Numerosity and tangibility in investment decisions

We will recruit participants from Prolific. They will participate in an online experiment that collects all independent as well as the dependent measures, i.e. participants’ decisions in 5 experimental conditions (between-subjects design).

We will measure their risk preference, risk perception, the variance and the economic rationality of their answers when confronted to actual currencies (GBP), and 4 different ECU frames.

2023-04-01

2023-10-01

1) Research question

How does the ECU framework affect risk taking in financial investment decisions?

With the ECU framework being defined as the exchange rate between Experimental Currency Units used during the experiment and actual currency used for payment.

2) Dependant variables

We first assess the participants' risk perception by presenting them a lottery and asking them to assess how risky they perceive this lottery with a Likert scale.

Then, in two investment tasks, the participants receive and ECU amount to invest.

In task 1, the participants receive an ECU endowment and invest five times in a simple two-outcome lottery (Gneezy and Potters, 1997). The dependent variable is the fraction invested - a measure of risk taking: f = Investment / Endowment. The higher the fraction invested, the higher the risk taken.

In task 2, the participants receive an ECU endowment and invest five times in a three lotteries portfolio decision. The dependent variables form different perspectives on the expected return (r) and the standard deviation (s) of the portfolio. On one hand we consider risk taking as measured by the sharp ratio and the standard deviation. On the other hand, we consider the deviation from the efficiency frontier.

3) Independent variables

Participants appointed to the following treatments receive the corresponding endowment:
o t0: 200 p (1 penny decision level)
o t1: 200 tokens (1 token decision level)
o t2: T1 x 10 = 2,000 tokens (10 token decision level)
o t3: T1 x 10^2 = 20,000 tokens (100 token decision level)
o t4: T1 x 10^3 = 200,000 tokens (1,000 token decision level)

4) Control variables

- Age
- Disposable income
- Gender
- Field of study
- Financial proficiency

5) Testable hypotheses

A) Tangibility: Participants have higher risk aversion with £ than with ECU, so they invest less in to than t1 (for both games).

B) Numerosity : Higher amounts increase numerosity which means that people become more uncertain dealing with higher amounts. Hence, we expect a reduction of participants' precision in all tasks (measured in standard deviation).

C) Non-constant relative risk aversion: Higher amounts induce participants to be more risk averse (Fehr-Duda et al., 2010). Hence, we expect a reduction in risk taking in all tasks when ECU frame increase.

D) Game-board effect: Extreme high amounts reduce understanding and value granted to ECUs by participants (Davis & Holt, 1993), we thus expect less risk aversion and higher standard deviation in the decisions they take.

1) Literature

A brief literature review on the topic of ECU effects in experiments can be found in Voslinsky & Azar (2021).

The first mention of the ECU methodology issue appears in (Davis & Holt, 1993). They present their point of view of the different advantages and drawbacks of using ECU without testing them as follow:

- Expected advantages
• Finer grid that matches better continuous models
• Possibility to vary conversion rates amongst participants
• Intercountry comparability
• Money illusion

- Expected drawbacks
• “Game-board effect”: artificial sense of speculative competitiveness

Because this last drawback, Davis and Holt advise to use experimental currencies only in cases where there are deemed necessary.
There has been only one study that experimentally compares the use ECU and actual currencies in the lab, and they strictly focus on second-price auctions (Drichoutis et al., 2015). No effect has been found in this particular article


2) Theoretical environment and predictions

Our research focuses on participants’ attitude towards risk when confronted to real money or ECU frames. We aim at investigating this phenomenon more in depth in the context of ECUs and aggregating the following existing theories:

- The tangibility effect: the fact that using actual cash versus displaying amounts on computers or forms in the lab changes participants’ behaviour. This effect is first showed by Reinstein & Riener (2012), who found that participants are less generous when using cash than when paid via a computer. In our context, using ECUs brings more distance between the participant and its payoff, causing lower risk aversion.

