table 3.1 here Synthesis step: In order to understand the synthesis step computationally we should first understand the representations used by the PLS1 programs and that used by the GA (which is similar to PLS1). The concept given by PLS1 is represented in the form of regions (Rendell[1986]), where each region is represented by conjunction of intervals for each attribute. In other words, the concept given by PLS 1 can be represented by P = P, v... v P m -< < d ll ^! < e n ) & ...& ( d lk v <1 21 & 1 22 &...& l 2k > v...v {!„,, & !„,,&...& 1^} where each Pj corresponds to a region and is represented as conjunction of intervals £^j corresponding to the attribute xj, j = 1,2,.. Jc. The representation used by GA is similar, except that instead of using the whole concept P as a single member in the population (for reasons explained later), it uses each region as an individual member. Thus, the concepts generated by the PLS1 programs are first broken into individual 17 disjuncts and then given to the GA as initial population. There are two ways of synthesizing the concepts generated by the PLS1 programs using a GA. One way is to let each member in the population represent a complete concept ( Pi v P2 v ... v Pm) generated by each PLS1 program. Alternatively, the other method is to let each member be just a single disjunct Pi = (§14 &... & |kt,i)- Since each PLS1, working on a different sample, gives a concept of different length (number of regions), hence using the first method results in a population for the GA wherein each member has a different length. Also, since we want to get a more concise concept, we would like to get more powerful disjuncts (or rules). Hence, to avoid the problem of variable length of each member and to make the disjuncts compete against each other, we use the second method in our GA, in which the GA, simulating the concept-synthesis step, tries to find the best possible disjunct (i.e., covering as many positive examples as possible) by recombining the disjuncts, representing the partial solutions, given by the PLS1 programs that play the role of agents. After it converges, the best disjunct found is retained and the positive examples which it covers are removed. The process is again repeated to find a new disjunct to cover as many of the remaining positive instances as possible. This process terminates after all the positive instances are covered. The final rule is then the disjunct of all the disjuncts found, which becomes the group solution synthesized from the partial solutions generated by the agents. In order to compare the concepts generated by the DLS system based on their generality, we define a generality index g as follows: Definition 4 : For any given concept or rule Q, the generality index gi is defined as the ratio of number of unseen examples it can cover to the number of all possible examples. In other words, gi = fraction of the instance space covered by Q. Thus, a concept Ci is more general, and hence with more predictive power, than C2, if g 1 > g 2- Time complexity: It would be desirable to have a linear time complexity for DLS with respect to the size of the problem (i.e. number of examples) regardless of the time complexity of the learning algorithm used. That this is indeed the case is explained below. Suppose that the learning agent used (let us call it 'A ') is linear in time complexity with respect to the number of the training examples. Assume that N is the total number of training examples available and n is the number of agents (each using the algorithm A ) used in DLS. Then 18 the time complexity of A is O(N) and that of DLS is 0(N+LP) 5 , where L is the length of each member of population in the GA which depends on the number of attributes (and hence is constant) and P is the population size which depends on the number of agents n (and hence is constant for a given n). Thus, the time complexity still remains linear. However if the algorithm A is not linear in time complexity then one would expect that DLS would also be not linear. But it turns out that it can still have linear time complexity provided we have the parallel architecture. This is so because with the increase in the number of training examples N, we can also linearly increase the number of agents n (keeping the number of training examples per agent as constant) and hence keep the time taken by the agents same. Then the only increase in time would be due to the complexity O(LP) of the GA, which is again linear in N. 6 Thus, the time complexity of DLS remains linear with respect to the number of training examples even if the learning algorithm it uses has exponential time complexity . 4. Computational Advantages of Group Problem-Solving: An Example In this section, we present an example to demonstrate the advantages of using a group problem-solving approach to the problem of rule learning. We use the DLS system, modelling the GPS approach, wherein the data set is decomposed into several sub-samples and given to different learning agents. The results of the learning agents are then synthesized into the group solution using a genetic algorithm. We show the improvement in performance obtained by using this multi- agent approach over that of the traditional approach of using a single agent. 4.1 Problem Description For the empirical analysis, a large set of real-world data from a chemical plant was used for all the experiments discussed in this and next section. The problem concerned with controlling a chemical process, producing a certain chemical product, with about 30 process variables. In the process of producing the product an undesirable byproduct was produced which was not measured directly. To remove this byproduct an expensive chemical was added in just sufficient quantities to chemically remove the byproduct from the product. The problem was to change the controllable process variables (9 out of the 30 variables) so that the usage of the expensive chemical was 5 Since each agent handles a fraction N/n of the data set and since there are total of n agents, hence their total time complexity is N and the time complexity of a GA is LP 6 Because P is linear in n, and n in turn is linear in N 19 minimized. Since there are no theoretical formulae linking the process variables with the amount of product produced, the only way to solve this problem is to induce the relationships based on a set of actual plant readings. The problem was formulated as a single concept learning problem 7 by considering the examples corresponding to large quantity of the expensive chemical used as being positive examples and the rest as negative examples. The data set had 572 instances of which 355 were