1 Introduction
(Reference Plummer, Hornik, Leisch and ZeileisPrelec et al., 2017) recently proposed the “surprisingly popular” method for aggregating multiple-choice decisions over a group of people. The method is motivated by the challenge of finding an accurate answer to a single question, especially in situations where many people in the group could believe in the wrong answer. For example, if asked whether Seattle is the capital city of the state of Washington, many people mistakenly answer “yes”. The key feature of the surprisingly popular method is that, as well as providing their answer, people are asked to estimate what percentage of other people they expect will also give the same answer. Thus, if somebody knows the capital of Washington is Olympia, but also realizes that most people mistakenly believe it is Seattle, both of these pieces of information can be expressed.
The surprisingly popular method combines the cognitive judgment (the basic decision) and the meta-cognitive judgment (the estimate of the decisions of others) by comparing the expected and observed proportions of people making a decision. The observed proportion is simply how many say “yes” to Seattle being the capital. The expected proportion combines the estimated percentages for those who say both “yes” and “no”. A person who says “yes” and expects 90% of people to agree contributes to the expected proportion in the same way as person who says “no” but expects only 10% to agree. The final decision made by the surprisingly popular method compares the observed and expected proportions, and chooses the answer that has more observed agreement than is expected (i.e., the answer that is “surprisingly popular”). Intuitively, people who believe Seattle is the capital of Washington will tend to believe others will say the same, while those who know it is not will expect others to disagree. Thus, the expected agreement is very high, but the observed agreement is lower, because some knowledgeable people say “no”. This means “no” will be the surprisingly popular answer.
Figure 1 provides a concrete demonstration of the operation of the surprisingly popular method, based on data for the Seattle question reported by (Reference Plummer, Hornik, Leisch and ZeileisPrelec et al., 2017). It uses a visual display that shows the decisions and meta-cognitive estimates of agreement that people made, and the expected and observed levels of agreement that produce the final decision. The two bar graphs show the distribution of meta-cognitive judgments. The distribution of estimated agreement from participants answering that Seattle is the capital of Washington is shown by the upper (blue) distribution, while the distribution of estimated agreement from participants answering that Seattle is not the capital of Washington is shown by the lower (gray) distribution. Each bar in these distributions correspond to a 10% range in the meta-cognitive estimates. The total of the upper bars corresponds to how many people answered “yes”, while the total of lower bars corresponds to how many people answered “no”.
An example of the surprisingly popular method choosing the correct minority answer for the question “Is Seattle the capital of Washington?” The upper (blue) distribution shows the meta-cognitive estimates of agreement, in 10% bins centered from 5% to 95%, provided by people who answered “yes”. The lower (gray) distribution shows, on a downward oriented y-axis, the meta-cognitive estimates of agreement provided by people who answered “no”. The proportion of observed “yes” answers and the proportion of expected “yes” answers based on these data are shown by vertical lines, and listed. Also listed is the answer of the surprisingly popular method, which is “no” because the observed proportion is less than expected. The tick mark indicates the answer is correct.
The solid and broken vertical lines in Figure 1 show, respectively, the expected “E” and observed “O” levels of agreement calculated from the decisions and meta-cognitive estimates. A majority of 68% of people answered that Seattle was the capital, but the expected agreement was 89%, leading the surprisingly popular method to give the correct answer that Seattle is not the capital of Washington. The key contribution to the relatively high expected value is evident from the distributions of meta-cognitive judgments. The large lower gray bar at 5% indicates that a significant number of people who answered “no” expected that fewer than 10% of others would agree with them. Intuitively, these people know that Seattle is not the capital, but also know they are in a minority. Their meta-cognitive estimates allow the surprisingly popular method to side with the minority.
(Reference Plummer, Hornik, Leisch and ZeileisPrelec et al., 2017) evaluate the accuracy of the surprisingly popular method in a number of domains, including trivia and general knowledge questions, medical diagnoses, and art price category evaluations. For all of these domains, the surprisingly popular method achieves impressive levels of accuracy, outperforming standard alternatives like the majority answer, and the answer in which people express the greatest overall confidence. While the trivia and general knowledge domains have limited real-world applicability — outside the confines of a trivia competition, it is possible simply to look up the answers — the medical diagnosis and art evaluation domains clearly could have real-world application. In both cases, an accurate decision based on cheap and simple behavioral judgments is a useful capability, since determining the true answer involves expensive medical testing in the first case, and time-consuming and complicated mechanisms like auctions in the second case.
