Maintaining Knowledge Distribution System's Sustainability Using Common Value Auction Anas Al-Tirawi Computer Science Department, Wayne State University, Detroit, MI 48202, U.S.A aaltirawi@wayne.edu Robert G. Reynolds Computer Science Department, Wayne State University, Detroit, MI 48202, U.S.A reynolds@cs.wayne.edu Abstract In Cultural Systems there are many ways to collect and environment so that it can maintain or improve its distribute problem solving knowledge within social performance over time [1]. networks. Such mechanisms include games, auctions, and A cultural system will devote some of its resources various voting mechanisms. Here, a new auction to each of these two properties. If too many resources mechanism, Common Value Auctions, is presented. In this are devoted to robustness in the short term, it may paper Common Value Auctions are used to distribute impact its ability to be resilient in the long term and problem solving knowledge within a given model of social vice versa. So there needs to be a balance between the systems. These mechanisms are compared with the other two in order for a system to be sustainable over the distribution mechanisms in the solution of dynamic real- long term. valued optimization problems. Specifically, their relative abilities to support the robustness and resilience of Cultural One key aspect of a Cultural System is how Systems in environments that vary in their dynamic information can be distributed throughout its social complexity from static to chaotic are assessed. The Cultural networks in order to support both robustness and Algorithms Toolkit (CAT) is used as a vehicle to generate resilience. In this paper, the impact that various real-valued dynamic problem landscapes of varying knowledge distribution mechanisms in a system will complexities. The results show that using the Common have on the systems robustness and resilience will be Value Auction in CAT4 has significant improvements over assessed. These mechanisms include voting schemes, Weighted Voting methods (CAT2) in terms of both auctions, games, and pure random processes. They robustness and resilience across complexities that range from static to chaotic. will be studied through the lens of a computational model of cultural evolution, Cultural Algorithms. Keywords—cultural algorithm, sustainability, evolutionary algorithm, common value auction, robustness, resilience. In the next section the basic knowledge distribution mechanisms currently available for I. INTRODUCTION Cultural Algorithms are discussed. In section III the new knowledge distribution mechanism, Common Cultural systems provide a framework for human Value Auction, is described. Section IV describes the existence. One key observation that can be made is that dynamic landscape in which the performance of the certain cultures are more sustainable over time than new mechanism will be assessed. Section V provides others. Robustness and reliance are key factors behind a description of the experimental framework through the sustainability of cultural systems. These two which the sustainability of the Cultural Algorithm factors are needed, so the system can handle a wide systems will be assessed. In the following section the range of inputs/ perturbations while maintaining its performance of Common Value Auctions will be integrity, structure, and reducing the severity of the assessed in terms of the systems relative sustainability. impact that these perturbations can have on a system. Section VII presents the conclusions and suggestions Robustness is the property of a complex system to for future work. withstand the impact of a dynamic change or perturbation in its environment. Like a boxer in the ring, robustness is the quality of a system to endure a series of blows but still continue to function at a certain level or above. Resilience on the other hand is the ability of the system to adapt to the dynamics of its This work was supported by NSF grant #1744367. II. KNOWLEDGE DISTRIBUTION in the population space. Fig. 2 shows the different MECHANISMS IN CULTURAL knowledge distribution mechanisms. It is important to ALGOGRITHMS note that the amount of information that a distribution algorithm knows about a problem solution the The Cultural Algorithm (CA) was introduced by fidelity), increases from left to right [8]. Reynolds [2] as a computational model of Cultural Systems and their Evolution. It has been applied to The first mechanism was called the Marginal many practical applications since then, one of which Value Approach by Peng [9]. Every individual was is: modeling the origins of agriculture in the valley of controlled or directed by one KS in each generation. Oaxaca, Mexico [3]. In addition, CAs have been Peng in her approach [10], integrated the five KSs in applied to concept learning [4], decision trees [5], the