positive and 217 negative examples and it was randomly broken up into a training set of 458 and a testing set of 1 14 examples. In this section we present an example to show the advantage of using a multi-agent (group) problem solving approach to inductive learning vis-a-vis that of using a single-agent approach. The concept learning problem (single-agent) can be stated as: based on a set of positive and negative examples of a concept infer a description or hypothesis of the concept which correctly explains all the positive examples without covering any of the negative examples. The group problem-solving (or multi- agent) approach to the same concept learning problem can then be termed as: distribute the data set among several agents; let them induce a concept description or hypothesis explaining their data sub-sets; and based on the individual results let them reach a group consensus about the final concept. The following parameter values were used for the DLS: n=5 and d=0.2. We present the detailed empirical results for different values of n and d in §5 where we show that best performance is obtained for these parameters values. 4.2 Discussion of Results In this sub-section we discuss the results obtained by using the group problem-solving approach, using multiple agents, and single-agent approach to inductive learning where we used the whole training set of 458 instances on the PLS1 program (with s=l) 8 . Figure 4.1 shows the learning result, in the form of concept description, of the single-agent approach, the results of the individual agents in multi-agent approach and the final results in the multi-agent approach obtained by synthesizing the results of individual agents. The learning performance of the system as a whole is measured by decision quality determined by prediction accuracy, rule-size, which is the length of the concept generated, and the generality index g (see definition in §3.3). We explain in detail each part of the computer output below. 7 Single concept learning problem is to find a concept which can explain (or cover) all the positive examples without covering any of the negative examples. However, because of the presence of noise in real world data, the constraint of not covering any negative examples is relaxed, allowing a few of them to be covered. 8 s is the significance level used by the PLS1 program which corresponds to the approximate noise level in the data. 20 Insert Figure 4.1 here As can be seen from Figure 4.1, the rule-size of the concept given by the single-agent approach was 17. The concept is therefore (ri V r2 V ...V v\-j) where each rule rj contains the interval ranges for the 9 variables (the complete range is 0-63) followed by the number of positive and negative examples covered from the training set and the number of positive and negative examples covered from the testing set. For example, the eight rule, rs, says that if the 9 variables are within their respective intervals then it correctly explains 176 of the 276 positive examples from the training set, at the same time wrongly covers 12 of the 182 negative examples, and correctly predicts 50 of the 79 positive examples from the test set while wrongly predicting 6 of the 35 negative examples. Thus, the result from single-agent approach is: (prediction) accuracy = 83.3% and rule-size =17. Figure 4.1 also shows the results given by the 5 individual agents, each working on 20% of the training data set and the final result given by the DLS, which takes the results given by the 5 agents and synthesizes them using GA. Table 4.1 shows the prediction accuracy and rule-size of each individual agent and that of the final result given by the DLS, together with the generalization index g of the best rule from each of them. Insert Table 4. 1 here As can be seen, the best result given by an individual agent (agent 3) is: accuracy = 79.8% and rule-size = 4, and the final result obtained by synthesizing the 5 results is: accuracy = 86% and rule-size = 3. Thus, the synthesizing step improves upon the individual results given by the agents by a minimum of about 9% in accuracy and 25% in rule-size and is better than the single-agent result by about 3.2% in accuracy and about 82% in rule-size. Also, the g index shows that the solutions generated by the agents are local in scope as their best rule covers about 1 % of the instance space on average as compared to about 3% covered by the best rule from DLS. This shows that the synthesis step (using a GA) has the ability of combining the local solutions reach a "group solution" by performing a more globally oriented search. Figure 4.2 summarizes the comparison of performance of the distributed approach vis-a-vis that of a single-agent. Insert Figure 4.2 here 21 5. Empirical Study on Distributed Learning: The Simulation of Group Problem Solving The example presented in the last section showed the phenomenon of "emergent intelligence," where the group performance was better than the mere sum of the performances of its individual members, thus confirming our proposition 1. However, in order to build information systems to aid group problem-solving, we have to understand the effect of different situational variables on the group performance. As mentioned before, research related to GDSS and DAI has identified the importance of group size, problem decomposition, type of information exchange, anonymity, and task confronting the group in determining decision quality, number of unique alternatives generated, time to decision, and participation. Based on the above, we had hypothesized in §2.3 that the performance of the group (of either artificial or human agents) would depend on the number of agents used, problem decomposition, and the diversity among the agents. We had also hypothesized that a genetic algorithm can be used to model the group synthesis of the individual solutions of the members. In this section we present an empirical study designed to test our model for applying GPS approach to rule learning based on the propositions discussed above and to gain theoretical insights into the type of relationship(s) between the situational variables and the group performance. 