Why not Barbell?
How bounded rationality causes the disfavour of Taleb’s strategy
Ruben Bieze
10359699
Abstract
According to Taleb (2007) people tend to systematically overestimate their ability to predict the future. He argues that in fact much is determined by unpredictable and unlikely events, coined Black Swans. Taleb (2012) proposes the Barbell strategy, which opens up to an unlimited gains potential at the cost of regular small losses. However, Kahneman (2013) argues that while this may lead to a good payoff, people are boundedly rational in such a way that they are disinclined to follow such a strategy. Loss aversion, present bias, diminishing marginal utility and most importantly the occasional disregard of small probabilities all lead to the unpopularity of the Barbell strategy. While most factors can be offset by increasing the scale of the Black Swan event, the disregard of small probabilities rules out the Barbell strategy in any context.
MSc in Econometrics Mathematical Economics track
Date: July 7, 2017
Supervisor: dr. Anita Kop´anyi-Peuker
Statement of Originality
This document is written by Ruben Bieze who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 3
2 Theory 5
2.1 Randomness and the effect of the improbable . . . 5
2.2 The Barbell Strategy . . . 8
2.3 Prospect Theory . . . 11 2.4 Present Bias . . . 14 3 The model 17 3.1 Game Structure . . . 17 3.2 Equilibria . . . 19 3.3 Generalisability of results . . . 27 4 Experiment 29 4.1 Design . . . 29 4.2 Results . . . 30 5 Discussion 33 Literature 35 Appendix 37 Instructions Experiment . . . 37
1
Introduction
People constantly make decisions about the future, based on their beliefs about the set of possible outcomes and their respective probabilities. In this forecasting effort they assess known probabilities and outcomes, and make assumptions about the uncertain ones. The quality of forecasts can be assessed only after events play out. Taleb (2007) argues that people overestimate their forecasting skills and instead much more is determined by random events than most would think. People’s inclination to fit a narrative to a course of events with hindsight makes the world look more predictable, he argues. Moreover, biases such as the survivorship bias, indicating that only lucky survivors remain in a sample, lead to a further trust in people’s predictive abilities (Taleb, 2007).
Most importantly, Taleb (2007) introduces Black Swan events, unlikely, and therefore unpredictable, but very impactful events. He argues that these events, due to their combination of unpredictability and impact, are of crucial importance. However, they fall beyond the range of any forecast, and are therefore very uncertain. Taleb (2007) holds that due to a combination of the probability distribution and the complexity of the payoffs, many economic variables are so susceptible to these Black Swan events that it is not sensible to try to predict them at all. Instead, by acknowledging the large role of randomness in the world people can protect themselves from falling prey to unpredictable calamities.
In this effort to shield oneself against unlikely adversity Taleb (2012) proposes a Barbell strategy, named after a weightlifter’s barbell as it is built up from two extremes. This strategy aims to incur regular, capped losses, in case no Black Swan event occurs and opens up to unbounded payoffs in case it does occur. The Barbell strategy is thus convex, rather than concave in its payoffs. This directly opposes a strategy based on forecasts, which maximises its payoffs under the forecasted events, while being vulnerable to extreme losses in case of an unlikely crisis. However, to Taleb’s chagrin people tend not to adopt this strategy in many cases.
According to Kahneman (2013) behavioural psychology suggests that people are not wired to appreciate such a strategy, even if the cumulative payoff is in fact larger. In this paper I evaluate whether observed boundedly rational behaviour, as explained in
prospect theory and present bias theory, contributes to people’s disinclination to adopt the Barbell strategy. In their expansion of prospect theory Kahneman and Tversky (1979) find that people are loss averse, have a decreasing marginal utility and do not weigh all outcome probabilities equally. This implies that the value function is convex in the losses and concave in the gains, as well as reference-dependent. In addition to prospect theory, present bias theory, which entails that people overvalue their preferences in the present (O’Donoghue and Rabin, 1999), leads people away from the Barbell strategy.
In this paper, I evaluate the performance of the Barbell strategy, compared to its antipode, in the context of expected rational utility as well as under the assumptions of boundedly rational preferences as identified by prospect theory and present bias theory. I find that under these assumptions the Barbell strategy is markedly less favourable than in the rational case. However, even under these assumptions, there are numerous cases where the Barbell strategy dominates. In order to model the ambiguity of the outcome magnitude as well as the probability of a Black Swan event, two unknown parameters are introduced. It turns out that the magnitude scaling factor is most impactful, given that the possibility of a Black Swan event is not disregarded altogether. Kahneman and Tversky (1979) find that people either overweigh very small probabilities, or neglect them entirely. While the overweighing is favourable for the Barbell strategy, the latter is completely dominated in case the Black Swan probability is neglected altogether. This may be the most important determining factor that leads to people’s reluctance to adopt the Barbell strategy.
In order to test some of these findings in practise I conducted a behavioural experiment in two groups of secondary school students. In the experiment, subjects play three of the games described in the theoretical section sequentially, with decreasing ambiguity measures. There is a significant preference for the Barbell strategy in the game setting with ambiguity for both payoff and probability, while this preference disappears as these parameters become known in the later games. Moreover, there is a weakly significant positive correlation between the preference for the Barbell strategy and the belief for the payoff scaling parameter. The Barbell strategy is favoured quite often in this setting, as compared to normally. A possible explanation for this is the explicit presence of the probability of a Black Swan event, which may create an availability bias. This again
points in the direction that in real life the Barbell strategy is overlooked because the small probability of a Black Swan event is ignored, unlike in the experimental setting.
The subsequent section covers the theoretical background of this topic, starting with an explanation of the Black Swan events and the Barbell strategy that Taleb (2012) proposes as an antidote against them. This is followed by an expansion of prospect theory and present bias theory. Next, in section 3, I discuss the games that I use in this paper to investigate the topic, as well as the equilibria and a generalisation of the results. Section 4 is dedicated to a behavioural experiment, where the games are used to test the results in an experimental setting. Section 5 concludes with a discussion and opportunities for further research.
2
Theory
2.1
Randomness and the effect of the improbable
When basing risky decisions on forecasts, the uncertainty involved with the forecast should be incorporated into the decision. People who consistently overestimate the accu-racy of their forecasts will make faulty decisions and therefore one’s own assessment of the quality of one’s forecasting is essential (Taleb, 2009). However, for a number of reasons, people might not actually be as good at forecasting as they think they are. Taleb (2001, 2007) argues that people tend to overestimate their predictive abilities, which causes them to be very vulnerable to rare, impactful misfortunes. He argues that in some cases it is better to base one’s decisions on the assumption that one lacks knowledge.
Taleb (2007) introduces Black Swans, events that are beyond the range of every prediction, have extreme impact yet are being explained with hindsight. He argues that precisely their unpredictability makes these events so impactful; just as people’s tendency to retrospectively generate a causal story that led to such an event causes their inflated confidence in their predictive abilities. This hindsight bias entails that people see patterns where there is only noise and fit stories to a series of random events, causing the past to look more explainable and leading to a perception of a much more predictable future. Our failure to take into account these unpredictable Black Swans makes for a substantial
error in the way we view uncertainty. Taleb (2001, 2007) argues that randomness has a much larger impact on the course of history and therefore the exclusion of outliers in one’s predictions does not only lead to mistakes, it is actually worse than making no predictions at all in many cases.