belief space into a single influence function as software testing [6] and other hybrid approaches [7]. shown in Fig. 3. Peng used a random process based on their relative performance to select a KS to influence an individual. A KS roulette wheel, with proportional areas, based on relative performance, allows for an informed random selection process. However, Peng did not account for the influence among neighbors of a social network. Fig. 1 Cultural Algorithm s framework [3] As shown in Figure 1, the CA is a knowledge intensive evolutionary framework. First, the individuals in the population space are evaluated in terms of their performance in a problem space. Next, Fig. 3. Integration of multiple KSs [10] a subset of individuals is selected via the acceptance function and their performance is uploaded into the Next, Ali and Reynolds [11], [12] developed a Belief Space which is a network of Knowledge majority win approach. In their approach the influence Sources (KS). After updating the Belief Space of the neighbors is considered when selecting a network, the KS’s can direct the next generation of KS. The social fabric is the connection between the individuals in the population space via the influence individuals in a population. A conflict resolution function. The knowledge sources (KS) utilize a variety process allows individuals to select the KS by which of distribution mechanisms in order to circulate their they are influenced, if their neighbors are influenced influences among the individual agents in the by one or more different ones. First, each individual population space. was assigned a direct influence based upon the relative performance of the KSs using a roulette wheel as suggested above by Peng. Next, Ali used a conflict resolution strategy based on majority win in order to calculate the controlling knowledge source for each individual. They summed up the direct influence of the adjacent individuals in the social fabric and those of its current neighborhood. After that, the KS with the majority of the votes won the influence over this individual in that generation of the systems. Another approach is the Weighted Majority Win proposed by Che [8]. Che used the average fitness of each KS to determine how much weight this KS deserves in the vote. The key to determining the Fig. 2. The big picture for all Knowledge Distribution Mechanisms weight for each KS is the average fitness value of the that utilize the CA. individuals that have recently been influenced by the Previous mechanisms have utilized CAs to knowledge source in the population. Figure 4 gives an distribute the influence of the KS over the individuals example of the weighted voting process. The individual, A0, has information about five competing As with the previous mechanisms the process for a KSs. They are represented in the figure as follows: S: generation starts with the assignment of a knowledge Situational, D: Domain, H: History, T: Topographical, source to each individual as their direct influence. The N: Normative. AO: represents the individual. The next step is the selection of the bidders, those KS’s number of votes for each KS is given as x (number of who will be participating in the auction, from the KSs votes). For Situational it is x3, or 3 individuals have it list. The KS’s compete to influence each individual as a direct influence. The weight along each arc is the (x). The algorithm currently only allows the normalized relative performance for each KS. immediate adjacent neighbors of the individual (x) to participate in the auction. The actual auction takes place as shown in Fig. 5, where the system requests the selected bidders to submit their bidding values. Each selected KS will spin its correspondent wheel to get the bidding value. Finally, the auction system will determine the winner and assign the winner KS to influence individual (x) It may take several iterations to do so as shown in Fig. 5. Fig. 4. Weighted Majority win in belief space through the social network [8] In Fig. 4, the winning KS is the Domain KS, even though it does not have the most votes (votes=2). However, (D) does have the greater weight, which is the key factor in the weighted-majority win approach. Che has used many network topologies in his system including LBest, Square, Hexagon, Octagon, Hex- decagon, and Gbest. In addition to that he has also tested his system on different problem complexity levels [13]. As a result, Che concluded that when the Fig 5. Conducting the Auction [14]. performance function is of higher fidelity, the weighted approach can spread new information faster In the auction mechanisms above the bidders did not through a population than the majority win approach. know anything about the properties of the individuals When the signal strength of information about the upon which they were making