Investigation of the Stochastic Choice under Risk using Experimental Data

We extend the analysis of the data from the experiment of Pradiptyo et al.’s (2015) to help explaining the subjects’ behaviour on decision under risk. We investigate the relative empirical performance of the two general models of the stochastic choice: the random utility model (RUM) and the random preference model (RPM). We specify these models using two preference functionals, expected utility (EU) and rank-dependent expected utility (RDEU). We then estimate parameters in each model using a maximum likelihood technique and run a horse-race using the goodness-of-fit between the models. The results show that the RUM better explains the subjects’ behaviour in the experiment. Additionally, the RDEU fits better than that of the EU in modelling the stochastic choice.


Introduction
The decision-making process is arguably the central part of economics. In the standard economic model, the decision maker (DM) is assumed to evaluate the utility correctly and to choose accordingly (Mas-Colell et al. 1995). The expected utility (EU) theory, coined by von- Neumann and Morgenstern (1944), has been widely accepted as the workhorse of the choice theory under risk and uncertainty. This is not always the case, however, in explaining the actual decision; the classic experimental studies which widely referred are Allais paradox and Ellsberg paradox. These findings have led to the development of the deterministic model to have a better explanation of the DM's behaviour under the assumption of maximising utility. 1 One of the prominent models in non-expected utility theory is the rank-dependent expected utility (RDEU) theory, introduced by Quiggin (1982). Apart from the utility maximisation, the intuition of this model is the DM may weigh the probability of the outcomes differently (Diecidue and Wakker 2001).
Experimental studies 2 have been conducted to infer the subjects' true preference. Most findings have found that decision is noisy though the subjects have a well-defined preference. It is inevitable in many situations, including in a controlled lab-experiment with some restrictive procedures. By this, we need to accommodate the stochastic process into the model to have a proper identification given the preference functional. To proceed with this objective we always use theory to investigate whether or not the hypothesis is true. It, rather technically, is crucial to have a correct specification of the stochastic process. This has been experimentally investigated by Hey (1995), Loomes and Sugden (1998) and Blavatskyy (2007) 3 to name a few. The use of the econometric approach in analyzing the experimental data may well motivate this interest. One advantage of this approach is to exteriorise different explanation of noise in the experimental data -of the fact that it is very rarely we can have perfect data without noise.
As we strongly hypothesise that the experimental data must be stochastic, we focus on the two alternative stochastic models to investigate that, namely the random utility model (RUM) and the random preference model (RPM). The idea of the RUM was firstly coined by Manski (1977) and McFadden (1981) in an attempt to characterise the inconsistent pattern of individual behaviour. This, later on, has been extensively used to estimate the preference functionals, including risk aversion (Cicchetti and Dubin 1994, Hey and Orme 1994, Holt and Laury 2002, Harrison et al. 2007, von Gaudecker et al. 2011, Toubia et al. 2013. We take the idea of the RPM from Loomes and Sugden (1995) who posit that the DM's preference is represented by some set of functions and the DM acts as if he or she picks one of those randomly. Some literature to use this model are Carbone (1997), Loomes et al. (2002) and Moffatt (2015).
We apply both models using the dataset from Pradiptyo et al.'s experiment (2015). The main difference between both approaches is on the source of the stochastic process. Within the RUM, the DM calculates his or her utility with noise in the preference function. Hence 1 For references see: Kahneman and Tversky (1979) for prospect theory and its extension (referencedependent, loss aversion and cumulative prospect theory); Machina (1982) for EU without independence axiom; Loomes and Sugden (1982) for regret theory; Quiggin (1982) for rank-dependent expected utility theory; Gilboa and Schmeidler (1989) for maxmin expected utility. 2 See Kahneman and Tversky (1979), Loomes and Sugden (1982), Quiggin (1982), Gilboa and Schmeidler (1989), Cicchetti  the utility is evaluated with noise and that we assume the noise is normally distributed. The DM then chooses accordingly to what he or she reveals from his or her calculation. Within the RPM, the DM draws parameter(s) in the preference function from a distribution. Here we assume the DM randomly draws a parameter in the preference function from the normal distribution and the utility is evaluated without error. This approach explains why the DM may behave differently on different occasions. This paper aims to answer the main question of which stochastic models, between the RUM and the RPM, to be the best-fit of the subjects' behaviour. This study shows that the RDEU is well-applicable, rather than the EU, to model the preference in our stochastic stories. The primary contribution of this study is that our models allow us to characterise and to identify the source of noise in the subjects' preference. This paper is organised as follows: section 2 discusses the preference functional and its assumption. Section 3 and 4 explain the preference functionals and the modelling stochastic choice, respectively. We then present the results and analyses in section 5 and concludes the paper findings in section 6.