Whether or not an erroneous prediction is worse than no prediction at all depends on two main factors, the distribution of the variable in question and the complexity of the payoff (Taleb, 2009). The measure of volatility is different for different variables. Taleb (2009) distinguishes fat-tailed and thin-tailed distributions. For fat-tailed distributions the volatility is higher, and the impact of the outlier increases markedly. They do not, or only very remotely, converge to normality. The trouble is of course, that in order to evaluate probabilities of rare events, one needs a huge sample size. What is more, apart from this small sample in the tails, there is also an extra lack of left-tail observations due to the survivorship bias. This is a bias that excludes certain cases from the sample as they are no longer visible. For example if one considers a sample of firms, the ones that have gone bankrupt will have left the sample (Taleb, 2009).
This survivorship bias is very relevant in the context of negative Black Swan events, since these are by definition the catastrophic events that would lead to bankruptcy. The survivorship bias thus makes past Black Swan events less visible, which in turn causes their probability of occurring to be underestimated or even ignored. Informational variables, as opposed to physically constrained variables, tend to be fat-tailed as there are no physical limitations. Socio-economic variables are often within the category of the fat-tailed distributions, for instance the inflation number in Zimbabwe or the rise of Google were not capped by any physical constraints.
The other aspect that is very influential on the applicability of forecasting is the structure of the payoff. Taleb (2009) distinguishes simple binary payoffs versus more complex payoffs. In the case of binary payoffs it is easy to see that being wrong a few times does not make a substantial difference as long as the sample size is large enough. However, when the size of the payoffs differs, a single mistake can already mean ruin. In this higher moment cases, not only the probability of an event is unknown, but also the magnitude of the impact. It is no longer important how many times one is right, but rather what the total impact is (Taleb, 2009).
Taleb (2009) defines a two by two matrix with the payoff structure and the fatness of tails on the respective axes. It is in the fourth quadrant of complex payoffs and fat-tailed distributions that the Black Swans reside. Most economic variables, such as commodity prices, currencies, inflation numbers, GDP and company performance, fall into this fourth quadrant, and are therefore extremely vulnerable to the Black Swan events. It is with these variables specifically that forecasting becomes especially troublesome (Taleb, 2009, Bundt and Murphy, 2006).
Taleb (2009) finds a huge excess kurtosis in a study of daily values of 43 economic variables over 29-40 years. In all cases the kurtosis contribution of the largest observation substantially exceeds the expected 0.6% of a normally distributed variable with n = 10000, while in the worst cases 70-90% of the excess kurtosis comes from a single day. This implies that a single observation in forty years, which is impossible to predict, completely disqualifies the normality assumption.
This difficulty with predicting under uncertainty, or in fact the difficulty with ac-knowledging that we cannot predict, can be attributed to some psychological features that are inherent to human beings. Taleb (2007) discerns several biases and fallacies that undermine people’s ability to evaluate uncertainty. For instance, the narrative fallacy or hindsight bias entails that people retrospectively connect what was really a random course of events into a causal story, which makes events appear predictable with hindsight.
To illustrate the idea of the Black Swan, Taleb (2007) considers the example of the inductive reasoning of a turkey. The turkey in question gets fed by its owner every day and with each day the trust in the owner increases, since it has an increasing number of past events that lead it to be more and more certain of its prediction that it will get fed again the next day; however, as thanksgiving approaches, all of a sudden the turkey is killed instead of being fed. This unpredictable event, which lies completely outside the prediction models of the turkey, is so impactful that it renders the prediction utterly useless.
Taleb (2007, 2012) stresses that by recognising the limitations of our predictive abil-ities, we can try to be less like the turkey. The following section is concerned with how this is to be done.
Figure 1: The payoff structure and respective probability distributions of fragile, robust and antifragile exposures respectively (Taleb, 2012, figures 20-22, p. 442)
2.2
The Barbell Strategy
Having established that forecasting in this fourth quadrant of fat-tailed distributions and higher order payoffs is error-prone, Taleb (2012) sets out to provide a strategy for dealing with these unpredictable variables. He argues that in the cases that fall into the fourth quadrant it is better to acknowledge one’s ignorance of the probability distribution and magnitude of unexpected outliers, and instead manage one’s exposure to them. By seek-ing optionality in asymmetric payoff structures that have capped losses and unbounded potential gains, a strategy can become antifragile, which implies that it benefits from volatility (Taleb, 2012).
Taleb (2012) explicitly discerns antifraglity from robustness; while robustness implies that a system or strategy does not experience serious harm from extreme events, the antifragile experiences sizeable gains when exposed to extreme events. In figure 1 the payoff structure and respective probability distributions for these cases are shown as compared to two types of fragility, namely asymmetric and symmetric fragility to disorder. The asymmetric kind has capped gains and is therefore not able to benefit from positive shocks, while the symmetric kind is subject to volatility in both directions.
Although it is by definition impossible to build a strategy that benefits from luck in a situation of pure randomness, the success of a strategy lies with the asymmetry of the payoffs. By seeking optionality in settings with positive asymmetry, one can build a strategy that is open to uncapped gains while limiting losses in the worst case. The
Figure 2: The probability distributions of variable x and convex transformation of the exposure f(x) to this variable (Taleb, 2012, figure 27, p. 452)
strategy involves picking out the situations with positive asymmetry and then exercising some options while cutting off the rest when losses are still small. These strategies are aimed at incurring frequent, but bounded losses, while exposing oneself to infrequent, but unbounded gains; crucially, they do not require sophisticated forecasts to function well. These options, which can be both financial options and options in a broader sense, benefit from volatility as they are exposed to the potential large upside while being sheltered from the downside. The price for this is regular losses in most cases when the unpredictable does not happen (Taleb, 2012).
Taleb (2012) defines Barbell strategies as bimodal strategies that limit the negative exposure to Black Swan events, while being exposed to positive Black Swans. This can be achieved by aiming for positively skewed payoffs; the most straightforward example from financial trading is to invest a large share of one’s portfolio in the safest imaginable stock, while investing the remainder in very risky products. This is where the barbell metaphor originates, since the investment strategy looks like a weightlifter’s barbell, with two distinct parts on both extremes of the riskiness spectrum. A more formal way to describe the barbell strategy is by a convex transformation of the exposure to a variable with unknown distributional properties. The convex transformation is represented in figure 2; as can be seen from this figure the variable x has a certain distribution that is a general fat tailed distribution with no particular skewness. The exposure f (x) to x on the other hand is positively skewed. Taleb (2012) maintains that it is much more
straightforward to manage the distribution of the exposure than of the variable itself. The trick of the Barbell strategy is to manage f (x) in such a way, that knowledge of the distribution of x is not relevant. An example of this is the function of the travel time from one side of the city to another, which may depend on the amount of cars on the roads at a particular time. The distribution of the amount of cars in this case is unknown. This is the x that can neither be managed nor predicted. However, one may base their travel strategy on the knowledge that this is the case and choose to take a bicycle instead. In this way the exposure to x can be managed, and the travel time is no longer too fragile to the amount of traffic (Taleb, 2012).