bids. Those properties problem becomes even stronger, the auction approach can be the location in the network, the number of can be effectively employed to find a immediate neighbors, and the strength of their solution. Kinnaird-Heether and Reynolds’ [14], [15] connections, what knowledge sources have influenced embedded auction mechanisms into a CA. The new it in the past, among others. In the next section an version was called CAT3. In CAT3, the production approach, the Common Value Auction, is discussed. of the bidding tokens is the starting point. The process This approach provides a common set of parameters starts by producing bidding tokens and uses them to that are available to all bidders. These parameters can form biddings wheels for each KS. These tokens are be used to condition the bids made by the participants. generated by listing all of the individuals that were III. THE COMMON VALUE AUCTION recently influenced by a KS over a given previous time DISTRIBUTION MECHANISM window, t.. The individual’s fitness values are the bidding tokens and are normalized so that each The new mechanism, Common Value Auction previous result takes up a relative proportion of the Toolkit (CAT 4) is an extension of CAT3. CAT4 token bidding wheel. The bid of a KS corresponds to propagates the influence using the Common Value the performance associated with the result of spinning knowledge. The Common Value knowledge is a set of the bidding wheel. Since the KS’s do not have specific parameters that every KS can know about the common knowledge about the location of the individuals in the social network. These include the individual in the population network, the process must individual’s location in the network and the KS(s) that be stochastic based upon past performances. influenced the individual in the past previous incentive for the KS that influenced the individual in approaches. the past and for those KSs that were able to influence the individual’s neighbors. The first step is to build the KS wheel (one wheel for all KSs), by normalizing the KS average score, In the fourth step, the influencers that satisfy their where every KS will have a wheel’s share that reflects bidding rules are then chosen to participate in the its average (score). Each KS will have a portion of the auction. The bidding wheel is spun for each to wheel that reflects the average performance of those determine their bid. In the experiments conducted here individuals who have been influenced by the each KS had a bidding wheel comprised of a single correspondent KS. Next the algorithm assigns a direct average value for the performance of the selected influencer KS randomly using the roulette wheel subset in order simplify computations at this stage. approach discussed previously to each individual in the population. This step is the same as that for all In the fifth step, the bidding strategy rules that are other mechanisms discussed so far. satisfied for a KS are then applied to the bid as shown in the rule above to give a final bid for that KS. The In the second step each KS constructs a bidding bids are then compared with each other and the winner strategy wheel that will be used later to determine their is selected to control the individual for that generation. bidding decision on a specific individual. This is done If there are no bidders, then the direct influencer of the by selecting a subset of recent individual performances individual is retained. This redistribution process is directed by that KS over a given past time window. A then repeated for all individuals in the network. wheel is constructed such that each score comprises an area that is proportional to its contribution to the total Fig 6 covers the big picture of CAT4. First, the KS score of the subset for the KS. In addition, a set of rules roulette wheel is spun to generate the direct influencer is selected to determine whether the KS will bid on an for each individual. If one or more of the individual’s individual based upon common value knowledge neighbors possess a different KS then each decides about the location of the individual in the social fabric. whether it wishes to bid for that individual using the common value information about individual. The Next, the direct influence for each individual in the selection process is governed by a rule-based expert population is compared against those of its neighbors, system associated with each KS. The selected KSs here, just the directly adjacent neighbors are used. If then participate in the bidding for the auction as the direct influence of an individual agrees with those described above. of all of its neighbors, its direct influence is then chosen to guide it during that generation. Otherwise an auction is conducted between those KS’s who directly influence that individual and its neighbors. In order