The Reference Point in an Information Context
Research Master Thesis
Mark van Oldeniel
∗University of Groningen
August 12, 2019
Abstract
While reference dependence is one of the core elements of prospect theory, it is de-bated how the reference point is determined. We designed an experiment in which we can distinguish between the predictions of models with different reference points. In our experiment, subjects participate in two monetary gain or loss lotteries and have to decide how they want to be informed about the outcome of these lotter-ies. Prospect theory with a status quo reference point predicts that people want to learn the outcomes of the gain lotteries separately while learning the outcomes of the loss lotteries clumped together. An expectations-based reference point predicts that subjects always prefer to learn the outcome of the lotteries clumped together. In this thesis, I present some preliminary findings of this experiment. This prelim-inary analysis shows no preference for either separation or integration in both the gain and the loss lotteries.
Keywords: Prospect Theory, Reference Point, Expectations, Information Prefer-ences
Acknowledgements
1
Introduction
One of the core principles of prospect theory (Kahneman and Tversky, 1979) is that individuals evaluate outcomes in terms of gains and losses compared to some reference point. How this reference point is determined, however, is debated. In the earlier days, the status quo was often being taken as the reference point. However, in more recent theoretical models, most notably in K˝oszegi and Rabin (2006, 2009), expectations were incorporated in the reference point. These models put some discipline on the formation of a reference point, and thereby address the criticism that prospect theory has an arbitrary reference point (Dhami, 2016). At the same time, these models impose a high level of cognitive sophistication on the decision-maker which also makes the models more complex. An important question is whether these more complex models are better able to predict behavior than a model with a ’simple’ status quo reference point.
This project aims to experimentally address whether a status quo reference point model or a more complex expectations based reference point model is better able to predict the preference of individuals in an information context. The information context is chosen as K˝oszegi and Rabin (2009) themselves derive the implications of their model for information preferences as an application of their model. Additionally, our experiment in the information context contributes to a line of experiments evaluating the reference point in different contexts, such as an effort provision context (Abeler et al., 2011), and an exchange context (Ericson and Fuster, 2011). Kahneman and Tversky (1979) already argued that while the reference point usually corresponds to the current asset position, it could be affected by expectations or the formulation of the offered prospect. This line of research can answer in which contexts exactly which reference point is most appropriate.
in two gain lotteries, the other half in two loss lotteries. If the reference point is the status quo, individuals engaging in mental accounting (Thaler, 2008) may want to separate learning the outcome of both gain lotteries, while integrating learning the outcomes of both loss lotteries. If the reference point is influenced by expectations, subjects may form beliefs about their earnings. Choosing to learn about the outcome of the lotteries separately will then expose subjects to more fluctuations in these beliefs than choosing to learn the outcomes combined together. K˝oszegi and Rabin (2009) predicts that individuals dislike these fluctuations and hence that subjects should prefer to learn the outcomes of the lotteries clumped together for both the gain and loss lotteries.
Hence, these two models make different predictions about the Gain treatment and make the same predictions about the Loss treatment. The Loss treatment was included to distinguish between these models and other models about information preferences, particularly from the model by Ely et al. (2015). In that model, agents like feelings of surprise and suspense, which leads to the prediction that subjects should prefer to separate learning the outcomes of the two lotteries in both the Gain and the Loss treatment.
In our experiment, the choice between separation and integration also results in differ-ences in whether the information about the outcome is provided sooner or later. Therefore, we added Sooner-Later treatments where subjects have to choose between learning the integrated outcomes of the two lotteries sooner or later. The choices between separation and integration are compared with the choices in these treatments to isolate preferences for separation and integration.
classified as a loss, but the context of an electric shock is also different than the context of a monetary lottery. In our experiment, we include a separate Gain and Loss treatment, both in the context of monetary lotteries. This allows us to distinguish between the pre-dictions of Ely et al. (2015), K˝oszegi and Rabin (2009), and of prospect theory with a status quo reference point. Our design allows us to test whether the distinction between gain and loss lotteries is irrelevant, as is implied by K˝oszegi and Rabin (2009) and Ely et al. (2015), or whether it matters to subjects.
In this thesis, I will present some of the initial results of this experiment. The data collection is not finished yet, and, hence, the results presented are preliminary. This preliminary data does not show a significant difference between the choices of individuals in the Gain and Loss treatments.
The remainder of this thesis is structured as follows: in section 2, I review the lit-erature, the experimental design is discussed in section 3. The predictions of different models are presented in section 4. In section 5, the data is discussed and in section 6 the preliminary analysis of this data is presented, before concluding in section 7.
2
Literature Review
Reference dependence is a key element in prospect theory (Kahneman and Tversky, 1979, 1992), and entails that utility is not derived from final wealth levels, but from changes (gains and losses) in wealth levels compared to some reference point. While the reference point is an essential element, Kahneman and Tversky (1979) do not specify how this ref-erence point is determined exactly (Barberis, 2013). Initially, the status quo was typically being taken as the reference point (for example in Kahneman et al., 1990).
preferences. In their model, the reference point of an individual is their rational expec-tation about outcomes. K˝oszegi and Rabin (2009) builds on this model and provides a model of reference-dependent consumption plans. Individuals form beliefs about present and future consumption and are loss-averse over belief changes.