Perhaps the most interesting and important potential application of the surprisingly popular method, however, is to situations requiring genuine prediction, such as geopolitical forecasting or sporting predictions (Reference SilverSilver, 2012, TetlockGardner, 2016), where the true answer is in principle not knowable at the time people make decisions. None of the domains considered by (Reference Prelec, Seung and McCoyPrelec et al., 2017) are of this type, unless the argument is made that the value of art does not exist until it is socially constructed by an auction or some other valuation mechanism. Given this gap in evaluation, our goal is to provide a direct predictive test of the surprisingly popular method, by evaluating its performance forecasting the outcome of the 256 games played in the regular season of the US National Football League (NFL) in the 2017-2018 season.
The structure of this paper is as follows. We first describe the empirical data, collecting people’s predictions, on which our evaluation of the surprisingly popular method is based, and the benchmark prediction data we use to assess performance. We then analyze the empirical and benchmark data from a number of perspectives, including overall accuracy, the relationship of the surprisingly popular method’s predictions to those made by other methods, and the calibration between confidence and accuracy. We discuss insights into the potential applicability of the surprisingly popular method suggested by our findings, and a number of avenues for extending its empirical evaluation as well as improving its ability to make predictions.
2 Data
The NFL regular season involves 256 games, with each of 32 teams playing 16 games over a 17-week season, and each team having one bye week. This means that there are between 13 and 16 games each week, depending on how many teams have byes that week.
2.1 Experimental Data
During the 2017–2018 season we collected predictions each week on Wednesday. These predictions were made about every game for that week, which were typically played from Thursday night to the following Monday night. Each week the predictions were made by 100 participants recruited using the Amazon Mechanical Turk (AMT) system, with data collected using the Qualtrics survey system. Different participants were recruited for each week and all were paid US $1.
Participants first provided basic demographic information including their gender, and their age range from the options 18–24, 25–34, 35–44, 45–54, 55–64, and 65+ years. They then rated their knowledge of NFL football on a 5-point scale: extremely knowledgeable, very knowledgeable, moderately knowledgeable, slightly knowledgeable, and not knowledgeable at all. To make predictions about each game, participants first made a prediction about which team they believed would win, then they rated their confidence in this prediction on a 5-point scale: guess, low confidence, moderate confidence, high confidence, and very high confidence. Finally, participants estimated the percentage of others they believed would agree with their prediction. This percentage estimation was done with a slider that showed the exact integer from 0 to 100 being selected as the slider was moved, as well as permanent indicators 0, 10, …, 100 on a scale.
Once a participant had completed these three behavioral responses — their prediction, confidence rating, and meta-cognitive estimation of agreement — for a game, they pressed an “advance” button to move to the next game. They could adjust the three behavioral responses for a specific game while it was the current game, but could not return to an earlier game once they had advanced. Each participant made predictions about the games in a randomly-determined order. A screenshot of the experimental interface is available in the on-line supplementary material.
2.2 Benchmark Data
We also collected benchmark predictions for the 256 games coming from two other sources. One benchmark source was the collation of media expert predictions provided by nflpickwatch.com. These predictions take the form of binary selections for each game. We use the data from 94 experts who provided predictions for all, or almost all, of the 256 games over the season.
The other benchmark data source is provided by the data-science forecasting website fivethirtyeight.com. These predictions are made by an algorithm that is a variantFootnote 1 of the Elo statistical method used to measure performance in chess (Elo, 1978). This benchmark is not based on human judgments, but on the history of game results for the NFL teams. The Elo predictions are probabilistic and take the form of winning percentages for each team.