5.1 Experiment Details We use the Distributed Learning System (DLS), an implementation of our process model for GPS approach to rule learning, for this empirical study where in the data set is decomposed into sub-problems and given to different PLS1 programs. The PLS1 programs work asynchronously on their subproblems and their results are then synthesized using a genetic algorithm. The same real world data set from the chemical process control problem was used for all the experiments discussed in this section. In each experiment the data set was randomly broken up into a training set of 458 and a testing set of 1 14 examples. All the results given are the average of 5 runs with a different training and testing set used in each run. Again, DLS is used to simulate the following group problem-solving task: based on the individual data sub-sets available to each agent of the group, reach a group decision on a concept description which correctly explains all the positive examples of the data set while covering as few of the negative examples as possible. The following parameter values for the GA were used for all the experiments: total number of generations used was 100, Baker's (1987) SUS algorithm was used for selection, uniform crossover operator was used with probability 0.7, and probability of mutation was 0.05. As 22 explained in §3 these can be viewed as the probabilities for combining and modifying the individual agent's results respectively. The population size corresponds to the number of unique alternatives generated by the group. The performance of DLS, simulating the group problem- solving task of rule learning, will be measured in terms of (1) decision quality, given by prediction accuracy, rule-size, and the generality index g, (2) time to decision, and (3) number of unique alternatives generated. The Distributed Learning System is implemented in Common Lisp on a TI- Explorer machine. 5.2 Empirical Results Effect of problem decomposition: Since the group size and problem decomposition had been identified as important situational variables (as envisioned by propositions 2 and 3), we first study their effect on group performance. Table 5.1 presents the results for different decomposition strategies and different number of agents used. Insert table 5.1 here The decomposibility index 'd' (see §3.2 for definition) corresponds to a particular decomposition strategy. It corresponds to the fraction of the resources (in our case, the number of training examples) allocated to each agent. Thus, for example, if there are a total of 'n' agents in the group, a value of d < 1/n corresponds to 'underutilization' of the available resources. As one would expect, we can see from table 5.1 that when the resources are underutilized the group performance suffers (for a fixed number of agents used), as measured by the decision quality (prediction accuracy) and the number of unique alternatives generated. On the other hand, a value of d > 1/n corresponds to overutilization of the resources and, as expected, considerably increases the time to decision for the group. This confirms the view reflected in proposition 3, that the group performance will depend on the decomposition strategy used. Moreover, the results show that, in order to be more efficient and maximize the productivity of the group, the best strategy is to equally divide the resources among the agents. So, if there are n agents and there are total of E training examples, then each agent gets E/n of the examples (corresponding to d=l/n). This unequivocally indicates that the distributed problem-solving approach can achieve more with the same amount of resources, which is another way to view emergent intelligence. 23 Effect of group size: Corresponding to the decomposition strategy of equally dividing the resources among the agents, lets consider the effect of varying the group size on the group performance measured in terms of decision quality, time to decision and number of unique alternatives generated. For the sake of convenience, the required information is extracted from table 5.1 and presented in table 5.2. Insert table 5.2 here The above results confirm Proposition 2 that the group performance will depend on group size. As can be expected, the time to decision increases as the group size increases, as does the number of unique alternatives generated. However, the marginal increase in the number of unique alternatives generated decreases as the group size increases. In terms of quality of the decision, although rule-size more or less remains the same, there is a distinct trend with respect to the prediction accuracy. The accuracy peaks when 5 agents are used and then consistently drops as the group size increases. This clearly shows the dependence of group performance on group size, but the question remains: why does the performance peak at a certain group size and then decrease? We will return to answer this question in the next section. Effect of group diversity: We had hypothesized in proposition 4 that the group performance would depend on the composition of the group and would be best when there is diversity among the group members. Since, as mentioned above, the best group performance was obtained when 5 agents were used, we tested the effect of varying diversity among a group of 5 agents. Table 5.3 shows the results of the experiment. Insert table 5.3 here The diversity number corresponds to the number of different agents used. For example, diversity number '3(1+2+2)* corresponds to having a total of 3 different agents in a group of 5 agents, with 2 agents being replicated. As hypothesized, the decision quality, as measured by prediction accuracy, does indeed depend on diversity. Moreover, there is a general trend of increasing accuracy with diversity, confirming our hypothesis that greater the diversity the better the group performance will be. Although, there is no specific trend in rule-size or the number of 24 alternatives generated, the time to decision for the group generally increase with diversity. This should be expected because as the diversity among the group increases, it takes more time for the group to come to a consensus. 6 Discussion of Results In the empirical study we use DLS to simulate group problem solving by a group of learning agents. The empirical results confirm the "emergent intelligence" behavior wherein the group performance is more than just the sum of individual performances of the group members. However, the group performance depends on the group size or number of agents used and the problem decomposition strategy. As mentioned above, in order to increase the efficiency the best strategy is to divide the resources among the agents. For this decomposition strategy it was found that the decision quality, measured by the prediction accuracy, peaked when 5 agents were used and then decreased as the group size increased. To. better understand the reason for such a behavior, consider once again the effect of group size on the group performance, as summarized in table 6.1. Insert table 6.1 here As can be seen, there are two opposing forces effecting the group performance as the group size increases. Since the resources