Experimental attempts to evaluate the role of expectations show a mixed picture. Abeler et al. (2011), for example, shows in the domain of effort provision that the ex-pectation of income serves as a reference point and influences effort provision. In their experiment, subjects have to perform an effort task. Subjects either receive a piece rate per completed task, or they receive a fixed fee, both with a 50 percent probability. Sub-jects complete more tasks when the fixed fee is increased, even though the piece rate remains the same. This evidence is consistent with the fixed fee influencing the expected earnings of subjects. These expected earnings serve as a benchmark, and subjects may be reluctant to end up below this amount. Hence, they increase their effort level if the fixed fee is increased.
Ericson and Fuster (2011) show in an exchange experiment that the probability of being allowed to trade an endowment affects the willingness to trade conditional on being allowed to trade. In this experiment, subjects get an item and the probability with which subjects are allowed to trade this item is varied. The higher the probability that subjects are allowed to trade, the higher their willingness to trade if they are allowed to trade. This evidence is in line with the probability to be allowed to trade affecting subjects’ expecta-tion about keeping their endowment, which subsequently influences subjects’ willingness to trade.
reference points.
2.1
Information Context
K˝oszegi and Rabin (2009) derive implications of their model about the information pref-erences of individuals. Agents derive utility from consumption and from changes in beliefs about current and future consumption. Information affects beliefs about consumption. Individuals are loss averse in belief fluctuations, which implies that information in ex-pectation hurts the individual. This causes individuals to prefer receiving information clumped together over receiving bits of information spread over time (piecewise infor-mation). Individuals dislike piecewise information as this exposes them to more belief fluctuations compared to receiving the information clumped together. Additionally, the model predicts that individuals will prefer to receive information sooner rather than later. This is caused by assuming that the impact of belief changes about near consumption is larger than the impact of belief changes about consumption in a more distant future. If the impact of belief changes is decreasing in time, the sooner individuals receive the information, the lower the negative impact of belief changes. In the next section, these predictions will be derived more formally.
individu-als to prefer piecewise information about potential gains and clumped information about potential losses. This prediction will also be derived more formally in the next section.
The empirical evidence regarding the preferences towards piecewise information is mixed. Kocher et al. (2014), for example, found that a great majority of their subjects preferred the separation of two lotteries. Zimmermann (2014) find no aversion nor pref-erences towards piecewise information about the outcome of a lottery in which subjects could win or lose money. In contrast, Falk and Zimmermann (2016) do find that sub-jects are averse to piecewise information about the outcome of a lottery that determined whether subjects would receive an electric shock.
In Kocher et al. (2014) subjects had to buy two tickets for a lottery drawing from an initial endowment. They could buy tickets either for today’s or tomorrow’s drawing. Around 70% of the subjects preferred to buy one ticket for each of the two drawings over buying two tickets for a single drawing.
The papers by Zimmermann (2014) and Falk and Zimmermann (2016) are most closely related to our experiment. In the main treatments of Zimmermann (2014), subjects got a starting endowment of 30 euros. They had to choose how they wanted to be informed about the outcome of a lottery. A die was thrown three times, and subjects won 50 euro if the sum of the three throws was larger than or equal to 13, and lost 15 euro if the sum was smaller than 13. Subjects could choose between clumped and piecewise resolution of the lottery. A choice for clumped resolution implied that subjects were informed about the outcome of the three dice throws on the day after the experimental session. A choice for piecewise implied that subjects were informed about the outcome of one dice throw on each of the three days following the experimental session. A fraction of 52% of the subjects chose for the clumped resolution of the lottery. Zimmermann (2014), hence, does not find an aversion to piecewise information as predicted by K˝oszegi and Rabin (2009).
In their experiment, subjects received information about the outcome of a lottery that determined whether they would receive a series of electric shocks. Subjects had to pick 5 out of 10 sealed envelopes. The sealed envelopes either contained a red or a blue card. The number of red cards in the 5 selected envelopes determined whether subjects received the electric shocks. Subjects had to choose how they wanted to be informed about the outcome of this lottery.
In the Clumped-Piecewise treatments of Falk and Zimmermann (2016), subjects could choose between clumped information, which implied that all envelopes were opened im-mediately, and piecewise information, which implied that the first envelope was opened immediately and that one more envelop was opened every 3 minutes. The series of electric shocks started after 15 minutes. A choice for piecewise information hence also implied a delay of information.
To separate preferences toward piecewise information from preferences for the tim-ing of information, the choices in the Clumped-Piecewise treatments were compared to choices in the Sooner-Later treatments. In the Sooner-Later treatments, subjects could choose between opening all envelopes immediately, similar to a choice for clumped reso-lution in the Clumped-Piecewise treatments, or opening all envelopes after 12 minutes. Within the Sooner-Later treatments, 76% of the subjects preferred sooner over later infor-mation. In the Clumped-Piecewise treatments, about 90% of subjects preferred clumped over piecewise information. The fraction of subjects preferring clumped resolution in the Clumped-Piecewise treatments is significantly higher than the fraction preferring sooner information in the Sooner-Later treatments.