- The numerosity effect: the fact that face value induces a confusion regarding the perceived actual value of a good. Numerosity perception is the elementary numerical ability in both humans and animals, representing nonverbal information of quantity without counting (Dehaene & Changeux, 1993; Feigenson, Dehaene & Spelke, 2004). This phenomenon has been identified in economics (Fehr & Tyran, 2007; Shafir et al., 1997; Shrivastava et al., 2017) and psychology (Bagchi & Davis, 2016), then extended to consumer behaviour research (Wertenbroch et al., 2007). In the case of ECUs, this theory implies that if participants play with high quantity of ECUs, their perception of actual value may be biased, causing more confusion and thus more variability in the data.

- On high ECU frames, we also follow the findings of Fehr-Duda et al. (2010) where relative risk aversion is not constant but increases with stake size. We thus expect risk aversion to increase with exchange rate.

- The “game-board” effect: proposed by Davis and Holt (1993), they argue that tokens may be less considered by participants than actual money, especially at very high amounts, and thus taken less seriously. Participants’ value perception is thus to be biased downwards, causing decreasing risk aversion and higher variance.

We expect these effects to be true at different stages of the experiments. If using pennies instead of ECUs, the tangibility would prevail and participants would get higher utility from actual money than ECUs, thus taking more serious decisions with actual money rather than ECUs. With ECUs, at intermediate exchange rate, non-constant relative risk aversion is expected to be stronger, as “face value” of the ECUs would bias participants’ perception upwards, bringing them to obtain more utility from tokens at constant actual value. Finally, at very high exchange rates, we expect Davis and Holt (1993) “game-board” theory to be stronger, subjects would then give less value to ECUs, considering them as petty money.

We assume that if participants receiving more utility from a form of currency would pay more attention to the games, play more seriously and thus achieve results closer to the economic optimum. This would also be measured by comparing variances between treatments.

 
3) Project academic and societal contribution

Academically, the project aims to compensate the current lack of knowledge concerning the use of experimental currency units and how these are perceived by the participants. Such an addition to the literature may provide more tools and better understanding to the experimentalists who use ECUs.
The societal contributions of this work could be first to isolate bias in risk decision-making when individuals take risky decisions with big numbers. We have identified that this contribution can be helpful in the fields of foreign exchange trading and gambling addiction prevention (Palmer et al., 2022).
A better understanding of numerosity could also change the choice architecture in currency trading.

