An ensemble method utilising multiple thinking styles that boosts the wisdom of the inner crowd effect

An ensemble method utilising multiple thinking styles that boosts the wisdom of the inner crowd effect | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article An ensemble method utilising multiple thinking styles that boosts the wisdom of the inner crowd effect Itsuki Fujisaki, Lingxi Yu, Yuki Tsukamura, Kunhao Yang, Kazuhiro Ueda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3971890/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 21 Aug, 2025 Read the published version in Scientific Reports → Version 1 posted 10 You are reading this latest preprint version Abstract Previous studies have demonstrated that individuals can utilize the wisdom of crowds, known as ‘the wisdom of the inner crowd’. This requires them to estimate a single question multiple times, and subsequently average these estimates. Although several methods have been proposed to achieve more accurate estimates, its efficacy remains relatively low. Therefore, this study proposes a method that assembles multiple independent methods to stimulate the wisdom of the inner crowd effect. Particularly, our method instructs participants to provide estimates five times. Through a behavioural experiment, we confirmed that our method can produce the wisdom of the inner crowd effect. Moreover, we found that our method produced more accurate estimates than a method that required participants to estimate five times without specific instructions. Furthermore, mathematical modelling demonstrated that the effectiveness of our method was greater than that of 1.5 persons. In sum, this study proposes a method to improve daily estimates. Biological sciences/Psychology Biological sciences/Psychology/Human behaviour The wisdom of crowds The wisdom of the inner crowd estimation Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction In daily life, people often need to estimate unknown events (e.g. the number of people attending a conference or the price of a car). How can people make accurate estimates for these types of questions? One promising approach is that of ‘the wisdom of the crowds’ 1 – 12 . In other words, the average estimate of a crowd of individuals yields surprisingly accurate estimates. Researchers have investigated this topic for over 100 years. However, it is also well known that the wisdom of crowds has an underlying problem: difficulty in collecting estimates from several people. Many studies have addressed this issue. Specifically, they have shown that even an individual could use the wisdom of the crowds (called ‘the wisdom of the inner crowd’ 13 – 30 ). In these studies, individuals were instructed to produce different estimates (mainly, twice) for a single question. In other words, they were expected to produce quasi-crowd estimates. The estimates were then averaged. The results showed that the average estimate was more accurate than the individual estimate (i.e. the first estimate). Thus, the wisdom of the inner crowd has the potential to improve estimates of daily life. However, it also has a fundamental problem in that its efficacy is relatively low. Specifically, previous studies 13 , 16 reported that the accuracy of the averaged estimate corresponded to only 1.1–1.4 persons’ first estimates. Therefore, this study aims to propose a method that can ‘boost’ 31 – 35 the wisdom of the inner crowd effect. To do so, we combined multiple methods proposed in previous studies. The first refers to using the forgetting power. For a single question, a previous study 13 provided participants with a timespan (two weeks) between two estimates. Subsequently, they discovered that the wisdom of the inner crowd emerged. The second approach uses the power of dialectics. Previous studies 14 , 19 asked participants to consider the opposite in their second estimates and showed that people could utilize the wisdom of the inner crowd (called ‘dialectical bootstrapping’). The third method involves perspective-taking 36 , 37 . It is well known that taking others’ perspectives changes various forms of cognition (e.g. stereotypic biases 38 ; preferential values 28 ; and egocentric thinking 39 ). Based on these findings, previous studies have asked participants to consider others’ perspectives in their second estimates. For accessing the perspectives of others, one study 30 used general crowds, and another 29 used a person who disagreed with the participants. Both previous studies reported that people can produce the wisdom of the inner crowd. As illustrated above, these methods seem to differ in how they instruct participants to produce estimates. Therefore, it seems possible that, by combining these methods, an individual can boost the wisdom of the inner crowd effect. The purpose of this study was to test this hypothesis. In addition, along with the above methods, this study also proposes a new method. For a single question, the method makes people think intuitively (called ‘Intuition’; for the full instruction, see Table 1 ) in the first estimate and think deliberately in the second estimate (‘Deliberation’). The two estimates are then averaged (for optimal weighting, see S1 for Supplementary Information). This method is based on the findings 40 – 44 that people’s inferences differ from using heuristics (i.e. assumed to correspond to Intuition in our method) and using some knowledge (i.e. Deliberation; see Discussion section for more detail). Table 1 Full instruction for each estimate. Type of estimate Instruction Intuition Please do not think deliberately. Answer quickly what come to mind intuitively. You have eight seconds to answer. Deliberation Please ignore your intuition. Think deliberately before answering. Utilize your knowledge and experience, while being aware of the basis of the estimate. This time have no time limit. Please use enough time to think the following question. Dialectic First, assume that your second estimate is off the mark. Second, think about a few reasons why that could be. Which assumptions and considerations could have been wrong? Third, what do these new considerations imply? Was the second estimate too high or too low? Fourth, based on this new perspective, make a third, alternative estimate. (The computer display shows the second estimate). General crowd's perspective How do you think people in general estimate the following question? Think and answer how people in general estimate this. Please do not answer yourself estimate. Disagree-other's perspective Now picture a friend whose views and opinions are very different from yours. To illustrate, when discussing politics, societies, and daily affair, you often find yourself disagreeing on various issues. How would he or she answer the following six questions? Answer these questions now as this friend. Please do not answer yourself estimates. Note: Only Dialectic estimate required the second (i.e. Deliberation) estimate to be displayed, while the other estimates did not. All instructions were translated into Japanese. We examined this method on the general knowledge questions (Table 2 ; see Method section for more detail) in Experiment 1 and found that the two average estimates were significantly more accurate than that in Intuition ( t = 2.97, p = 0.0084; Fig. 1). We also found that the average of the two estimates was more accurate than that in Deliberation, although the difference was marginally significant ( t = 2.31, p = 0.055). Subsequently, we can regard this method as inducing the wisdom of the inner crowd, although it did not include the participants’ own estimates. -----Figure 1 about here----- Table 2 Questions and correct answers used in the experiments. Number Question Answer 1 What percent of the world’s roads are in India? 9.7% 2 What percent of the world’s telephone lines are in China, USA, or the European Union? 52.0% 3 Saudi Arabia consumes what percentage of the oil it produces? 72.1% 4 What percent of the world’s population is between 15 and 64 years old? 65.2% 5 What percent of the world’s population is Christian? 31.4% 6 What percent of the worldwide labour force works in the service sector? 50.6% 7 What percent of the worldwide gross domestic product (GDP) is re-invested? 25.2% 8 What percentage of Japanese adult males smoke? 27.1% 9 What percentage of Japanese households have a fixed-line phone? 69.0% 10 What percentage of the world’s countries have a higher life expectancy than the United States? 22.6% Note: We checked all the answers on 2022/10/27. We used the answers in The CIA World Factbook 57 , World Bank Open Data 58 , and the data from Japan's Ministry of Internal Affairs and Communications 59 , based on previous studies 19 , 30 . Experiment 1 used all the questions. Experiment 2 used Questions 2, 3, 4, 5, 6, and 10. All questions were translated into Japanese. Based on these findings, we propose a new method that combines the aforementioned methods. This method consists of making five estimates in response to a single question (Fig. 2 and Table 1 ): 1) making an estimate intuitively, 2) making an estimate deliberately, 3) considering the opposite (i.e. dialectical bootstrapping), 4) taking the general crowd’s perspective, and 5) taking the disagree-other’s perspective. The estimates are then averaged (for optimal weighting, see S2 for Supplementary Information). Because this study aimed to propose a method that could be used for everyday estimation, we excluded methods that required a two-week timespan 13 . Importantly, previous studies have attempted to develop methods that consist of making five estimates for a single question. In Rauhut and Lorenz (2011) 16 , participants made five estimates without specific instructions. In Fujisaki et al. (2023) 30 , participants first provided their own estimates and then four estimates from the perspective of general crowds. However, the results indicated that these methods were ineffective. In particular, the average of the five estimates was not more accurate than that of the two estimates. Therefore, the key question in this research is whether our method (hereafter, ‘Ensemble method’) could produce accurate averaged estimates. In the following sections, through a behavioural experiment, we confirm that the Ensemble method can produce estimates whose accuracy increases monotonically (i.e., over the two estimates). Moreover, we compared the Ensemble method condition with the control condition, which asked participants to estimate five times without specific instruction (‘Repeated condition’, Table 3 ). We found that the Ensemble method was more effective than the method used in the control condition. -----Figure 2 about here----- Table 3 Full instructions in the Repeated condition. Estimate number Instruction 1st estimate (No specific instruction) 2nd estimate (No specific instruction) 3rd estimate (No specific instruction) 4th estimate (No specific instruction) 5th estimate (No specific instruction) Note: As shown, there were no specific instructions. Results Main results Figure 3 presents the results of this analysis. The mean squared error (MSE) was used as an index of estimate accuracy. We then calculated the MSE for each participant for all questions. To do so, we averaged the estimates as follows: Let us define the correct answer as θ . For example, when the number of estimates was three, we calculated the MSE as follows: $$\varvec{M}\varvec{S}\varvec{E}= \left\{\begin{array}{c}{\left(\varvec{\theta } -\frac{\varvec{I}\varvec{n}\varvec{t}\varvec{u}\varvec{t}\varvec{i}\varvec{o}\varvec{n} + \varvec{D}\varvec{e}\varvec{l}\varvec{i}\varvec{b}\varvec{e}\varvec{r}\varvec{a}\varvec{t}\varvec{i}\varvec{o}\varvec{n} + \varvec{D}\varvec{i}\varvec{a}\varvec{l}\varvec{e}\varvec{c}\varvec{t}\varvec{i}\varvec{c}}{3}\right)}^{2} \left(\varvec{i}\varvec{n} \varvec{E}\varvec{n}\varvec{s}\varvec{e}\varvec{n}\varvec{b}\varvec{l}\varvec{e} \varvec{c}\varvec{o}\varvec{n}\varvec{d}\varvec{i}\varvec{t}\varvec{i}\varvec{o}\varvec{n}\right)\\ {\left(\varvec{\theta } -\frac{\varvec{F}\varvec{i}\varvec{r}\varvec{s}\varvec{t} \varvec{E}\varvec{s}\varvec{t}\varvec{i}\varvec{m}\varvec{a}\varvec{t}\varvec{e}+ \varvec{S}\varvec{e}\varvec{c}\varvec{o}\varvec{n}\varvec{d} \varvec{e}\varvec{s}\varvec{t}\varvec{i}\varvec{m}\varvec{a}\varvec{t}\varvec{e}+ \varvec{T}\varvec{h}\varvec{i}\varvec{r}\varvec{d} \varvec{E}\varvec{s}\varvec{t}\varvec{i}\varvec{m}\varvec{a}\varvec{t}\varvec{e}}{3}\right)}^{2} \left(in Repeated condition\right)\end{array}\right.