The results of Falk and Zimmermann (2016), however, are also in line with the reference point being the status quo (not receiving an electric shock) and individuals wanting to integrate information about a potential loss (electric shock). The results by Zimmermann (2014) cannot be explained by K˝oszegi and Rabin (2009) since that model predicts a general aversion to piecewise information. Since subjects could either win or lose money in Zimmermann (2014), the predictions of prospect theory with a status quo reference point are a bit less clear. The difference in findings between Kocher et al. (2014) and Falk and Zimmermann (2016) could result from the electric shock being perceived as a potential loss and the monetary lotteries being perceived as a potential gain. However, the difference could also result from information preferences in the context of electric shocks being different from preferences in the context of monetary lotteries.
Our experimental design allows us to clearly distinguish between the predictions of prospect theory with a status quo reference point and the model by K˝oszegi and Rabin (2009). Additionally, it allows us to distinguish between the predictions of these two models and another model of information preferences by Ely et al. (2015). Ely et al. (2015) provides a model of suspense and surprise. Individuals’ utility is increasing in suspense and surprise. A period is suspenseful if the variance of the next period’s belief is higher. Receiving information piece by piece is perceived as being exciting or entertaining in this model. This causes agents to prefer piecewise information over clumped information.
3
Experimental Design
potential outcomes, A and B. The two lotteries are carried out independently of each other. Outcome A occurs with a 90% probability and outcome B with a 10% probability. Subjects have to choose how they want to get informed about the outcomes of the two lotteries and can choose between option X and option Y .
In all treatments, option X entails that a subject will learn the outcomes of the two lotteries combined together on the day after the experimental session. Option Y is varied in the treatment variations. In the Clumped-Piecewise (CP) treatment variations, sub-jects have to choose between learning the outcome of the two lotteries combined together on the day after the experimental session (X = Clumped1) or learning the outcomes of the two lotteries separately, such that the outcome of the first lottery is reported to them on the first day after the experimental session and the outcome of the second lottery on the second day after the experimental session (Y = Piecewise).
Within the CP treatments, a choice for Piecewise not only implies that subjects sepa-rate how they learn the outcome of the two lotteries, but also that they delay learning the outcome of the second lottery. To separate preferences for the timing of information from preferences for integration or segregation, we included Sooner-Later treatment variations, as was done by Falk and Zimmermann (2016). In the Sooner-Later (SL) treatment vari-ations, subjects face the choice between learning the sum of the two lottery outcomes on the first day after the experimental session (X = Clumped1) or learning the sum of the lottery outcomes on the second day after the experimental session (Y = Clumped2).
these Loss treatments first get a starting balance of 52 euro. In each loss lottery, subjects have a 90% chance to lose 22 euro (A = −22) and a 10% chance to lose 4 euro (B = −4). Subjects participating in two loss-lotteries thus have an 81% chance to lose 44 euro, an 18% chance to lose 26 euro, and an 1% chance to lose 8 euro. These losses are deducted from the starting balance of subjects. If we include the starting balance, final payoffs and probabilities are identical in the Gain and Loss treatments. The expected earnings for subjects are 11.60 euro.
The experimental design is presented in Table 1.
Gain Loss Clumped-PiecewiseA: 4 B: 22 A: -22 B: -4
Y: Piecewise Y: Piecewise Sooner-Later A: 4 B: 22 A: -22 B: -4
Y: Clumped2 Y: Clumped2
Table 1. Experimental Design
The experiment consists of an experimental session of about 30 minutes. During this session, subjects are informed about the lotteries and have to choose how they want to be informed about the outcomes of the lotteries. Additionally, they are informed about a small online verification task that they have to perform on an online platform on both the first and the second day following the experimental session. Subjects are informed that they will only get paid if they successfully complete the experimental session and the two small online verification tasks. The verification task entails that subjects have to verify that they have read the information presented to them on that day on the online platform. This online platform could be accessed from home or from anywhere where subjects had internet access.
Subject’s choice Information on day 1 Information on day 2 Clumped1 Sum of gains (losses) Unrelated information Clumped2 Unrelated information Sum of gains (losses) Piecewise Gain (loss) from Lottery 1 Gain (loss) from Lottery 2
Table 2. Information presented on the online platform
unrelated information entails that we post a random letter for subjects. If a subject chooses to receive the outcomes of the two lotteries piecewise, the first task on the first day after the experimental session is to verify that they have read their gain (loss) in the first lottery. On the second day, this subject has to verify that they read their gain (loss) in the second lottery. If a subject chooses to receive the outcomes clumped together on the first day (Clumped1), the first task entails verifying that they have read how much they gained (lost) in the two lotteries together. On the second day, the subject has to verify that they have read the random letter that we posted. Table 2 gives an overview of the information that is presented to the subjects on the online platform.
Our set-up ensures that subjects have to observe some information and complete a task on both the first and second day after the experimental session, regardless of their choice. Hence, the choice of subjects does not influence their workload or the moment at which they have to perform a task. Additionally, the moment of payment was not influenced by the choices of the participant. On the third day after the experimental session, the experimenter entered the earnings for all participants who successfully completed the two online tasks in the university’s automated payment system. Payments went through this payment system in approximately 2 working days.
