**1. Introduction**

Distributed financial platforms are on the rise, ranging from Decentralized Autonomous Organizations [1], crowd-sourced prediction systems [2] to the very recent events during which retail investors self-organized using social media and drove up asset and derivative prices [3,4]. In this work, we investigate how financial agents process information from one another and predict-individually and collectively—the future prices of real assets. Specifically, we are interested in understanding the computational models they use to update their beliefs after information exposure and how different social vs. non-social belief update strategies lead to trade-offs in prediction performance.

Here, we expand the typical definition of performance for collective prediction to include the concept of risk. Typically, the prediction performance of collectives and swarms is measured mostly by the accuracy of the group over collections of tasks [5–7]. However, it has been shown theoretically [8,9] and observed in a variety of applications [10,11] that there is a fundamental trade-off between prediction accuracy (average error) and prediction risk (variance of error).

This means that for any prediction system, risk will always be present, and that maximizing accuracy will come at the expense of increased risk. Hence, the performance of the system will always exist within a pre-defined Pareto frontier [12,13] which is the curve containing all possible system performance parametrizations (here, pairs of possible accuracy and risk values). Therefore, a platform designer will need to make trade-offs between risk and accuracy and cannot achieve arbitrarily combinations of risk and accuracy. Treating risk and accuracy as equally important for prediction is standard in

**Citation:** Adjodah, D.; Leng, Y.; Chong, S.K.; Krafft, P. M.; Moro, E.; Pentland, A. Accuracy-Risk Trade-Off Due to Social Learning in Crowd-Sourced Financial Predictions. *Entropy* **2021**, *23*, 801. https:// doi.org/10.3390/e23070801

Academic Editor: Stanisław Drozd˙ z˙

Received: 25 May 2021 Accepted: 17 June 2021 Published: 24 June 2021

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statistical [8–10] and financial [14–16] forecasting applications and literature because it allows for prediction systems to be calibrated and deployed with regard to specific accuracy and risk profiles [17–21].

However, characterizing the performance of crowd-based prediction systems regarding both accuracy and risk is not common and such a Pareto frontier has not been observed in crowd-sourced financial asset price prediction. We are therefore interested in investigating if a Pareto frontier exists and what the causes are behind this trade-off. From the perspective of crowd-sourced financial platform designers, understanding the trade-off between accuracy and risk and how to select subsets of predictions that achieve a certain accuracy and risk is useful to fit a required risk profile. This, in turn, allows for more sophisticated and versatile applications of crowd-sourced predictions such as hedging risks over portfolios of prediction tasks.

To test our hypothesis that a Pareto frontier exists between risk and accuracy and that it is mediated by social learning, we designed our collective prediction experiments as a series of Wisdom of the Crowd (WoC) tasks. For background, the Wisdom of the Crowd [22,23] is a popular domain within the collective intelligence literature where participants (the 'crowd') are asked to make predictions of a certain quantity, such as the future price of an asset on the stock market [24] or the caloric content of food items [25]. Prior work in the WoC literature [25–27] has focused on maximizing the average accuracy of collectives with little regard to the risk of the predictions.

The structure of this paper is as follows: we do a short literature review of the connections of this work to research on collective intelligence and the accuracy-risk tradeoff in Section 2. We discuss our materials and methods (experimental design, data collection, and modeling and estimation) in Section 3. We present our results (belief update modeling, accuracy-risk trade-off and prediction under high uncertainty during Brexit) in Section 4. We discuss the implications and limitations of our work in Section 5.

### *Contributions*

Our work makes the following novel contributions:


These results are not only important for the practical deployment of distributed financial prediction platforms but also expand our understanding of how financial agents process information and make distributed predictions.