$$ Subsequently, we manipulated the number of estimates from one to five. As shown in Fig. 3, the MSE in the Repeated condition remained flat, as reported in previous studies 16 , 30 . In other words, the accuracy did not tend to increase even if the number of estimates increased. In contrast, in the Ensemble method condition, the larger the number of estimates, the more the MSE decreased, although the slope tended to be slightly attenuated. First, we verified whether the Ensemble method condition reproduced the wisdom of the inner crowd. We could assume the first estimate in the Repeated condition as the participants’ ‘own estimate’. Therefore, we used this estimate in our analysis. We found that the MSE of the average of the five estimates in the Ensemble method condition was lower than that of the first estimate in the Repeated condition ( t = 2.14, p = 0.033). The results suggest that the Ensemble method could emerge the wisdom of the inner crowd. Second, and more importantly, the MSE of the average of the five estimates in the Ensemble method condition was lower than that of the average of the five estimates in the Repeated condition ( t = 2.42, p = 0.016). Thus, the Ensemble method condition was more effective than the Repeated condition. Moreover, the results showed that the Ensemble method produced estimates with monotonically increasing accuracy. We applied the Jonckheere-Terpstra trend test and found that the MSE in the Ensemble method condition decreased monotonically as the number of estimates increased ( JT = 230072, p < 0.01). By contrast, we confirmed that the MSE in the Repeated condition neither increased nor decreased monotonically ( JT = 260606, p = 0.91). -----Figure 3 about here----- Comparison with the wisdom of (the outer) crowd Further analyses were performed to address the effectiveness of the Ensemble method. In particular, we compared the Ensemble method condition (and the Repeated condition) with the wisdom of the crowd. In the Repeated condition, participants first provided their own estimates. Therefore, to collect these estimates, we got the wisdom of crowd effect (in a general sense) (hereafter called ‘Repeated-first group’). We employed a nonlinear mixed model and conducted a Bayesian parameter estimation (for more details, see S6 in the Supplementary Information). We assumed that the relationship between the number of estimates ( T ) and MSE could be represented by a hyperbola as follows: $$\varvec{M}\varvec{S}\varvec{E} = \varvec{a} / \varvec{T} + \varvec{b}$$ where a represents the magnitude of the wisdom of (the inner) crowd effect and b represents the error when the number of estimates is infinite. Figure 4 presents the results of this analysis. In the Repeated-first group, the larger the group size, the smaller the MSE. Thus, we confirmed that the wisdom of the crowd effect emerged in the Repeated-first group. Compared to the Repeated-first group, the Ensemble method condition had a more gradual slope. In other words, the Ensemble method condition had a weaker wisdom of the crowd effect than the Repeated-first group. Subsequently, we performed a quantitative comparison. We used T T , which represented the number of people in the Ensemble method condition corresponding to the Repeated-first group. For example, when T T = 1.3, the Ensemble method condition corresponded to 1.3 persons in the Repeated-first group. Table 4 presents the results of the analysis. When T was 5, T 5 was over 1.5 (i.e. 1.51). In other words, the effect of the Ensemble method was larger than 1.5 persons of the Repeated-first group. To the best of our knowledge, this value is larger than that of the method tested in previous studies, especially those focusing on the general knowledge question 13 , 16 . However, even if the number of estimates was infinite, T T could not exceed two in the Repeated-first group. Thus, the effectiveness and limitations of the Ensemble method can be observed. -----Figure 4 about here----- Table 4 Results of T T . Repeated Ensemble method T 2 1.02 1.18 T 3 0.99 1.33 T 4 0.99 1.40 T 5 0.96 1.51 T ∞ 0.97 1.90 Note: In the Ensemble method condition , T T was over 1.5. On the contrary, in the Repeated condition, the values were around 1 irrespective of T T . Decomposition of the error How does the Ensemble method condition produce more accurate estimates than the Repeat condition? It is well known that the wisdom of the crowd effect can be represented mathematically 5 . Note that we translated the equation into our context (i.e. wisdom of the inner crowd). Let us define Ei as the estimate of group member i , as its average over an individual’s estimates, and θ as the correct answer. Then, the equation is: $${\left( - \varvec{\theta }\right)}^{2} = - <{\left(\varvec{E}\varvec{i} - \right)}^{2}>$$ Here, we refer to the left side of the equation as the collective error, which indicates the MSE. Therefore, a lower collective error value indicates an accurate estimate. We refer to the first term on the right-hand side as the expected squared error. A lower value of the expected squared error represents an accurate estimate. The second term on the right side represents diversity. As the equation shows, higher diversity leads to better collective performance. Table 5 presents the results of the analysis. As mentioned above, the Ensemble method condition showed a lower collective error than the Repeated condition. Importantly, the Ensemble method condition showed a higher expected squared error than the Repeated condition. In other words, the estimate in the Ensemble method condition tended to be less accurate than that in the Repeated condition (see also S5 for Supplementary Information). However, the Ensemble method condition had a significantly larger diversity than the Repeated condition (see S4 for Supplementary Information). Consequently, the Ensemble method condition had a lower collective error (i.e. better performance) than the Repeated condition. Table 5 also displays the results of the Repeated-first group analysis. For this group, we defined Ei as the first estimate of group member i and as its average over all group members. As shown in the table, the Repeated-first group exhibited a lower expected squared error than the Ensemble method. In addition, the Repeated-first group had approximately the same diversity as the Ensemble method. As a result, the Ensemble method was not more effective than the Repeated-first group. Table 5 Results of the decomposition of the error. Collective error Expected squared error Diversity Ensemble method 225.8 438.8 212.9 Repeated 315.0 359.0 43.9 Repeated-first 108.2 300.3 192.0 Discussion This study proposes a method that boosts the wisdom of the inner crowd effect. Our method requires participants to provide five estimates in response to a question:1) making an estimate intuitively (intuition); 2) making an estimate deliberately (deliberation); 3) considering the opposite (dialectic); 4) taking the general crowd’s perspective (general crowd’s perspective); and 5) taking the disagree-other’s perspective (disagree-other’s perspective). It then averaged the five estimates. We first confirmed that our method recorded higher accuracy than the first estimate and the average of the five estimates in the control condition (i.e. the Repeated condition). Moreover, the results show that our method produced estimates whose accuracy increased monotonically with the increasing number of estimates. Furthermore, through mathematical modelling, we found that the estimation accuracy of our method was higher than 1.5 persons’ estimates. This study also makes two significant contributions to the following literature. First, in Experiment 1, we found that the average of the two estimates was more accurate than estimates based only on intuition or deliberation. This method is based on the claim 40 – 45 that people’s inferences differ when using heuristics and when using some knowledge. The results of the present study support this claim. Second, more importantly, the results also contribute to the discussion on ‘rationality’. So far, numerous studies have investigated this issue. Some researchers 42 , 44 have shown that intuition can cause bias, and others 40 , 41 , 45 argued that intuition includes rationality. In this respect, the results demonstrated that the combination of intuition and deliberation was more ‘rational’, at least in our context. Hereafter, we discuss related studies. In Experiment 1, we found that the average of Intuition and Deliberation was (marginally) significantly more accurate than Intuition and Deliberation on their own. This experiment is related to the study by Keck and Tang (2020) 10 , wherein participants were instructed to provide estimates either intuitively or deliberately. The findings revealed that the combination of intuition and deliberation enhances the wisdom of the crowd effect. In particular, the group in which half of the participants thought intuitively and the other half thought deliberately recorded higher accuracy than the groups in which all the participants thought intuitively or deliberately. Therefore, Experiment 1 replicated the results of Keck and Tang 10 at an individual level. In Experiment 2, we instructed participants to take others’ (i.e. the general crowd and disagree-other) perspectives under the Ensemble method condition. The experimental settings were similar to those of meta-prediction methods 7 , 46 – 49 , that instruct participants to predict what others would predict. For example, Prelec et al. (2017) 7 examined this method. In particular, they proposed an algorithm, which selected the answer that was more popular than predicted by people. The results showed that this method can correct the bias in crowds and enhance the wisdom of the crowd effect. As illustrated above, the proposed method differs from meta-prediction methods in several respects. First, our method aims to improve an individual’s (not a