Day 0 Day 1 Day 2 Day 3 GrEELab At home At home
Experimental session Task 1 Task 2 Payment Instruction Decision: how informed
about outcome lotteries? Report information Report information
Table 3. Timeline Experiment
4
Predicitions
In this section, I will derive the predictions of different models about our experiment. First, I will discuss what is predicted if preferences for information are purely driven by time preferences. Second, I will discuss the implications of prospect theory with a status quo reference point. Then, I will discuss the implications of the model of reference-dependent consumption plans (K˝oszegi and Rabin, 2009). Lastly, I will discuss the impli-cations of another model about information preferences by Ely et al. (2015).
4.1
Time Preferences
In the CP treatments, choosing for Piecewise not only implies the separation of learning the lottery outcomes, but also the delay of learning the outcome of the second lottery. Subjects may have preferences over the timing of learning the outcomes of the lottery.
preference is due to anticipated thrill. Not resolving the lottery keeps the dream of winning alive. Loewenstein (1987) showed that individuals in certain situations may want to delay pleasurable consumption events to savor positive anticipation, while moving forward averse events, such as receiving an electric shock, to reduce the dread associated with contemplating the future.
Hence, there are multiple reasons why individuals could have preferences regarding the timing of information. Within the SL treatments, the choice between Clumped1 and Clumped2 only reflects a choice about the timing of the information. If people’s choices are purely driven by a preference over the timing of information, choices in the CP and SL treatments should be similar. Hence, there should be a similar amount of participants choosing Clumped1 in the SL and the CP treatments.
Prediction 1. If choices are purely driven by time preferences, the share of subjects choosing Clumped1 should be equal in the Gain-SL and the Gain-CP treatments. Also, the share of subjects choosing Clumped1 should be equal in the Loss-SL and the Loss-CP treatments.
In this prediction, I do not make any assumption about the direction of the time preferences, or how they may differ in the Gain and Loss treatments. I do not make any predictions or assumptions about how the share of subjects choosing Clumped1 in the Gain-CP (SL) and Loss-CP (SL) treatment may differ. This prediction only states that the fraction of subjects choosing Clumped1 should be similar in the Gain (Loss) CP and the Gain (Loss) SL treatments if only time preferences play a role.
4.2
Status Quo Reference Point
not have preferences over the timing of information. I will come back to this assumption later.
In the CP treatments, subjects in our experiment get a choice between learning the outcome of the two lotteries combined together, or separated over two days. If participants separate learning the outcomes of the lotteries, they can experience a gain (loss) on two different occasions. Hence, subjects have a choice between experiencing one integrated gain (loss) or two separate gains (losses).
In prospect theory, a decision-maker aims to maximize a value function. If a subject in our experiment chooses for separation of the two lotteries, the corresponding value is equal to twice the value of a single lottery. The value function associated with two separate lotteries is given by:
V (L) = 2 ∗ (πAv(A) + πBv(B)) (1)
This equation states that the value (V ) of lottery L is determined by the utility level corresponding to lottery outcome A, v(A), weighted by a decision weight (πA) plus the
utility level corresponding to lottery outcome B, also weighted by a decision weight. The value associated with the separated two lotteries is equal to twice the value associated with the single lottery.
If a subject chooses to learn the outcome of the two lotteries clumped together, there are three potential outcomes: winning A in both lotteries, winning B in both lotteries, and winning A in one lottery and B in the other.1 The value associated with integration
1In principle winning A in the first lottery and B in the second is a different outcome than winning B in
is given by:
V (L) = πAAv(A + A) + πBBv(B + B) + πABv(A + B) (2)
Which states that the value associated with the integrated two lotteries is equal to the utility levels corresponding to the three outcome possibilities, weighted by a corresponding decision weight.
We follow Kahneman and Tversky (1992) in assuming the following utility function:
v(x) = xα if x ≥ 0 −λ(−x)βif x < 0 (3)
Where x stands for the (monetary) outcome relative to the reference point (which is normalized to 0), and λ represents the coefficient of loss aversion. The curvature of the utility function in the gain and loss domain is represented by α and β respectively. With a status quo reference point, the revelation of the lotteries will result in experienced gains for subjects in the ’Gain’ treatments and experienced losses for subjects in the ’Loss’ treatments.
In our Gain treatments, the value associated with separation is:
V (L) = 2 ∗ (πA∗ 4α+ πB∗ 22α) (4)
The value associated with integration in the Gain treatments is:
V (L) = πAA∗ 8α+ πBB∗ 44α+ πAB ∗ 26α (5)
0.1, πAA= 0.81, πBB = 0.01, πAB = 0.18). In Appendix A, I will show that the predictions
derived here still hold for reasonable parameter values if probability weighting is added.