Black Swan events lead to nonlinearity in the payoffs. The direction of this nonlin-earity determines whether a certain variable is fragile to volatility; concave variables, such as travel time, can only be badly affected by randomness, while being bounded on the upside. Convex variables on the other hand, have capped possible losses, and unlimited potential, and are thus antifragile to volatility. This difference follows from Jensen’s inequality, stating that the average of the function is greater than the function of the average for strictly convex functions and the other way around for strictly concave functions (Jensen, 1906).
In figure 3 the distribution of a fragile and an antifragile exposure are being compared. The big difference is the negative or positive skewness respectively as the fat tail is located on opposite sides. The extreme events are unpredictable and in a way invisible in the model. Hence the mean will be overestimated in the fragile case, while being underestimated in the antifragile case. For instance, book sales or biotech inventions are antifragile; there is a limited investment of time and money to be lost in the worst case, while at the same time opening up to the opportunity to create the next Harry Potter or malaria medicine (Taleb, 2012).
Taleb (2012) thus recommends a bimodal Barbell strategy, which embraces uncer-tainty and aims to gain from the unpredictable, while at the same time incurring limited losses regularly. Although the reasoning is sound and the expected payoff is arguably rationally superior to the common concave payoff strategies, this volatility embracing strategy does not rhyme with people’s experienced utility as observed in behavioural psychology. In practise, people tend not to adopt this barbell strategy in many cases.
Figure 3: The probability distributions of fragile and antifragile exposures to unlikely events (Taleb, 2012, figure 35, p. 462)
An explanation for this can be sought in different directions; in this paper two angles are considered, prospect theory and present bias. Prospect theory considers the way people diverge from rational decisions when choosing under risk and are thus boundedly rational. Present bias entails that people overvalue the present disproportionally over the future. Both are being considered in the following sections.
2.3
Prospect Theory
Daniel Kahneman and Amos Tversky, over the course of their lifelong careers, defined the field of prospect theory, which studies choice under uncertainty. They question the assumption from expected utility theory that people behave rationally when making de-cisions. Kahneman and Tversky (1979) instead argue that several biases lead people to behave in a boundedly rational way. They find that subjects of their numerous exper-iments have certain inconsistencies in their preferences and behaviour, indicating that people consider their true utility in a markedly different way than suggested by previous theory.
Prospect theory is a descriptive, rather than normative way to describe this boundedly rational behaviour. Tversky and Kahneman (1992) rebuke the two main counterargu-ments against prospect theory, namely that rationality is required to function well in a competitive environment and that once rationality is relaxed, behaviour does not retain any structure and becomes chaotic. They argue that survival in a competitive setting is possible, and indeed very common, without being perfectly rational. Moreover, while not
behaving rationally, people are boundedly rational in a quite orderly way. Thus, the re-laxation of rationality does not descend into inauspicious chaos (Tversky and Kahneman, 1992). Some of the particulars of prospect theory in fact point towards such boundedly rational behaviour that it appears at odds with Taleb’s (2012) Barbell strategy.
Kahneman and Tversky (1979) discern several inconsistencies of risk preferences that preclude capturing preferences in a single utility function. In an attempt to overcome this they define a value function that defines risk preferences as a joined function of the value of outcomes and the decision weights. Hence it allows for nonlinear preferences, as utility of prospects is nonlinear in the probabilities of the outcomes (Tversky and Kahneman, 1992).
Axioms by Von Neumann and Morgenstern (1947) such as substitution, transitivity and dominance tend to be violated in experiments. Kahneman and Tversky (1979) dis-tinguish some effects that seem particularly at odds with these axiom. Substitution is violated by what they define the certainty effect, the apparent result that people tend to overweight certainty, leading to risk aversion in the gains domain and risk seeking in the losses domain for large probabilities. Interestingly, the effect is exactly opposite for small probabilities, indicating a different experience of possibility, probability and certainty. In general, small probabilities tend to be overweighted or disregarded altogether (Kahneman and Tversky, 1979).
Said effects tend to exactly oppose each other in the gains and losses domains re-spectively, indicating that people are risk seeking in the losses domain for the same probabilities where they would be risk averse in the gains domain. This reflection effect disqualifies uncertainty aversion as sole determinant of the observed behaviour (Kah-neman and Tversky, 1979). The invariance axiom is violated not only by the differing decision weights, but also as a consequence of framing effects. The way a certain prospect is phrased is a strong indicator of its preferability (Kahneman and Tversky, 1984).
Furthermore, Kahneman and Tversky (1979) specify the isolation effect, which com-prises that people tend to cancel out common attributes from a decision between two prospects. This leads to inconsistencies between choices with exactly the same end states, thus opposing a final state evaluation of prospects. Moreover, they find that people sim-plify choices by rounding up probabilities and cancelling extremely unlikely effects. The
Figure 4: An example of a possible value function (Kahneman and Tversky, 1979)
latter effect helps to explain the notable cutoff between overweighted small probabilities and ignored ones.
The overweighting of small probabilities leads to risk seeking in the gains domain and risk aversion in the losses domain, explaining the prevalence of lotteries and insurance. However, as is relevant for Taleb’s (2012) Barbell strategy, this overweighting effect of small probabilities does not always hold. Latent small probabilities, such as the often un-seen Black Swan event probabilities, may fall into the ignored category, which potentially explains peoples’ disregard for them.
With prospect (x, p; y, q), the prospect of obtaining outcomes x and y with probabili-ties p and q respectively, Kahneman and Tversky (1979) define the value function in the following way:
V (x, p; y, q) = π(p)v(x) + π(q)v(y)
Where π(·) is the subjective decision weighting function of a given probability and v(·) is the subjective value function of a given outcome. Decision weights need not be the same as probabilities, and often turn out to be markedly different, particularly around the edges (Kahneman and Tversky, 1979). In addition to this incorporation of the weighting function, the value function considers changes rather than end states. In these two ways it differs most markedly from the preceding models.
An example of a value function by Kahneman and Tversky (1979) is shown in figure 4. The three main effects that can be observed from the graph are that the value function is convex in the losses and concave in the gains, has a kink at the origin and is reference-dependent. The difference between concave and convex sections indicates differences in risk aversion versus risk seeking for losses and gains for the majority of the probability spectrum. The kink at the origin indicates an asymmetry between gains and losses. A steeper function in the losses domain reflects the common finding that losses loom larger than gains (Kahneman and Tversky, 1979). People tend to be loss averse and disfavour mixed prospects of equal probabilities.
Lastly, the value function shown in figure 4 encompasses reference-dependence, which involves that people think in terms of gains and losses rather than absolute levels of wealth. The direction of the movement of one’s wealth level turns out to be more impor-tant than the actual level. This has some telling implications for the Barbell strategy, which involves many losses and few gains, be it of notably different magnitude.
While Kahneman and Tversky (1979) develop a model for risky choices, they limit their work to the area of known probabilities. In later work, Tversky and Kahneman (1992) incorporate decisions with ambiguous probabilities in their model for choice under uncertainty. They find that not all sources of uncertainty have the same effect on people’s preferences. For instance, while being generally ambiguity averse, individuals preferred bets with ambiguous odds in areas of their expertise over matching chance events with known probabilities that were outside their expertise.
2.4
Present Bias
An interesting extension to the aforementioned prospect theory is to consider uncertain decision making over time. When posed with a choice not only between prospects, but also between different moments in time to choose from, people usually assign higher value to immediate payoffs, as compared to those in the future. While most economic theory takes into account a discounting factor to discount the value of payoffs that are further away, it is generally assumed that this discounting is time-consistent. However, people tend to be time-inconsistent in their preferences in a way that they overvalue the present.