are equally divided among the group members, as the group size increases the amount of resources available to each member decreases and so the quality of output from individual members reduces. On the other hand, as the group size increases so does the number of unique alternatives generated by the group. Because of these opposing effects of decreasing quality of the individual outputs and increasing number of the alternatives generated, the group performance peaks at a certain group size and starts to decrease. Since the group performance peaks at an optimum size, n*, a group with size other than n* needs to use a GDSS to achieve the same peak performance. The results also confirm that a genetic algorithm can effectively model the group processes involved in group interaction and co-ordination. One of the important features of a GA is its ability to perform globally oriented search without getting trapped in local optimum solutions. This is especially important in our case because the individual solutions given by the agents are based on a local view of the problem which have to be combined to get a complete global solution. However, the GA also suffers from the problem of premature convergence (Booker [1987]), where the presence of a very strong individual in the population, at the beginning, leads to its dominance by 25 having more number of its copies and the population eventually converges to it. This is also true even in group processes involving humans, where in the majority opinion tries to regulate the social norm of the group. "Once people get into collective settings, they appear only too ready to conform to the majority in the group and to abandon their own personal beliefs and opinions" (Brown [1988], p.90). Also, research related with GDSS has identified that the common problems experienced by decision-making groups include dominance of discussion by one or more members; lack of acknowledgement of ideas of low-status members; low tolerance of minority opinions (DeSanctis and Gallupe [1987]). One of the ways to avoid this premature convergence is diversity maintenance (Goldberg [1989]), where in the strong individual in the population is not allowed to dominate and the weaker members are allowed to survive, atleast in the beginning. In other words, enforcing diversity maintenance is akin to encouraging equal participation in a group process. As mentioned in §2.3, the importance of maintaining diversity has been demonstrated in many different fields (most notably the avoidance of bandwagon effect in delphi technique). Our results have also shown the importance of maintaining the diversity among the members of the group. This is consistent with the GDSS results, such as discussed in Hubert (1984) and Nunamaker (1988) that one of the major contributing factors of a GDSS to group problem-solving is enforcing equal participation. Thus, to summarize, we have shown empirical evidence to support the propositions 1-5, for the problem domain of inductive learning, and have uncovered new relationships between the group parameters and group performance. Specifically, the findings are: (1) Group problem-solving approach, using multiple agents, with appropriate parameter values gives better results than single-agent approach with the same amount of resources ; (2) The group performance in multi-agent approach depends on the problem decomposition; (3) The group performance depends on the number of agents used and peaks at a certain optimum number of agents for a given group; (4) The group performance is best when there is maximum diversity among the agents ; and (5) Genetic algorithm effectively models the group processes involved in solution synthesis. Implications to developing DAI Systems: Two of the important problems in the design of a DAI system are (i) the problem 26 decomposition and (ii) synthesis of the local solutions generated by the individual agents. Though (i) was straightforward in the example presented, since it just involved breaking up the data set, it is not true in general that a given problem can be decomposed in this way. The question of problem decomposibility in general is domain or problem specific and so will have to be assumed given if we are to have a domain-independent system. Thus the most important implication of the DLS system to the DAI approach is a general solution to (ii) i.e., the feasibility of using a GA as a method for synthesizing the local solutions generated. Since the concept underlying the working of a GA is based on the building block hypothesis , wherein the GA tries to locate good building blocks (in terms of schemata, Goldberg [1989]) which can be combined to get a near optimal solution, it lends itself easily to the synthesis of partial solutions generated by the agents. One of the problems in a DAI approach is the conflict of interest between an individual agent and the system as a whole. In other words, since the agents usually work on sub-problems, they tend to have a local view of the problem and getting an optimum solution based on the local view does not always lead to a globally optimum solution. Hence, what is needed for synthesizing these local solutions is a method which can take these solutions and combine them using a global search. GA has this unique ability to perform a globally oriented search using the building blocks (which in this case are provided by the agents) and so should be the logical choice for synthesizing the solutions. This is also confirmed in the case of distributed learning by the empirical results presented in §5, with the major difference being that the solutions provided by the agents were complete but with local view rather than being partial with local view. Thus, as a DAI research, the DLS system (I) shows the feasibility of using genetic algorithm as a means of synthesizing individual agent solutions, and (2) describes the multi-agent learning mechanisms for DAI systems. Implications for GDSS: The characteristics of group problem solving revealed in this study can help delineate the necessary designs needed for using computer-based systems for supporting group problem solving. Unlike most of the empirical studies done in the GDSS area, we attempted to describe group problem solving by a computational model. This effort is parallel with the way the General Problem Solver was developed to simulate (single-agent) problem solving(Newell and Simon, 1972). Our findings indicate that the group performance depends on the group size, the diversity among the agents, the task distribution, and mechanisms for focusing and selecting the relevant information. These findings computationally confirm the importance of