crowd’s) estimate. Second, our method averages the estimates. Nevertheless, we assume that our method and the meta-prediction methods together demonstrate that we can enhance the wisdom of (the inner) crowd by regulating people’s thinking. We can also connect this study to previous studies on social circle 50 – 54 . These studies indicate that by using an individual’s knowledge of their social circle (e.g. friends), we can improve predictions such as those referring to the results of a political election 50 . We can assume that a prediction based on knowledge of the social circle is similar to the estimate from the crowd and disagrees with others’ perspectives. However, the aims of the studies were different. In contrast to previous research, our method aims to obtain an accurate average estimate. In other words, for our method, the estimates from the crowd and disagreeing others’ perspectives are not necessarily accurate (see S5 for Supplementary Information). Since Vul and Pashler (2008) 13 and Herzog and Hertwig (2009) 14 demonstrated that individuals could use the wisdom of crowds, many studies have focused on this. However, its shortcomings have been highlighted. Rauhut and Lorenz (2011) 16 pointed out that people cannot produce accurate estimates by increasing the number of estimates. Nevertheless, for over a decade, effective methods to overcome this shortcoming have not been proposed. In this paper, we propose an effective method that combines multiple methods proposed in previous studies. By utilising this method (i.e. the Ensemble method), we can significantly enhance the accuracy of our estimates. Limitation of the study Finally, we describe the limitations of this study. We developed the Ensemble method, especially the order of the five estimates, for the following reasons. First, we could not set the Dialectic as the first estimate method because it would make an individual ‘re-consider’ the previous estimate. Second, we considered that we should set the Intuition as the first estimate because it was the estimate in which an individual answered what came to mind first. However, we do not claim that the order of the estimates in our method is the only one which can boost the wisdom of the inner crowd effect. Therefore, further studies using this methodology should be conducted. For example, we excluded the timespan method proposed by Vul and Pashler (2008) 13 from our method because it required participants to commit for a long time (two weeks). Thus, by adding the timespan method to our method, an individual may enhance the wisdom of the inner crowd effect. Moreover, in Experiment 2, we compared our method with a control condition (i.e. Repeated condition) that did not include specific instructions. However, we did not directly compare our method with those used in previous studies 13 , 14 ,, 30 . Therefore, further studies comparing these methods are warranted. Another limitation of this study is the type of questions. We used questions with the percentage of correct answers based on previous studies 13 , 17 , 18 , 22 , 29 , 30 . However, other types of questions exist. For example, some studies used the years of historical events 14 , 17 , whereas others used numerical estimation tasks 16 , 17 , 21 , 29 . A review study 18 reported that the wisdom of the inner crowd effect was maintained across different types of questions. However, the manner in which our method works for other types of problems remains unclear. Notably, our method may also enhance the accuracy of choice tasks. Because it requires participants to answer five times, it is possible that the aggregation rules, as majority rules, work effectively. For example, in the binary choice task, even if Intuition and Deliberation were incorrect, we could perform accurate inference by majority rule (i.e. when dialectic, crowd perspective, and disagree-other perspective were correct). Further research should be conducted to generalise the findings of this study. Methods Two experiments were conducted using Qualtrics software. All the participants provided informed consent before participating in the study. The experimental protocol was approved by the Research Ethics Committee of the university to which the last author belongs and was conducted in accordance with the latest version of the Declaration of Helsinki. Details of Experiment 1 Participants. The participants were 64 Japanese undergraduate and graduate students (24 women, 39 men, and one did not want to respond; M age = 21.25 and SD age = 2.24). After the experiment, they received a flat fee of 1,000 Japanese yen (approximately $ 9.17 at the currency rate at the time) for participation. Stimulus. Based on previous studies 18 , 30 , we prepared ten questions about general knowledge (Table 2 ). Procedure. We set only one condition in this experiment: all participants provided two estimates for each question (Sets 1–2). In Set 1, participants answered all ten questions intuitively. In this set, we also set a time limit of eight seconds, based on previous studies 54 , 55 that manipulated participants’ thinking styles. After answering each question, participants rated their confidence (see S3 for Supplementary Information). Between Sets 1 and 2, we set a 30-minute time interval because of the experimental design. During this period, the participants performed an irrelevant task. In this task, participants were instructed to make a binary choice task as for consumer products. In Set 2, participants answered all ten questions deliberately. In this set, we did not set any time limits. After answering all the questions, participants answered questions about their thinking styles in this set. We randomised the order of the questions for each participant. The randomised order of the questions remained constant across the (two) sets. Details of Experiment 2 Participants. The participants were 76 Japanese undergraduate students. In this experiment, we set two conditions: the Ensemble method condition and the Repeated condition. Participants were randomly assigned to one of the two conditions (Ensemble method condition: n = 38, M age = 19.58, SD age = 0.86, 24 women, 14 men; Repeated condition: n = 38, M age =19.74, SD age = 0.92, 23 women, 14 men, and one did not want to respond). This experiment did not include a participation fee because it was conducted as part of a psychology class. Stimulus. We prepared six questions on general knowledge (Table 2 ) based on those used in Experiment 1. Procedure. In the Ensemble method condition, the participants answered each question five times based on the instructions. They performed Intuition, Deliberation, Dialectic, General crowd’s perspective, and Disagree-others’ perspective estimates in this order. We randomised the order of the questions for each participant. Across the five estimates, the randomised order of the questions remained constant. Note that in the Ensemble method condition, we set a 30-minute time interval between the Dialectic and General crowd conditions. During this period, the participants performed an irrelevant task. In this task, participants were instructed to make a binary choice task as for consumer products. In the Repeated condition, participants were instructed about the estimation task, but without any instruction on how to estimate, which was different from the Ensemble method condition. They were told that this experiment would reward them depending on the accuracy of each estimate based on the previous study 16 , although it did not include an actual participation fee. After the instructions, the participants answered each question five times. We randomised the order of the questions for each participant. Across the five estimates, the randomised order of the questions remained constant. Note that in the Repeated condition, participants performed the same irrelevant task as in the Ensemble method condition for 30 minutes after completing the estimation task. Mixed-effect analysis All mixed-effects analyses 56 were conducted using the R (4.1.1) packages lme4 and lmerTest . We selected the best model and computed all the statistical values using the step() function for the full model with random participants and stimulus intercepts. All multiple comparisons were performed using the R packages lsmeans and pbkrtest . Declarations AUTHOR CONTRIBUTIONS IF, LY, and KU developed the study concept and contributed to the study design. LY and YT collected data. IF, LY, and YK analysed the data. IF wrote the manuscript, with feedback from YK and KU. KU won funding. DATA AVAILABILITY The R-code and the three datasets analysed during the current study are available in the Mendeley Data: https://data.mendeley.com/datasets/yx7xscnxg8/1 ACKNOWLEDGMENTS This research was supported by JST CREST, Grant Number JPMJCR19A1. DECLARATION OF INTERESTS The authors declare no competing interests. References Surowiecki, J. (2004). The wisdom of crowds . Anchor. Lorenz, J., Rauhut, H., Schweitzer, F., and Helbing, D. (2011). How social influence can undermine the wisdom of crowd effect. Natl. Acad. Sci. 108 , 9020–9025. (doi:10.1073/pnas.1008636108) Hertwig, R. (2012). Tapping into the wisdom of the crowd--with confidence. Science 336 , 303–304. (doi:10.1126/science.1221403) Becker, J., Brackbill, D., and Centola, D. (2017). Network dynamics of social influence in the wisdom of crowds. Natl. Acad. Sci. 114, E5070-E5076. (doi:10.1073/pnas.1615978114) Jayles, B., Kim, H., Escobedo, R., Cezera, S., Blanchet, A., Kameda, T., et al. (2017). 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(doi:10.1073/pnas.1804002115) Galinsky, A. D., and Moskowitz, G. B. (2000). Perspective-taking: Decreasing stereotype expression, stereotype accessibility, and in-group favoritism. Pers. Soc. Psychol. 78, 708–724. (doi: 10.1037/0022-3514.78.4.708) Yaniv, I., and Choshen-hillel, S. (2012). When guessing what another person would say is better than giving your own opinion : Using perspective-taking to improve advice-taking. Exp. Soc. Psychol. 48, 1022–1028. (doi:10.1111/j.1467-9280.2006.01704.x) Gigerenzer, G., Todd, P., and the ABC Research Group. (1999). Simple heuristics that make us smart. Oxford University Press: New York. Goldstein, D.G., and Gigerenzer, G. (2002). Models of ecological rationality: the recognition heuristic. Rev. 109, 75–90. Kahneman, D., and Frederick, S. (2005). A model of heuristic judgment. In: Holyoak, J., and Morrison, R. G. (eds) The Cambridge handbook of thinking and reasoning. Cambridge University Press: New York, pp. 267–293. Hertwig, R., Herzog, S. M., Schooler, L. J., and Reimer, T. (2008). Fluency heuristic: a model of how the mind exploits a by-product of information retrieval. Exp. Psychol. Learn. Mem. Cogn. 34 , 1191–1206. Kahneman, D. (2011). Thinking, fast and slow. Macmillan: New York. Honda, H., Matsuka, T., and Ueda, K. (2017). Memory-based simple heuristics as attribute substitution: Competitive tests of binary choice inference models. Sci. 41 , 1093-1118. Palley, A. B., and Soll, J. B. (2019). Extracting the wisdom of crowds when information is shared. Sci. 65 , 2291–2309. (doi: 10.1287/mnsc.2018.3047) Himmelstein, M., Budescu, D. V., and Ho, E. H. (2023). The wisdom of many in few: Finding individuals who are as wise as the crowd. Exp. Psy. Gen. 152 , 1223–1244. (doi: 10.1037/xge0001340) Wilkening, T., Martinie, M., and Howe, P. D. (2022). Hidden experts in the crowd: Using meta-predictions to leverage expertise in single-question prediction problems. Sci. 68, 487-508. (doi: 10.1287/mnsc.2020.3919) Palley, A. B., and Satopää, V. A. (2023). Boosting the wisdom of crowds within a single judgment problem: Weighted averaging based on peer predictions. Sci. 69, 5128-5146. (doi: 10.1287/mnsc.2022.4648) Galesic, M., Bruine de Bruin, W., Dumas, M., Kapteyn, A., Darling, J.E., and Meijer, E. (2018). Asking about social circles improves election predictions. Hum. Behav. 