to do this the bidding strategy for each of the competing KSs is checked to see if it will bid on that individual based upon the common value information. The rule set associated with the KS is checked to see if taken together they support a bid on the current individual. This “expert system” can technically be comprised of many rules. For the experiments here, the same one rule is used for all KSs. To do so, the following distributing mechanism based upon just one subset of common values, the extent to which the individual and its neighbors have been influence by the KS in the past: If KS[j] has influenced individual (i) in the past m generation or If KS[j] is influencing currently the neighbors of individual Then increase the bidding value by a bonus as shown in the equation below: Bidding value= KS’s bidding value + 0.5 Fig. 6. Big picture of CAT4 Algorithms (Boost) (1) This is where the Common Value information is used to determine the winner. We simply give IV. THE DYNAMIC PERFORMANCE ENVIRONMENT: As the value of A increases, the system generates THE CONES WORLD more complicated behavior. Figure 8 shows how Y will change as a result of A for a sequence of To analyze the results and test the performance on landscapes. The x-axis gives the number of the different levels of complexity, a robust problem generations, the z axis gives the A value, and the Y generator (Cones World) was used in both CAT2 and axis gives the Y-value produced over the given CAT4. The Cones world framework was inspired by generations for a specific A. Each of the Y trajectories the work of Morrison and De Jong [16]. This tool has is color coded with the A value that produces it. The the ability to generate dynamic problem environment color code is in the legend on the right side of the over various landscape complexities. A given cone graph. Low values of A produce gradual linear world configuration can be described as follows: changes while high values produce wildly oscillating 2 values for Y. In the next section we discuss the f(⟨x1 ,x2 ,…,xn ⟩)= max (Hj -Rj ∙√∑ni=1(xi -Cj,i ) ) (1) experimental framework of CAT4. Also, we explain j=1,k how the dynamic environment can affect the learning Where: K: the number of the Cones. Hj: the cone curve of the whole system and consequently the height, Rj: the cone slope, N: the dimensionality. Cj, i: produced results. Coordinates of the cone j in dimension i., (Xi, Yi): determine the location of the cones on the landscape. The values for the cone height, slope, and coordinates can be assigned randomly through the problem generator or logistic function. However, the values would be selected from the ranges below: Hj ∈ (Hbase, Hbase +Hrange); Rj ∈ (Rbase, Rbase +Rrange); and Cj,i ∈ (-1,1). The Max function here is used to handle the combination of the cones when they overlap. For example, if two cones overlap, the Max function will choose the height of the combined cone to be the height of the highest cone for the two overlapped cones. Fig 7 shows how the landscape looks like with the following parameters: k = 15, Hbase = 1, Hrange = 9, Rbase = 8, and Rrange = 12. To determine the dynamic changes of the system Morrison and De Jong used the logistics function below: Yi = A ∗ Yi−1 ∗ (1 − Yi−1) (2) A= Constant value, Yi = is value of Y at iteration i. Fig. 8. The value for Y (on the Y-axis as a function of A (z axis) over the number of generations, x axis. The Y curves are color coded with A-value that generated them. The color code is on the right. V. DYNAMIC EXPERIMENTAL FRAMEWORK In these experiments, the performance of the Common Value Auction was compared with the Weighted Majority algorithm for three complexity levels of A= {1.01, 3.35, and 3.99}. These three A- Values were selected because they represented a wide spectrum of complexities over which to test CAT4 against. The full list of experimental framework parameters are summarized in the table below. The Fig.7. an Example Landscape In two-dimensional space (n = 2) types of social fabrics are explained here [17]. bound by x ∈ (-1.0, 1.0), y ∈ (-1.0, 1.0) with k = 15, H ∈ (1, 20), and R ∈ (8, 20) [9]. TABLE I. EXPERIMENTAL FRAMEWORK be used to affect bidding strategies, the focus here will PARAMETERS be in just a single set of factors, the KS previously used to influence an individual and its neighbors. The goal Parameter Name Value will be to show that the addition of just this new Complexity Class 1.01, 3.35, and 3.99 information can make a substantial difference in the performance of the cultural system. Number of Runs Per 300 Complexity The first dynamic landscape to be assessed was that produced by A=1.0. As seen in the previous Number of landscapes 50 section, that landscape involves a series of small linear Max. number of generation per 800 shifts in the locations of the cones. Fig. 9 gives the landscape