{'fraction_non_alphanumeric': 0.02700785610085788, 'fraction_numerical': 0.010291893628126715, 'mean_word_length': 5.103577846757066, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Behavioral Economics for Human-in-the-loop Control Systems Design: Overconfidence and the hot hand fallacy

Successful design of human-in-the-loop control systems requires appropriate models for human decision makers. Whilst most paradigms adopted in the control systems literature hide the (limited) decision capability of humans, in behavioral economics individual decision making and optimization processes are well-known to be affected by perceptual and behavioral biases. Our goal is to enrich control engineering with some insights from behavioral economics research through exposing such biases in control-relevant settings. This paper addresses the following two key questions: 1) How do behavioral biases affect decision making? 2) What is the role played by feedback in human-in-the-loop control systems? Our experimental framework shows how individuals behave when faced with the task of piloting an UAV under risk and uncertainty, paralleling a real-world decision-making scenario. Our findings support the notion of humans in Cyberphysical Systems underlying behavioral biases regardless of -- or even because of -- receiving immediate outcome feedback. We observe substantial shares of drone controllers to act inefficiently through either flying excessively (overconfident) or overly conservatively (underconfident). Furthermore, we observe human-controllers to self-servingly misinterpret random sequences through being subject to a"hot hand fallacy". We advise control engineers to mind the human component in order not to compromise technological accomplishments through human issues.


I. INTRODUCTION
Our century is bringing interesting challenges and opportunities that derive from the way that digital technology is shaping how we live as individuals and as a society. A key feature of a number of engineered systems is that they interact with humans. Rather than solely affecting humans, often people make decisions that have an effect on the engineered system. As an example, when driving our cars we often decide to take a route which differs from that suggested by the navigation system. This information is fed back to the service provider and henceforth used when making route suggestions to other users. The analysis and design of such Cyber-physical Human Systems (CPHSs) would benefit from an understanding of how humans behave. However, given their immense complexity, it is by no means clear how to formulate appropriate models for human decision makers, especially when operating in closed loop systems.
CPHSs fit into the general context of human-machine systems and pose multidisciplinary challenges that have been tackled in a number of domains. For example, interesting surveys on signal processing approaches include [1] and [2], whereas [3] adopts a computer science viewpoint. More focused on specific applications, the articles in the special issue [4] as well as [5] study the interaction between humans and vehicles. The paper [6] surveys systems comprised of humans and robot swarms and [7] reviews telemanipulation by small unmanned systems. Also within the control systems community significant advances have been made, see, e.g., [8], [9], [10], [11]. Here it is common to model the effect of humans as a limited actuation resource or unknown deterministic dynamics to be identified. This opens the door to use various robust control and game theoretic methods to design closed loop systems. However, most paradigms adopted in the systems control literature so far hide the decision capability of humans and limitations to their cognitive and computational capabilities. Notable exceptions to the literature include [12] and [13] where human decision making is characterized via a Partially Observed Markov Decision Process (MDP) [14] with states representing basic physical human aspects, such as operator fatigue or proficiency. In particular, [13] investigates a stochastic two player game, resulting from the interaction between a human operator and an unmanned aerial vehicle (UAV). Here the UAV is allowed to react to the decisions made by the operator, to compensate for potential non-cooperative behaviour. For a CPHS with multiple human decision makers, [15] models limitations in their decision making capabilities using "level-k reasoning" and associated game theory [16]: Each agent optimizes a cost function using only reduced knowledge of the other agents' policies.
In the present work we investigate human decision making from a behavioral economics perspective. We pose the question how behavioral biases affect decision making, since we expect behavioral influences to potentially hamper technological optimization efforts. More generally, we aim to direct attention to the human as a poorly observed uncertainty source in CPHS. We thus consider humans as being "bounded rational", i.e., they are strategic thinkers who want to maximize their benefits but, at the same time, have only limited cognitive and computational capabilities. We distinguish between closed loop scenarios, where the decision maker receives feedback about the success of its actions, and open loop situations, where no such feedback information is available. Here, we pay special attention to the role played by feedback on arXiv:2007.15869v1 [math.OC] 31 Jul 2020 behavioral biases in form of overconfidence [17] and the so-called hot hand fallacy [18]. The latter effect may arise when a decision maker becomes aware of her past success and overestimates future success probabilities. We will next introduce some prominent findings from behavioral economics that can be regarded as influential for CPHSs. These will be subsequently applied to an experimental human-in-the-loop framework inspired by the UAV piloting scenario of [13]. Throughout our presentation we will highlight the value of giving greater consideration to human behavior and behavioral biases in control engineering. The central outcome of our study is that frequent feedback to human decision makers may lead to sub-optimal results. This stands in contrast to situations wherein computers carry out optimizations and more information is always beneficial [19], [14], [20]. As we shall see, when humans act as decision makers, then the situation is more ambiguous.

II. CONTRIBUTIONS FROM BEHAVIORAL ECONOMICS
Behavioral economics is a comparably young research field that is concerned with the overarching question of how humans actually do behave in economic decisions, in contrast to how they are prescribed to behave by economic theory [21]. Therefore, traditional economic assumptions and models are revised and enriched by insights from psychology. This serves to improve the understanding and predictability of human behavior, especially regarding recurring errors and biases in decision making [22]. The perception of humans thereby shifts from the assumption of a rationally optimizing Markov decision maker towards regarding humans as rather bounded rational agents. Humans constantly use cognitive mechanisms of simplification, namely decision heuristics, in order to process information and make decisions under uncertainty [21], [23]. Such heuristics are used subconsciously due to individuals not managing to adequately process the complexity of a decision problem or to take all relevant information into account [22]. In fact, when making judgments or estimations of events, frequencies or probabilities under uncertainty, individuals do not always obey Bayesian rules and statistical logic, as they are meant to do in models assuming perfectly rational agents and Markov decision makers. Such heuristic simplifications sometimes yield reasonable judgments, but may also lead to severe and systematic errors [24], [25], [26]. Interestingly, even experienced researchers and professionals often underlie the same judgment heuristics and biases as laypersons do [25].

A. Overconfidence
Overconfidence is well-known to be one of multiple behavioral stylized facts influencing and thereby biasing human decision making. Overconfidence can be defined as a general miscalibration in beliefs [27], more specifically, the discrepancy between confidence and accuracy [17]. In being overconfident, people overestimate their own capabilities, their knowledge or the general favorability of future prospects [28]. The authors of [29] go as far as labeling overconfidence the most robust finding in the psychology of judgment.
Early contributions to research on overconfidence found that people systematically tended to be unrealistically optimistic about their future, in judging themselves to be more likely to experience a variety of positive events, and to be less likely to experience negative events, compared to others. This pattern has been traced back mainly to the degree of desirability influencing the perceived probability of such events [30]. In the original study [31] over 80% of the subjects regarded themselves as more skillful and less risky car drivers than the average driver. Moreover, one half of the subjects estimated themselves to be amongst the top 20% of the sample, vividly illustrating overconfidence as the discrepancy between confidence and accuracy (belief and reality) in doing so. Meanwhile, evidence on overconfidence does not depend on whether or not experimental subjects are familiar to the given tasks or actions [32]. The tasks difficulty, on the other hand, is known to have an impact on overconfidence, as more difficult tasks were shown to facilitate overconfidence rather than easy tasks [27], [33]. Overconfidence can express itself, and consequently has been studied, in multiple ways, like 'better-than-average' beliefs, e.g. [32]), or 'overprecision', e.g. [34]). The facet of overconfidence most fitting for the purpose of our current work appears to be overestimation, described as self-servingly overestimating the likelihood of desirable outcomes, supposedly fueled by wishful thinking for such desirable outcomes to occur * [36], [37]. An experimental approach for studying overconfidence was requested in [32], as most research in this field merely relies on verbal statements or subjective estimations, rather than monitoring human decisionbehavior. Our current analysis makes a contribution to this call. Overconfidence is connected to a variety of other behavioral phenomena, which will be revised in the sequel.