2, 187–193. (doi: 10.1038/s41586-021-03649-2) Bruine de Bruin W., Parker, A. M., Galesic, M., and Vardavas, R. (2019). Reports of social circles’ and own vaccination behavior: A national longitudinal survey. Health Psychol. 38 , 975–983. (doi: 10.1037/hea0000771) Bruine de Bruin, W., Galesic, M., Parker, A.M., and Vardavas, R. (2020). The role of social circle perceptions in “False consensus” about population statistics: evidence from a national flu survey. Decis. Making 40 , 235–241. (doi: 10.1177/0272989X20904960) Bruine de Bruin, W., Galesic, M., Bååth, R., de Bresser, J., Hall, L., Johansson, P., et al. (2022). Asking about social circles improves election predictions even with many political parties. J. Public. Opin. Res. 34 :edac006. (doi: org/10.1093/ijpor/edac006) Dane, E., Rockmann, K.W., and Pratt, M. G. (2012). When should I trust my gut? Linking domain expertise to intuitive decision-making effectiveness. Behav. Hum. Decis. Process 119, 187–194. (doi: 10.1016/j.obhdp.2012.07.009 Evans, A. M., Dillon, K. D., and Rand, D. G. (2015). Fast but not intuitive, slow but not reflective: Decision conflict drives reaction times in social dilemmas. Exp. Psychol. Gen. 144 , 951-966. (doi: 10.1037/xge0000107) Bates, D., Mächler, M., Bolker, B. M., and Walker, S. C. (2015). Fitting linear mixed-effects models using lme4. Stat. Softw. 67 , 1-48. (doi:10.18637/jss.v067.i01) Central Intelligence Agency (2020). The CIA World Factbook 2020-2021. https://data.worldbank.org/indicator/AG.LND.AGRI.ZS. https://www.soumu.go.jp/johotsusintokei/whitepaper/ja/r02/html/nd252110.html Additional Declarations No competing interests reported. Supplementary Files XiwisSup11.docx Cite Share Download PDF Status: Published Journal Publication published 21 Aug, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 31 Jul, 2024 Reviews received at journal 24 Jul, 2024 Reviews received at journal 29 Jun, 2024 Reviewers agreed at journal 16 Jun, 2024 Reviewers agreed at journal 12 Jun, 2024 Reviewers invited by journal 19 Mar, 2024 Editor assigned by journal 19 Mar, 2024 Editor invited by journal 29 Feb, 2024 Submission checks completed at journal 29 Feb, 2024 First submitted to journal 20 Feb, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3971890","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":276097974,"identity":"7eda8db0-6e65-4062-8257-9ffaad2c2bdb","order_by":0,"name":"Itsuki Fujisaki","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYBACAyBm/MDAwMwP4iUUEKmFWQKIJRtAWgyI1MLAA2IcgHMJAHPp7jQJyR217MbnVyd+eGDAIM8vdgC/Fss5ZzcbFJ45zmx24+1mCaDDDGfOTiDgsBu5Gx9Ith0Dajm7AaQlweA2YS0bDvACtRjPOLv5B7FaNj7gbathNuDv3UacLSC/GEu2HWCWuMG7zSLBQIKwX8yle7dJfmyrS+bvP7v55o8KG3l+aQJaGCTA5OFkBokEBJcYLXV2DPwHiFA9CkbBKBgFIxIAANUgRp1w8P0DAAAAAElFTkSuQmCC","orcid":"","institution":"Tohoku University","correspondingAuthor":true,"prefix":"","firstName":"Itsuki","middleName":"","lastName":"Fujisaki","suffix":""},{"id":276097975,"identity":"d64484d5-714b-466b-aab7-7fb3e78ededd","order_by":1,"name":"Lingxi Yu","email":"","orcid":"","institution":"The University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Lingxi","middleName":"","lastName":"Yu","suffix":""},{"id":276097976,"identity":"335f6457-1bb1-4139-bb67-e8f95174cef7","order_by":2,"name":"Yuki Tsukamura","email":"","orcid":"","institution":"The University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Yuki","middleName":"","lastName":"Tsukamura","suffix":""},{"id":276097977,"identity":"62a49701-b608-475b-978c-e15f6db08e20","order_by":3,"name":"Kunhao Yang","email":"","orcid":"","institution":"Yamaguchi University","correspondingAuthor":false,"prefix":"","firstName":"Kunhao","middleName":"","lastName":"Yang","suffix":""},{"id":276097978,"identity":"e1b77d69-ab19-4a98-9c9b-95a605a7f5b1","order_by":4,"name":"Kazuhiro Ueda","email":"","orcid":"","institution":"The University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Kazuhiro","middleName":"","lastName":"Ueda","suffix":""}],"badges":[],"createdAt":"2024-02-20 06:22:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3971890/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3971890/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-14740-3","type":"published","date":"2025-08-21T16:29:14+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":52108322,"identity":"7568ca4d-8270-470d-bf25-42f67bcd3d0d","added_by":"auto","created_at":"2024-03-06 20:12:02","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":136389,"visible":true,"origin":"","legend":"\u003cp\u003eResults of Experiment 1\u003c/p\u003e\n\u003cp\u003eAverage indicates the average of the estimates in Intuition and Deliberation. As the figure shows, the estimate in Average was (marginally) significantly accurate compared to those in Intuition and Deliberation.\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/fb52ba9956ad785d2c66829c.png"},{"id":52107951,"identity":"67d1c098-16bc-42d9-b9d3-6d94c8e5e489","added_by":"auto","created_at":"2024-03-06 20:04:02","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":133246,"visible":true,"origin":"","legend":"\u003cp\u003eAn illustration of our method\u003c/p\u003e\n\u003cp\u003eFor a single question, a participant estimates five times. The order of the estimates is shown in this figure.\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/24ab789ea93d8d1396d64624.png"},{"id":52107953,"identity":"a33e4d28-2ee7-4205-909d-bab2d45ee70d","added_by":"auto","created_at":"2024-03-06 20:04:02","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":67914,"visible":true,"origin":"","legend":"\u003cp\u003eResults of Experiment 2\u003c/p\u003e\n\u003cp\u003eWhen the number of estimates was 1, the Ensemble method condition had larger MSE than the Repeated condition. However, the larger the number of estimates, the smaller the MSE in the Ensemble method condition. In contrast, the MSE in the Repeated condition remained flat. As a result, the Ensemble method condition had a smaller MSE than the Repeated condition when the number of estimates was 5 (p = 0.016; comparing the average estimates when the number of estimates was 5).\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/8bd280405245a22ce592e911.png"},{"id":52107950,"identity":"c72b916a-c8b1-4f5e-8338-e03dfa78106a","added_by":"auto","created_at":"2024-03-06 20:04:02","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":62485,"visible":true,"origin":"","legend":"\u003cp\u003eResults of the model fitting\u003c/p\u003e\n\u003cp\u003eThe points indicate actual values and the solid lines represent the estimated hyperbolas. To compute TT, we first projected the MSE in the Ensemble method condition onto the hyperbola of the Repeated-first group. Subsequently, we observed the x value and regarded it as TT. To compute TT, we used parameter b since b represents the MSE when the number of estimates was infinite. Subsequently, we projected b onto the hyperbola of the Repeated-first group and observed T∞.\u003c/p\u003e","description":"","filename":"Figure4.png","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/795bdd81ed90822a9b636ed2.png"},{"id":89847233,"identity":"72bdc1c3-44b8-4dd6-8d8d-2628c9c86f4f","added_by":"auto","created_at":"2025-08-25 16:42:23","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1297093,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/b39a8a4d-4b18-40cc-a30d-eb224373ec8c.pdf"},{"id":52107954,"identity":"62318445-9525-443c-bcd5-300649bd73dd","added_by":"auto","created_at":"2024-03-06 20:04:03","extension":"docx","order_by":6,"title":"","display":"","copyAsset":false,"role":"supplement","size":1860565,"visible":true,"origin":"","legend":"","description":"","filename":"XiwisSup11.docx","url":"https://assets-eu.researchsquare.com/files/rs-3971890/v1/0e9907ab8a20fb0155bec2b5.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"An ensemble method utilising multiple thinking styles that boosts the wisdom of the inner crowd effect","fulltext":[{"header":"Introduction","content":"\u003cp\u003eIn daily life, people often need to estimate unknown events (e.g. the number of people attending a conference or the price of a car). How can people make accurate estimates for these types of questions? One promising approach is that of \u0026lsquo;the wisdom of the crowds\u0026rsquo;\u003csup\u003e\u003cspan additionalcitationids=\"CR2 CR3 CR4 CR5 CR6 CR7 CR8 CR9 CR10 CR11\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e. In other words, the average estimate of a crowd of individuals yields surprisingly accurate estimates. Researchers have investigated this topic for over 100 years.\u003c/p\u003e \u003cp\u003eHowever, it is also well known that the wisdom of crowds has an underlying problem: difficulty in collecting estimates from several people. Many studies have addressed this issue. Specifically, they have shown that even an individual could use the wisdom of the crowds (called \u0026lsquo;the wisdom of the inner crowd\u0026rsquo;\u003csup\u003e\u003cspan additionalcitationids=\"CR14 CR15 CR16 CR17 CR18 CR19 CR20 CR21 CR22 CR23 CR24 CR25 CR26 CR27 CR28 CR29\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e). In these studies, individuals were instructed to produce different estimates (mainly, twice) for a single question. In other words, they were expected to produce quasi-crowd estimates. The estimates were then averaged. The results showed that the average estimate was more accurate than the individual estimate (i.e. the first estimate).\u003c/p\u003e \u003cp\u003eThus, the wisdom of the inner crowd has the potential to improve estimates of daily life. However, it also has a fundamental problem in that its efficacy is relatively low. Specifically, previous studies\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e,\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e reported that the accuracy of the averaged estimate corresponded to only 1.1\u0026ndash;1.4 persons\u0026rsquo; first estimates. Therefore, this study aims to propose a method that can \u0026lsquo;boost\u0026rsquo;\u003csup\u003e\u003cspan additionalcitationids=\"CR32 CR33 CR34\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e the wisdom of the inner crowd effect.