allowing for equal participation by the members in a GDSS, so that more diversity can be incorporated. Equal participation in a GDSS situation also enables the group to pursue multiple solution paths, just as 27 in our simulation, which can help avoid premature convergence into bad solutions. The findings also show the benefit of using mechanisms for iteratively elicitating views of the issues, evaluating these views, and selecting candidates for solutions. We have in effect developed a system capable of simulating group problem-solving in a specific task domain: rule induction. Table 6.2 below summarizes the important correspondences between the various parameters and mechanisms involved in a GDSS session and our process model for group problem-solving. Insert Table 6.2 here The results also underscore the importance of a GDSS in providing the support needed to aid the group problem solving as the group size increases. Our findings indicate that the group performance reaches a peak at a certain group size, n*, and then decreases. In other words, the best group performance is attainable at n* and if that level of performance is to be maintained, it is necessary to support the group by getting the members to focus more quickly and precisely on the issues. That is why the need for GDSS becomes more important when the group gets larger. In a fundamental way, our results also help justify the development of GDSSs. The advantages of having a group of people working together is more than pooling together their knowledge and expertise; something new could be generated synergistically not attainable by the individual agents if they were working alone. We refer to this as emergent intelligence. In his book, the Fifth generation, Feigenbaum refer to this as " Knowledge Fusion"(Feigenbaum and McCorduck, 1984). Implications for Machine Learning Methods: As mentioned before, the DLS also provides an efficient computational procedure for the problem of rule learning. Although, a particular learning algorithm, PLS1, was used in the implemented system, it is possible to replace it with any learning algorithm. For example, ID3 programs can be used as individual agents in which case the DLS takes different trees generated by the ID3 programs and combines them using a GA to give a more concise and accurate tree. Comparison of DLS with ID3 (Sikora&Shaw[1990]) shows that DLS produces better results than ID3, hence it is possible to obtain a more concise tree generation program by using ID3 programs in DLS. In fact, it is even possible to use different learning algorithms as different agents in the DLS provided we make sure that the representations used are made compatible with that of the GA. 28 Since the concept of bias 9 is an important one in inductive learning and since different algorithms use different biases (either implicitiy or explicitiy), using different algorithms in DLS as different agents provides a unique approach of using multiple biases. This also has important implications in terms of the solution (hypothesis) quality and efficiency because usually it is not known a priori which bias is suitable for the problem at hand and use of wrong bias can sometimes make the problem unlearnable or very inefficient. Thus, combining different algorithms can be an insurance against the above problems. Summarizing, for the inductive learning area: (1) DLS brings the potential of more problem solving power in the form of distributed systems, (2) it is sufficiently general to be applied with any learning algorithm, and more importantly (3) it improves the overall performance of the learning algorithm . Implications for Problem-Solving Strategies: Although DLS serves as a model demonstrating the salient features of group problem-solving behavior, it also is an embodiment of two fundamental problem-solving strategies which have been widely used elsewhere. There are two important reasons for using a distributed approach to problem-solving: first, the problem at hand may be too large and complex for a single agent to solve; second, the chance of successfully getting the solution can be improved by searching through multiple paths for the solution. These two strategies - identified as decomposition and diversification - have been used previously for problem-solving. The decomposition used in DLS is basically a divide-and-conquer method to break the problem into simpler sub-problems to tackle. In a similar vein, it has been used quite extensively in solving large-scale optimization problems. Interestingly, just as the way we aim at gaining insight into group problem-solving systems through the distributed approach, the decomposition approaches to solving optimization problems are, besides being used as a solution strategy for solving large-scale mathematical programming problems (Dantzig&Wolfe[1961], Lasdon[1968]), also used to model decision making within decentralized organizations. On one hand it helps solve difficult problems by transforming them into several sub-problems which have simpler structures and are easier to solve; on the other hand, the decomposition process and the way partial solutions are transferred correspond nicely to the way problems are solved in a decentralized organization, where the individual departments are assigned with tasks/resources and coordinated by a pricing mechanism called transfer prices (Burton et al.[1974]). 9 Bias is related to the representational language used by a learning algorithm which allows it to constrain the search space of all possible hypothesis. 29 As mentioned, another advantage of taking the distributed problem-solving approach, as in a human group situation, is that more diverse viewpoints and sources of ideas are present in guiding the problem-solving activities, thereby increasing the chance of success. An interesting source of literature related to the analysis on the diversity among a group of individuals and its impact on group performance comes from the study on strategies for biological evolution. Brady(1985), for instance, showed that strategies used in biological evolution - such as speciation, wherein only individuals within a species compete against each other - can also be used as strategies for solving optimization problems. Brady also articulated the importance of diversification in solving optimization problems such as the traveling salesman problem. He showed that the performance can be improved simply by dividing the computer time equally between two independent trial solutions (i.e., two agents) and selecting the best. This strategy can be generalized into the N-agent case - according to Brady's analysis, even though these agents have to share the computation time and resources, they still result in better performance on balance. This finding is quite consistent with our empirical results reported in §5. 