B. Underestimation of systematic risks
Analogous to overestimating favorable outcomes, overconfidence also implies underestimating risks, or rather the variance of risky processes, indicating a too narrow distribution in one's subjective probability beliefs [38]. Often, humans assess probabilities incorrectly or rather draw incorrect conclusions from them through not adhering to Bayesian rules or neglecting base-rates [24], [39]. In doing so, smaller probabilities are usually overestimated while larger probabilities are underestimated [40], [41]. Vice versa, decision makers are also found to underestimate the overall likelihood of the occurrence of risks with small single probabilities of occurrence in some instances. Especially risks of disqualification [42] through lowprobability events generate less concern than their probability warrants on average [43]. A cognitive process behind this underestimation of especially very low probability risks, like being involved in car crashes or natural disasters, can be characterized as subconsciously approximating low probabilities with zero in order to not having to mentally deal with them anymore [44].
Consequences of this issue can be observed in various areas of economic and social decision making. Overconfident individuals, who underestimate health, financial or driving risks, are more likely to have insufficiently low insurance coverage against such risks [45]. Furthermore, employees are observed to underestimate the risk of their own company's stock and being overly optimistic about its future performance. They tend to include such stocks too heavily into their retirement plans, despite the respective stock being riskier than the overall market, as a consequence of excessive extrapolation of positive past performance [46]. Money being at stake does neither change overconfident behavior, nor does it lead to better estimation results concerning abilities, probabilities or risk. Experimental evidence was obtained on overconfidence leading to overly rushed market entries, that often precede business failures, due to entrepreneurs overestimating their relative chances to succeed [47]. Overconfident top-managers were further shown to underestimate the volatility of cash flows of S&P 500 companies, resulting in erroneous investment and financing decisions, in a large-scale survey over 10 years [38]. Especially managers competing for leadership positions display overconfidence by tending to take on riskier projects due to underestimating their risks [48]. Overconfident CEOs also overestimate the positive impact of their leadership and their ability to successfully complete a merger to generate future company value. These CEOs are found to execute valuedestroying mergers, thereby exposing the affected companies to high financial or even existential risks [49]. Multiple experimental studies found that overconfident investors trade more excessively than others, overestimate the impact of little information they have [50], [51], and do not adjust their trading volume to new negative information [52]. These findings, which are inconsistent with the Markov assumption and Bayesian updating, contradict theoretical expectations of rationally optimizing economic agents and re-emphasize the notion of individuals as bounded rational decision makers [21].

C. Attribution theory
Another mechanism contributing overconfident decision making is a biased perception of causality in a self-serving way. Accordingly, individuals express the tendency to attribute positive past outcomes to themselves and their abilities, while blaming negative outcomes on external circumstances [53]. Successes are internalized, failures are externalized [54]. In an early experiment, teachers attributed learning successes of their pupils to their teaching skills, while blaming bad learning performances on the pupils themselves [55]. This self-serving attribution bias was identified to be the major driver of the aforementioned CEO overconfidence leading to risky mergers [56]. In [57] overconfidence evolves when traders get feedback about their ability through experience in a multiperiod market model and overweigh the role of their ability on prior success. Overconfident investors subsequently tend to become even more overconfident in their future investments. Meanwhile, financial losses are rather attributed to environmental circumstances, like unfavorable macroeconomic developments or simply bad luck, with the investors degree of overconfidence remaining constant [58], [59]. Also, professionals were found to be as likely as laypersons to express overconfidence in making economic decisions as well as in re-evaluating the quality of their own previous decision in hindsight [60].
People overestimate their own capabilities as well as their control over future events that are actually determined by external factors, especially chance. This was illustrated in an experimental study in which participants were either given a lottery ticket (control group) or allowed to select a lottery ticket themselves (treatment group). Subjects who had chosen their ticket themselves were, on average, demanding four times more money than those who were simply given their tickets, when asked to name a price they would sell it for. Those who chose the ticket themselves mistakenly assumed they would therefore be more likely to win, although the winners would be determined entirely by chance. This mechanism has been labeled the "illusion of control" [54], although critics of this concept have argued that it rather identifies a pattern of people overestimating their ability to predict future outcomes than of people overestimating their ability to control future outcomes [61]. Both interpretations are regarded as sufficient for the purpose of our current analysis.

D. The role of feedback in overconfidence
Providing feedback is commonly regarded as an efficient learning mechanism in Markov Decision Processes in order to compute optimal behavior [62], [14], [19] and offers great value for deriving logical inference rules in accordance with Bayesian reasoning in machine learning [63], [64]. While these points may hold true for cyber-physical systems without human decision makers, humans are often observed to react to feedback differently. Contradicting Markovian assumptions, behavioral economic research constantly observes errors in updating the information set (e.g., [65], [66]). This indicates that providing more feedback information to humans does not necessarily lead to better estimations and decisions. The importance of outcome feedback in counteracting certain behavioral biases, through continuous "monitoring of progress through a judgment-action-outcome loop" [67] and offering corrective adjustments, has been pointed out before [68], but it may not apply equally to every type of bias.
While research studying the connection of high information supply to overconfidence is comparably scarce, in various instances a higher amount of feedback was not found to be able to improve behavioral rationality with regards to overconfidence, as it would be expected by theory. Both overconfidence and underconfidence were shown to persist in employees' choices of incentive schemes although receiving clear feedback revealing the optimal alternative for them [69]. Experiments on sports outcome-estimation showed that overconfidence did not decrease but rather increase with additional information, as the subjects confidence increases more than their accuracy does [70]. Similarly, CEO overconfidence in forecasts was found to persist against corrective feedback [71] and venture capitalists were shown to be more overconfident when having access to more information [72]. In general knowledge tasks that are related to each other, corrective outcome feedback has shown able to mitigate underconfidence but not overconfidence [73]. It turns out that feedback appears to affect behavioral biases rather ambiguously with overconfidence tending to largely persist, or even growing, in response to additional information.