\u003c/p\u003e \u003cp\u003eTo do so, we combined multiple methods proposed in previous studies. The first refers to using the forgetting power. For a single question, a previous study\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e provided participants with a timespan (two weeks) between two estimates. Subsequently, they discovered that the wisdom of the inner crowd emerged. The second approach uses the power of dialectics. Previous studies\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e asked participants to consider the opposite in their second estimates and showed that people could utilize the wisdom of the inner crowd (called \u0026lsquo;dialectical bootstrapping\u0026rsquo;). The third method involves perspective-taking\u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e. It is well known that taking others\u0026rsquo; perspectives changes various forms of cognition (e.g. stereotypic biases\u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e; preferential values\u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e; and egocentric thinking\u003csup\u003e\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u003c/sup\u003e). Based on these findings, previous studies have asked participants to consider others\u0026rsquo; perspectives in their second estimates. For accessing the perspectives of others, one study\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e used general crowds, and another\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e used a person who disagreed with the participants. Both previous studies reported that people can produce the wisdom of the inner crowd.\u003c/p\u003e \u003cp\u003eAs illustrated above, these methods seem to differ in how they instruct participants to produce estimates. Therefore, it seems possible that, by combining these methods, an individual can boost the wisdom of the inner crowd effect. The purpose of this study was to test this hypothesis. In addition, along with the above methods, this study also proposes a new method. For a single question, the method makes people think intuitively (called \u0026lsquo;Intuition\u0026rsquo;; for the full instruction, see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) in the first estimate and think deliberately in the second estimate (\u0026lsquo;Deliberation\u0026rsquo;). The two estimates are then averaged (for optimal weighting, see S1 for Supplementary Information). This method is based on the findings\u003csup\u003e\u003cspan additionalcitationids=\"CR41 CR42 CR43\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e that people\u0026rsquo;s inferences differ from using heuristics (i.e. assumed to correspond to Intuition in our method) and using some knowledge (i.e. Deliberation; see \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003eDiscussion\u003c/span\u003e section for more detail).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFull instruction for each estimate.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eType of estimate\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInstruction\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIntuition\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePlease do not think deliberately. Answer quickly what come to mind intuitively. You have eight seconds to answer.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDeliberation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePlease ignore your intuition. Think deliberately before answering. Utilize your knowledge and experience, while being aware of the basis of the estimate. This time have no time limit. Please use enough time to think the following question.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDialectic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFirst, assume that your second estimate is off the mark. Second, think about a few reasons why that could be. Which assumptions and considerations could have been wrong? Third, what do these new considerations imply? Was the second estimate too high or too low? Fourth, based on this new perspective, make a third, alternative estimate. (The computer display shows the second estimate).\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGeneral crowd's perspective\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHow do you think people in general estimate the following question? Think and answer how people in general estimate this. Please do not answer yourself estimate.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDisagree-other's perspective\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNow picture a friend whose views and opinions are very different from yours. To illustrate, when discussing politics, societies, and daily affair, you often find yourself disagreeing on various issues. How would he or she answer the following six questions? Answer these questions now as this friend. Please do not answer yourself estimates.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"2\"\u003e\u003cb\u003eNote: Only Dialectic estimate required the second (i.e. Deliberation) estimate to be displayed, while the other estimates did not. All instructions were translated into Japanese.\u003c/b\u003e\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWe examined this method on the general knowledge questions (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e; see Method section for more detail) in Experiment 1 and found that the two average estimates were significantly more accurate than that in Intuition (\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.97, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0084; Fig.\u0026nbsp;1). We also found that the average of the two estimates was more accurate than that in Deliberation, although the difference was marginally significant (\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.31, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.055). Subsequently, we can regard this method as inducing the wisdom of the inner crowd, although it did not include the participants\u0026rsquo; own estimates.\u003c/p\u003e \u003cp\u003e-----Figure 1 about here-----\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eQuestions and correct answers used in the experiments.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAnswer\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the world\u0026rsquo;s roads are in India?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e9.7%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the world\u0026rsquo;s telephone lines are in China, USA, or the European Union?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e52.0%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSaudi Arabia consumes what percentage of the oil it produces?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e72.1%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the world\u0026rsquo;s population is between 15 and 64 years old?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e65.2%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the world\u0026rsquo;s population is Christian?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e31.4%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the worldwide labour force works in the service sector?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e50.6%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percent of the worldwide gross domestic product (GDP) is re-invested?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e25.2%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percentage of Japanese adult males smoke?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e27.1%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percentage of Japanese households have a fixed-line phone?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e69.0%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhat percentage of the world\u0026rsquo;s countries have a higher life expectancy than the United States?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e22.6%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"3\"\u003e\u003cb\u003eNote: We checked all the answers on 2022/10/27. We used the answers in The CIA World Factbook\u003c/b\u003e\u003csup\u003e\u003cb\u003e\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/b\u003e\u003c/sup\u003e, \u003cb\u003eWorld Bank Open Data\u003c/b\u003e\u003csup\u003e\u003cb\u003e\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e\u003c/b\u003e\u003c/sup\u003e, \u003cb\u003eand the data from Japan's Ministry of Internal Affairs and Communications\u003c/b\u003e\u003csup\u003e\u003cb\u003e\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e\u003c/b\u003e\u003c/sup\u003e, \u003cb\u003ebased on previous studies\u003c/b\u003e\u003csup\u003e\u003cb\u003e\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/b\u003e\u003c/sup\u003e. \u003cb\u003eExperiment 1 used all the questions. Experiment 2 used Questions 2, 3, 4, 5, 6, and 10. All questions were translated into Japanese.\u003c/b\u003e\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eBased on these findings, we propose a new method that combines the aforementioned methods. This method consists of making five estimates in response to a single question (Fig.\u0026nbsp;2 and Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e): 1) making an estimate intuitively, 2) making an estimate deliberately, 3) considering the opposite (i.e. dialectical bootstrapping), 4) taking the general crowd\u0026rsquo;s perspective, and 5) taking the disagree-other\u0026rsquo;s perspective. The estimates are then averaged (for optimal weighting, see S2 for Supplementary Information). Because this study aimed to propose a method that could be used for everyday estimation, we excluded methods that required a two-week timespan\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eImportantly, previous studies have attempted to develop methods that consist of making five estimates for a single question. In Rauhut and Lorenz (2011)\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e, participants made five estimates without specific instructions. In Fujisaki et al. (2023)\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e, participants first provided their own estimates and then four estimates from the perspective of general crowds. However, the results indicated that these methods were ineffective. In particular, the average of the five estimates was not more accurate than that of the two estimates.\u003c/p\u003e \u003cp\u003eTherefore, the key question in this research is whether our method (hereafter, \u0026lsquo;Ensemble method\u0026rsquo;) could produce accurate averaged estimates. In the following sections, through a behavioural experiment, we confirm that the Ensemble method can produce estimates whose accuracy increases monotonically (i.e., over the two estimates). Moreover, we compared the Ensemble method condition with the control condition, which asked participants to estimate five times without specific instruction (\u0026lsquo;Repeated condition\u0026rsquo;, Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). We found that the Ensemble method was more effective than the method used in the control condition.\u003c/p\u003e \u003cp\u003e-----Figure 2 about here-----\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFull instructions in the Repeated condition.