4) References

Albert, S., & Duffy, J. (2012). Differences in Risk Aversion between Young and Older Adults. Neuroscience and Neuroeconomics, 1, 1–7. https://doi.org/10.2147/NAN.S27184.Differences
Bagchi, R., & Davis, D. F. (2016). The role of numerosity in judgments and decision-making. Current Opinion in Psychology, 10, 89–93. https://doi.org/10.1016/j.copsyc.2015.12.010
Borghans, L., Golsteyn, B. H. H., Heckman, J. J., & Meijers, H. (2009). Gender differences in risk aversion and ambiguity aversion. Journal of the European Economic Association, 7(2–3), 649–658. https://doi.org/10.1162/JEEA.2009.7.2-3.649
Charness, G., Gneezy, U., & Halladay, B. (2016). Experimental methods: Pay one or pay all. Journal of Economic Behavior and Organization, 131, 141–150. https://doi.org/10.1016/j.jebo.2016.08.010
Davis, D., & Holt, C. A. (1993). Experimental economics. In Princeton university press. https://doi.org/10.1053/j.semvascsurg.2005.04.008
Dehaene, S., & Changeux, J. P. (1993). Development of elementary numerical abilities: A neuronal model. Journal of Cognitive Neuroscience, 5(4), 390–407. https://doi.org/10.1162/jocn.1993.5.4.390
Erdfelder, E., FAul, F., Buchner, A., & Lang, A. G. (2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41(4), 1149–1160. https://doi.org/10.3758/BRM.41.4.1149
Fehr-Duda, H., Bruhin, A., Epper, T., & Schubert, R. (2010). Rationality on the rise : Why relative risk aversion increases with stake size. Journal of Risk and Uncertainty, 40, 147–180. https://doi.org/10.1007/s11166-010-9090-0
Fehr, E., & Tyran, J. R. (2007). Money illusion and coordination failure. Games and Economic Behavior, 58(2), 246–268. https://doi.org/10.1016/j.geb.2006.04.005
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://doi.org/10.1016/j.tics.2004.05.002
Füllbrunn, S., & Luhan, W. (2020). Responsibility and limited liability in decision making for others – An experimental consideration. Journal of Economic Psychology, 77. https://doi.org/10.1016/j.joep.2019.06.009
Gneezy, U., & Potters, J. (1997). An experiment on risk taking and evaluation periods. Quarterly Journal of Economics, 112(2), 631–645. https://doi.org/10.1162/003355397555217
Hartog, J., Ferrer-i-Carbonell, A., & Jonker, N. (2002). Linking measured risk aversion to individual characteristics. Kyklos, 55(1), 3–26. https://doi.org/10.1111/1467-6435.00175
Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the mean-variance model for portfolio selection. Organizational Behavior and Human Decision Processes, 42(3), 388–410. https://doi.org/10.1016/0749-5978(88)90007-6
Palmer, L., Cringle, N., & Clark, L. (2022). A scoping review of experimental manipulations examining the impact of monetary format on gambling behaviour. International Gambling Studies, 1–23. https://doi.org/10.1080/14459795.2022.2041067
Reinstein, D., & Riener, G. (2012). Decomposing desert and tangibility effects in a charitable giving experiment. Experimental Economics, 229–240. https://doi.org/10.1007/s10683-011-9298-0
Rieger, M. O., Wang, M., & Hens, T. (2015). Risk preferences around the world. Management Science, 61(3), 637–648. https://doi.org/10.1287/mnsc.2013.1869
Shafir, E., Diamond, P., & Tversky, A. (1997). Money illusion. Quarterly Journal of Economics, CXII(2), 341–374.
Shrivastava, S., Jain, G., Nayakankuppam, D., Gaeth, G. J., & Levin, I. P. (2017). Numerosity and allocation behavior: Insights using the dictator game. Judgment and Decision Making, 12(6), 527–536. https://doi.org/10.1017/s1930297500006665
Voslinsky, A., & Azar, O. H. (2021). Incentives in experimental economics. Journal of Behavioral and Experimental Economics , 93(March), 101706. https://doi.org/10.1016/j.socec.2021.101706
Wertenbroch, K., Soman, D., & Chattopadhyay, A. (2007). On the perceived value of money: The reference dependence of currency numerosity effects. Journal of Consumer Research, 34(1), 1–10. https://doi.org/10.1086/513041

Following the literature, we expect:
- Risk aversion to be higher for female participants (Borghans et al., 2009),
- Risk aversion to increase with participants’ age (Albert & Duffy, 2012),
- Risk aversion to increase with disposable income (Hartog et al., 2002),
- Country of birth: we expect subjects from countries with more individualistic cultures to be more risk averse (Rieger et al., 2015),
- Variance to increase for non-native English speakers (as understanding of the instructions may decrease),
- Variance to increase for people studying non-economic topics.

Design description

Participants will be recruited via Prolific. We may also recruit participants from Portsmouth University and Radboud University subject pools (except for members of the Faculty of Business and Law), or MTurk to verify the validity of our findings for different types of crowds. We will use the software Qualtrics to code the experiment.

Step 1: Presentation of consent form. Participants refusing to sign will stop the experiment at this stage, and be paid the show-up fee.

Step 2: Randomization into treatments (see treatments detail below) when sending the session link to participants. Each session corresponds to a treatment, following a between subject approach.

Step 3: General instructions and payment information. Details are given on experiments duration, payment procedures and anonymity.

Step 4: Control task to determine participants' risk perception.

Step 5: Treatment instructions. Participants are presented the instructions for the respective treatment that they were randomly allocated into.
- Game 1: simple Investment game
- Game 2: portfolio building game

Step 6: Demographic questionnaire

Design description

Participants will be recruited via Prolific. We may also recruit participants from, Portsmouth University and Radboud University subject pools (except for members of the Faculty of Business and Law), or MTurk to verify the validity of our findings for different types of crowds. We will use the software Qualtrics to code the experiment.