E. Misperceptions of random sequences
With regards to human sequential decision processes, another potential threat of high information supply leading to biased decision making, is the danger of falling subject to the hot hand fallacy. The study of this error in the assessment of past outcomes originates in observations from basketball. If this player repeatedly scored on his past throws, he is commonly judged to have a hot hand by his teammates and the audience. Therefore, the player's likelihood of scoring on future throws is judged to increase by the audience, even though future throws are independent of past throws. In doing so, the player's objective probability of scoring is overestimated based on his past successes [18]. This hot hand fallacy results from misjudging statistically independent favorable events to be connected to one another, implying a positive autocorrelation between them. Repeated past positive outcomes are therefore erroneously expected to occur again in the near future, thereby overestimating their objective probability of occurrence. The hot hand fallacy is a very vivid illustration of biased probability judgments through presumed representativeness of recent information. Misinterpretations of random sequences through such extrapolation of recent outcomes into the future were empirically found to apply to several contexts. Regarding basketball, the findings by [18] were supported by studies on basketball betting market odds, although only small effects were found [74], [75]. In a laboratory experiment, simulating a blackjack game, gamblers were observed to bet more money after a series of wins than after a series of losses [76]. In more sensitive contexts, individual decisions show signs of the hot hand fallacy as well. For instance, financial investors who had negative prior experiences with low stock returns, exhibit a decreased willingness to take financial risks in future investments [77]. Attribution theory, especially the illusion of control mentioned above, may factor into the hot hand assumption as well, since subjects are more likely to attribute recent sequences with low alternation to human skilled performance, while sequences with high alternation are rather perceived as chance processes [78].
Before proceeding, we note that the hot hand fallacy's opposing bias, the so-called gamblers fallacy, presents the erroneous expectation of systematic reversals in stochastic processes. These are grounded in the belief that a small sample should be representative for the underlying population, i.e., that a Bernoulli random process should balance out even across few rounds [79]. Both phenomena have recently been strongly disputed in their status as biases [80], [81].

HYPOTHESES
Our experimental design for analyzing behavioral biases in CPHSs builds upon a basic setup involving UAVs that interact with human operators, as discussed in [13] and which Fig. 1. Illustration of a road network for UAV missions, adapted from [13].
is graphically illustrated in Fig. 1. Partly autonomous UAVs are used for road network surveillance inside a network that connects multiple traffic junctions. The drone pilot's objective is to gain the maximum information about the traffic at all junctions using a single drone which follows a predesigned path, see Fig. 1. To increase the picture quality (and thereby the information obtained), the operator may choose to fly up to 8 rounds over each junction j, each round taking an additional photo. The individual photos are combined into a higherresolution image, whose total value will be denoted I j . This total value depends on the number of flights, say f j , and on the random additional information content of each photo taken. The situation is modelled via the random process σ j (i) which quantifies the information content of the combined photo of junction j at round i. The process is initiated at σ j (1) = 25 and through iteration of Here, the additional information content of the photo taken at round i is quantified by the concave function ρ(⋅) which describes a decreasing marginal yield: The above reflects the fact that each additional picture often comes with less information value added, since a rough image of the traffic flow is already gained by the earlier pictures, and subsequent pictures only help further sharpening the image. In (1), s j (i) ∈ {0, 1} is an i.i.d. Bernoulli random process, with probability P [s j (i) = 1] = p. This models instances wherein the new picture offers no information value added over the previous one due to, for instance, bad weather causing poor visibility, strong wind or lacking flow of traffic.
Whilst taking more photos will often lead to combined images of higher quality and also with more information content, after each individual flight there exists a (small) probability of r of the drone crashing, leading to a loss of D (the value of the UAV) and the inability to continue flying again over the current or later junctions. where if the drone is still intact after flying i rounds at junction j, 0, otherwise.

(4)
The total value gained by the operator at the end of the mission is given by:

A. Decision problem
Given the possibility of the drone crashing, rather than simply intending to fly the maximum number of 8 rounds over each junction, the operator is faced with the decision of how many rounds to fly over each junction in order to maximize its profit V . This amounts to a sequential stopping rule problem defined over a finite horizon. To make decisions when to fly (or switch) to the next junction, in addition to knowing the system model and its probabilities, the operator receives feedback from the drone, see Fig. 2. Every time the UAV passes over a junction, it sends back σ j (i), the information gain obtained so far over the current junction. Further the operator, is informed whether the drone has crashed. Using this information, the human operator is faced with the task of designing a closed loop flight policy.
The exact stopping rule can, in principle, be derived using dynamic programming [20], [82]. However, the value function depends not only on the value σ j (i), but also on the current round i as well as the junction j, with later junctions having less value. Thus, instead of pursuing an optimal strategy, a 1-stage look-ahead rule within the current junction becomes a reasonable alternative. Using such a myopic policy, the operator chooses to flight another round over the current junction j if and only if the drone is intact (c j (i) = 1), i ≤ 8 and the marginal gain of flying one more round is positive: where we have used (3), see also Fig. 3.
In the sequel we will fix the parameters to D = 400, p = 0.5 and r = 0.02, so that (2) provides This makes flying until an information value σ j (i) = 70 is accumulated in the current junction the optimal choice. From the fourth node onwards, the marginal gain from flying another round is lower than the marginal loss, therefore a rational (and risk-neutral) agent should refrain from flying further rounds.
In addition to this full feedback case, we will also consider a situation wherein the drone does not transmit the feedback information σ j (i) to the operator. This requires the operator to design an open loop policy [19], as detailed below.