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEstimate number\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInstruction\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1st estimate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(No specific instruction)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2nd estimate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(No specific instruction)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3rd estimate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(No specific instruction)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4th estimate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(No specific instruction)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5th estimate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(No specific instruction)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"2\"\u003e\u003cb\u003eNote: As shown, there were no specific instructions.\u003c/b\u003e\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eMain results\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;3 presents the results of this analysis. The mean squared error (MSE) was used as an index of estimate accuracy. We then calculated the MSE for each participant for all questions. To do so, we averaged the estimates as follows: Let us define the correct answer as \u003cem\u003eθ\u003c/em\u003e. For example, when the number of estimates was three, we calculated the MSE as follows:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\varvec{M}\\varvec{S}\\varvec{E}= \\left\\{\\begin{array}{c}{\\left(\\varvec{\\theta } -\\frac{\\varvec{I}\\varvec{n}\\varvec{t}\\varvec{u}\\varvec{t}\\varvec{i}\\varvec{o}\\varvec{n} + \\varvec{D}\\varvec{e}\\varvec{l}\\varvec{i}\\varvec{b}\\varvec{e}\\varvec{r}\\varvec{a}\\varvec{t}\\varvec{i}\\varvec{o}\\varvec{n} + \\varvec{D}\\varvec{i}\\varvec{a}\\varvec{l}\\varvec{e}\\varvec{c}\\varvec{t}\\varvec{i}\\varvec{c}}{3}\\right)}^{2} \\left(\\varvec{i}\\varvec{n} \\varvec{E}\\varvec{n}\\varvec{s}\\varvec{e}\\varvec{n}\\varvec{b}\\varvec{l}\\varvec{e} \\varvec{c}\\varvec{o}\\varvec{n}\\varvec{d}\\varvec{i}\\varvec{t}\\varvec{i}\\varvec{o}\\varvec{n}\\right)\\\\ {\\left(\\varvec{\\theta } -\\frac{\\varvec{F}\\varvec{i}\\varvec{r}\\varvec{s}\\varvec{t} \\varvec{E}\\varvec{s}\\varvec{t}\\varvec{i}\\varvec{m}\\varvec{a}\\varvec{t}\\varvec{e}+ \\varvec{S}\\varvec{e}\\varvec{c}\\varvec{o}\\varvec{n}\\varvec{d} \\varvec{e}\\varvec{s}\\varvec{t}\\varvec{i}\\varvec{m}\\varvec{a}\\varvec{t}\\varvec{e}+ \\varvec{T}\\varvec{h}\\varvec{i}\\varvec{r}\\varvec{d} \\varvec{E}\\varvec{s}\\varvec{t}\\varvec{i}\\varvec{m}\\varvec{a}\\varvec{t}\\varvec{e}}{3}\\right)}^{2} \\left(in Repeated condition\\right)\\end{array}\\right.$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubsequently, we manipulated the number of estimates from one to five. As shown in Fig.\u0026nbsp;3, the MSE in the Repeated condition remained flat, as reported in previous studies\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e. In other words, the accuracy did not tend to increase even if the number of estimates increased. In contrast, in the Ensemble method condition, the larger the number of estimates, the more the MSE decreased, although the slope tended to be slightly attenuated.\u003c/p\u003e \u003cp\u003eFirst, we verified whether the Ensemble method condition reproduced the wisdom of the inner crowd. We could assume the first estimate in the Repeated condition as the participants\u0026rsquo; \u0026lsquo;own estimate\u0026rsquo;. Therefore, we used this estimate in our analysis. We found that the MSE of the average of the five estimates in the Ensemble method condition was lower than that of the first estimate in the Repeated condition (\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.14, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.033). The results suggest that the Ensemble method could emerge the wisdom of the inner crowd.\u003c/p\u003e \u003cp\u003eSecond, and more importantly, the MSE of the average of the five estimates in the Ensemble method condition was lower than that of the average of the five estimates in the Repeated condition (\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.42, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.016). Thus, the Ensemble method condition was more effective than the Repeated condition.\u003c/p\u003e \u003cp\u003eMoreover, the results showed that the Ensemble method produced estimates with monotonically increasing accuracy. We applied the Jonckheere-Terpstra trend test and found that the MSE in the Ensemble method condition decreased monotonically as the number of estimates increased (\u003cem\u003eJT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;230072, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.01). By contrast, we confirmed that the MSE in the Repeated condition neither increased nor decreased monotonically (\u003cem\u003eJT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;260606, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.91).\u003c/p\u003e \u003cp\u003e-----Figure 3 about here-----\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eComparison with the wisdom of (the outer) crowd\u003c/h2\u003e \u003cp\u003eFurther analyses were performed to address the effectiveness of the Ensemble method. In particular, we compared the Ensemble method condition (and the Repeated condition) with the wisdom of the crowd. In the Repeated condition, participants first provided their own estimates. Therefore, to collect these estimates, we got the wisdom of crowd effect (in a general sense) (hereafter called \u0026lsquo;Repeated-first group\u0026rsquo;).\u003c/p\u003e \u003cp\u003eWe employed a nonlinear mixed model and conducted a Bayesian parameter estimation (for more details, see S6 in the Supplementary Information). We assumed that the relationship between the number of estimates (\u003cem\u003eT\u003c/em\u003e) and MSE could be represented by a hyperbola as follows:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\varvec{M}\\varvec{S}\\varvec{E} = \\varvec{a} / \\varvec{T} + \\varvec{b}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ea\u003c/em\u003e represents the magnitude of the wisdom of (the inner) crowd effect and \u003cem\u003eb\u003c/em\u003e represents the error when the number of estimates is infinite.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;4 presents the results of this analysis. In the Repeated-first group, the larger the group size, the smaller the MSE. Thus, we confirmed that the wisdom of the crowd effect emerged in the Repeated-first group. Compared to the Repeated-first group, the Ensemble method condition had a more gradual slope. In other words, the Ensemble method condition had a weaker wisdom of the crowd effect than the Repeated-first group.\u003c/p\u003e \u003cp\u003eSubsequently, we performed a quantitative comparison. We used \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e, which represented the number of people in the Ensemble method condition corresponding to the Repeated-first group. For example, when \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e = 1.3, the Ensemble method condition corresponded to 1.3 persons in the Repeated-first group.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents the results of the analysis. When \u003cem\u003eT\u003c/em\u003e was 5, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e5\u003c/em\u003e\u003c/sub\u003e was over 1.5 (i.e. 1.51). In other words, the effect of the Ensemble method was larger than 1.5 persons of the Repeated-first group. To the best of our knowledge, this value is larger than that of the method tested in previous studies, especially those focusing on the general knowledge question\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e,\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e. However, even if the number of estimates was infinite, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e could not exceed two in the Repeated-first group. Thus, the effectiveness and limitations of the Ensemble method can be observed.\u003c/p\u003e \u003cp\u003e-----Figure 4 about here-----\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults of \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sub\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRepeated\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnsemble method\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.18\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.33\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e4\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.40\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e5\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.51\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u0026infin;\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.90\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"3\"\u003e\u003cb\u003eNote: In the Ensemble method condition\u003c/b\u003e, \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003eT\u003c/b\u003e\u003c/sub\u003e \u003cb\u003ewas over 1.5. On the contrary, in the Repeated condition, the values were around 1 irrespective of\u003c/b\u003e \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003eT\u003c/b\u003e\u003c/sub\u003e.\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eDecomposition of the error\u003c/h2\u003e \u003cp\u003eHow does the Ensemble method condition produce more accurate estimates than the Repeat condition? It is well known that the wisdom of the crowd effect can be represented mathematically\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e. Note that we translated the equation into our context (i.e. wisdom of the inner crowd). Let us define \u003cem\u003eEi\u003c/em\u003e as the estimate of group member \u003cem\u003ei\u003c/em\u003e, \u0026lt;\u003cem\u003eEi\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;as its average over an individual\u0026rsquo;s estimates, and \u003cem\u003eθ\u003c/em\u003e as the correct answer. Then, the equation is:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$${\\left(\u0026lt;\\varvec{E}\\varvec{i}\u0026gt; - \\varvec{\\theta }\\right)}^{2} = \u0026lt;{\\left(\\varvec{E}\\varvec{i} - \\varvec{\\theta }\\right)}^{2}\u0026gt; - \u0026lt;{\\left(\\varvec{E}\\varvec{i} - \u0026lt;\\varvec{E}\\varvec{i}\u0026gt;\\right)}^{2}\u0026gt;$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, we refer to the left side of the equation as the collective error, which indicates the MSE. Therefore, a lower collective error value indicates an accurate estimate. We refer to the first term on the right-hand side as the expected squared error. A lower value of the expected squared error represents an accurate estimate. The second term on the right side represents diversity. As the equation shows, higher diversity leads to better collective performance.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e presents the results of the analysis. As mentioned above, the Ensemble method condition showed a lower collective error than the Repeated condition. Importantly, the Ensemble method condition showed a higher expected squared error than the Repeated condition. In other words, the estimate in the Ensemble method condition tended to be less accurate than that in the Repeated condition (see also S5 for Supplementary Information). However, the Ensemble method condition had a significantly larger diversity than the Repeated condition (see S4 for Supplementary Information). Consequently, the Ensemble method condition had a lower collective error (i.e. better performance) than the Repeated condition.