7. Conclusions In this paper we have tried to bridge the gap between the fields of DAI and GDSS by extracting a set of core group parameters and group processes involved in a group problem-solving situation. Based on the identified set of parameters and processes, we presented the design, implementation, and evaluation of the Distributed Learning System (DLS), a system modelling the group problem-solving approach to the task of rule learning. One of the central question addressed by the above experiments was whether group problem-solving strategy, using multiple agents, is in general a better strategy than a centralized, single-agent approach. The task of rule learning (or more generally inductive learning) was chosen as the domain of the system because of its importance in automating the construction of knowledge-based systems. We showed empirical evidence supporting the emergent intelligence behavior, wherein the group can achieve more than the sum of its individual members using the same amount of resources. We also addressed the related issues concerning the effect of the situational variables like the group size, diversity within the group etc. on group performance measured in terms of solution quality - measured by prediction accuracy, rule-size, and generality of the concept generated; time to decision; and number of unique alternatives generated. We presented empirical evidence which showed that the performance of DLS was depended on the group size and reached a peak at an optimum group size of n*. This was explained by the interaction of two opposing forces acting on the group performance: the number of unique alternatives and the quality of the alternatives 30 generated by the group. We also showed the importance of maintaining diversity among the group for better performance. In view of the importance of information sharing in a group problem- solving situation, a group mechanism must be used to help the group sort through the pool of hypotheses generated, evaluating them, and making decision on what the correct concept description should be in the group solution. In DLS we incorporated a solution-synthesis procedure using a genetic algorithm, which plays an important role in generating the group solutions based on the individual agents' results. This synthesis step should be generalizable to other distributed problem-solving domains. These results not only help shed light on the fundamental components and mechanisms involved in successful group problem solving situations, they also help provide guidelines for developing DAI systems, GDSSs, machine learning methods, and problem-solving strategies. 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D., "Models for Inferring Relationships Between Group Size and Potential Group Productivity," Behavioral Science, 11 (1966), 273-283. WHITE, S.E., J. E. DITTRICH AND J. R. LANG, "The Effects of Group Decision Making Process and Problem Situation Complexity on Implementation Attempts," Admin. Sci. Quart. , 25, 3 (1980), 428-440. WOO, C.C. AND LOCHOVSKY, F.H., "Supporting Distributed Office Problem Solving in Organizations," ACM Trans, on Office Information Systems, July 1986, pp. 185- 204. 34 Appendix A: PLS1 Algorithm The inductive process followed by the PLS 1 algorithm starts with the entire space of possible events (the 'feature space'). The space is then further split into two 'regions,' those of which have a greater likelihood to being in a specific class (positive events) and those which have a greater likelihood to being in the other classes (negative events). The process of splitting continues, each split using only one attribute that is chosen according to an information-theoretic approach, until a stopping criterion is satisfied. In each iteration, the region R in the feature space can be defined by the tuple (r,u,e ), where r is region or disjunct represented as conjunction of conditions (similar to the representation given in sec. 4.2; a disjunction of regions would then constitute a concept or hypothesis); u is the utility function giving the fraction of positive events to the total events covered by the disjunct, and e is the error rate allowed by the disjunct which is based on the number of positive events covered by the disjunct as compared to the total number of positive events. Since the purpose of the algorithm is to maximize the dissimilarity between the disjuncts, the split is made based upon maximizing the difference in the utilities of the two disjuncts (known as the distant function). Each disjunct, also called a hyper-rectangle, is also associated with its error measure e . In proportion to the number of positive events covered, e has a lower value. The distant function (d ) is defined as follows d = | log u i - log u 2 | - t * log(e i*e 2) where, u \ 7 u 2 - utilities for a tentative region dichotomy, e \, e 2 - respective error factors, t - a constant representing degree of confidence. Larger values of d correspond to higher dissimilarity. Let S be the set of positive and negative training events and R as the hyper-plane that contains all events in E , the PLS1 algorithm can be summarized as follows: ALGORITHM PLS 1: While any trial hyper-plane remains untested, do Begin 1. Choose a hyper-plane not previously selected to become a tentative boundary for two subregions of R, r \ and r% 2. Using the events from S , determine the utilities u \ and u 2 of r \ and r 2, and their error factors e \ and e 2. 3. If this tentative dichotomy produces a dissimilarityc/ larger than any previous 35 value for d then : create two permanent regions R \= (r \,u \,e \) and R 2 = (r 2,« 2^ 2 ) having the (previously recorded) common boundary that gives the most dissimilar probabilities; else : place R in the defined region set R to be output, and quit. End. 36 Appendix B: Genetic Algorithm Genetic Algorithms (GAs) are adaptive search algorithms which have the properties of parallel search and ability to locate global maxima without getting trapped in local maxima. Goldberg(1989), describes GA as search algorithms based on the mechanics of natural selection and natural genetics. They combine survival of the fittest among string structures with a structured yet randomized information exchange to form a search algorithm with some of the flair of human search. In every generation a new set of artificial creatures(strings) is created using bits and pieces of the fittest of the old; an occasional new part is tried for good measure. While randomized, genetic algorithms are no