B. Experimental drone framework
In our behavioral economic engineering project (see Sidebar "Behavioral Economic Engineering"), we abstract the framework into an economic experiment (see Sidebar "Experimental research method and Induced Value theory") in order to gain insights on actual individual behavior in human-in-the-loop control. The experiment translates the CPHS concepts of open versus closed loop control into a sequential decision making game, framed in a context in which the subjects are owners to an UAV with a photo function. They are told that they were hired by the local city administration to support the city's traffic surveillance by taking photos of ten important traffic junctions. As outlined above, the subjects' task is to decide, how many rounds the UAV should fly over each of the 10 traffic junctions (up to a maximum of 8 times), while the drone would fly and take pictures autonomously. Subjects are incentivized, as they are paid according to the amount of information value gained through the pictures their drones take in the fictitious currency "Taler", with one Taler equaling one unit of information value gathered. Taler are exchanged into Euro, after all parts of the experiment are completed. However, during each round in which a UAV flies, it faces a constant crash risk of r = 2%. At the end of this job, i.e., after all ten traffic junctions, subjects are able sell their drone for an additional fixed payment of D = 400 Taler, as long as it is still intact at this point, see (5).
Each traffic junction represents one stage of the experiment. For the sake of simplicity, subjects do not have to bear any costs from flying the drone. Further, the drone's value does not diminish from usage and subjects are told that batteries for the drones as well as (ground) transport between the traffic junctions is provided free of charge by the city administration. One battery charging allows the drone to fly up to eight rounds before the drone must land to recharge. Therefore, subjects are able to decide for a maximum of eight rounds per traffic junction and consequently take up to eight pictures. This corresponds to a maximum of eight nodes that can be reached per junction. In the first round flown over each traffic junction, an information value of σ j (1) = 25 is gained with certainty, as any picture taken represents an increase in information value over no picture taken at all. From the second round onwards, the information value potentially gained per round decreases, see (2). For the sake of simplification, a crash can only happen after a picture is taken. As per (5), the information value gained at the junction where the crash occurs up to the point of it, still qualifies for the payment, as subjects were told that the drone's memory chip would survive the crash.
The operators need to decide how many rounds a UAV should fly over each traffic junction in order to maximize the total gain V in (5) and consequent payoff. In order to do so, some of them receive feedback on the results of the previous rounds -depending on the treatment -which they can base their decision on. As noted before, the resulting optimization problem belongs to the class of general stopping problems. Its solution, both with and without feedback information, can in principle be derived, but requires careful computations. Since individuals are well-known to have limited cognitive and computational capabilities, they will not be able to compute a globally optimal strategy. Instead, optimizing subjects will use the decision heuristic in (6), i.e., the fourth node is identified as the one to reach.
The above leads to two decision heuristics, depending on whether feedback information is available or not. In the former (closed loop heuristic), at every junction the UAV should aim to fly as many rounds as needed until reaching an information value of σ j (i) = 70. To be more precise, introduce a j (i) ∈ {0, 1}, where a j (i) = 1 corresponds to the decision to fly another round over the current junction, then the closed loop heuristic can be stated via: a j (i) = 1, if and only if σ j (i) < 70 and i ≤ 8 and the drone is intact. (8) Note that in this case, the total number of rounds flown over a junction j, namely f j = ∑ a j (i), depends on the sequence of increases and non-increases the subject experiences.
In the absence of feedback from the drone, a suitable open loop heuristic amounts to attempting to fly f j = 5 times at every junction: a j (i) = 1, if and only if i ≤ 4 and the drone is intact. (9) This yields the expected result of three increases in information value, with the first one being granted with certainty in the first round. The above rules of thumb present reasonable heuristic approximations to the task's optimal solution when implemented for all junctions. A rational and risk-neutral decision maker, who acts according to expected utility theory [83], would follow these heuristics. Deviations from these decision-rules upwards or downwards represent indicators of overconfidence or underconfidence respectively, depending on the direction.
The experiment ends either after all junctions are completed and the drone is sold or once the drone has crashed. A crash could be caused by external factors like weather conditions, the drone hitting some object or animal, or technical failure. This risk of a total loss of the drone's value in case of a crash imposes the prospect of high one-time costs onto the participants and thus prevents them from simply choosing the maximum number of rounds possible at every traffic junction. As subjects are paid according to the information value gained, striving for the optimal information value (using the above heuristic) presents the rational strategy. Always flying the maximum number of 8 rounds is not efficient, as gaining information value has to be weighed against maintaining the chance of being able to sell the drone at the end of the experiment, as indicated in (5).

C. Hypotheses
The variety of behavioral phenomena and stylized facts regarding decision biases and errors in relation to overconfidence presented in the preceding section, suggests inefficient human decision making. In fact, individuals are commonly observed to act overconfidently. In doing so, they tend to underestimate systematic risks, insufficiently process joint and conditional probabilities, self-servingly misattribute causal relationships and erroneously expect favorable outcome sequences to continue in the future. These aspects should be particularly relevant in our example, since the probability of the drone crashing may appear small for one specific round (2%), while the overall risk of the UAV crashing at any time over the course of the experiment is substantially higher. Modeling the crash probability as a Bernoulli process, the chance of a crash when attempting to fly the maximum number of rounds at each of the ten junctions would be 1 − 0.98 80 > 0.8. A narrow focus on the low single-decision crash probability [84] could therefore lead subjects to underestimate the overall crash probability and consequently take higher risk. Providing feedback is commonly assumed to diminish these behavioral biases but known to not being able to entirely resolve them, and in some instances even aggravating them. Subjects may rather attribute positive feedback to their own decision performance to themselves instead of realizing that it is caused by chance. All these aspects point in the direction of expecting overconfident drone piloting. Manager overconfidence in economic decision making can be summarized to originate in either overestimating expected cash flows or underestimating risk [85]. Translated to our design, the drone pilots' overconfidence could either result from overestimating the likelihood of information value improvements or from underestimating the risk of the UAV crashing. While the task is framed in a quite understandable context, it is non-trivial in its solution. Most subjects are expected to approach the task rather intuitively, i.e., not solving it systematically through comparing marginal gain and marginal costs, and consequently act overconfidently through one of its facets described above, if not through both. According to the general consensus of literature on overconfidence in human decision making, the following hypothesis is posed: a) H1: Drone pilots will generally act overconfidently in flying more than the optimal number of rounds, regardless of feedback.
As is also known from the literature, individuals are prone to unreflectively repeat decisions that previously yielded them positive outcomes, even though such past successes entirely depended on chance and have no impact on future outcomes at all. This decision heuristic, the hot hand fallacy already discussed, presents a major misinterpretation of random sequences. While the hot hand fallacy in regard to the original study on basketball throws [18] is subject to criticism today (with the data being re-analyzed and the conclusion being reversed [80]), the general idea of individuals misjudging the meaning of random outcome sequences remains to be tested in the context of CPHS. For sequential decision problems featuring feedback from the system, decision makers appear especially vulnerable to fall victim to the hot hand fallacy. Translated to our drone framework, it is therefore expected that those drone pilots who immediately reach the optimal information value of a junction through experiencing gains in repeated rounds in the closed loop system (as the possibility is only given there), perceive themselves to be "on a roll", i.e., as having an equivalent to a hot hand † . Since decision makers are tempted to expect such apparent trends to continue, they will likely fly beyond the optimal number of rounds, overconfidently making their operation inefficient with regards to risk and reward. This way, subjects fall victim to the hot hand fallacy ‡ Compared to the gamblers fallacy, a hot hand is considered more suitable to portray drone controllers behavior in his attempt to gather information value, with humans more prone to expect favorable outcomes to repeat under their apparent control [78]. § . This leads to our second hypothesis: b) H2: Sequences of immediate positive feedback will lead to drone pilots falling victim to the hot hand fallacy.
These two hypotheses were tested in an experiment involving students at Paderborn University in September 2019. Details are given next.

IV. EXPERIMENTAL DESIGN
In order to be able to compare behavioral effects in closed and open loop operation, a between-subject design experiment is conducted, meaning each subject can only participate in exactly one treatment with the groups being compared afterwards.