Concavity of the value function in the gain domain implies that 0 < α < 1. This concavity causes individuals to prefer separation over integration in the Gain treatments:
Proof. V (Integration) < V (Separation)
V (integration) = 0.81 ∗ 8α+ 0.01 ∗ 44α+ 0.18 ∗ 26α
V (integration) < 0.81 ∗ 8α+ 0.01 ∗ 44α+ 0.18 ∗ 22α+ 0.18 ∗ 4α
0.81 ∗ 8α+ 0.01 ∗ 44α+ 0.18 ∗ 22α+ 0.18 ∗ 4α = 0.81 ∗ 2α4α+ 0.18 ∗ 4α+ 0.01 ∗ 2α22α+ 0.18 ∗ 22α 0.81 ∗ 2α4α+ 0.18 ∗ 4α+ 0.01 ∗ 2α22α+ 0.18 ∗ 22α< 1.80 ∗ 4α+ 0.20 ∗ 22α = V (separation) V (integration) < V (separation)
In the Loss treatments, the value associated with separation is:
V (L) = 2 ∗ −λ(πA∗ 22β+ πB∗ 4β) (6)
The value associated with integration in the Loss treatments is:
V (L) = −λ(πAA∗ 44β + πBB∗ 8β + πAB ∗ 26β) (7)
Convexity of the value function in the loss domain implies that 0 < β < 1. This convexity causes individuals in the loss-domain to prefer integration over separation. The proof is omitted, as it is very similar to the proof in the gain-domain.
treatments, subjects only choose between sooner or later information and not between integration and separation. Since prospect theory makes no direct predictions regarding these preferences, all potential outcomes in the SL treatments could be consistent with individuals behaving in line with prospect theory. In the CP treatments, individuals not only face a choice between integration and separation but also about the timing of the information.
The SL treatments were included to prevent ourselves from wrongly concluding that a preference for separation or integration is driving our results. To illustrate the impor-tance of these treatments, consider the following example: Suppose that no one would have preferences in line with prospect theory. However, suppose that a large part of the population would like to delay positive information to savor the anticipatory thrill, as in Kocher et al. (2014). Similarly, suppose that a large part of the population would like to move forward negative news to prevent the dread of not knowing how much they will lose. These preferences would result in a large fraction of subjects choosing separation in the Gain-CP treatment, while a large fraction would choose integration in the Loss-CP treatment. Without the SL treatments, we would wrongly conclude in this example that subjects want to separate their gains and integrate their losses.
We isolate preferences for integration and separation from pure time preferences by comparing choices in the CP treatments to the choices in the SL treatments. If a con-siderable part of the population has preferences in line with prospect theory, the share of subjects choosing Clumped1 in the Gain-CP treatment should be lower than the share in the Gain-SL treatment. Similarly, the share of subjects choosing Clumped1 should be higher in the Loss-CP than in the Loss-SL treatment.
4.3
Reference-Dependent Consumption Plans
In the model by K˝oszegi and Rabin (2009), agents derive instantaneous utility in every period t from both their consumption in period t and from the changes in beliefs about current and future consumption. Equation 1 on page 912 of K˝oszegi and Rabin (2009) states that instantaneous utility in period t is:
ut= m(ct) + T
X
τ =t
γt,τN (Ft,τ|Ft−1,τ) (8)
where m(ct) reflects the consumption utility, N (Ft,τ|Ft−1,τ) reflects the gain-loss utility
resulting from the change in belief about period τ consumption. The γt,τ parameter
indicates the weight attached to this gain-loss utility. Additionally, they normalize γt,t= 1
In Appendix A of Zimmermann (2014) and Appendix B of Falk and Zimmermann (2016), the predictions of the model by K˝oszegi and Rabin (2009) for their experiments are formally derived. The predictions for our experiment are derived by applying the approach of Falk and Zimmermann (2016) to our specific case.
In our experiment, subjects choose how to receive information about lottery outcomes. Hence at period t = 0, they choose what information to receive at t = 1 and t = 2. Payments will be made to the subjects at t = 3. The payment amount and the date of payment is independent of the choices made by the subject at t = 0.2
At t = 3, agents receive money, but their beliefs do not change, as agents already know the outcomes of the two lotteries after t = 2. Hence, at t = 3 instantaneous utility only consists of the consumption utility associated with getting the money. This consumption utility will for simplicity be assumed to be equal to: m(c3) = c3, where c3 is the amount
2I assume here that agents both receive and immediately spend all their money at t = 3, which is not
of money received at t = 3.
At t = 1 and t = 2, the agent receives information about the outcomes of the lotteries, but the agent does not receive any money. Hence, instantaneous utility in both periods is only determined by changes in beliefs about period 3 consumption.
I follow Falk and Zimmermann (2016) in assuming that belief changes about consump-tion in the near future have a larger impact than belief changes about consumpconsump-tion in the far future. Hence, γt,τ is increasing in t: γ0,3 < γ1,3 < γ2,3. K˝oszegi and Rabin (2009)
also state that they find this the most plausible assumption about γ.
I follow Falk and Zimmermann (2016) in assuming that the gain-loss utility is given by:
N (Ft,3|Ft−1,3) = µ(πt,3− πt−1,3) (9)
where πt,τ reflects the period t belief about consumption in period τ . This µ() function
is a gain-loss utility function and assumed to be linear: µ(x) = ηx if x > 0, and µ(x) = ηλx if x < 0.