standard deviation of the two systems over all three environments. For the linear dynamic landscape, the Number of cones 100 two systems each was perturbed by an average of around 85 generations for each landscape change. So Number of agents 50 their relative level of robustness is about the same for Social fabrics {L-Best, Square, Hexagon, this environment. Octagon, Sixteengon, Global} Max fitness value 20 CAT4 vs CAT2 Standard Precision of solution 0.001 Deviation Comparison 100 The key hypothesis to be tested here is whether the needed to find solutions for each landscapes Common Value Auction mechanism is able to produce STD DEV of Average number of gernations 90 a more sustainable cultural system than the weighted majority voting mechanism. The extent to which this 80 is accomplished will be observed in terms of the two 70 system’s relative robustness and resilience over the course of 40,000 generations for each of the 300 runs 60 for the 3 complexity classes. 50 STD DEV of CAT4 Robustness will be assessed in terms of the ability 40 of the system to bounce back after each of the 50 STD DEV of landscape shifts for a given run. The standard 30 CAT2 deviation over the set of 300 runs will provide an 20 indicator of the need for each system to bounce back from a landscape change. Resilience on the other hand 10 will be observed in terms of the extent to which the systems are able to adapt to these landscape shifts by 0 reducing the time needed to achieve the optimum in the next landscape. The systems will then be compared in terms of how the complexity of the environment impacts their relative sustainability as the Fig. 9 CAT4 vs CAT2 standard deviation comparison. environmental complexity shifts from static, then to cyclic, and finally to chaotic. The relative resilience of each of the two systems in the linear landscape is illustrated in Fig. 10 and 11. VI. A COMPARISON OF THE RELATIVE SUSTAINABILITY OF THE COMMON VALUE AUCTION Both systems are able to significantly reduce the AND THE MAJORITY WIN KNOWLEDGE number of generations needed to find the new DISTRIBUTION MECHANISMS optimum over time. The CAT4 system was able to produce a correlation of (0.662) between the number The main difference between the two algorithms is that CAT4 uses information about the individuals of generations needed to solve the changed landscape before the auction starts. The CAT4 algorithm is an and landscape number. The corresponding coefficient informative algorithm that provides crucial of determination, the percentage of the total variance, information for the bidders about past behavior of the explained by the correlation is (0.43). CAT2 exhibited individuals in the population space. This information a coefficient of determination of (0.289). As shown in is used to trigger bidding strategies for each of the Table II the correlations were significantly different knowledge sources. While many different factors can from each other at the (0.05) level of significance. So On the other hand, CAT4 improved on its ability CAT4 was able to do a better job of adapting to the to adjust to the change in landscapes as reflected in an changing linear environment than CAT2. improved correlation coefficient (0.73) and coefficient of determination (54%) as shown in Table II. The While CAT4 exhibited a significant level of Weighted Majority system exhibited a much lower learning within an environment with linear dynamics, overall coefficient of determination, (0.17). Again, the the next question is how it would adapt to an two systems exhibited a significant difference in environment in which the changes were non-linear adaptability over time, but now in a nonlinear from landscape to landscape. A nonlinear shift in cone environment. location was produced by the landscape generated for Overall when the environment switched from a A=3.35 as shown in Fig. 8 above. The relative change linear to a non-linear one the CAT4 mechanism in robustness produced by the shift to a non-linear produced a distinctly more robust and resilient dynamic for the two systems is given in Fig. 12 and behavior than CAT2. The next question is how the two Fig.13. CAT4 exhibited an approximately 15 systems would adapt to an extremely “chaotic” generation increase in terms of its response to a environment that was characterized by the perturbation compared to CAT2. That is a significant superposition of numerous non-linear patterns of difference in its ability to rebound from a perturbation behavior? in this environment. On the one hand, a non-linear environment required CAT4 to respond more robustly than before. 