A. Treatments
To evaluate the effect of feedback in human-in-the-loop control, two treatments are presented: Closed Loop and Open Loop. The groups differ only in the fact that subjects in one of them receive immediate outcome feedback after every decision made, while subjects in the other group do not receive any feedback until the end of the experiment.
In the Closed Loop Treatment, subjects receive feedback directly after each round of each junction on whether an increase in information value was gained in this specific round, see Fig. 3. They were also informed about the current total information value gained for the respective junction and whether the drone was still intact. Subsequently, they were asked whether they wanted to fly another round. The framing in the instructions said that the drone would transmit the pictures taken to the drone pilots laptops immediately, so feedback was given just in time for the next rounds decision.
In the Open Loop Treatment, subjects are provided with no feedback at all. Subjects make decisions on how many rounds the drone should fly over each junction entirely in advance. Afterwards, they are informed about the total information value they obtained and whether the drone is still intact † In our experiment, we define a hot hand as three consequent increases from the very start of a junction. With the junctions being distinct from each other and the first success being certain anyway, there is no possibility of pattern overlays [81]. Therefore, as well as due to the outcome of a fourth flight being unimportant for judging the decision inefficient, we do not consider our results subject to "streak selection bias" [80]. ‡ Literature observes a shift from hot hand beliefs for shorter outcome streaks towards a gamblers fallacy pattern for increasing streak length [86], [79]. Drone controllers who experience multiple increases in a row would therefore begin to overestimate the probability of a non-increase at some point and stop flying to prevent a crash accordingly. § Also, a hot hand fallacy can be considered the more problematic error compared to the gamblers fallacy, since in the case of the latter, subjects would stop earlier and not risk a drone crash through flying in periods that yield marginal losses (although sacrificing marginal gain). and sold at the end of the experiment. The framing in the instructions states that the UAV has to be deconstructed in a complex procedure to be able to extract and read out the memory chip to check on the pictures taken. Therefore, no feedback on the pictures would be provided until either all junctions were completed or the drone had crashed.

B. Procedure
The experiment was computerized using the experiment software oTree [87] and hosted centrally on a university server, so subjects could remotely access the experiment and did not have to come to the laboratory physically. 500 subjects who previously had enrolled voluntarily into the BaER-Lab ¶ student participant pool were chosen randomly, with 250 each being randomly assigned to the two treatments in advance. These subjects were contacted via the online recruitment system ORSEE [88] and invited to participate within the following five days.
The invitation email included a hyperlink that directed the subjects to their respective treatment, where they received the detailed instructions for the experiment. The instructions of both treatments only differed regarding the provision of feedback, as explained above. To be able to progress to the drone flying task, subjects had to correctly solve four control questions. The questions revolved around the central parameters of the experimental design in order to assure that the subjects had read and understood the instructions in its critical parts. Subsequently, subjects advanced to the drone flying task, in which they had to make their decisions through clicking single choice buttons. Subjects would receive one unit of the fictitious currency Taler for each unit of information value gained as pay for flying the drone. During the whole experiment, one subject's payoffs did not depend on the decisions of any other subject. After completing the drone flying task, a result screen presented the total information value accrued over all traffic junctions, the state of the drone, i.e., whether it was still intact and consequently sold to earn 400 additional Taler or not, as well as the corresponding payoffs the subject generated in Euros. Payoffs were translated from Taler to Euro at an exchange rate of 1 Euro per 120 Taler. Subjects were then asked to answer a standardized questionnaire, that included questions on demographics, perceived task difficulty and the subject's reasons for choosing the numbers of rounds the way they did. Afterwards, subjects were presented with the multiple price list of [89] (see Sidebar "Multiple Price List Format"), in order to measure their risk preferences.
Filling out the price list was incentivized, as for every fifteenth subject one out of the list's twenty rows was randomly selected and the payoffs that the subjects chosen alternative row yielded in this row was added to the his total payoffs. In case Alternative A had been selected by the subject, the fixed amount stated in the respective row was simply added to its payoffs. In case Alternative B had been selected, the lottery was automatically played out and either e0 or e30 were added to the subject's payoffs. After completing the price ¶ BaER-Lab stands for "Business and Economic Research Laboratory". For further information visit www.baer-lab.org. list, subjects were informed about whether the list came into effect for their payoffs, and in case it did which row had been selected. On a final screen, subjects were informed about their total payoffs in Euros and thanked for their participation. A subject's total payoff function can consequently be formalized as ∆m = 1 120 where V is the total value, as in (5). The term d( ) represents the additional payoff from the multiple price list, conditional on the selected row of the list that a subject was paid in case of being a fifteenth participant, i.e., a subjects participant's ID being a multiple of 15. Subjects were able to collect their payoffs in cash at the Chair's secretariat following their participation by stating a unique eight-figure ID-Code they had to create in the beginning of the experiment. That way the correct payoffs could be handed out to each subject, while maintaining anonymity about the subject's decisions made.

V. EXPERIMENTAL RESULTS
Out of the 105 subjects who finished the online experiment, 57 participated in the Closed Loop and 48 in the Open Loop treatment respectively. The average age of the subjects was 23.87 years and varied by about half a year between treatments. 31 subjects (29.52%) were female, while one subject classified itself as "diverse". On average, subjects earned e4.89 from the experiment, plus the amount they would earn from filling out the multiple price list in case they were paid due to being a fifteenth participant.
Subjects in the Closed Loop treatment flew, on average, 5.89 rounds over each traffic junction, thereby exceeding the average number of rounds from the Open Loop treatment by 0.79 rounds per junction. Over all rounds flown, subjects in the Closed Loop treatment gained a cumulative information value of 651.84, on average. This represents a 27.83% surplus compared to the average information value gained in the Open Loop treatment. Similarly, a higher drone crash rate of nearly 70% was observed in the Open Loop group. These two dimensions generally interact, since the information value is fixed, once a drone crashes. Averages of rounds flown and accumulated information value as well as average earnings and crash rates for each treatment are displayed in Table I. As could be expected from control theoretic results [19], the Closed Loop system is clearly observed to be more effective than the Open Loop piloting system, with regards to accumulating information about traffic conditions in our framework. Considerably more information value was aggregated with a closed loop of immediate outcome feedback being implemented, which also translates to a higher monetary payoff for the subjects. However, subjects displayed behavioral biases as we will see. Indeed, the average number of rounds in the Closed Loop treatment being nearly six already hints that several subjects tended to fly beyond the heuristically optimal number of rounds. In doing so, they would have taken inefficiently high risks, since marginal losses exceeded marginal gains in later rounds of each junctions. For instance, instead of aiming to add five or ten more units of information value when already sitting at 70, subjects should stop and continue with the next traffic junction to obtain twenty-five units of information value with certainty at the same crash risk.