This implies that a choice between today and next week is not the same as a choice between three weeks from now and four weeks from now. O’Donoghue and Rabin (1999) term this particular time-inconsistency present bias, implicating that people relatively overweigh the earlier moment as this moment draws closer.
O’Donoghue and Rabin (1999) conduct an experiment where people choose to perform a single action at a particular time. They can choose the time to perform a certain action, which involves a particular cost and payoff structure for each moment in time. They find, unsurprisingly, that people tend to procrastinate tasks with immediate costs and preproperate tasks with immediate benefits. This means that on the whole, we overestimate our well-being at the moment and delay tasks that we do not really want to do, and instead do activities with immediate payoffs. Based on these results they develop a utility function that overweighs well-being in the present, and treats preferences as time-consistent otherwise. In extension to the more commonly used discount factor δ, they introduce a parameter β ∈ (0, 1] which assigns a relative weight to future utilities compared to the present, thus representing discounted utility over time as a two parameter function (O’Donoghue and Rabin, 1999).
Ut(ut, ut+1, ..., uT) = δtut+ β T
X
τ =t+1
δτuτ (1)
with δ, β ∈ (0, 1]. β = 1 implies that there is no present bias, whereas a β close to zero means that this bias is very large. Likewise, a small δ means that there is a lot of discounting and a large δ implies little discounting. The larger δ the more patient the person is. In this representation, time-consistent preferences exist outside the present.
O’Donoghue and Rabin (1999) distinguish two types of present-biased subjects, so-phisticated and naive. The soso-phisticated subjects are aware of their present bias, as well as their future selves’ present bias, and act accordingly. This implies that they perform both favourable tasks and unfavourable tasks earlier because they know that their fu-ture selves will also be inclined to preproperate, or procrastinate, them. Interestingly, sophisticated subjects perform all acts earlier, because they are pessimistic about their future selves, thus they compound self control problems (O’Donoghue and Rabin, 1999). Naive subjects on the other hand, are under the impression that their future selves are
time-consistent and base their actions on this. With immediate costs, it is thus better to be sophisticated, but with immediate gains, it is better to be naive. While the former seems intuitive, the latter is not clear immediately. It means that sophisticated people are aware of their present bias also in future periods, and will thus give up any ambition to save the good for later, as they know that their future selves will not do so either.
Notably, O’Donoghue and Rabin (1999) find that if costs and rewards can be arbi-trarily large, the consequences of the present bias can be arbiarbi-trarily huge welfare losses even from one-shot decisions. They focus their work on the bounded version, but in the context of Taleb (2010, 2012) this is an interesting sidestep. Present bias may in fact be part of the reason why the Barbell strategy (Taleb, 2012) is not favoured by people in many cases. The Barbell strategy after all is aimed at incurring small losses in the present with very large likelihood, in order to get large gains in an uncertain time in the future. As O’Donoghue and Rabin (1999) point out, people tend to be biased against immediate losses and in favour of immediate gains. Hence, present bias may partially explain the unfavourable attitude towards the Barbell strategy.
In conclusion of this section, Taleb (2012) proposes a bipolar strategy in dealing with decisions under uncertainty. In doing so one opens up to large potential gains, while incurring regular losses. In the long run, unexpected outliers will enter the sample and this strategy will pay off. However, psychologically it may be hard to bear. Kahneman and Tversky (1979) reason that people are loss averse and that reference-dependence of payoffs causes people to strongly dislike incurring regular losses even if it does lead to a cumulatively better payoff. What is more, their evaluation of small probabilities is such that they overestimate small probabilities within a certain range, and then suddenly disregard them. Present bias, as posed by O’Donoghue and Rabin (1999) poses another threat to the Barbell strategy. This entails that people overvalue their present well-being over its future equivalent, thus leading to a bias that is at odds with the Barbell strategy. The next section considers a set of games for which an attempt is made to isolate these effects, and thus to investigate which of these factors underlie the unpopularity of the Barbell strategy.
100wt pt −1 1 − pt B −100wt pt 1 1 − pt A
Figure 5: General one-shot structure of the game
3
The model
3.1
Game Structure
The games are binary choices between two strategies, yielding exactly opposing payoff functions, one of them convex, and the other concave. The first strategy gives regular small wins, while being vulnerable to a large loss in case of a Black Swan event, which has an unknown, small probability. Contrarily, the second strategy is structured in such a way that regular small losses are incurred, but in the case of a Black Swan event a large amount is gained.
In figure 5 the general structure of the game is given. Players choose either strategy A or B, which represent the situation of concave, versus convex payoffs. Strategy A represents Taleb’s (2007) turkey’s strategy and B represents the Barbell strategy. After choosing either strategy, Nature decides whether a Black Swan event occurs in this round with probability pt. The payoffs are given by a normalised gain or loss of 1 in the regular
case, or a reverse effect of much larger magnitude in the case of a Black Swan event. In order to play with the level of ambiguity the probability pt ∈ (0, 1) as well as the
payoff scaling factor wt∈ [1, 100] can either be kept unknown or given. If these parameters
are unknown, they can take on different values each time period. Consequently, subjects cannot update their beliefs about the parameter values in any sensible way. When known, the payoff scaling factor is kept at wt = 1 ∀t, in order to show the lower bound of the
... Figure 6: Game type α: players can choose a strategy at the start, and are unable to alter it henceforth
...
Figure 7: Game type γ: players can choose a strategy at the start, and are able to alter it after every round
results, since the Barbell strategy is more favourable for a larger wt. This captures to some
extent Taleb’s (2001) point that in the real world, neither the probability of an unlikely event, nor its actual magnitude can be known. Strategies A and B can be viewed in the notation of Kahneman and Tversky (1979) as two prospects (1, (1 − pt); −100wt, pt) and
(−1, (1 − pt); 100wt, pt).
The game in figure 5 is the one-shot version of the actual game that is being used. In fact the same game is played sequentially for twenty rounds, this number of rounds is chosen to have a finite game within the event horizon of the subjects. In figure’s 6 and 7 the sequential game is represented in two versions, figure 6 showing a game where the strategy can be chosen at the beginning only, and figure 7 representing a game where the strategy can be altered at each round. Each square represents the decision as shown in figure 5, and each circle represents a choice by Nature, or chance event, on which the players have no influence. I evaluate both versions as choices may change with the evaluation period due to present bias or other effects. While it seems more desirable to be able to change at all times, it may turn out to be better to commit to a strategy.
Thus these three binary options yield a total of eight possible games, with differing levels of ambiguity, as well as different choice moments. In table 1 the prospects are shown for each level of ambiguity, with game 1 a setting of pure risk and game 4 a setting of maximum ambiguity. In games 2 and 3, the ambiguity lies with the magnitude of the outcome in case of a Black Swan event and the probability of a Black Swan event respectively. Each of these games can be played both in a setting where players choose a strategy at the beginning only, and where they are allowed to change strategies after each round. Thus, games 5, 6, 7 and 8 are the single-choice equivalents of games 1 to 4,
Games Probability Magnitude Prospect A Prospect B
1,5 0.01 1 (1, 0.99; −100, 0.01) (−1, 0.99; 100, 0.01)
2,6 0.01 wt (1, 0.99; −100wt, 0.01) (−1, 0.99; 100wt, 0.01)
3,7 pt 1 (1, (1 − pt); −100, pt) (−1, (1 − pt); 100, pt)
4,8 pt wt (1, (1 − pt); −100wt, pt) (−1, (1 − pt); 100wt, pt)
Table 1: Prospects of one-shot game for different levels of ambiguity
as is shown in table 1.