The experimental design and the data
In this paper, we use the dataset from Pradiptyo et al.'s (2015) experiment, with 245 subjects but we only use data from 242 subjects due to the incomplete data of 3 subjects. All subjects were traders from traditional markets in Yogyakarta, Island of Java (122 subjects) and Pontianak, Island of Kalimantan (120 subjects), Indonesia. The experiment was conducted in two different settings: laboratory (Yogyakarta, Java) and lab-in-thefield (Pontianak, Kalimantan) typical on-campus-lab. We refer to Charness et al. (2013) that calls this type of experiment as an extra-laboratory experiment. This practice must be an appropriate experiment for Pradiptyo et al. (2015) given the main purpose of the study and the subjects' background.
The subjects were invited manually 4 and they were asked to fill in a short questionnaire regarding their personal information prior to the experiment. In addition, they were given written instructions and were shown presentation slides of the instruction before the experiment. All instructions and presentation slides were in Indonesian. The subjects completed all tasks in an electronic tablet. Ten assistants were available to aid the subjects only if they had difficulties in operating the electronic touchscreen tablet. The assistants' particular task was to minimise any confusion among participants as they might have not been familiar with such experiment or in operating the electronic tablet. The assistants, however, did not assist the subjects in making any decision during the experiment. 4 The common alternative practice in the experimental economics research is to invite subjects randomly using a database in which any subjects expresses their interest to be a participant in an experimental research. In Pradiptyo et al. (2015), the subjects were recruited manually due to the intended characteristic in a subject.
The experiment was constructed under two main purposes: an opinion survey and a decision making under risk and uncertainty experiments. There were 4 main sessions with the first two main sessions are the opinion survey, and the other 2 main sessions are the experiments. There were 75 problems in total. The opinion survey has 39 questions in a total of two sessions, whereas the decision-making sessions has 36 questions spread over the last two sessions. Subjects were given a break after completing the 2 nd session and they started the 3 rd session together. The subjects were allowed to finish all questions within Session 1 and 2, and also within Session 3 and 4, anytime they wished to. In addition, each main session was preceded by a practice session.
We particularly use the dataset from the decision-making experiment (3 rd main session). There were 20 pairwise choices consisting of 10 positive pairwise-choice problems and of another 10 negative pairwise-choice problems -subjects were asked to choose between two alternatives (Option A and Option B). The design follows Kahneman and Tversky (1979). Due to the technical specification issue, we only use six positive pairwise choices that capture three axioms in the decision theory: common consequence, common ratio, and substitution. The pairwise choice problems used are described in Appendix 1. Furthermore, we pooled all the observations from all subjects for the purpose of estimation.
Random lottery mechanism was used to determine the subject's payment for this experiment. Every subject picked a random number which corresponds to the number of respective questions (in all sessions) as the basis of his or her payment. If, in particular, a subject got a question in the 3 rd session, he or she would play the lottery for real according to his or her answer. A show-up fee of IDR25,000 was given as an endowment in each of the 3 rd and 4 th session -an additional IDR125,000 was given as the subjects also faced the negative pairwise choices. 5 This design ensures the subjects to maximise their preference in the 3 rd session. The payment range that the subjects could earn from the experiment was IDR0 to IDR300,000; The experiment software was written (mostly by Ali Faiq) in PHP script.