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e also displays the results of the Repeated-first group analysis. For this group, we defined \u003cem\u003eEi\u003c/em\u003e as the first estimate of group member \u003cem\u003ei\u003c/em\u003e and \u0026lt;\u0026thinsp;\u003cem\u003eEi\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;as its average over all group members. As shown in the table, the Repeated-first group exhibited a lower expected squared error than the Ensemble method. In addition, the Repeated-first group had approximately the same diversity as the Ensemble method. As a result, the Ensemble method was not more effective than the Repeated-first group.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults of the decomposition of the error.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCollective error\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eExpected squared error\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDiversity\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEnsemble method\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e225.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e438.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e212.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRepeated\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e315.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e359.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e43.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRepeated-first\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e108.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e300.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e192.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThis study proposes a method that boosts the wisdom of the inner crowd effect. Our method requires participants to provide five estimates in response to a question:1) making an estimate intuitively (intuition); 2) making an estimate deliberately (deliberation); 3) considering the opposite (dialectic); 4) taking the general crowd\u0026rsquo;s perspective (general crowd\u0026rsquo;s perspective); and 5) taking the disagree-other\u0026rsquo;s perspective (disagree-other\u0026rsquo;s perspective). It then averaged the five estimates.\u003c/p\u003e \u003cp\u003eWe first confirmed that our method recorded higher accuracy than the first estimate and the average of the five estimates in the control condition (i.e. the Repeated condition). Moreover, the results show that our method produced estimates whose accuracy increased monotonically with the increasing number of estimates. Furthermore, through mathematical modelling, we found that the estimation accuracy of our method was higher than 1.5 persons\u0026rsquo; estimates.\u003c/p\u003e \u003cp\u003eThis study also makes two significant contributions to the following literature. First, in Experiment 1, we found that the average of the two estimates was more accurate than estimates based only on intuition or deliberation. This method is based on the claim\u003csup\u003e\u003cspan additionalcitationids=\"CR41 CR42 CR43 CR44\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e that people\u0026rsquo;s inferences differ when using heuristics and when using some knowledge. The results of the present study support this claim. Second, more importantly, the results also contribute to the discussion on \u0026lsquo;rationality\u0026rsquo;. So far, numerous studies have investigated this issue. Some researchers\u003csup\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e,\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e have shown that intuition can cause bias, and others\u003csup\u003e\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e,\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e,\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e argued that intuition includes rationality. In this respect, the results demonstrated that the combination of intuition and deliberation was more \u0026lsquo;rational\u0026rsquo;, at least in our context.\u003c/p\u003e \u003cp\u003eHereafter, we discuss related studies. In Experiment 1, we found that the average of Intuition and Deliberation was (marginally) significantly more accurate than Intuition and Deliberation on their own. This experiment is related to the study by Keck and Tang (2020)\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e, wherein participants were instructed to provide estimates either intuitively or deliberately. The findings revealed that the combination of intuition and deliberation enhances the wisdom of the crowd effect. In particular, the group in which half of the participants thought intuitively and the other half thought deliberately recorded higher accuracy than the groups in which all the participants thought intuitively or deliberately. Therefore, Experiment 1 replicated the results of Keck and Tang\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e at an individual level.\u003c/p\u003e \u003cp\u003eIn Experiment 2, we instructed participants to take others\u0026rsquo; (i.e. the general crowd and disagree-other) perspectives under the Ensemble method condition. The experimental settings were similar to those of meta-prediction methods\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e,\u003cspan additionalcitationids=\"CR47 CR48\" citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e\u003c/sup\u003e, that instruct participants to predict what others would predict. For example, Prelec et al. (2017)\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e examined this method. In particular, they proposed an algorithm, which selected the answer that was more popular than predicted by people. The results showed that this method can correct the bias in crowds and enhance the wisdom of the crowd effect. As illustrated above, the proposed method differs from meta-prediction methods in several respects. First, our method aims to improve an individual\u0026rsquo;s (not a crowd\u0026rsquo;s) estimate. Second, our method averages the estimates. Nevertheless, we assume that our method and the meta-prediction methods together demonstrate that we can enhance the wisdom of (the inner) crowd by regulating people\u0026rsquo;s thinking.\u003c/p\u003e \u003cp\u003eWe can also connect this study to previous studies on social circle\u003csup\u003e\u003cspan additionalcitationids=\"CR51 CR52 CR53\" citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u003c/sup\u003e. These studies indicate that by using an individual\u0026rsquo;s knowledge of their social circle (e.g. friends), we can improve predictions such as those referring to the results of a political election\u003csup\u003e\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e. We can assume that a prediction based on knowledge of the social circle is similar to the estimate from the crowd and disagrees with others\u0026rsquo; perspectives. However, the aims of the studies were different. In contrast to previous research, our method aims to obtain an accurate average estimate. In other words, for our method, the estimates from the crowd and disagreeing others\u0026rsquo; perspectives are not necessarily accurate (see S5 for Supplementary Information).\u003c/p\u003e \u003cp\u003eSince Vul and Pashler (2008)\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e and Herzog and Hertwig (2009)\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e demonstrated that individuals could use the wisdom of crowds, many studies have focused on this. However, its shortcomings have been highlighted. Rauhut and Lorenz (2011)\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e pointed out that people cannot produce accurate estimates by increasing the number of estimates. Nevertheless, for over a decade, effective methods to overcome this shortcoming have not been proposed. In this paper, we propose an effective method that combines multiple methods proposed in previous studies. By utilising this method (i.e. the Ensemble method), we can significantly enhance the accuracy of our estimates.\u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003eLimitation of the study\u003c/h2\u003e \u003cp\u003eFinally, we describe the limitations of this study. We developed the Ensemble method, especially the order of the five estimates, for the following reasons. First, we could not set the Dialectic as the first estimate method because it would make an individual \u0026lsquo;re-consider\u0026rsquo; the previous estimate. Second, we considered that we should set the Intuition as the first estimate because it was the estimate in which an individual answered what came to mind first. However, we do not claim that the order of the estimates in our method is the only one which can boost the wisdom of the inner crowd effect. Therefore, further studies using this methodology should be conducted. For example, we excluded the timespan method proposed by Vul and Pashler (2008)\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e from our method because it required participants to commit for a long time (two weeks). Thus, by adding the timespan method to our method, an individual may enhance the wisdom of the inner crowd effect.\u003c/p\u003e \u003cp\u003eMoreover, in Experiment 2, we compared our method with a control condition (i.e. Repeated condition) that did not include specific instructions. However, we did not directly compare our method with those used in previous studies\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e,\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e. Therefore, further studies comparing these methods are warranted. Another limitation of this study is the type of questions. We used questions with the percentage of correct answers based on previous studies\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e,\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e,\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e,\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e,\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e. However, other types of questions exist. For example, some studies used the years of historical events\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e, whereas others used numerical estimation tasks\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e,\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e,\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e,\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e. A review study\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e reported that the wisdom of the inner crowd effect was maintained across different types of questions. However, the manner in which our method works for other types of problems remains unclear. Notably, our method may also enhance the accuracy of choice tasks. Because it requires participants to answer five times, it is possible that the aggregation rules, as majority rules, work effectively. For example, in the binary choice task, even if Intuition and Deliberation were incorrect, we could perform accurate inference by majority rule (i.e. when dialectic, crowd perspective, and disagree-other perspective were correct). Further research should be conducted to generalise the findings of this study.