500 Average Number of Generations Needed to Find the Optimum 500 450 y = -3.2381x + 243.66 R² = 0.2892 450 y = -3.7082x + 250.04 400 R² = 0.4382 Average Number of Generations Needed to Find the Optimum 400 350 300 350 250 300 200 250 150 200 100 150 50 100 0 0 10 20 30 40 50 50 Landscape Number 0 Fig. 11. CAT2 Regression line over 50 runs for complexity A = 1.0 0 10 20 30 40 50 Since the generating process was deterministic in Landscape Nnmber nature, all of the information needed to provide a perfect prediction of the environment’s dynamics is there, it is just a matter of extracting all of the Fig. 10. CAT4 Regression line over 50 runs for complexity A = 1.0 intertwined threads. 300 environment. While both system’s behavior is now clearly nonlinear, the regression line provides a general indicator of the additional stress that is placed on each system over time. In the first two environments, the systems were not y = -3.3423x + 195.2 only able to survive the perturbations but to adapt to 250 R² = 0.5413 them. This produced a strong sense of sustainability. Average Number of Generations Needed to Find the Optimum Of the two, CAT4 was more able to exploit the nonlinear environment. In the chaotic environment the theme was less on adaptability but survivability over time. Both systems displayed symptoms of stress over 200 time. 300 150 y = -1.0428x + 113.68 R² = 0.1706 250 Average Number of Generations Needed to Find the Optimum 100 200 50 150 0 0 10 20 30 40 50 Landscape Number 100 Fig. 12. CAT4 Regression line over 50 runs for complexity, A=3.35 As demonstrated in Fig. 14 and 15, the robustness of the CAT4 system is still significantly greater than that for CAT2. The difference in the number of 50 additional generations needed to response to a perturbation is now 10. That is down from 15 before, but still a significant difference in system robustness. In such a chaotic environment learning is less of an issue than sustainability. As shown in Table II the two 0 systems now exhibit a much lower level of resilience. 0 10 20 30 40 50 The coefficient of determination for CAT2 is now Landscape Number significantly greater than that for CAT4 but notice that the relation between the number of generations needed to solve the problem is now increasing with increased Fig. 13. CAT2 Regression line over 50 runs for complexity, A = 3.35 landscape number. The rate of increase is now higher for CAT2 than CAT4 which means that its performance is more susceptible to degradation in this 300 TABLE II. COMPARING CAT4 AND CAT2 REGRESSION Average Number of Generations Needed to Find the THROUGH DIFFERENT A-VALUES COMPLEXITIES y = 1.4781x + 49.985 R² = 0.1476 A- R CAT4(R2) CAT2(R2) Sig F 250 Value Change Static 1.01 0.662 0.438 0.289 0.000 200 Periodic 3.35 0.736 0.541 0.170 0.000 Optimum 150 Chaotic 3.99 0.384 0.147 0.212 0.006 Another way of comparing the two algorithms is to 100 compare the standard deviation for average number of generation needed to find the solution for given problem. The three tables below are showing the 50 comparison for the three different complexity values {A=.101, 3.35, 3.99}. As showing in Table III, CAT4 needed less number of generations to find the solution 0 for the same number of problems. Except for Octagon 0 10 20 30 40 50 topology, CAT4 was more efficient than CAT2. Landscape Number TABLE III: COMPARING THE STANDARD DEVIATION FOR CAT2 VS CAT4, A = 1.0 Fig. 14. CAT4 Regression line over 50 runs for complexity, A = 3.99 A-Value=1.01 300 y = 1.4956x + 50.809 200 Average Number of Generations Needed to Find the Average number of generation needed to find a solution R² = 0.2123 180 250 160 140 200 120 100 Optimum 150 80 60 100 40 20 50 0 0 -10 10 30 50 CAT4 Overall STD Dev Landscape Numbers CAT2 Overall STD Dev Fig. 15. CAT2 Regression line over 50 runs for complexity, A = 3.99 When the complexity increases to A=3.35, CAT4 was TABLE V: COMPARING THE STANDARD DEVIATION FOR CAT2 VS CAT4, A = 3.99 outperformed by CAT2.With the exception of the first two topology (L-Best, and Square), CAT2 was more efficient. CAT2 needed less number of generations to solve the same given problems when compare with A-Value=3.99 200 CAT4. Average number of generations needed to find a solution TABLE IV: COMPARING THE STANDARD DEVIATION FOR 180 CAT2 VS CAT4, A = 3.35 160 A-Value=3.35 140 200 Average number of generations needed to find a solution 120 180 100 160 80 140 120 60 100 40 80 20 60 0 40 20 0 CAT2 CAT4 VII. CONCLUSIONS AND FUTURE WORK CAT4 Overall STD Dev In society, there are many ways to collect and CAT2 Overall STD Dev distribute problem solving knowledge. Such mechanisms include games, auctions, and various voting mechanisms. Previous work has focused on Interestingly for Complexity level A=3.99, When the Independent value auctions. KSs did not have system is in complete chaotic situation, CAT4 knowledge about the individuals on who they were outperform CAT2. bidding and did not have consistent bidding strategies. In this paper, Common Value Auctions were presented. 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