While the payoff structure of the respective games can be found in table 1 the utility depends on subjective valuations of the probabilities and the payoffs. For the one-shot utility, Kahneman and Tversky’s (1979) value function is used as a basis. This is then extended to utility over time using O’Donoghue and Rabin’s (1999) two parameter rep-resentation. Thus the utility of a prospect over time is given by
ut(x, 1 − pt; y, pt) = πt(1 − pt)vt(x) + πt(pt)vt(y) Ut(ut, ut+1, ..., uT) = δtut+ β T X τ =t+1 δτuτ
where ut(x, 1 − pt; y, pt) is the utility of a prospect with payoff y with probability pt and
payoff x with probability 1 − pt in a given period t, with y = {−100wt∪ 100wt} and
x = {1 ∪ −1} for the respective strategies. Ut(u
t, ut+1, ..., uT) is the utility of the same
prospect over T + 1 = 20 periods. In the next subsection equilibria of the different games are being considered.
3.2
Equilibria
Using the aforementioned utility function, equilibria of the eight games are considered. The rational expected utility equilibria are calculated first to function as a benchmark, after which the assumption of rationality is being relaxed, as suggested by prospect theory (Kahneman and Tversky, 1979) and present bias theory (O’Donoghue and Rabin, 1999). Assumptions on the parameter values based on their experimental findings are made, in order to reach a numerical solution, since both utility functions depend on
Games Probability Magnitude U0 A(pe, we) UB0(pe, we) 1,5 0.01 1 −0.088 0.088 2,6 0.01 wt 8.78(0.99 − we) 8.78(−0.99 + we) 3,7 pt 1 8.78(1 − 101pe) 8.78(−1 + 101pe) 4,8 pt wt 8.78(1 − (1 + 100we)pe) 8.78(−1 + (1 + 100we)pe)
Table 2: Rational utility of the games with wt ∈ [1, 100], pt ∈ (0, 1), a linear value
function and discount factor δ = 0.9
subjective valuations of the parameters. For the games where subjects are allowed to change strategies each period, as is depicted in figure 7, only the one-shot utility needs to be considered. This is as both pt and wt can change over time and no information is
available about their distribution. Hence, subjects cannot do Bayesian updating to alter their expectations about these parameter values. On the other hand, for the games in figure 6 the discounted utility of both prospects at the moment of choice needs to be compared. As subjects do not have information about the distribution of wtand pt, they
use their subjective expectations we and pe.
For the simplest rational utility, π(pt) = pt, v(x) = x and there is no present bias.
Utility over time is discounted by factor δ. For this section δ = 0.9 is used, as typically discount factors are somewhat close to, but not equal to 1 (O’Donoghue and Rabin, 1999). The one-shot utility is equal to the expected value of the prospects. ut= ptx + (1 − pt)y
and the discounted utility over twenty periods becomes equal to U0(δ = 0.9) = 19 X τ =0 δτuτ = 1 − δ 20 1 − δ u e = 8.78ue
with ue the subjective expectation of the utility based on the subjective expectation of the parameters.
The utilities of the respective games are given in table 2, where games 1 to 4 are the games where a change of strategy is allowed at each period and games 5 to 8 are the single-choice games. By construction UA(pe, we) < UB(pe, we) holds for all eight games,
Figure 8: The value function that is used here
implying that the Barbell strategy B is preferable to the concave alternative.
The simplest case of rational preferences with a linear utility function form a good starting point to move into the boundedly rational preferences. Since there are several subjective functions of the given probability and payoff parameters assumptions need to be made to consider the equilibrium outcomes. Assumptions on the behaviour of the weighting function π(pt) and the value function v(x) are based on experimental results
from Kahneman and Tversky (1979), while assumptions on the present bias parameter β are based on empirical findings by O’Donoghue and Rabin (1999). Naturally, both functions are conjured up based on the established properties, but are otherwise entirely hypothetical. As soon as the assumption of rationality is relaxed some arbitrariness necessarily arises, since there are legion of ways to subjectively evaluate probabilities and values. The generalisability of these results is considered in the subsequent section.
As is being discussed in further detail in section 2.3, Kahneman and Tversky (1979) develop a value function v(x) based on their numerous behavioural experiments, with a few specific characteristics. In figure 4 it is visible that this function is convex in the losses domain and concave in the gains domain, indicating different risk preferences for
Figure 9: The weighting function that is used here
gains and losses. What is more, the function is steeper for losses than for gains, reflecting the higher relative importance of losses as compared to gains. Based on these findings a value function can be constructed, which may lie closer to the truth than the linear value function. I use the following function, which has said properties, and is shown in figure 8. v(x) = −2√−x for x < 0 √ x for x ≥ 0
Likewise, for the weighting function the assumption of rationality is relaxed as π(pt) 6=
pt. Kahneman and Tversky (1979) establish that the weighting function is not very
well behaved around the edges, as people either overweigh or completely ignore small probabilities. Hence the value function is somewhat S-shaped and cut off near the edges. The crux in the case of the games considered here, is for a large part whether the small probability of a Black Swan event falls within the ignored domain or in the overestimated domain. Therefore, both cases are being considered here. Firstly, I consider a weighting function that disregards the probability of a Black Swan. The following function, as depicted in figure 9, has the necessary properties and is being used here
π(pt) = 0 for pt< 0.02 1−cos(πpt) 2.2 + 0.05 for 0.02 ≤ pt≤ 0.98 1 for pt> 0.98
The present bias parameter is set equal to β = 12, following O’Donoghue and Rabin (1999). This indicates that subjects value the present twice as much as any moment in the future. Combining these assumptions the utility function becomes equal to
U0(u0, u1, ..., u19) = π(1 − p0)v(x) + π(p0)v(y) +1 2 19 X τ =1 0.9τ(π(1 − pe)v(x) + π(pe)v(y))
with pt the probability of a Black Swan event, x the payoff in the regular case and y the
payoff in case of a Black Swan event. If ptis so small that it falls before the break-off point
in the weighting function, then π(pt) = 0 and π(1 − pt) = 1, which simplifies the utility
function substantially. Now the utility at the start of the fifth game, with pt = 0.01,
wt= 1 ∀t, and no opportunity to change is now equal to
UA0(pe, we) = v(1) + 1 2 19 X τ =1 0.9τv(1) =√1 + 1 2 0.9 − 0.920 1 − 0.9 √ 1 = 1 + 5(0.9 − 0.920) = 4.89 UB0(pe, we) = v(−1) +1 2 19 X τ =1 0.9τv(−1) = −9.78