simple random walk, they efficiently exploit historical information to speculate on new search points with expected improved performance. A GA should be equipped with the following four components (1) a chromosomal representation of solution to the problem. (2) a way to create an initial population of solutions. (3) an evaluation function that rates the solutions in terms of their "fitness". (4) genetic operators that alter the composition of solutions during reproduction In addition, in applying GA, one needs to decide the various values for the parameters that the genetic algorithm uses, such as the population size, the number of generations, and the probability of mutations. Since the GA works with string structures (analogous to chromosomes in biological systems), the solutions should be encoded and represented in a string form. This low level representation with which a GA works is called genotype , and the corresponding set of apparent characteristics is called phenotype. The individual elements of the genotype are called genes , and their possible values are alleles. The GA's work with a population of solutions at a time, the number of solutions being a parameter of choice. Each solution is evaluated using the fitness function and a "fitness" score is assigned to it. Starting from an initial population of solutions, the GA exploits the information contained in the present population and explores new solutions by generating a new population of solutions from the old population through application of recombination or genetic operators. The genetic operators most often used are: (a) reproduction; (b) crossover, and (c) mutation. The reproduction operator just duplicates the members of the population to be used to derive new members. The number of copies that each member (solution) gets is proportional to its fitness score. Thus the fitness of an individual is clearly related to its influence upon its future development of the population. In other words, solutions which have high fitness compared to 37 others, have more influence in altering the other solutions and solutions which have very low fitness usually get replaced by better solutions. After reproduction, new individuals are generated by selecting two individuals at a time from the resulting population and applying the operator of crossover. Crossover exchanges the genes between the two selected individuals(parents) to form two different individuals. Crossover is the key to the power of the GAs as it helps in combining information from different solutions to discover more useful solutions. Usually crossover is applied with a constant probability P c . The mutation operator randomly changes some of the genes in a selected individual and is applied at a much lower rate P m (i.e., Pm « 1 ) . The basic GA can thus be described by the following procedure: PROCEDURE GA (population size n, max. number of generations Ng) begin; select an initial population of n genotypes {g}; no-of-generations = 0; repeat; for each member b of the population; compute f( g), the fitness measure for each member; /* evaluation */ repeat; stochastically select a pair of genotypes gi, g2 with probability increasing with their fitness f; /* reproduction *l using the genotype representation of gi and g2, mutate a random bit with probability p u ; /* mutation */ randomly select a crossover point and perform crossover on gi and g2 to give new genotypes g\ and g' 2 ; /* crossover */ until the new population is filled with n individuals g'j; no-of-generations = no-of-generations + 1; until all the members converge; /* termination */ end; 38 FIGURES: Figure 2.1 Phases of Distributed Problem Solving Figure 3.1 The Distributed Learning System using Group Problem Solving Figure 4.1 The results of single-agent and multi-agent approaches to inductive learning Figure 4.2 Comparison of multi-agent vis-a-vis single-agent approach TABLES: Table 3.1 Correspondence between mechanisms involved in group decision making and in aGA Table 4.1 Results of the 5 individual agents together with the final synthesized result Table 5.1 DLS results for different values of n and d Table 5.2 Effect of group size on group performance Table 5.3 Effect of group diversity on group performance Table 6.1 Dependence of group performance on group size Table 6.2 Correspondences between the parameters and mechanisms involved in a GDSS and the proposed model of group problem-solving SUBPROBLEM SOLUTION PROBLEM DECOMPOSITION o ANSWER SYNTHESIS Figure 2.1 Phases of Distributed Problem Solving / p (training data set) (learning agents) PROBLEM DECOMPOSITION TASK ASSIGNMENT INDUCTIVE LEARNING INDUCTIVE LEARNING INDUCTIVE LEARNING LOCAL PROBLEM SOLVING (group adaptation) SOLUTION SYNTHESIS SOLUTION SYNTHESIS (learned concept) Figure 3.1 The Distributed Learning System Using Group Problem Solving Nov 12 1900 1990 mult 1 -agent . example Page 1 (genetics) NULTI-ACENT VS SINGLE-AGENT PROBLEM SOLVING) USING SIG. LEVEL OF 1 FOR DLS AND USING LEAVE OUT 80» ONLY FOR PLS PART OF DLS ) USING 5 DIFFERENT SAMPLE RESULTS FROM PLS1 TO GET THE INITIAL POP.) MUTATION RATE OF 5»0.01) USING 100 GENERATIONS) TESTING SAMPLE HAS 79 »VE AND 35 -VE TRAINING SAMPLE HAS 276 »V£ AND 182 THE SINGLE AGENT RESULTS ARE) RULE SIZE • 17) (2 63) (2 63) (8 61) (3 63) (10 21) (2 63) (2 63) (8 61) (3 63) (10 11) (2 63) (2 63) (8 IS) (3 63) (22 63) (2 6) (2 63) (16 61) (3 22) (22 63) (2 52) (2 63) (16 61) (23 63) (22 63 (2 6) (2 63) (16 61) (23 63) (22 63) (7 52) (2 63) (50 61 (2 52) (2 63) (16 61 (2 52) (2 63) (16 61 (2 52) (2 63) (16 61 (2 52) (2 63) (16 61 (2 52) (2 63) (16 61 (2 31) (2 63) (16 47 (2 31) (2 63) (16 61 (2 11) (2 63) (16 61 (32 63) (2 63) (16 37) (3 63) (22 50 (2 63) (2 63) (38 61) (3 63) (22 63) 23 63) (22 63 3 63) (22 63) 63) (22 50) 13) (22 63) 63) (22 50) 14) (51 63) 63) (28 63) 63) (28 63) 63) (28 63) ECS) VE ECS) (5 63) (0 (5 63) (39 (22 63) (0 (10 63) (0 ) (10 63) (10 63) ( ) (10 63) (10 63) ( (10 63) ( (5 63) (0 (5 63) (1 (5 63) (1 (23 63) ( (5 26) (4 (27 63) ( ) (5 63) ( (5 63) (0 38) (0 63) ( 63) ( 35) ( (0 35) 35) (0 35) 35) 36 36) 18) ( 9 36) 9 36) 37 40) 1 63) 41 63) 37 63) 63) ( 63) ( 63) 63) 55) (0 30 (31 55 (31 5 (0 55) (0 55 56 60) (56 60 (56 60 (0 45 (0 45) (0 45 (0 60 61 63) 28 50)) - ( (28 50)) - (28 50)) - (28 34)) - ) (28 34)) ) (28 34)) 5) (28 34)) (35 50)) - ) (28 50)) (28 50)) - ) (28 50)) ) (28 50)) ) (28 50)) (28 50)) ) (28 50)) ) (28 50)) (28 50)) 4 0) - (0 (1 1) - ( (2 2) - ( (1 0) - ( - (12 0) -(4 0)- - (1 0) (176 12) - (13 4) (1 1) - - (16 3) "(10)- - (« 1) - (30 4) - -(4 0)- "(2 0)- (2 1) " 0)) 0)) D) 0)) - (5 0)) (1 0)) - (0 0)) - (50 6)) - (0 0)) (0 0)) - (5 1)) (0 0)) (2 0)) (6 3)) (2 0)) (0 