Step 1: Presentation of consent form. Participants refusing to sign will stop the experiment at this stage, and be paid the show-up fee.

Step 2: Randomization into treatments (see treatments detail below) when sending the session link to participants. Each session corresponds to a treatment, following a between subject approach.

Step 3: General instructions and payment information. Details are given on experiments duration, payment procedures and anonymity.

Step 4: Control task to determine participants' risk perception.

Step 5: Treatment instructions. Participants are presented the instructions for the respective treatment that they were randomly allocated into. In treatments t1 to t4, participants are explained the ECUs mechanisms, as well as the respective exchange rate to actual currency. After that, they will play to games 1 and 2, 5 times each. The same amount is given as endowment for every round of each game. One round will ne randomly selected for each game, and will constitute participants’ payoff (Charness et al., 2016). These endowments have the same value (2£), but different ECU/£ exchange rates across treatments. For each game, subjects are presented with the instructions of the game, they answer a comprehension test and then can make their decision.

- Game 1: Investment game (Gneezy & Potters, 1997), repeated five times. An endowment of £2 is granted to participants. They can invest any amount of this endowment into a risky ‘project’ which can either be a success or a failure. If the project is successful, they get the investment back, plus an additional 3.5 times of it. If the project is a failure, they lose the investment. Anything that is not invested in the asset is risk free, and is paid out. To determine whether the project is a success or a failure, a dice is thrown by the computer. The project is successful is the result is a 6, i.e. with a probability of 1/6.

An attention check will be performed at the end of round 2 of this game.

- Game 2: Portfolio building game (based on Kroll et al., 1988), repeated five times. An endowment of £2 is granted to participants. Participants must constitute a portfolio by investing their endowment in shares depending of the treatment decision level (see below) in three different assets with the following degrees of risk and return:

Asset 1 yields two times the amount invested when a virtual coin lands heads and zero otherwise. (1/2 winning chance)
Asset 2 yields five times the amount investedn when a virtual dice lands 1 and zero otherwise. (1/6 winning chance)
Asset 3 yields forty times the amount invested when two virtual dices show a one and a six (1/18 winning chance) or zero otherwise.

We calculate the payment as follows:
- a random draw of five periods,
- a virtual coin toss, a virtual throw of one dice, and a virtual throw of two dices determine the payment based on the amount invested.

An attention check will be performed at the end of round 3 of this game.

Step 6: Participant answer a demographic questionnaire for age, gender, and income.

We differentiate between real currency (RC) and experimental currency units (ECU). The tasks are organized such that the RC payment structure remains constant while the ECU payment structure changes. Hence, higher ECU endowments means a higher ECU-to-RC exchange rate.

Here is the detail of the treatments:

Treatment Endowment Decision level
t0 200 p 1 p
t1 200 tk 1 tk
t2 2,000 tk 10 tk
t3 20,000 tk 100 tk
t4 200,000 tk 1,000 tk

Decision level is kept constant in terms of value to avoid any distortion in the results that would be caused by differences of granularity.

All participants take part in the same Qualtrics survey. The survey randomly allocates treatments to the participants keeping the observations per treatment at the same size.

Individual.

No

None.

500 participants in total.

100 participants per treatment.

Stat test: Mann-Whitney Number of tails: 2 α: 0.05 1-β: 0.80 d: 0.5 Participants per treatment: 67 Total required number of participants: 335 Following the hypothesis of an inverted u-shape utility curve for tokens, two-tails analysis is necessary. We picked d=0.5 for detection of medium effects, following existing experiments on investment games (Füllbrunn & Luhan, 2020). This power calculation has been computed with the software G*Power (Erdfelder et al., 2009). According to this figure, we will need 67 participants per treatment, with one baseline and 4 treatments, that makes a total of 335 participants. We decided to plan for a margin in order to still have enough observations in the case participants fail attention check. So, we increased our total number of observations to 500.