A. Overconfidence and underconfidence
To determine whether a subject acts overconfidently, optimally or even underconfidently at a certain traffic junction, the number of rounds flown is compared to the heuristics discussed before, see (6). If the subject flew beyond the heuristically optimal number of rounds, the decision at this junction is noted as overconfident. If the subject decides to fly less rounds than optimal the junction's decision indicates underconfidence. In case the heuristic is met the subject acts optimizing.
As mentioned before, flying until the fourth node is reached (associated with an information value of 70) is a suitable heuristic for a risk neutral decision maker. This induces different strategies for the treatments. In the closed loop system, it is optimal for the subjects to fly as many rounds as necessary to reach the fourth node. The actual number of rounds needed depends on the sequence of increases and nonincreases experienced, as illustrated in Fig. 3. In the open loop system, meanwhile, flying five rounds is a suitable heuristic. As subjects have to decide on the number of rounds to fly for all traffic junctions in advance, this strategy applies to all of them, as explained before.
An indicator for overconfidence on the subject-level is created by calculating the quotient of the number of overconfident junctions and the total number of junctions per subject, since the number of traffic junctions played differed between subjects due to some drones crashing prematurely. This relative frequency of overconfident junctions by a subject will be labeled "overconfidence degree". Degrees of underconfidence and optimizing behavior are computed analogously, so all three degrees add up to 1. The average degrees of each of these behavioral tendencies are displayed in Table II.
The average overconfidence degree of 0.44 in the closed loop system means that subjects, on average, decided overconfidently for 44% of the traffic junctions. Comparatively, overconfidence degrees do barely differ between the two treatments with subjects deciding overconfidently for over 42% of the junctions. The average underconfidence degree was higher in the Open Loop treatment. Degrees of overconfidence and underconfidence are relatively close to each other for the open loop, while overconfidence characterizes the predominating behavioral tendency in the closed loop system. The optimization degree drops off in both treatments, with subjects only deciding optimally at one fourth and one fifth of the junctions respectively. This gap also becomes apparent in the graphic illustration of the average degrees of overconfidence, underconfidence and optimization displayed in Fig. 4. Degrees of overconfidence did not differ significantly by gender in either treatment.
Seven subjects in the Closed Loop and eight subjects in the Open Loop group behaved overconfidently in every single phase they flew in, resulting in the maximum overconfidence degree of 1. On the other side, only three subjects in the closed loop system and two in the open loop system decided optimally for every junction, while three subjects in each treatment acted underconfidently at every junction. Among those junctions in which subjects act overconfidently, they flew, on average, 2.88 (SD = 1.36) more rounds after reaching the optimal node in the Closed Loop treatment. In the Open Loop group, subjects flew on average 1.88 (SD = 0.85) rounds beyond the optimum respectively. It appears that those subjects, who decide to fly more rounds than optimal, they do it clearly, i.e., usually by more than one round. Also, subjects exceed the optimal number of rounds by, on average, one more round in the closed loop compared to the open loop system once they fly beyond the optimum. The difference is statistically significant in a Mann-Whitney U-Test (Z = 7.393, p = 0.000) and match the observation of subjects in the Closed Loop treatment flying more rounds overall and accruing more information value in total.
To gain a more precise impression about the distribution of subjects deviations from optimal behavior within each treatment, we classify all subjects into broad categories. Subjects that act in accordance with the strategy of flying until reaching the fourth node at all traffic junctions are classified as optimizing. A subject whose overconfidence degree exceeds one third up to 0.5 is classified as rather overconfident. Once a subject flies beyond the optimal number of rounds in more than half of the junctions (i.e., displaying an overconfidence degree over 0.5), this subject is considered strongly overconfident. Analogously, the categories rather underconfident and strongly underconfident are defined using the same thresholds in terms of underconfidence degrees as the ones for overconfidence. The sixth category, mixed, defines scenarios, in which subjects show overconfident or underconfident behavior in at least one junction but in less than one third of all junctions they played (equaling overconfidence or underconfidence degrees between 0 and 0.333). Since these subjects act optimizing in some junctions but not in all, their behavior still contradicts the notion of a strictly optimizing agent, although no clear dominant decision-pattern can be assigned to this behavior. The distribution of subjects into these categories by treatment is displayed in Table III. || According to Table III, only few subjects in each treatment act optimally throughout. Rather, individual behavior deviates from the optimal number of rounds in both directions. Approximately half of the subjects in both groups (Closed Loop: 51.79%, Open Loop: 46.81%) can be considered rather overconfident or strongly overconfident. Additionally, large shares of rather underconfident or strongly underconfident individuals || Two subjects from the Closed Loop treatment were excluded, because they crashed in the very first round at the first traffic junction, wherefore no statement on their decision making behavior can be made.
(Closed Loop: 37.50%, Open Loop: 44.68%) are observed as well. From the assumptions of MDPs and standard economic theory follow notions of individuals as rationally optimizing decision makers that leave no room for overconfident or underconfident behavior. Consequently, our empirical results do not meet these theoretical propositions. In fact, according to a Binomial Test of the data for the Closed Loop treatment the probability of observing 29 overconfident (rather or strongly overconfident) subjects out of a sample of 56 is virtually zero (p < 0.000001), given the assumption of optimizing behavior. The same result is obtained for overconfidence in the open loop systems. Consequently, we can fully support H1, indicating a strong general tendency of individuals to act overconfidently in CPHSs, regardless of whether receiving immediate outcome feedback or not. While not part of the hypothesis, we also observe the equivalent results for underconfident behavior in both treatments.

B. Hot Hand Fallacy
In contrast to an open loop, only a closed loop system provides the threat of sequences of immediate outcome feedback luring subjects into a hot hand fallacy. In our experiment, a subject in the Closed Loop treatment is defined to have a "hot hand" once it reaches the optimal node through obtaining three increases in information value in a row even though the first increase was certain. This corresponds to the common perception of three repeated outcomes as a streak [90]. In this case, an optimizing subject would stop flying when following the one-stage optimal strategy, since three increases are necessary to reach the optimum and flying beyond the optimum yields a negative marginal gain. The outcome of prior rounds should have no relevance for the subject's decision, since the rounds outcomes are independent from each other. If a subject decides to continue flying though, it violates the heuristically optimal decision rule, falling victim to the hot hand fallacy. Overall, at 84 junctions the respective subject experienced three continuous increases in information. In 70 out of these 84 cases (83.33%), subjects decided to fly at least one more round thereby falling victim to the hot hand fallacy. This share of hot hand fallacies among situations that pose the threat of falling victim to it is highly statistically significant: Under the assumption of subjects being optimizing Markov decision makers, the probability of such a high proportion disobeying the optimal strategy is practically zero (Binomial Test: Pr(all subjects act optimally 83.33% of the hot hand situations result in the hot hand fallacy): p < 0.0000001). Comparing non-overconfident and overconfident decision making between junctions with and without a hot hand in Table IV, we observe those subjects who experience a hot hand situation (70 out of 84, 83.33%) to have a significantly higher chance to act overconfidently at that junction, compared to those subjects who do not reach the optimum through three consecutive gains in information value (123 out of 260, 47.31%). Running a Chi-Squared Test to test the hypothesis statistically, we obtain a highly significant relationship between hot hand situations and subsequent overconfident behavior, i.e., the hot hand fallacy,  IV  HOT HAND FALLACY -CHI-SQUARED TABLE. THE TEST CAPTURES EVERY   INSTANCE IN WHICH A SUBJECT STARTED FLYING AT ANY TRAFFIC   JUNCTION IN THE CLOSED LOOP TREATMENT.   Not overconfident Overconfident  Total  No hot hand  137  123  260  Hot hand  14  70  84  Total  151 193 344 with an error probability p of virtually zero (χ 2 (df = 1) ∶ 33.46, p = 0.000). Consequently, H2 is strongly supported.