Since the Black Swan events are now entirely ignored, the concave strategy that is vulner-able to them is attributed a much larger utility than the Barbell strategy, which benefits from them. Hence the disregard of small probabilities may form an explanation of the lack of popularity of the Barbell strategy. This result is straightforward since a constant payoff of −1 is compared to a constant payoff of 1, with no uncertainty. The same re-sult holds for any weighting function that satisfies the property of disregarding the small probability of a Black Swan event. Next, the case is considered where the probability of a Black Swan event falls just outside the disregarded domain of the weighting function. As pt= 0.01 ∀t for the first game, the weighting function is now set at
π(pt) = 0 for pt < 0.005 1−cos(πpt) 2.2 + 0.05 for 0.005 ≤ pt≤ 0.995 1 for pt > 0.995
Consequently, the utility at the start of the fifth game, with pt = 0.01, wt= 1 ∀t and no
opportunity to change is now equal to
UA0(0.01, 1) =√1 1 − cos (0.99π) 2.2 + 0.05 − 2√100 1 − cos (0.01π) 2.2 + 0.05 + 1 2 19 X τ =1 0.9τ √ 1 1 − cos (0.99π) 2.2 + 0.05 − 2√100 1 − cos (0.01π) 2.2 + 0.05 = 4.89 √ 1 1 − cos (0.99π) 2.2 + 0.05 − 2√100 1 − cos (0.01π) 2.2 + 0.05 = −0.22 UB0(0.01, 1) = 4.89 −2√1 1 − cos (0.99π) 2.2 + 0.05 +√100 1 − cos (0.01π) 2.2 + 0.05 = −6.92
For the other games, outcomes evidently depend on both pe and we. Respective utilities
for some different values of these parameters are shown in table 3 as an illustration. The Barbell strategy has a lower expected utility for the simplest case of games 1 and 5. However, when either pe or we increases, performance of the Barbell strategy goes up, as
compared to the concave alternative A. This holds since ∂ ∂we(U 0 B(p e, we) − U0 A(p e, we)) = ∂ ∂we4.89 −2√1 1 − cos ((1 − p e)π) 2.2 + 0.05 +√100we 1 − cos (p eπ) 2.2 + 0.05 − 4.89 √ 1 1 − cos ((1 − p e)π) 2.2 + 0.05 − 2√100we 1 − cos (p eπ) 2.2 + 0.05 = 37.01 − 33.34 cos(πp e) √ we > 0 ∀ w e, pe | π(pe) 6= 0 ∂ ∂pe(U 0 B(p e , we) − UA0(pe, we)) = (209.49√we+ 20.94) sin(πpe) > 0 ∀ we, pe | π(pe) 6= 0
If pt is kept fixed at pt = 0.01 ∀t, both strategies have equal utility for we = 3.645.
Likewise, when wt is kept fixed at wt = 3 ∀t, both strategies have equal utility for
pe = 0.048. I choose to fix w
t = 3 as for this value there is a sensible equilibrium value
of pe to be found. For smaller w
t this equilibrium probability becomes too big and for
Game Probability Magnitude U0 A(pe, we) UB0(pe, we) 1 0.01 1 −0.22 −6.92 2 0.01 100 −44.33 15.18 6 0.01 4 −5.14 −4.47 7 0.005 1 −0.21 −6.93
Table 3: Boundedly rational utility of the games for several values of wt ∈ [1, 100] and
pt∈ (0, 1)
UB0(0.01, we) − UA0(0.01, we) > 0 if we> 3.645 UB0(pe, 3) − UA0(pe, 3) > 0 if pe > 0.048
UB0(pe, we) − UA0(pe, we) > 0 ∀ pe | π(pe) 6= 0 if we > 3.679
In the preceding equations the pivot points after which the utility of the Barbell strategy is higher than that of the alternative are being evaluated. The difference U0
B(pe, we) −
UA0(pe, we) increases in pe as well as in we, as can be derived from the strictly positive derivatives. In the first two functions one of both parameters is kept fixed, in order to find the pivot value of the other for this fixed parameter value. In the third equation neither parameter is kept fixed. It follows that the attributed probability weight does not matter, as long as the payoff scaling factor is larger than 3.679, given that the probability is not disregarded altogether. Already for we= 3.7, or any larger wethe Barbell strategy
dominates the concave alternative, regardless the attributed probability weights. This seems attainable for games 2, 4, 6 and 8 since wt ∈ [1, 100]. Since there are many occasions
where we ≥ 3.7, this emphasises the importance of the assumption that π(pe) 6= 0 for the
Barbell strategy to dominate. The cutoff value of pe for which π(pe) = 0 turns out to be
crucial.
The way the games are constructed make the existence of a present bias hard to verify, since the expected payoff is the same across time periods. As a consequence, up to here all results are interchangeable between the versions of the games with repeated decisions and games with a single decision at the start. Hence, in order to evaluate the effect of present bias, the games need to be altered slightly. Specifically, I consider game 6 with pt = 0.01 ∀t known, and wt = 4.5 ∀t here, for two reasons. Firstly, for this game the
Barbell strategies dominates the concave strategy, under the assumptions made so far. Secondly, the setting with only a single choice at the start of the game is required here, instead of the games where changes can be made every round, since otherwise each round would just be the new present.
In order to evaluate the present bias the prospects are tweaked slightly such that A = (1, 1) and B = (−1, 1) in the first round and A = (1, 0.99; −450, 0.01) and B = (−1, 0.99; 450, 0.01) afterwards. This is a reasonable change, since it makes sense that the very near future is less uncertain than the further future. In this way, prospect A is clearly preferable in the present, while prospect B dominates in the other rounds. The nontrivial weighting function with π(pt) 6= 0 for pt ∈ [0.005, 0.995] is being used. The
discount factor remains constant at δ = 0.9 and the present bias parameter β is varied. utA(0.01, 4.5) = π(0.99) ∗ v(1) + π(0.01) ∗ v(−450) = −1.172 ∀t > 0 utB(0.01, 4.5) = π(0.99) ∗ v(−1) + π(0.01) ∗ v(450) = −0.823 ∀t > 0 UA0 = 1 + β 19 X τ =1 δτuτA= 1 + βδ − δ 20 1 − δ u t A= 1 + 7.784βu t A UB0 = −1 + β 19 X τ =1 δτuτB = −1 + βδ − δ 20 1 − δ u t B = −1 + 7.784βutB Thus UB0 − U0 A= −2 + 7.784β(u t B− u t A) = −2 + 2.715β > 0 if β > 0.74
Thus for this particular prospect, or another equivalent set of payoff and probability structure there is a cutoff point for β = 0.74 between the two strategies. Hence, present-biased subjects, who might have β = 0.5 as in O’Donoghue and Rabin (1999), may be inclined to go for strategy A, while others prefer strategy B. To this extent, present bias possibly plays a role in the bias in favour of strategy A.
In conclusion of this section, there are multiple limitations of rationality that may con-tribute to the unfavourable position of the Barbell strategy. The characteristics of the weighting function, the value function and the present bias each potentially contribute to this. However, the intensity of the respective effects differs considerably. The effect of the present bias appears to be comparatively small. The concavity of the value function in the gains domain, the convexity in the losses domain and the kink at the reference point
also have an adverse effect on the favourability of the Barbell strategy, but are eliminated for large enough we.
The weighting function may be the only limitation of rationality that can cause the Barbell strategy to be dominated for each value of we, as the size of the scaling factor
does not matter once the probability of the event is equated to zero. The Barbell strategy is dominated in all cases if π(pe) = 0. Hence, the disregard of very small probabilities
(Kahneman and Tversky, 1979) appears to be the most important factor causing the unpopularity of the Barbell strategy.