3 Results
We first analyze the overall accuracy of our AMT participants relative to the media expert benchmark predictions. We find that the sub-group of our participants who self-rated their NFL knowledge as “extremely knowledgeable” have a similar distribution of accuracy to the media experts. Accordingly, we apply the surprisingly popular method, and the other prediction methods based on human judgments, to this sub-group of expert AMT participants. We then compare the effectiveness of these predictions to the media experts and Elo benchmarks, considering first overall accuracy, then patterns of agreement and disagreement between predictions, and finally the calibration of the predictions.
3.1 Accuracy by Expertise
Figure 2 examines the behavior of the AMT participants, in terms of the accuracy of their predictions, the relationship between their confidence ratings and accuracy, and the calibration of their meta-cognitive estimates of agreement.
The left panel shows the distribution of prediction accuracy for six different groups of people. At the top is the distribution for media experts and below are the distributions for AMT participants with different levels of self-rated overall knowledge. The middle panels shows the relationship between accuracy and confidence ratings for each self-rated knowledge group. The right panels show the relationship between the meta-cognitive estimate and true level of agreement in each prediction for each self-rated knowledge group.
The left panels show the distribution of accuracy over all 256 games for six different groups. At the top is the distribution for the media experts. Below are the distributions for the AMT participants, divided into the five categories of self-rated knowledge of NFL football. Within each panel, the area of each square is proportional to the number of people with that level of accuracy. The total area of the squares in each distribution is proportional to the number of participants in each category showing, for example, that more AMT participants rated themselves as “very knowledgeable” or “moderately knowledgeable” than “not knowledgeable at all”.
The middle panels of Figure 2 show the mean accuracy of predictions made with different levels of confidence, again broken into the self-rated knowledge groups. Thus, for example, the top-most panel in the middle column shows how accurate decisions made with confidence ratings of 1,…,5 are on average, for just those AMT participants with the highest self-knowledge rating. These results show that, in general, more confident predictions were more accurate for all of the knowledge levels. The calibration of confidence and accuracy is most impressive for the “extremely knowledgeable” group. It is nearly linear and has no departures from monotonicity.
The right panels of Figure 2 show the relationship between the meta-cognitive estimate of agreement and the true level of agreement, again broken into the self-rated knowledge groups. The correlations between the estimate and the truth are around 0.3 for all of the groups, except for the “not knowledgeable at all” group, which has a lower correlation.
Collectively, the results in Figure 2 led us to focus, in an exploratory way, on the group of AMT participants who rated themselves as “extremely knowledgeable”. The left panels show that this group makes predictions with levels of accuracy that are most similar to those of the media experts.Footnote 2 The middle panels show that they also supply meaningful confidence ratings for their individual decisions, with a monotonic relationship between increased confidence and increased accuracy. The right panels show that, while none of the groups is especially good at making these judgments, the “extremely knowledgeable” group is at least as good as the others.
Given the goal of testing the ability of the surprisingly popular method to make accurate predictions, it makes sense to focus on those participants who are evidently supplying good predictions, calibrated confidence judgments, and reasonable meta-cognitive estimates. Thus, the following results focus exclusively on the AMT participants who self-rated as “extremely knowledgeable” about the NFL. The ideal experimental test of the surprisingly popular method would have been to ask the media experts to provide meta-cognitive judgments. Obviously, this was not possible, and so the group of AMT participants who most closely match the media experts in terms of prediction accuracy serves as a sensible surrogate group. The number of “extremely knowledgeable” AMT participants varied from week to week, from a minimum of 12 participants, to a maximum of 33 participants, with a mean of 18 participants.
It is worth noting that the basis for identifying this group is self-rating data that were available before the games were played, and so could form part of a real-world prediction method. We emphasize, however, that there is an exploratory aspect to this method of analysis. In particular, although we posted all of our raw data before games were played, the analysis method was not pre-registered. Thus, it is important to replicate our findings to test the efficacy of self-rating in identifying accurate participants. For completeness, in the supplementary information, we present analyses of the surprisingly popular method based on all of the AMT participants. In the on-line supplementary material, we also provide the results of another variant suggested by a reviewer that uses all but the least knowledgeable AMT participants. In both cases, the prediction accuracy is much less impressive.