C. Risk Preferences
Subjects' individual risk preferences are broadly classified to be risk neutral, risk averse or risk seeking, see [89] and Sidebar "Risk Preferences". Given that the presented strategy is based on the assumption of a risk neutral decision maker, the observed distribution of risk preferences in each treatment is displayed in Table V. ** Consistent with the literature, the clear majority of around two thirds of the subjects in both treatments displays risk aversion. Overall, the distribution of risk preferences appears very similar between the treatments, with the relative shares of each risk attitude varying only slightly between the groups. This impression is supported by a two-sided Kolmogorov-Smirnov test, in which the null hypothesis that the risk preference distribution in the two treatments is not statistically different from each other cannot be rejected (D = 0.030, p > 0.1). Also, degrees of overconfidence and underconfidence did not differ significantly in between-treatment comparisons for each risk attitude using Mann-Whitney U-tests.
Further, we did not observe risk seeking subjects to display significantly more overconfident behavior (Open Loop: H = 5.437, p = 0.0660) † † . On the other hand, in the closed loop system, we find risk averse subjects to be significantly more underconfident, compared to the risk neutral and risk seeking subjects (∅ underconfidence degree: 0.3480(SD = 0.3018), Kruskal-Wallis equality of populations test: H = 8.891, p = 0.0117), while no effect was found in the open loop system.
Beside overconfidence, risk seeking preferences could in theory present an alternative explanation for drone pilots flying beyond the optimal number of rounds, with subjects primarily aiming to achieve a large information value while just hoping for their drones not to crash. However, we do not find substantial differences in overconfidence conditional on risk attitude between treatments and overall mainly observe individuals who can be classified as risk averse. This would theoretically predict subjects to decide rather conservatively through settling for a lower number of rounds in order to remain in the game and to protect the drone from crashing, even at the cost of missing out higher payments from information value gains. Since the experimental results display a large share of subjects flying beyond the optimum, the case for individual overconfidence as the dominant explanation for inefficient drone piloting beyond the optimum is strengthened. Consequently, we conclude risk preferences to not being able to explain overconfident behavior although it appears to explain the observed degrees of underconfidence).

VI. DISCUSSION AND CONCLUSION
Our experimental drone framework allows to observe how individuals behave when faced with the task of piloting an UAV under risk and uncertainty, paralleling a real-world decision-problem. Even though a closed feedback loop was identified to be the more successful system compared to open loop operation, we still observed inefficient drone piloting from the vast majority of subjects. Individuals expressed both overconfident and underconfident behavioral tendencies, regardless of receiving immediate outcome feedback. Specifically, our results indicate that immediate outcome feedback, that is originally intended to support optimal decision making, can turn out to be rather counterproductive in this regard. Overconfident decisions and consequently inefficient drone piloting can be facilitated by the hot hand fallacy as a misinterpretation of random sequences of immediate feedback on positive outcomes, since subjects fail to realize such sequences to be caused by chance and therefore being history independent. In fact, a handful of subjects in the closed loop treatment stated in the questionnaire that their decision strategy was to fly as long as they achieved steady increases in information value, trying to exploit an apparent hot hand. The fact that the possibility for this fallacy is only given in a closed loop system, presents an obvious weakness that should be considered in designing such feedback policies.
In general, the current work exposes the human as an under observed source of errors in human-in-the-loop control systems. We thus advise researchers and practitioners to carefully account for the behavioral component in the control of cyberphysical systems and the potential problems that arise from it, besides mathematical model optimization only. In particular, our findings illustrate the impact of behavioral biases regarding effects of immediate feedback and the (miss-)understanding of history independence in chance processes.
While more information is commonly regarded to result in better decisions in cyber-physical systems, human susceptibility for perceptual biases in response to high information supply has to be taken into account. Therefore, identifying an optimal quantity and frequency of feedback remains a goal for future research. We expect a carefully crafted intermittent feedback to be better suited for this purpose and stress the need for an intelligent feedback design that adapts to an individual's rationality in order provide suitable amount of information.
Generally, considering the effect of humans in control loops more seriously presents an important issue for research and practice. Overall, humans were shown to mostly not act optimizing in the given decision-problem in our experimental framework, which strongly puts the Markov decision maker as an adequate characterization of human decision making in question. Models of human decision processes should be revisited to account for limits of cognitive capacities and behavioral biases that result from them, in order to not jeopardize technological accomplishments through erroneous human decisions. Otherwise, individuals in human-in-the-loop control might take unnecessarily high risk and render thoughtfully designed policies inefficient, as seen for highly frequent feedback in the case of the hot hand fallacy.
Lastly, our study further provides a methodological contribution to research on CPHSs, making a first approach to incorporate insights from behavioral economics into control engineering. Further it introduces incentivized economic experiments as a viable option to reveal how individuals actually behave, in contrast to how they are theoretically prescribed to behave. We present an experimental UAV framework featuring a sequential decision-problem, with a focus on behavioral biases in relation to feedback policies. Future experimental research in this area may intensify efforts of incorporating various other behavioral phenomena and stylized facts into control engineering by building upon this framework, in order to design or test behavioral interventions that are able to proactively counteract them.

A. Sidebar: Behavioral Economic Engineering
The deviation between theoretically prescribed behavior and actual human decision making is often labeled behavioral messiness [91] and poses a huge challenge for organizational policy makers. Experiments thereby represent a tool for bridging economic theory and real-world institutional design in order to generate practical value from applied economic science [91]. The laboratory basically acts as a wind tunnel for practice in order to test behavioral reactions to institutional interventions in a controlled environment, before implementing them into the real world. Projects following this approach are summed under the label of behavioral economic engineering [91]. Prominent examples include the implementation of optimized retirement savings plans [92] and the redesign of matching algorithms for American physicians [93].