3.3
Generalisability of results
The results of the previous section are necessarily based on certain assumptions in order to find explicit numerical results. While the used functions and parameters have the characteristics that result from behavioural experiments (Kahneman and Tversky, 1979, O’Donoghue and Rabin, 1999), they are not unique in this. There may be other value functions and weighting functions that are endowed with the same properties. In this section I discuss the extent to which the results are generalisable.
While the chosen value function is just a single example of a function that satisfies Kahneman and Tversky’s (1979) properties, the results hold for any value function that lies above this value function on the positive part of the domain, or below it on the negative part, while keeping the rest constant. This holds because it implies a better appreciation of large gains in the gains domain or a worse experience of large losses, compared to the current situation, both of which favour the Barbell strategy.
For other value functions, for which the opposite holds, the pivotal parameter values of wt and pt, or we and pe in the ambiguous case, for which the strategies are equivalent,
are necessarily higher, weakening the results. For certain value functions that lie below the one used here in the gains domain, or above it in the losses domain, there are no pivotal parameter values at all, and the Barbell strategy is always dominated. This is the case for functions that are not rising above a certain asymptotical value in the gains domain, or below it in the losses domain. An example of the latter is the hyperbolic
tangent function, which does have the required properties v0(x) = 2 tanh(x) for x < 0 tanh(x) for x ≥ 0
Since tanh(10000) = 1, tanh(1) = 0.76, 2 tanh(−1) = −1.52 and 2 tanh(−10000) = −2 it is trivial to see that the Barbell strategy is dominated in any version of the game in this case. Therefore it can be concluded that there are potential value functions that meet the criteria of prospect theory (Kahneman and Tversky, 1979), for which the Barbell strategy is never preferable.
The results of the previous section are completely dependent on the weighting function in so far that the cutoff value of pt, or pe, for which π(·) = 0 is of crucial importance,
as was stated previously. However, other than this, the exact structure of the weighting function does not have an important impact on the results. In general, less overweighting around the edges would lead to a slightly higher threshold value of we for which both strategies yield equivalent results, but this influence is relatively minor. This follows from the previous section, where I found that variations of pt, or pe, have a very small
influence on the outcomes. Moreover, while probability weight attribution may differ to some extent, it is necessarily bounded, unlike the outcome.
Lastly, in the evaluation of the effect of present bias the payoff structure of the games was modified slightly, such that the payoff in the first period is non-stochastic. In this case there is no chance of a Black Swan event in the present, which implies that the Barbell strategy is less favourable.
UB0(pe, we) − UA0(pe, we) = −2 + 3.892(ueB− ue A) > 0 if ueB− ue A> 2 3.892 UB0(pe, we) − UA0(pe, we) > 0 ∀ pe | π(pe) 6= 0 if we > 5.11
Now the threshold value of weis higher than in the case where there is a chance of a Black Swan event in all periods, but there is still a threshold. This implies that the Barbell strategy is slightly weaker in this context, but there are still many situations where it dominates for this payoff structure too. In conclusion, there is a range of value functions and weighting functions that satisfy the qualities, for which there are equilibrium values
of we and pe for which the strategies perform equally well, if not all value functions.
Also, for the slightly adapted version of the payoff structure, equilibria exist, although they are slightly less advantageous for the Barbell Strategy. Hence, while the assumptions in section 3.2 may seem quite arbitrary, the results are generalisable to some extent.
4
Experiment
4.1
Design
The experiment was conducted at the Gemeentelijk Gymnasium Hilversum, a secondary school in Hilversum, where experiments took place in a classroom setting. Subjects were students of the school in the fifth grade, with age sixteen or seventeen years old. Cumulatively, 32 subjects participated in 2 Mathematics classes, of the highest and second highest levels respectively. In the first class of the higher level there were 11 subjects, while there were 21 in the second class. Students received instructions on paper, as well as oral instructions before starting the experiment, both of which were in Dutch. A translated version of the instructions on paper has been included in the appendix. Students did not participate more than once.
At each session participants played three of the eight games as described in section 3.1. They were not asked to play all eight games, as time and concentration was limited. From the previous section I conclude that the main effect lies with the ambiguity of the probability and scaling parameters pt and wt and not with the present bias. For these
reasons, one group was invited to play games 8, 7 and 5 while the other group was invited to play games 8, 6 and 5. These are the games where subjects commit to a strategy for all rounds at the first period. This particular sequence is important to prevent priming of the parameter values, as the games move from less information to more information.
What is more, in order to prevent priming or updating of prior beliefs as much as possible, subjects did not receive any feedback after completing the first two games and were shown the instructions for the next game only after completing the previous one. The instructions took a few minutes and the actual experiment took just over ten minutes, leading to a total of fifteen minutes per classroom session. For each class, game scores were
8 7 6 5 Frequency 8 7 6 5 Frequency B B B 45.5% B B B 38.1% A B B 9.1% B A B 14.3% A A A 18.2% A A A 9.5% B B A 27.3% A B A 4.8% B B A 4.8% B A A 28.6%
Table 4: Relative frequency of combinations of strategies for different games, the remain-ing six combinations were not chosen at all
used for a weighted lottery, determining who won the prize, which was a tompouce. After finishing the experiment both groups were told the context and goal of the experiment, and they were shown the simulation mechanism that led to a prize winner.
4.2
Results
Having conducted the experiment, I considered whether there are any patterns to be distinguished in the chosen strategies. I deal with within-subject differences between the chosen strategies in the different games as well as between-subject differences. The frequency of each combination of strategies is shown in tabel 4, apart from the six other possible combinations that were not chosen at all.
In games 8 and 7 strategy B is chosen significantly more than strategy A, which I establish with a binomial probability test with a ratio of 0.5 as a null hypothesis. This indicates that subjects neither chose randomly, nor favoured A. In games 6 and 5 on the other hand, there is no significant difference, as is shown in table 5. However, after having played game 8 non-randomly first, it is unlikely that this is due to a random choice. In fact the rational payoffs of A and B for game 5 are not far apart, indicating that an even distribution is not completely unexpected. Using the Wilcoxon signed-rank test I establish that there is a significant within-subjects difference between games 8 and 5, subjects choose B more often in game 8, as is shown in table 6. Hence strategy B is preferred more often in the case with the most ambiguity, compared to the case with
Game Class 1 Class 2 5 0.55 0.50 6 0.48 7 0.82*** 8 0.73** 0.86*** ***1%, **5% and *10% significance, n1= 11, n2= 21
Table 5: Ratio of strategy B for both classes
the least ambiguity. This difference is significant also for the second class separately, but not for the first class. The latter insignificance may possibly be attributed to the smaller sample of the first class.
There is no significant difference between the outcomes of game 5 for the different classes, as shown in table 6. There are two potential drivers of a difference between these results. Firstly, there is a chance of between-group heterogeneity, as the first class was a group of students with exceptional maths skills. Secondly, the first group was shown game 7, while the second group was shown game 6, before playing game 5. Hence, this could have changed their priors for game 5, the unambiguous game, which was played last. In order to isolate the potential heterogeneity effect I consider the between-subjects difference by class for game 8, the fully ambiguous game where neither group has been primed by any previous games, or given parameter values. The difference is not significant here either, as can be seen in table 6. Hence, neither between-group heterogeneity, nor playing a different second game caused significant differences between the groups in either the ambiguous or the unambiguous games.