When making their choice at t = 0, the agent aims to maximize the (expected) sum of instantaneous utilities: U0 = E( 3 X t=0 ut) (10)
Agents choosing to learn the outcomes of the two lotteries at t = 1 (Clumped1) will only experience belief changes at t = 1, while an agent choosing for Piecewise will experience belief changes at t = 1 and t = 2. In the SL treatments, a choice for Clumped2 will only expose an agent to belief fluctuations at t = 2.
lotteries. Expected utility at t=0 of choosing for Clumped1 is given by:
E0[γ1,3µ(π2− π0) + c3] (11)
Before the resolution of any lottery, the agent believes that they have an 81% chance to receive 8 euro at t = 3, an 18% chance to receive 26 euro and an 1% chance to receive 44 euro. This belief is the same in the Gain and Loss treatments. Hence, π0 =
0.81 ∗ 8 + 0.18 ∗ 26 + 0.01 ∗ 44. An agent choosing for Clumped1 learns at t = 1 whether they will receive 8, 26 or 44 euro at t = 3. If an agent learns that they will receive 8 euro, they will experience a loss compared to receiving 26 euro and a loss compared to receiving 44 euro. This experienced loss takes into account that the chance of receiving 26 or 44 euro was relatively low. His experiences loss is equal to 0.18 ∗ (26 − 8) + 0.01 ∗ (44 − 8). If an agent learns that they will receive 26 euro, they will experience a loss compared to receiving 44 euro, but a gain compared to receiving 8 euro. This experienced loss is equal to 0.01 ∗ (44 − 26) and this gain is equal to 0.81 ∗ (26 − 8). Similarly, an agent learning that they will receive 44 euro will experience a gain of 0.81 ∗ (44 − 8) + 0.18 ∗ (44 − 26). Before the resolution of the lotteries, the agent knows the potential outcomes, the associated experienced gains and losses, and the associated probabilities with which they occur.
Hence, the expected utility of choosing Clumped1 at t = 0 is:
Since agents are loss averse (λ > 1) in belief changes, receiving information is utility decreasing in expectation.
In the SL treatments, choosing for Clumped2 will expose the individual to exactly the same belief fluctuations, only a day later. Hence the expected utility of choosing for Clumped2 is U0
Clumped2 = γ2,3(−2.9808η(λ − 1)) + c3. Since the impact of belief changes
is lower if the consumption date is further away, γ1,3 < γ2,3. Hence, the expected utility
of Clumped1 is higher than the expected utility of choosing Clumped2:
Prediction 3. In the Sooner-Later treatments, Clumped1 should be preferred over Clumped2.
In the CP treatments, choosing for Piecewise will expose the individual to belief fluc-tuations at two different moments.
Expected utility from choosing Piecewise is given by:
E0[γ1,3µ(π1− π0) + γ2,3µ(π2− π1) + c3] (12)
In what follows, I will work with the Gain lottery for expositional simplicity, but the results are exactly the same for the Loss lottery. At t = 1, the individual learns whether they won 4 or 22 euro in the first lottery. If the individual wins 4 euro, the individual’s belief about consumption at t = 3 will change to 0.9 ∗ 8 + 0.1 ∗ 26. Compared to their initial belief this entails a loss of 0.09 ∗ (26 − 8) + 0.01 ∗ (44 − 26). Similarly, if the agent wins 22 euro, they will experience a gain of 0.81 ∗ (26 − 8) + 0.09 ∗ (44 − 26). The expected utility of learning the outcome of the first lottery is equal to γ1,3(−1.62η(λ − 1)). Similar
calculations show that the expected utility of learning the outcome of the second lottery is equal to γ2,3(−1.62η(λ − 1)). Hence the expected utility of choosing for piecewise is:
Since γ1,3 < γ2,3:
UP iecewise0 < −3.24γ1,3η(λ − 1) + c3
Comparing U0
P iecewiseand UClumped10 it is easy to see that the expected utility of choosing
Piecewise is lower than the expected utility of choosing Clumped1. Both an aversion to piecewise information and a preference for sooner information drive this prediction. Compared to the SL treatments, there is thus an extra reason to expect subjects to choose Clumped1 in the CP treatments.
Prediction 4. In the CP treatments, Clumped1 is preferred over Piecewise. The share of subjects choosing Clumped1 in the Gain-CP treatment is larger than the fraction choosing Clumped1 in the Gain-SL treatment. The share of subjects choosing Clumped1 is higher in the Loss-CP than in the Loss-SL treatment.
4.4
Alternative Model of Information Preferences
Looking at the predictions of prospect theory versus the predictions of K˝oszegi and Rabin (2009), prospect theory with the status quo reference point predicts separation of the gain lotteries, where K˝oszegi and Rabin (2009) predicts integration. Both models predict integration of the loss lotteries. However, it is important to include the Loss treatments to separate between these models and other models of information preferences, and specif-ically a model of Ely et al. (2015). This model predicts a preference for Piecewise in both the Gain and Loss treatment. In the Gain treatments, this model makes similar predictions as prospect theory with a status quo reference point.
lotteries is the same. I will derive the prediction resulting from this model in a similar way as in Appendix B of Falk and Zimmermann (2016).