B. Sidebar: Experimental research method and Induced Value Theory
Behavioral economic findings are often based on experimental research, especially controlled laboratory experiments, as "experimental control is exceptionally helpful for distinguishing behavioral explanations from standard ones" [22]. Laboratory experiments, which had been considered unfeasible for economic disciplines and privilege of natural sciences up until the late 20th century [94], aim to parallel real-world situations in laboratory settings, while abstracting from environmental factors. This means, experiments isolate certain variables of interest from more complex real-world contexts, while simultaneously controlling for conditions of the subjects' economic and social environments [95]. Participants, commonly referred to as "subjects", work on computerized or analogue tasks, like solving math problems [96] or assembling Lego figures [97], accompanied by anonymously making decisions, which are observed by the experimenter. Economic theories or concepts that the experimenter wants to test usually underlie the decision to make in the experiment often in relation to a given task. A constituting factor of experiments consists in them being incentivized, i.e., the participants are paid for solving tasks and making decisions by the experimenter, with the concrete amount of payments depending on the respective experimental design [95]. Subjects know about the payments they are able to receive beforehand, as they receive instructions containing the experimental procedure and the payoff function in the beginning of every experiment. Herein lies the biggest advantage of experimental economic research: actual human behavior can be observed with the subjects actions having actual monetary consequences for their payoffs in the end, so they have to "put their money where their mouth is". This is not the case for other research methods used in economic or social science like surveys or scenario studies, in which participants only state how they would behave in certain situations, while no information is gained whether they would actually behave the way they stated when faced with the decision-problem in reality. The problems of intentionbehavior gaps [98] and giving socially desirable answers [99] are well-known.
As a basic principle of economic experiments, there is no deception by the experimenter. Subjects will be asked to do exactly what is stated as their task in the instructions they receive before the experiment and they will be paid exactly according to the payoff function given in the instructions as well [100]. Experiments can be used for various purposes like testing economic theories and theoretical equilibria, testing policies and environments, establishing phenomena, stylized facts and new theories or deriving political recommendations [95]. Furthermore, experimental evidence can be replicated and re-evaluated by other experimenters using the same experimental setup and instructions, which are usually published alongside the results [101].
In order to experimentally test certain theories or economic interventions, subjects are randomly assigned into one control group and (at least) one treatment group, in order to avoid (self-)selection biases, with a treatment being the intervention that is going to be tested. Differences between control group and treatment group must thereby be reduced to exactly one factor, namely the so-called treatment variable, i.e., the intervention that the experimenter wants to evaluate [100]. Herein lies another major advantage of controlled economic experiments as they allow causal inferences. Due to the experimental environment being held constant and environmental factors being controlled for all groups, they cancel out through comparative statics between groups. Therefore, outcome in a treatment group compared to a control group can only be caused by an exogenous change, i.e., by the treatment intervention itself since it represents the only factor that differs between groups [102], [22], [100].
As mentioned above, a constituent factor of economic experiments consists in the provision of decision-dependent monetary incentives. Economic experiments underlie the general assumption that any kind of utility an individual experiences can be expressed by an equivalent monetary incentive. Through these incentives, the experimenter is able to induce a certain utility function U (x) onto the subjects, in order to neutralize the subjects inherent preferences for the duration of the experiment. This "induced value theory" was originally introduced by Nobel Memorial Prize in Economic Sciences laureate Vernon Smith [103]. According to him, subjects should derive all utility from the monetary incentives provided in the experiment itself, during their participation.
While the internal validity of experiments is usually recognized to be very strong, the external validity of these experiments, i.e., the transferability of their results to the world outside the laboratory, the external validity, is commonly discussed and criticized by opponents of this methodology [44], [100]. For instance, in the context of perceptional and judgmental biases, laboratory settings are accused of changing the environment in which a human makes efficient decisions to an artificial one. However, it can be argued the other way around in that laboratory experiments show people at their best through providing all information necessary and eliminating distractions [44]. If people fall to biases in this isolated, save environment, they will likely do so outside of it as well.

C. Sidebar: Risk Preferences
Individual risk preferences represent a factor commonly influencing human behavior that should necessarily be considered in order to explain individual decision making under risk and uncertainty. Human risk preferences are generally differentiated into risk averse, risk neutral and risk seeking behavior. They are usually determined by the individual's subjective evaluation of a probabilistic payment from a lottery compared to a certainty equivalent, i.e., a safe fix payment [104], [89]. If an individual subjectively judges the utility from the expected value E of a payment X as higher than the expected utility of such payment, namely u(X), then this individual is classified as risk averse [105]: As a consequence of Jensen's inequality, a risk averse individual's personal utility function is concavely shaped. This means that the individual is willing to sacrifice the chance of higher but uncertain payments in order to receive a smaller but certain payment, the certainty equivalent, thereby paying the so-called insurance premium [105]. In an example by [106] displayed in Fig. S1, an individual with e10 at his disposal has to decide whether or not to participate in a lottery in which e5 are gained with a probability of 50% and e5 are lost with a probability of 50%. For a risk averse individual the utility of the expected value u(10) is greater than the expected utility of wealth 0.5u(15) + 0.5u (5).
The other way around, if an individual derives less utility from the expectated value of X than the Xs expected utility then this individual is considered risk seeking. Its subjective utility function is convex (see the analogous example in Fig.  S2). Risk seeking individuals are willing to give up the certainty equivalent in order to get the chance to play the lottery through which they may obtain a higher payment, i.e., they favour gambling. The payment obtainable beyond the certainty equivalent through playing the lottery is called risk premium. In case an individual is indifferent between both alternatives, the individual can be considered risk neutral, with risk and insurance premium cancelling out [105].
Standard economic theories, in the tradition of the homo oeconomicus, are usually based upon the explicit or implicit assumption of individual risk neutrality, reflecting notion of individuals as rationally optimizing ("Markovian") decision makers [107]. Contrarily, behavioral economic research usually observes risk averse decision makers, in the laboratory [108], [109], as well as in the field [110].

D. Sidebar: Mulitiple Price List Format
Multiple price lists formats offer a simple but effective tool to reveal subjects' actual risk preferences. The basic concept was designed with the goal of testing the standard economic assumption of risk neutrality as well as the behavioral assumption of constant relative risk aversion. The list consisted of a table with two columns that represented two lotteries the subjects had to choose between for ten rows featuring varying probabilities of winning certain amounts of money that remained constant for all rows [104]. As the original price list contained monetary values with two decimals and varying probabilities that had to be weighed, it can be assumed that it was difficult for subjects to compare the different options. The subsequent work [89] designed a more clear-cut version of the multiple price list (see Table S1) in which subjects had to choose between a safe payoff that increased by one Euro with each row and a fair lottery that remained constant for all rows.
Subjects are asked to indicate for each row whether they prefer the offered safe payment (Option A) or the offered fair  [89] Option A Option B 1) e0 safe e30 with a prob. of 50%, e0 with a prob. of 50% 2) e1 safe e30 with a prob. of 50%, e0 with a prob. of 50% 3) e2 safe e30 with a prob. of 50%, e0 with a prob. of 50% 4) e3 safe e30 with a prob. of 50%, e0 with a prob. of 50% ⋮ ⋮ ⋮ 19) e18 safe e30 with a prob. of 50%, e0 with a prob. of 50% 20) e19 safe e30 with a prob. of 50%, e0 with a prob. of 50% lottery (Option B). The safe payment equals the number of the respective row minus one (in Euro), starting at e0 in the first row and increasing up to e19 in the twentieth row. The lottery is the same in each row, containing a 50% chance to win e30 and a 50% chance of winning nothing. A subject's risk attitude can be determined by observing the row in which his choice changes from B to A. When continuously choosing B until row 16 and continuously choosing A afterwards, a subject is classified as risk neutral, as both options yield the exact same expected payoffs of e15 in row 16. At this point a risk neutral subject is indifferent between the receiving the safe payment and playing the lottery. If the change from B to A occurs before the sixteenth row, then a subject is classified as risk averse, as it is willing to sacrificing the chance to play a lottery that yields a higher expected payoff in favor of a safe payment of a lower amount than such expected payoff. If the change occurs after the sixteenth row, a subject is regarded as risk seeking. Contrarily to risk averse subjects, risk seeking subjects are willing to sacrifice a safe payment in order to be able to play a lottery that yields an expected payoff that is lower than safe payment but yields a higher maximum payoff. After making their choices for every row, a fraction of the participant group (e.g., every seventh [89] or fifteenth [114] subject) is chosen at random and actually paid for one randomly chosen row of the table, according to their respective choice made in this row. This procedure incentivizes subjects to choose in accordance with their true preferences in each row, since they would forfeit expected utility if they did not, thereby revealing them to the experimenter [89]. Therefore, the multiple price list format presents an incentive-compatible instrument to measure individual risk preferences.