3.2 Accuracy of the Surprisingly Popular Method and Other Predictions
Figure 3 summarizes the overall performance of the surprisingly popular method, in the context of the performance of the individual media experts and other comparison methods. The stick figures show the distribution of the total number of games in the season correctly predicted by the 94 individual media experts. The worst-performed experts correctly predicted the outcome of around 150 of the 256 games, the best-performed experts correctly predicted about 180 games, and most experts were correct for between 155 and 175 of the games.
The number of games correctly predicted by the surprising popular (sp), confidence-weighted tally (conf), mode of AMT participants (mode), mode of media experts (media), and Elo (elo) methods, and for 94 individual media experts. The stick figures represent the distribution of the number of games correctly predicted by the media experts. The labeled lines show the number of games correctly predicted by the methods.
Note that, while the individual media experts made forced choices between the two teams in a game, the aggregate methods can all produce tied outcomes that favor neither team. The surprisingly popular method can produce equal expert and observed percentages, although that never happened for the current data. The confidence-weighted tallies can be equal for both teams, which happened for five games. The number of people favoring each team can be equal in determining a mode, which happened for nine games for the AMT expert participants, and once for the media experts. Finally, the Elo method can give a 50% winning percentage to both teams, which occurred for five games. For these games, the overall accuracies we report count these predictions as half a game.
The surprisingly popular method, based on the group of AMT participants self-rated as “extremely knowledgeable”, predicted 174 games correctly. This performance was inferior to five of the media experts, the same as two more, and superior to the remaining 87. The surprisingly popular method also slightly outperformed the modal (most common) predictions of the “extremely knowledgeable” AMT participants and the modal predictions of the media experts. These mode-based predictions were both correct for 1721/2 of the 256 games. The confidence-weighted tally of the “extremely knowledgeable” AMT made correct predictions for 166 games. These predictions were based on the team with the larger sum of confidence ratings across all of the participants favoring that team. The team with the greater Elo predicted winning percentage was correct for 1641/2 games.
3.3 Examples of the Surprisingly Popular Method
Figure 4 presents three examples of the performance of the surprisingly popular method, and the confidence-weighted tally method, aimed at giving some insight into the relative success of the surprisingly popular method. The detailed results for every game are available in the supplementary material. The three examples in Figure 4 were chosen because the surprisingly popular and confidence-weighted tally methods make different predictions and because, in two cases, the surprisingly popular prediction does not follow the majority. It is the ability of the surprisingly popular method to predict against confidence-based and majority opinion that makes it theoretically interesting.
Illustrative examples of the surprisingly popular and confidence-weighted tally methods for three NFL games. The three games correspond to the rows of panels, with the left-hand panel corresponding to the surprisingly popular method, and the right-hand panel corresponding to the confidence-weighted tally method. For the surprisingly popular method, as in Figure 1, the distributions of meta-cognitive estimates of agreement are shown for people choosing each team, and the observed and expected percentages of first-named home-team prediction are detailed, along with the answer of the method and its accuracy. For the confidence-weighted tally method, the distributions of confidence on a 5-point scale are shown for people choosing each team, and the confidence tallies are detailed, along with the answer of the method and its accuracy.
For each of the games in Figure 4 — between the Green Bay Packers and Seattle Seahawks in week 1, the Denver Broncos and Dallas Cowboys in week 2, and the Seattle Seahawks and Philadelphia Eagles in week 15 — the surprisingly popular method is shown on the left, and the confidence-weighted tally method is shown on the right. The surprisingly popular method uses the same display format used in Figure 1. The confidence-weighted tally method is shown by histograms representing the distribution of confidence ratings on the five-point scale. The confidence distribution for predictions favoring the first-named team is above and the confidence distribution for predictions favoring the second-named team is below. The tally of these confidence ratings is displayed as is the prediction made by the method.