The utility of suspense in a given period is given by:
Ususp = u(Et(˜πt+1− πt)2) (14)
where ˜πt+1 captures potential beliefs in the next period as compared to the current
belief (πt). Furthermore, u(·) is an increasing strictly concave function. Et(˜πt+1− πt)2 is
the variance in next period’s belief given the information of today. Hence Et(˜πt+1− πt)2 =
σt2.
Within the SL treatments, the only difference between resolution at day 1 (Sooner) and resolution at day 2 (Later) is the day at which the lottery is resolved, and consequently σ2
S = σ2L. In the model, the moment at which utility is generated is irrelevant. Hence,
subjects should be indifferent between sooner and later resolution of the lottery.
Within the CP treatments, individuals have a choice between receiving information that changes beliefs on one occasion (Clumped1) or two occasions (Piecewise), and hence between one and two moments of suspense.
Choosing to reveal the two lotteries on day 1 (Clumped), will result in belief changes only at t=1. The expected utility of choosing Clumped1 at the moment of choice (t = 0) is given by:
E0[U (σC2)] (15)
where σ2
C represents the variance in day 1 beliefs resulting from choosing Clumped1.
day 1 and day 2. Hence, the expected utility of suspense at the moment of choice is given by:
E0[U (σ12) + U (σ22)] (16)
The outcome of the lotteries is not influenced by the choice of the individual. There-fore, the sum of variances in the case of piecewise resolution is equal to the variance in the case of clumped resolution:
E0[σ12+ σ 2
2] = E0[σC2]
Since u(·) is strictly concave, E0[U (σ12 + σ22)] < E0[U (σ12) + U (σ22)]. Therefore, the
expected utility of choosing Piecewise is higher than the expected utility of choosing Clumped1.
Prediction 5. In the CP treatments, piecewise resolution of the lotteries should be pre-ferred over clumped resolution. The share of subjects choosing Clumped1 should be lower in the Gain-CP than in the Gain-SL treatment. The share of subjects choosing Clumped1 should be lower in the Loss-CP than in the Loss-SL treatment.
In this model, suspense about belief changes is utility increasing, and hence, receiving information is utility increasing in expectation. This is in stark contrast with K˝oszegi and Rabin (2009), where individuals are loss averse in belief changes, and receiving information is utility decreasing in expectation.
4.5
Overview Predictions
predictions if only time preferences play a role, the predictions of Kahneman and Tversky (1979) with the status quo reference point (KT), the predictions of the model by K˝oszegi and Rabin (2009) (KR), and the predictions of Ely et al. (2015) (EFK).
Prediction about share choosing Clumped1 Model GainCP vs. GainSL LossCP vs. LossSL CP SL Time Preferences GainCP =GainSL LossCP =LossSL - -KT GainCP <GainSL LossCP >LossSL - -KR GainCP >GainSL LossCP >LossSL CP>0.5 SL>0.5 EFK GainCP <GainSL LossCP <LossSL CP<0.5 SL=0.5
Table 4. Predictions about the share of subjects choosing Clumped1
5
Data
In April and May 2019, a total of 24 sessions were conducted. The sessions were run in GrEELab (Groningen Experimental Economics Laboratory). A total of 272 subjects participated in these sessions, of which the majority were male. All participants were students at the University of Groningen, and almost all participants were undergraduate students. These undergraduate students came mainly from faculties on campus (Faculty of Science and Engineering, the Faculty of Economics and Business, and the Faculty of Spatial Sciences). The average age of participants was about 20 years old.
As mentioned before, data collection is not finished yet. Since my Research Master thesis needs some empirics, I will present some preliminary findings in the next section. These results should not be used to draw definitive conclusions.
6
Analysis
0 .2 .4 .6 mean of Clumped1 Gain Loss CP SL CP SL
Figure 1. Percentage of individuals choosing Clumped1
treatments, about 53% of subjects chose for ’Clumped1’ in the CP and the SL treatment. In the Gain treatments, there is a larger difference between the CP and SL treatment. In the Gain-CP treatment, about 46% of the subjects chose Clumped1, while about 53% chose Clumped1 in the Gain-SL treatment.
(1) (2) Clumped1-Gain Clumped1-Loss GainCP -0.0537 (0.0943) LossCP -0.0139 (0.0768) Constant 0.0394 -0.0139 (0.317) (0.579) Observations 136 136 R-squared 0.072 0.120 Controls YES YES
Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
Table 5. Regression estimates for (1) Gain and (2) Loss treatment: CP versus SL treatment
Clumped1 in the Loss-CP treatment is compared with choices in the Loss-SL treatment. The share of subjects choosing Clumped1 is slightly lower in the Loss-CP treatment, but this is also insignificant. These regressions were run with the standard errors clustered at the session-level. Additionally, I added control variables for gender, age, session day and study field.