There is no significant effect of either age or gender on any of the results, which has been tested with the Wilcoxon ranksum test. For age, this was to be expected, since all subjects were either sixteen or seventeen years old. For gender on the other hand, some literature suggests that there could be an effect.
In order to verify whether there was an effect of the beliefs on the strategies played, a logit regression of game output on beliefs pe and we was performed. Game 6, with
Test p-value Test p-value
Game 8 vs game 5 0.0067*** Game 6 vs game 5 if game 6 = A 0.0833*
Game 8 vs game 5, class 1 0.3173 Game 8 vs game 5, class 2 0.0082*
Gender in game 8 0.7361 Gender in game 5 0.2721
Game 5 vs game 5 by class 0.9087 Game 8 vs game 8 by class 0.3789
***1%, **5% and *10% significance, n1= 11, n2= 21
Table 6: Within- and between-subjects tests of strategy choices
Variable Coefficient Std dev Loglikelihood Observations
Game 6 on we cons 0.9191 0.5966 -11.2596 20
we -0.0463 0.0295 -11.2596 20
Game 7 on pe cons -2.8091** 1.3735 -4.0559 11
pe 27.8628 21.1405 -4.0559 11
***1%, **5% and *10% significance
Table 7: Logit regression of games 6 and 7 on we and pe
and game 7, with ambiguity of the probability, was regressed on the belief pe, with the outcome of the games equal to 1 if subjects chose strategy A, and 0 otherwise. Beliefs we have a mean of 19.2 and a standard deviation of 31.58, and beliefs pe have a mean of 0.038 and a standard deviation of 0.041. The results show suggestive evidence of a higher likelihood of choosing strategy B if one’s belief we is higher, however due to a relatively small sample size this effect is not significant, as can be seen in table 7. There is no significant effect of belief pe on the choice of strategy in game 7 in the first class.
There is suggestive evidence of inconsistency, on a ten percent confidence level for the second class, since three out of eleven people who chose strategy A in game 6 do not do so in game 5. This was tested with a Wilcoxon signed-rank test and is shown in table 6. This indicates inconsistency since game 5 is a border case with wt = 1 ∀t, and thus
strategy A is relatively better in game 5 than in game 6, where wt ∈ [1, 100]. Hence
choosing A in the latter case, but not in the prior indicates inconsistency. Due to the small sample size this result is not significant at a lower percentage level.
5
Discussion
In making decisions about the future, some assumptions need to be made about the range of potential outcomes and their respective probabilities. Taleb (2007) argues that people overestimate their predictive abilities, and that in fact many influential events are random and completely unpredictable under uncertainty. In order to guard oneself against this randomness, and in fact to profit from it, Taleb (2012) proposes the Barbell strategy, that bets against predictability and takes advantage of unforeseen shocks, which he coins Black Swan events. In doing so, the Barbell strategy incurs regular small losses in case no Black Swan event occurs. However, while this strategy may have a positive expected payoff in many cases, some of the ways in which people are boundedly rational pose a threat to its adoption.
Kahneman and Tversky (1979) find that people are loss averse, and have a diminishing marginal utility, which are two effects that impede the Barbell strategy. Moreover, they find that people either overestimate or completely ignore small probabilities. If the prob-ability of a Black Swan event is disregarded altogether, the Barbell strategy is thwarted in any context, so this effect can be very influential on the choice of strategy. Another limitation of human rationality is the present bias (O’Donoghue and Rabin, 1999), which entails that people overvalue the present in their decisions.
Under the assumptions that I do in this paper, these limitations of rationality do make it harder for the Barbell strategy to dominate, but not impossible, with the exception of the disregard of small probabilities. The ambiguity of the scale of the payoff is more impactful than that of the probability, which endorses Taleb’s (2012) point that one does not need to know anything about the probability distribution, as long as one believes in the possibility of a large enough Black Swan event. In the experimental setting the Barbell strategy was favoured significantly in the case of ambiguity of the payoff and probability, and neither strategy was chosen significantly more often in the unambiguous case. This means that the Barbell strategy was chosen a lot more in this experiment than it is outside of it, which is most likely due to the particular experimental setting.
In my opinion the most important limitation of the experiment is the explicit presence of the probability of a Black Swan event. Taleb (2007) argues that this probability is
in fact often unobserved. Tversky and Kahneman (1975) define the availability bias, the effect that people overestimate the probability of the observable. This availability bias most likely nudges many subjects in the direction of overestimating, rather than disregarding the probability of a Black Swan event, causing their frequent choice of the Barbell strategy.
Another factor causing the frequent choice of the Barbell strategy may be found with the choice of games. Only the games with a single decision at the start were considered, as under the assumptions made here, these games have the same payoff as the games with a choice each period. However, this may not always hold. Benartzi and Thaler (1995) define myopic loss aversion, indicating that people’s risk aversion increases as the frequency of their evaluating the asset goes up due to mental accounting. Expanding on this finding Gneezy and Potters (1997) perform an experiment in order to evaluate the impact of the evaluation period on on their risk aversion. They find that indeed the more frequently the returns are evaluated the more risk averse the subjects are. Hence, the choice of the games with a single evaluation may have led to the relative popularity of the Barbell strategy.
Apart from this frequency of the choice for the Barbell strategy, many of the results were not significant. The internal validity may be jeopardised by the relatively small sample size. For instance the effect of the belief of the scaling parameter would be expected to have a positive effect on the likelihood of a choice for the Barbell strategy. While the results indeed suggest this direction, they are not significant. Furthermore, the external validity of this experiment can be questioned, as all subjects were youths with some talent for mathematics from the same area. Thus results may not be directly generalisable to people with a different culture, age or eduction level.
In the context of this paper the theoretical and experimental parts indicate that the overestimation or disregard of the probability of a Black Swan is the most influential factor causing the choice for or against the Barbell strategy. The present bias was not evaluated in the experiment. An interesting direction for future research would be to extend the experiment to different subjects, in order to establish better external validity. Moreover, substantially lengthening the duration of the experiment would likely provide interesting insights in subjects’ time consistency and myopic loss aversion. Most
interest-ing would be to try to neutralise the availability bias, but this is practically impossible in an experimental setting.
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Appendix
Instructions Experiment
version 1
Number: ... Class: ... Age: ... Gender: ...You play a game with two possible strategies, with different payoffs. By choosing a strategy you can earn points, which contributes to your chance of winning the prize. The game consists of twenty time periods, in which each time two things can happen. Either nothing happens, which is most likely, or a shock occurs, which is unlikely, but possible. The payoff depends on whether or not a shock occurs. You pick a strategy at the beginning, which remains in place for all twenty periods.
Game 1
The strategies are as follows
Strategy Payoff in case of no shock Payoff in case of a shock
A 1 −100wt
B −1 100wt
Here wt is an unknown number between 1 and 100, which can vary across periods. The
probability of a shock is pt. The exact probability is unknown, but it is given that it is
small. Likewise pt can vary across periods.
Number: ... Class: ... Age: ... Gender: ...
Now we consider the same game, but this time w = 1 is known.
Game 2
The strategies are as follows
Strategy Payoff in case of no shock Payoff in case of a shock
A 1 −100
B −1 100
The probability of a shock is pt. The exact probability is unknown, but it is given
that it is small. Likewise pt can vary across periods.
Circle your strategy of choice: A / B