In the Packers versus Seahawks game, the predictions of the participants are evenly split between the two teams. The surprisingly popular method predicts a Packers victory because many participants favoring the Packers believe relatively few others will agree with them. This means the relatively high percentage of participants predicting a Packers victory is above the expectation. The confidence-weighted tally method, however, predicts a Seahawks victory, because participants favoring the Seahawks tended to do so with higher confidence. This game provides a good example of the different mechanisms by which these two methods break a 50–50 tie in the binary predictions. As the tick and cross marks in Figure 4 indicate, the winner of the game was the Packers, and so the surprisingly popular method made the correct prediction.
In the Broncos versus Cowboys game, the surprisingly popular method chooses against the majority prediction. A total of 53% of participants favored the Broncos, but many participants favoring the Cowboys expected relatively few others to agree with them, leading to a higher expectation of 63%. Thus, the observed percentage in favor of the Broncos fell short of the expectation, and the “underdog” Cowboys were predicted. Meanwhile, the confidence-weighted tally method predicted a Broncos victory by some margin, since participants favoring the Broncos generally expressed high confidence while participants favoring the Cowboys generally expressed low confidence. As it turned out, the Cowboys won the game, so the confidence-weighted tally method made the correct prediction.
Finally, in the Seahawks versus Eagles game, there is an expectation that the Eagles are heavy favorites. Most participants who predict an Eagles win believe the majority of others will agree with them. Most participants predicting a Seahawks win believe the majority will disagree with them, but will instead favor the Eagles. Thus, even though a minority of 42% of participants predicted the Seahawks to win, this is higher than than the 26% expectation, and the surprisingly popular method correctly predicts the eventual Seahawks win.
3.4 Overlap in Predictions
Figure 5 provides a summary of the relationship between the predictions of the different methods and their accuracies. The panels correspond to all 10 possible pairings of the five aggregation methods. Within each panel, a 2×2 table is shown counting how often the pair of methods both made correct predictions, both made incorrect predictions, or one method was correct and the other was not. To measure these counts, we treated the tied predictions, in which a method did not make a clear prediction, as errors. The left column corresponds to correct predictions for the method labeled at the top, and the top row corresponds to correct predictions for the method labeled at the left. Within each cell, the area of the square is proportional to how often the pattern of correct and incorrect predictions across the pair of methods was observed, and the actual count is also displayed.
Relationship between pairs of methods and the accuracy of their predictions. Each panel corresponds to one of the 10 unique pairings of the five methods: surprising popular (sp), confidence-weighted tally (conf), mode of AMT participants (mode), mode of media experts (media), and Elo (elo). Within each panel, correct predictions are labeled as “c” and incorrect predictions are labeled as “i”. The areas of squares and overlain numbers show counts of games in which both methods made the same correct prediction (top-left), the same incorrect prediction (bottom-right), the left-labeled row method made a correct prediction but the top-labeled column method did not (top-right), or the top-labeled column method made a correct prediction but the left-labeled row method did not (bottom-left)
The results in Figure 5 show that the surprisingly popular method and the confidence-weighted tally method tend to produce more similar predictions than other pairings of methods. The two mode-based methods using expert judgments, for the AMT participants and the media experts, are also quite similar in their predictions. Meanwhile, the surprisingly popular method has the least overlap in its predictions with those made by other data sources: the media experts and the Elo method. Collectively, these results suggest that both the data source and the aggregation method contribute to the patterns of predictions.
Because the results show that the five methods have considerable variety in their predictions, it is possible that further aggregation across models could be useful. This is a familiar approach from machine learning, known as boosting (Reference Hastie, Tibshirani and FriedmanHastie et al., 2001), which often leads to improved predictions. As it turns out, the modal prediction of the five methods used in Figure 5 is correct for 178 out of the 256 games, which is more accurate than any of the individual methods. This result is obviously highly exploratory, and needs to be replicated. It is also the case that the five methods used were not chosen in an attempt to maximize the accuracy of this type of aggregation, and it is possible that using more or different methods would be a better approach.
3.5 Calibration
Our final analysis considers the calibration of predictions made by the surprisingly popular and other methods. Ideally, a method would always make correct predictions, but, since errors are inevitable, it is important to know how much confidence should be placed in a prediction. Confidence guides decisions and actions that are made based on predictions, such as betting. We examine the calibration of each method in terms of a calibration curve that relates the confidence in predictions to their empirical accuracy (Reference Lichtenstein, Fischoff, Phillips, Kahneman and TverskyLichtenstein et al., 1982).