Based on these regression estimates, I cannot reject the hypothesis that preferences are purely driven by time preferences. In the gain domain, the fact that fewer people chose Clumped1 in the CP treatments points toward some people preferring separation as predicted by prospect theory with a status quo reference point, although the estimate is insignificant. In the Loss treatment, there is no evidence that people prefer to integrate their lotteries, the point estimate is of the opposite sign as would be predicted by prospect theory with a status quo reference point. Similarly, these regression estimates do not show a significant aversion to piecemeal information as predicted by K˝oszegi and Rabin (2009), nor does it show a preference for piecemeal information as predicted by Ely et al. (2015).
data from the Gain and Loss SL treatments. About 53.3% of the subjects in the SL treatments preferred Clumped1 over Clumped2. I ran a binomial test to test whether more than 50% preferred Clumped1 over Clumped2. I cannot reject the null that 50% or less of the subjects preferred Clumped1 over Clumped2 (p = 0.25). This finding is in line with the indifference prediction by Ely et al. (2015), but not in line with the prediction by K˝oszegi and Rabin (2009) that subjects should prefer sooner over later information.
I also pooled the data of the CP treatments. About 49.3% of the subjects in these treatments preferred Clumped1 over Piecewise. Hence, I cannot conclude that Clumped1 is preferred over Piecewise as predicted by K˝oszegi and Rabin (2009). A binomial test finds no evidence that Clumped1 is more than 0.5 (p=0.57). Similarly, I cannot conclude that Piecewise is preferred over Clumped1 as predicted by Ely et al. (2015) (p=0.50).
(1) VARIABLES Clumped1 CP -0.0312 (0.0637) Constant 0.0919 (0.315) Observations 272 R-squared 0.073 Controls YES
Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
Table 6. Regression estimate SL versus CP treatment
age, gender, study field, and session day were added. The regression estimate shows no significant difference in the fraction of subjects choosing Clumped1 in the SL and CP treatments. Hence, no support for either the prediction of K˝oszegi and Rabin (2006) or the prediction of Ely et al. (2015) is found.
7
Conclusion
While reference dependence is a key component of prospect theory, it is debated how this reference point is determined. While in the earlier days the status quo was often taken as the reference point, more recent theoretical models, most notably K˝oszegi and Rabin (2006, 2009), have included expectations in the reference point.
In this thesis, I have presented an experiment evaluating the reference point in the information context. In our experiment, prospect theory with a status quo reference point makes different predictions than the model by K˝oszegi and Rabin (2009). This experiment contributes to a line of research evaluating the reference point in different contexts. Additionally, the information context was put forward by K˝oszegi and Rabin (2009) as one of the applications of their theoretical model with an expectations-based reference point. This application is put to the empirical test.
the outcome of the second lottery on the second day (Piecewise). In the Sooner-Later variation, subjects have a choice between Clumped1 and Clumped2, where Clumped2 entails that subjects will learn the outcome of the two lotteries combined together on the second day after the experimental session. This Sooner-Later variation allows us to separate the preferences for piecewise information from time preferences.
On the one hand, prospect theory with a status quo reference point yields the predic-tion that, relative to the Sooner-Later treatments, subjects should prefer Piecewise in the Gain treatment and Clumped1 in the Loss treatment. This is due to mental accounting, as a result of which people want to separate multiple gains while integrating multiple losses.
On the other hand, a model with an expectations based reference point put forward by K˝oszegi and Rabin (2009) predicts that subjects should prefer Clumped1 in all varia-tions and that this preference should be strongest in the Clumped-Piecewise treatments. This prediction is caused by agents being loss averse over belief fluctuations and by in-formation about near consumption events having a larger impact than inin-formation about consumption events further away.
Additionally, our set up allows us to distinguish between these two models and a model about information preferences by Ely et al. (2015). That model predicts that individuals should prefer Piecewise in both the Gain and Loss treatments. This is caused by agents having a preference for feelings of suspense. Piecewise resolution of the lottery exposes subjects to more suspense.
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Appendix A
Prospect Theory with Probability WeightingIn this appendix, I will show that the predictions of prospect theory with a status quo reference point continue to hold if we add probability weighting. Individuals in prospect theory engage in probability weighting and use decision weights instead of actual proba-bilities. The decision-weights are calculated by first ranking all outcomes from lowest to highest and next by transforming the objective probabilities into decision weights. We followed the approach of cumulative prospect theory (Kahneman and Tversky, 1992) in calculating these decision weights.
The probability weighting function used by Kahneman and Tversky (1992) is:
w(p) = p
τ
[pτ + (1 − p)τ]1τ
(17)
Where p is the objective probability and τ reflects the curvature of the weighting function, and w(p) is the probability weight. Kahneman and Tversky (1992) estimate τ to be 0.61 in the case of gains and 0.69 in the case of losses.
To derive the predictions of prospect theory with a status quo reference point including probability weighting, we also used the parameter estimates of Kahneman and Tversky (1992) about the curvature of the value function and the degree of loss aversion: α ' β = 0.88 and λ ' 2.25.
We used these parameter estimates to calculate the value associated with separation and integration in both the Gain and Loss treatment. This entailed calculating the correct decision weight and plugging the parameter estimates in the equations which state the value associated with separation and integration for the Gain and the Loss treatment.
treatment. In the Loss treatment, the value of integration is higher than the value of separation. Hence, the prediction that, relative to the Sooner-Later variation, Piecewise should be preferred in the Gain treatment and Clumped1 should be preferred in the Loss treatment still holds if probability weighting is added.