0)) (0 0)) " — — ) 5 AGENTS WHICH ARE USED AS INITIAL POPULATION BY THE GA ARE) (0 56) (10 58) (13 63) (13 57) (0 37) (0 55) (32 38)) - (188 40) - (54 8)) (0 56) (10 58) (13 63) (28 57) (0 37) (56 63) (32 38)) - (9 1) - (0 0)) (0 56) (10 58) (40 63) (13 57) (38 63) (0 63) (32 38)) - (20 6) - (7 3)) (0 56) (10 58) (13 63) (13 57) (60 63) (0 63) (32 38)) - (1 0) - (0 0)) (15 58) (4 63) (25 56) (3 57) (0 57) (24 58) (34 40)) - (177 48) - (54 9)) (15 58) (4 63) (25 50) (3 57) (0 57) (24 58) (34 40)) - (40 15) - (12 6)) (8 12) (7 52) (10 26) (5 63) (0 57) (24 63) (30 39)) - (2 2) - (0 0)) (8 12) (7 52) (27 63) (5 63) (0 57) (24 63) (30 39)) - (2 4) - (0 0)) (19 61) (7 52) (27 63) (5 63) (0 35) (24 45) (30 39)) - (155 20) - (49 4)) (19 61) (7 52) (58 63) (5 63) (36 46) (24 45) (30 39)) - (2 2) - (1 0)) (19 61) (7 52) (27 63) (5 63) (47 57) (24 45) (30 37)) - (19 4) - (4 1)) (19 61) (7 52) (27 50) (5 63) (0 57) (46 60) (30 39)) - (34 13) - (10 3)) (15 58) (4 52) (20 59) (3 63) (0 38) (24 60) (29 50)) - (162 35) - (49 4)) (15 58) (4 52) (20 50) (3 63) (0 38) (24 60) (29 50)) - (47 12) - (12 4)) (15 58) (4 52) (20 59) (3 63) (0 38) (61 63) (29 36)) - (2 4) - (0 0)) (15 58) (4 52) (20 59) (25 63) (39 41) (24 63) (29 50)) - (3 1) - (0 1)) (0 58) (4 52) (20 59) (3 63) (S3 S3) (24 63) (29 50)) - (4 0) - (1 0)) (21 56) (7 SS) (13 34) (3 63) (0 34) (0 58) (28 SO)) - (9 0) - (1 0)) <21 561 (7 551 (35 63) (3 63) (0 37) (0 55) (28 50)) - (193 37) - (55 6)) (37 56) (7 55) (35 63) (3 63) (38 57) (0 55) (28 50)') - (25 14) - (7 6)) (21 34) (7 SS) (33 63) (3 63) (0 57) (56 58) (28 50)) - (3 4) - (0 0)) RESULTS POPSIZE ACENT 1 (2 42) (2 42) (2 11) (12 42 AGENT 2 (2 36) (37 63 (2 63) (2 63) AGENT 3 (2 63) (2 63) (2 63) (2 63) AGENT 4 (2 36) (37 63 (2 63) (2 63) AGENT 5 (2 63) (2 42) (2 42) (2 31) (2 42) OF THE - 21) ) (2 42) (2 42) (2 42) ) (2 42) ) (2 63) ) (2 63) (2 63) (2 63) ) (2 63) (2 63) (2 63) (2 63) ) (2 63) ) (2 63) (2 63) (2 63) ) (2 63) (2 42) (2 42) (2 42) (2 42) THE FINAL RESULTS AFTER SYNTHESIZING TME ABOVE RESULTS ARE) (2 54) (2 43) (21 58) (7 55) (9 63) (11 63) (0 36) (0 55) (28 50)) - (203 30) - (57 4)) (2 31) (2 63) (18 61) (3 60) (27 59) (1 63) (39 63) (28 47) (24 38)) - (34 7) - (9 2)) (34 62) (2 42) (0 56) (2 58) (13 63) (4 57) (16 41) (56 62) (32 38)) - (20 7) - (5 2)) Fjiterlp-} nrlhble Re*d- Eval -Print l^sop. Type (DRIBBLE) to exit. Figure 4.1 The Results of Single-agent and Multi-agent approaches to Inductive Learning 20% of the complete ► Single-agent data set accuracy=83.3% rule size=17 Agent 1 accuracy=74.6% GA accuracy=86% data set 20% of the ^ rule size=4 Agent 2 accuracy=75.4% data set 20% of the ► rule size=4 Agent 3 accuracy=79.8% data set 20% of the rule size=4 rule size=3 Agent 4 accuracy=77.2% data set 20% of the *> w rule size=4 Agent 5 accuracy=76.3% data set ► rule size=5 Figure 4.2 Comparison of Multi-agent vis-a-vis Single-agent approach. Mechanisms involved in a group decision making Operators involved in a genetic algorithm evaluation of alternatives fitness function selection of promising alternatives and elimination of less promising ones reproduction modification of individual alternatives mutation combination of different alternatives to get new ones crossover Table 3.1 Correspondence between mechanisms involved in group decision making and in a GA Agents Prediction accuracy Rule size g Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 74.6% 75.4% 79.8% 76.3% 76.3% 4 4 4 4 5 0.00739 0.00655 0.00621 0.0184 0.01293 Synthesis 86% 3 0.0296 Table 4.1 Results of the 5 individual agents together with the final synthesized result. n=l d Prediction accuracy Rule-size Time to decision No. of unique alternatives generated 8 t 1.0 85.4% 17.4 lhr. 17.4 0.0658 n =2 d Prediction Rule-size Time to No. of unique 8 t accuracy decision alternatives generated 0.05 70.4% 2.2 lOmin. 5.4 0.0124 _ 0.1 72.1% 2.4 12 min. 5.6 0.0345 - 0.2 79.8% 3.2 36min. 11.4 0.0332 - 0.4 81.1% 3.6 1 hr. 16 0.0998 - 0.5 85.1% 3.4 1.42 hrs. 21.4 0.081 0.6 83.9% 2.8 1.4 hrs. 21.8 0.1152 0.2N 0.8 80.5% 2.8 1.72 hrs. 25 0.1467 0.6N n =5 d Prediction Rule-size Time to No. of unique g t accuracy decision alternatives generated 0.05 73% 2 16 min. 9 0.0121 - 0.1 78.8% 2.8 37 min. 14.4 0.0314 - 0.15 81.6% 3 1.07 hrs. 20.8 0.0816 - 0.2 86.9% 3.2 1.57 hrs. 29.2 0.0764 0.4 84.1% 2.8 2.5 hrs. 44 0.1052 N 0.6 81.9% 2 3.25 hrs. 49.6 0.1604 2N 0.8 82% 2.8 5.5 hrs. 50 0.1145 3N Table 5.1 DLS results for different values of n and d n =7 d Prediction Rule-size Time to No. of unique 8 t accuracy decision alternatives generated 0.05 77.2% 3 40min. 15.4 0.0428 0.1 82.6% 3.4 1.25 hrs. 23.2 0.0389 - 0.15 80.5% 3 1.6 hrs. 29.6 0.0414 0.1N 0.2 80.7% 2.8 1.9 hrs. 35.2 0.0508 0.4N 0.4 80.9% 2.8 3.15 hrs. 50 0.0816 1.8N 0.6 81.6% 3 4.2 hrs. 50 0.1346 3.2N n= =10 d Prediction Rule-size Time to No. of unique 8 t accuracy decision alternatives generated 0.05 77.4% 2.8 lhr. 22.6 0.0262 _ 0.1 78.4% 3.8 1.75 hrs. 31.6 0.0582 0.15 81.2% 3 2.2 hrs. 41.8 0.0928 0.5N 0.2 80% 3 2.8 hrs. 47.6 0.0807 N 0.4 82.3% 2.8 3.6 hrs. 50 0.0797 3N 0.6 80.5% 3.2 5.2 hrs. 50 0.1148 5N 0.8 81.6% 3.6 6.5 hrs. 50 0.125 7N Table 5.1 DLS results for different values of n and d(Continued) No. of Prediction Rule-size Time to No. of unique agents accuracy decision alternatives generated 1 85.4% 17.4 lhr. 17.4 2 85.1% 3.4 1.42 hrs. 21.4 5 86.9% 3.2 1.57 hrs. 29.2 7 80.5% 3 1.6 hrs. 29.6 10 78.4% 3.8 1.75 hrs. 31.6 Table 5.2 Effect of group size on group performance r i=5, d=0.2 Diversity Prediction Rule-size Time to Number of number accuracy decision alternatives generated 1 79.3% 2.8 lhr. 26 2 (2+3) 81.1% 2.2 1.1 hrs. 27.6 2 (1+4) 81.2% 2.2 1.3 hrs. 31.4 3 (1+1+3) 83.5% 2.8 1.5 hrs. 30.4 3 (1+2+2) 84.4% 4 1.5 hrs. 26.2 4(1+1+1+2) 82.3% 2.8 1.5 hrs. 31 5 86.9% 3.2 1.57 hrs. 29.2 Table 5.3 Effect of group diversity on group performance Number of Amount of resources No. of unique prediction agents available to each agent alternatives accuracy (as fraction of total resources) generated 1 1 17.4 85.4% 2 0.5 21.4 85.1% 5 0.2 29.2 86.9% 7 0.15 29.6 80.5% 10 0.1 31.6 78.4% Table 6. 1 Dependence of group performance on number of agents GDSS Process model of group problem-solvinp group size number of agents allowing anonymity for equal participation maintaining diversity to avoid dominance systematically directing the pattern, timing, or content of discussion providing mechanisms for the evaluation, selection, and generation of alternatives displaying individuals inputs on a public screen and facilitating message passing facilitating information sharing Table 6.2 Correspondences between the parameters and mechanisms involved in a GDSS and the proposed model of group problem-solving