For the Elo method, the confidence in a decision is given directly by the winning probabilities it produces. For the methods based on the majority prediction across people, the proportion of people in the majority is a natural measure of confidence (Reference Grofman, Owen and FeldGrofman et al., 1983,KerrTindale, 2004). For the confidence-weighed tally method, an obvious measure of confidence is the tally in favor of the team predicted to win, expressed as a proportion of the sum of both tallies. For example, in the Houston Texans versus Tennessee Titans game in the top-right panel of Figure 4, the Titans are predicted to win, since their tally of 27 is greater than 22. The confidence in this prediction, according to the measure we use, is 27/(27+22)≈ 55%. For the surprisingly popular method, one possible way to measure confidence is the difference between the expected and observed percentages for a team, since it is this difference that determines the prediction. We add this difference to the chance probability of 50% to determine the confidence the surprisingly popular method has in each prediction it makes. For example, for the same Houston Texans versus Tennessee Titans game, the Texans are predicted to win, since the observed percentage of people predicting them to win is 47%, which is greater than the expected 45%. The confidence in this prediction, according to the measure we use, is 50+|47−45|= 52%.
To examine the calibration relationship between confidence and accuracy, we use the statistical approach developed by (LeeLee, 2017), which is summarized in the supplementary information. This approach infers calibration curves based on logistic growth functions. It aims first to infer what type of growth function best describes the calibration curve and then infers the parameters of the appropriate function. Both the type of curve and the parameters have meaningful psychological interpretations, which help to characterize and measure the calibration properties of a method. Figure 6 shows the results of the calibration analysis for the five methods. For each method, the inferred type of calibration curve is shown by the inset histogram. The curve relates the confidence a method has in a prediction on the x-axis to the probability the prediction is correct on the y-axis. The inferred posterior distribution for the calibration curve is shown by the lines, with the posterior for individual predictions shown by circular markers.
Results of the calibration analysis for the five aggregation methods. In each panel, the inset histogram shows the posterior probability of the 5 possible logistic growth models (“c” = chance, “d” = deterministic, “sd” = shifted deterministic, “p” = probabilistic, “sp” = shifted probabilistic) used by (LeeLee, 2017). The most likely model is labeled in bold. The lines show samples from the posterior distribution of the most likely calibration model, and the circular markers show samples from the joint distribution of confidence and accuracy for the predictions of the NFL games.
The inferred parameters from the (LeeLee, 2017) analysis for each method are also shown in Figure 6. The parameter β measures how quickly additional confidence leads to more accurate predictions (i.e., the slope of the calibration curve); α measures the upper-bound on accuracy (i.e., how accurate decisions made with 100% confidence are expected to be); µ measures the mean level of confidence a method has across all of its predictions; and σ measures the standard deviation of the distribution of confidence. The Elo method stands out as being extremely well calibrated. It has a slope near β=1. It is inferred to be deterministic, meaning that in theory a 100% confident prediction will be 100% accurate. The mean level of confidence is about 67% and rarely are predictions made with more than 80% certainty. This caution contrasts with the high confidence of the media modal decision, which has a mean confidence of µ=0.99 but has an upper-bound on accuracy of α=0.80. The participant mode and confidence-weighted tally calibration curves fall somewhere between these extremes. Both show a broader distribution of confidence. They have similar upper bounds on accuracy to the media mode method.
It does not make sense to compare the calibration curve of the surprisingly popular method to the calibration curves of the other methods, even though the surprisingly popular method is our focus. While the difference between expected and observed proportions is the obvious way to generate a measure of confidence, it is not obvious how this difference should be scaled. The current scaling assumes a difference of 50% is maximal, which seems reasonable, but not the only possible choice. In addition, it is not obvious that the scaling should be linear, as is currently assumed. Theoretical development is needed to provide a strong test of the calibration properties of the surprisingly popular method. What the results in Figure 6 do suggest, however, is that the surprisingly popular method is at least capable of reasonable calibration, in the sense that increasing differences between the observed and expected agreement correspond to more accurate decisions.