\u003c/p\u003e \u003c/div\u003e"},{"header":"Methods","content":"\u003cp\u003eTwo experiments were conducted using \u003cem\u003eQualtrics\u003c/em\u003e software. All the participants provided informed consent before participating in the study. The experimental protocol was approved by the Research Ethics Committee of the university to which the last author belongs and was conducted in accordance with the latest version of the Declaration of Helsinki.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003eDetails of Experiment 1\u003c/h2\u003e \u003cp\u003eParticipants. The participants were 64 Japanese undergraduate and graduate students (24 women, 39 men, and one did not want to respond; \u003cem\u003eM\u003c/em\u003e \u003csub\u003e \u003cem\u003eage\u003c/em\u003e \u003c/sub\u003e = 21.25 and \u003cem\u003eSD\u003c/em\u003e \u003csub\u003e \u003cem\u003eage\u003c/em\u003e \u003c/sub\u003e = 2.24). After the experiment, they received a flat fee of 1,000 Japanese yen (approximately \u003cspan\u003e$\u003c/span\u003e9.17 at the currency rate at the time) for participation.\u003c/p\u003e \u003cp\u003eStimulus. Based on previous studies\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e, we prepared ten questions about general knowledge (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eProcedure. We set only one condition in this experiment: all participants provided two estimates for each question (Sets 1\u0026ndash;2).\u003c/p\u003e \u003cp\u003eIn Set 1, participants answered all ten questions intuitively. In this set, we also set a time limit of eight seconds, based on previous studies\u003csup\u003e\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e,\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/sup\u003e that manipulated participants\u0026rsquo; thinking styles. After answering each question, participants rated their confidence (see S3 for Supplementary Information). Between Sets 1 and 2, we set a 30-minute time interval because of the experimental design. During this period, the participants performed an irrelevant task. In this task, participants were instructed to make a binary choice task as for consumer products.\u003c/p\u003e \u003cp\u003eIn Set 2, participants answered all ten questions deliberately. In this set, we did not set any time limits. After answering all the questions, participants answered questions about their thinking styles in this set.\u003c/p\u003e \u003cp\u003eWe randomised the order of the questions for each participant. The randomised order of the questions remained constant across the (two) sets.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eDetails of Experiment 2\u003c/h3\u003e\n\u003cp\u003eParticipants. The participants were 76 Japanese undergraduate students. In this experiment, we set two conditions: the Ensemble method condition and the Repeated condition. Participants were randomly assigned to one of the two conditions (Ensemble method condition: \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;38, \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eage\u003c/em\u003e\u003c/sub\u003e = 19.58, \u003cem\u003eSD\u003c/em\u003e\u003csub\u003e\u003cem\u003eage\u003c/em\u003e\u003c/sub\u003e = 0.86, 24 women, 14 men; Repeated condition: \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;38, \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eage\u003c/em\u003e\u003c/sub\u003e =19.74, \u003cem\u003eSD\u003c/em\u003e\u003csub\u003e\u003cem\u003eage\u003c/em\u003e\u003c/sub\u003e = 0.92, 23 women, 14 men, and one did not want to respond). This experiment did not include a participation fee because it was conducted as part of a psychology class.\u003c/p\u003e \u003cp\u003eStimulus. We prepared six questions on general knowledge (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) based on those used in Experiment 1.\u003c/p\u003e \u003cp\u003eProcedure. In the Ensemble method condition, the participants answered each question five times based on the instructions. They performed Intuition, Deliberation, Dialectic, General crowd\u0026rsquo;s perspective, and Disagree-others\u0026rsquo; perspective estimates in this order. We randomised the order of the questions for each participant. Across the five estimates, the randomised order of the questions remained constant. Note that in the Ensemble method condition, we set a 30-minute time interval between the Dialectic and General crowd conditions. During this period, the participants performed an irrelevant task. In this task, participants were instructed to make a binary choice task as for consumer products.\u003c/p\u003e \u003cp\u003eIn the Repeated condition, participants were instructed about the estimation task, but without any instruction on how to estimate, which was different from the Ensemble method condition. They were told that this experiment would reward them depending on the accuracy of each estimate based on the previous study\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e, although it did not include an actual participation fee. After the instructions, the participants answered each question five times. We randomised the order of the questions for each participant. Across the five estimates, the randomised order of the questions remained constant. Note that in the Repeated condition, participants performed the same irrelevant task as in the Ensemble method condition for 30 minutes after completing the estimation task.\u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eMixed-effect analysis\u003c/h2\u003e \u003cp\u003eAll mixed-effects analyses\u003csup\u003e\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e\u003c/sup\u003e were conducted using the \u003cem\u003eR\u003c/em\u003e(4.1.1) packages \u003cem\u003elme4\u003c/em\u003e and \u003cem\u003elmerTest\u003c/em\u003e. We selected the best model and computed all the statistical values using the \u003cem\u003estep()\u003c/em\u003e function for the full model with random participants and stimulus intercepts. All multiple comparisons were performed using the \u003cem\u003eR\u003c/em\u003e packages \u003cem\u003elsmeans\u003c/em\u003e and \u003cem\u003epbkrtest\u003c/em\u003e.\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAUTHOR CONTRIBUTIONS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIF, LY, and KU developed the study concept and contributed to the study design. LY and YT collected data. IF, LY,\u0026nbsp;and YK analysed the\u0026nbsp;data. IF wrote the manuscript, with feedback from YK and KU. KU won funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDATA AVAILABILITY\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe R-code and the three datasets analysed during the current study are available in the Mendeley Data: https://data.mendeley.com/datasets/yx7xscnxg8/1\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eACKNOWLEDGMENTS\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research was supported by\u0026nbsp;JST CREST, Grant Number JPMJCR19A1.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDECLARATION OF INTERESTS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eSurowiecki, J. (2004). \u003cem\u003eThe wisdom of crowds\u003c/em\u003e. Anchor.\u003c/li\u003e\n\u003cli\u003eLorenz, J., Rauhut, H., Schweitzer, F., and Helbing, D. (2011). How social influence can undermine the wisdom of crowd effect. \u003cem\u003e Natl. Acad. 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(doi:10.18637/jss.v067.i01)\u003c/li\u003e\n\u003cli\u003eCentral Intelligence Agency (2020). \u003cem\u003eThe CIA World Factbook\u003c/em\u003e 2020-2021.\u003c/li\u003e\n\u003cli\u003ehttps://data.worldbank.org/indicator/AG.LND.AGRI.ZS.\u003c/li\u003e\n\u003cli\u003ehttps://www.soumu.go.jp/johotsusintokei/whitepaper/ja/r02/html/nd252110.html\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"The wisdom of crowds, The wisdom of the inner crowd, estimation","lastPublishedDoi":"10.21203/rs.3.rs-3971890/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3971890/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePrevious studies have demonstrated that individuals can utilize the wisdom of crowds, known as \u0026lsquo;the wisdom of the inner crowd\u0026rsquo;. This requires them to estimate a single question multiple times, and subsequently average these estimates. Although several methods have been proposed to achieve more accurate estimates, its efficacy remains relatively low. Therefore, this study proposes a method that assembles multiple independent methods to stimulate the wisdom of the inner crowd effect. Particularly, our method instructs participants to provide estimates five times. Through a behavioural experiment, we confirmed that our method can produce the wisdom of the inner crowd effect. Moreover, we found that our method produced more accurate estimates than a method that required participants to estimate five times without specific instructions. Furthermore, mathematical modelling demonstrated that the effectiveness of our method was greater than that of 1.5 persons. In sum, this study proposes a method to improve daily estimates.\u003c/p\u003e","manuscriptTitle":"An ensemble method utilising multiple thinking styles that boosts the wisdom of the inner crowd effect","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-06 20:03:57","doi":"10.21203/rs.3.rs-3971890/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-07-31T04:59:23+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-07-24T09:51:30+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-06-29T17:57:17+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"153644739702717349747423511942531378965","date":"2024-06-16T21:36:26+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"16114244921360231130465735229788109071","date":"2024-06-12T11:33:18+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-03-19T13:55:36+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-03-19T13:54:42+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-02-29T10:06:00+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-02-29T10:03:18+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-02-20T06:20:40+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b09b7b89-5806-4a12-b1e1-1fa1fe966f9d","owner":[],"postedDate":"March 6th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":29101806,"name":"Biological sciences/Psychology"},{"id":29101807,"name":"Biological sciences/Psychology/Human behaviour"}],"tags":[],"updatedAt":"2025-08-25T16:33:51+00:00","versionOfRecord":{"articleIdentity":"rs-3971890","link":"https://doi.org/10.1038/s41598-025-14740-3","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-08-21 16:29:14","publishedOnDateReadable":"August 21st, 2025"},"versionCreatedAt":"2024-03-06 20:03:57","video":"","vorDoi":"10.1038/s41598-025-14740-3","vorDoiUrl":"https://doi.org/10.1038/s41598-025-14740-3","workflowStages":[]},"version":"v1","identity":"rs-3971890","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3971890","identity":"rs-3971890","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}