4 Discussion
Our results suggest there is promise in applying the surprisingly popular method to predicting the outcomes of NFL games. The surprisingly popular method outperformed most of the media experts, the majority opinion of these experts, the Elo method, and other measures based on our AMT participant data. While the improvement in accuracy was small, the nflpickwatch.com and Elo measures provide genuine real-world benchmarks. If the current level of improvement was replicated for other seasons, it would be have implications for real-world prediction and betting.
Despite this promise, we recognize the limitations of evaluating of any prediction method based on only one season, comprising a few hundred binary outcomes. Accordingly, we view this study as a motivating demonstration of the applicability of the surprisingly popular method to making predictions, with a particular focus on the important class of predictions represented by sporting contests. It seems clear that people were able to make decisions and provide meta-cognitive judgments in a prediction setting as naturally as they are able in previously studied non-prediction settings like general knowledge questions. Indeed, as the examples in Figure 4 highlight, the basic insight of the surprisingly popular method can generalize to predictive settings. If a subset of people have insight into a surprise winner, the surprisingly popular method provides a simple and effective way to capture and use that knowledge.
The key question, therefore, is whether and how often such subsets of people exist. Our study provided some first suggestive evidence that they can exist and suggests that domain knowledge, as measured by self-rating in our case, may be an important factor. Future work should try and isolate the type of expertise, or the types of games, that are likely to have the private knowledge or insight needed. As we mentioned, the ideal test is how well the surprisingly popular method performs when media experts provide meta-cognitive estimates of agreement, without being aware of the predictions others are making.
Another direction for future research is suggested by the meta-cognitive judgments that are the novel empirical feature of the surprisingly popular method. It is clear from the illustrative examples in Figure 4 that the distributions of these meta-cognitive estimates are generally broad. Whether this arises because of a individual differences in opinion about the percentage, individual differences in the cognitive processes in the way in the estimates are generated, or both, is an interesting cognitive modeling question. Potentially, a model-based account of these data could be incorporated into the surprisingly popular method to improve the precision of the meta-cognitive estimates (McCoyPrelec, 2017). Ideally, differences due to the cognitive processes involved in meta-cognition could be “factored out”, leaving the key signal of expected agreement.
The surprisingly popular method is a clever and simple approach to combining people’s judgments to make accurate group decisions, based on the insight that meta-cognitive information is important. Our results suggest that it may be useful in forecasting applications involving genuine predictions, as an alternative or complementary method to more elaborate approaches like prediction markets (Reference SilverSilver, 2012). We hope that our results encourage the further investigation of the surprisingly popular method, both in terms of its real-world performance, and the cognitive modeling challenges it poses for understanding the relationship between choice and meta-cognition.
Supplementary Information
Results For All AMT Participants
Our application of the surprisingly popular method relies on just those AMT participants who self-rated as being “extremely knowledgeable” about NFL football. The rationale for limiting the analysis to these participants is to use them as surrogate experts, from whom the meta-cognitive estimation of agreement was available. It is natural, however, also to consider the performance of the surprisingly popular method based on all of the available AMT participants. These results are summarized in Figure 7. It is clear the surprisingly popular method performs much more poorly, with an accuracy well below the mode of the media experts, and in the middle of the distribution of the individual accuracies for these experts. Presumably, this decrease in accuracy is the result of the less accurate judgments made by the non-expert AMT participants, as evident in the left-hand panels of Figure 2.
The number of games correctly predicted by the surprising popular (sp) , confidence-weighted tally (conf), mode of AMT participants (mode), based on all of the AMT participants. The number of games correctly predicted by the mode of media experts (media) and Elo (elo) methods, and for 94 individual media experts, are shown.
All Predictions
Figure 8 details the accuracy of every aggregate prediction method for every game. Each panel corresponds to a week, labeled “w1” for week 1, and so on. Within each panel, rows correspond to methods, and columns to games, ordered from left to right from best overall predicted to worst overall predicted. A dark blue circle indicates a correct prediction; a light orange circle indicates an incorrect prediction; a gray circle indicates neither team was favored by the method. Not surprisingly, there is often strong agreement across the methods. Most weeks include a number of games where all the methods make the same prediction: more often than not this prediction is correct, leading to a column of blue circles, but sometimes it is incorrect, leading to a column of orange circles.
The accuracy of the predictions made by the surprising popular (sp), confidence-weighted tally (conf), mode of AMT participants (mode), mode of media experts (media), and Elo (elo) methods, for every game of the NFL season. Panels correspond to weeks of the seasons, rows to methods, and columns to games. Dark blue circles indicate a correct prediction; light orange circles indicate an incorrect prediction; and grey circles indicate neither team was favored.
Calibration Modeling
Our calibration analysis is a natural extension of the approach developed by (LeeLee, 2017). Their approach applies directly only to the prediction methods based on the majority decision over a set of individuals, but can be generalized straightforwardly to include methods with other measures of decision confidence. Formally, in the (LeeLee, 2017) approach, there are n decisions, and the ith decision has k i people making the majority decision out of n i people, with an accuracy of y i=1 if the majority decision is correct, and y i=0 otherwise. The proportion θi represents the size of the majority and the probability φi representing the probability the decision correct.
This approach applies directly to the majority decisions based on AMT participants and media experts. The observed majority is assumed follows a binomial distribution with respect to the underlying majority proportion and the total number of people, so that k i ∼ Binomial(θi,n i). For the surprisingly popular, confidence-weighted tally, and Elo methods, which produce a continuous measure of confidence, c i for the ith decision, we assume this observed confidence reflects the underlying confidence with a small amount of noise, so that c i ∼ Gaussian(θi,1/(0.01)2). In both cases, the observed accuracy is a Bernoulli draw with respect to the underlying accuracy, y i ∼ Bernoulli(φi) and the distribution of confidence across all decisions is modeled as coming from an over-arching truncated Gaussian distribution, so that θi ∼ Gaussian (0,1)(µ,1/σ2) where µ∼Uniform(1/2,1) and σ∼Uniform(0,1/2) are the mean and standard deviation, respectively.
To model the relationship between confidence and accuracy, (LeeLee, 2017) rely on a logistic growth model of the form
with a growth parameter β, a bound parameter α, and a shift parameter δ. This model is shown in the central panel of Figure 9. The upper bound on accuracy is controlled by α. As the confidence in a decision increases, its accuracy increases at a rate controlled by β, with β=1 corresponding to a linear increase, values β>1 corresponding to faster increases, and values β<1 corresponding to slower increases. In the limit β=0, there is no growth in accuracy. The shift of the growth curve is controlled by δ. When δ=0 a decision made with no confidence has chance accuracy.
Logistic growth calibration curves relating confidence on the x-axis to accuracy on the y-axis. The central panel shows the general model, with a growth parameter β, a bound parameter α, and a shift parameter δ. Surrounding panels show specific-case nested models with natural interpretations. Based on (LeeLee, 2017, Figure 3).
(LeeLee, 2017) also consider four special cases of the full shifted probabilistic model, shown in the surrounding panels of Figure 9. The deterministic model in the top-left corresponds to setting δ=0, α=1, and allowing only β to vary. The deterministic shift model in the bottom-left corresponds to setting just α=1, allowing for worse-than-chance performance. The probabilistic model in the top-right corresponds sets δ=0 so that accuracy starts at chance and grows to some upper bound α as confidence increases. Conceptually, α corresponds to the upper bound measuring the inherent (un)predictability of the domain, and β corresponds to how quickly that limit is approached. Finally, the chance model in the bottom-right corresponds to setting α=1/2, β=0, and δ=0, which reduces the model to θ=1/2 for all levels of confidence, so that accuracy is always at chance.
We follow (LeeLee, 2017) by implementing this model as a graphical model in JAGS (Plummer, 2003), which allows for fully Bayesian inference using computational sampling methods. The full shifted probabilistic model, and its four special cases, are treated as components of a latent-mixture model, allowing both the type of calibration curve and its parameters to be inferred.
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