Modelling the preference functional
We focus on the two different alternative stochastic choice theories under risk to model the subjects' preference, the RUM and the RPM. The former assumes the DM has a set of fixed parameters in the preference function when making the decision for all problems. The DM, however, evaluates the utility with noise. We assume the noise is normally distributed with the mean 0 and the standard deviation sv . In addition, we specify this model with a tremble parameter to capture the subjects' mistake. The RPM model assumes that the parameter(s) in the DM's preference function is drawn from a distribution. The DM evaluates the utility without noise. This means the subjects may have different parameter scores when making the decision in every problem. The parameter score is drawn from a distribution. Additionally, we assume that the other parameter(s) in the preference function remains constant across all problems. What the parameters mean and how they are specified will be explained in the following section.
We use the EU and the RDEU to model the preference. Both theories are arguably considered as two of the prominent theories in decision making under risk and uncertainty. We now will explain how we construct the model.
Recall that there are five basic outcomes in the problems used. The lowest outcome is IDR0 and the highest outcome is IDR100,000. We denote the outcomes as We then normalise the outcomes so u ( An option in a pairwise choice problem is a probability distribution over those five outcomes so we have {p 1 , ..., p 5 } as the corresponding probabilities of the Option A and {q 1 , ..., q 5 } as the corresponding probabilities of the Option B.
First, we look at the EU specification. The EU has two general properties, a set of true probabilities and a set of the utility functions. The key point of the EU is the linearity in probability and the utility function satisfies the von Neumann-Morgenstern expected utility function. The general form of the EU is: is the EU value of choosing an option, z i is a vector of probabilities of the corresponding outcomes (which is a set of the true probabilities), and u i is a vector of utility indices of the corresponding outcomes. Essentially, the EU has one key element that is the utility function.
We now turn to the RDEU specification. The RDEU has the identical utility function as interpreted for the EU, however, the key distinct feature of the RDEU is it is not linear in probability. The DM may not see the set of probabilities (z i ) as the true probabilities hence the DM transforms it in a specific way through the probability weighting function w (z i ). This means the RDEU has two key elements: the utility function and the probability weight function.
The general formulation of the RDEU is: is the RDEU value of choosing an option, Z i is a vector of the weighted probabilities, and u i is a vector of utility indices of the corresponding outcomes. The weighted probabilities are non-negative values and adapted to one. The probability transformation function enters Z i so the RDEU allows for a non-linearity in probability. A crucial intuition of the RDEU is that the DM ranks the outcomes in order to weigh the corresponding probabilities. The implication is that Z i is not the true probabilities despite the fact that the probabilities are given in the experiment. Given the setting that x i is the best outcome and x 5 is the worst outcome, we can define the weighted probability Z i as: . Hence it is going to be Z 1 = w (z 1 ). Note that w (z i ) is monotonically increasing in the area of [0, 1], with w (0) = 0 and w (1) = 1, and the RDEU reduces to the EU if w (z i ) = z i everywhere.
We look further into the specification of the probability weighting function for the RDEU. We use two common forms of the probability weighting function, namely the Power function and the Quiggin function. The functions are written as the following: where g is the parameter of the probability weighting function and it determines the shape of the indifference curve and explains the behaviour implication (Starmer 2000). Conte et al. (2011) suggest that g = 0.279095 unless the function is not monotonic. The function is an inverted S -shape for g < g < 1 and an S -shape for g > 1. 6 Note that both probability weighting functions lead the RDEU to reduce to the EU when g = 1.
Now we turn into the specification of the utility function. We use two forms of the utility function -the constant relative risk aversion (CRRA) and the constant absolute risk aversion (CARA). The application of CRRA and CARA under normalization therefore is: The CRRA and the CARA specifications can take any value of r between −∞ and ∞. A positive value of r indicates a risk-averter, a negative r indicates a risk-seeker, and r = 0 indicates a risk-neutral agent.

Stochastic specifications
To start our specification, we assume that the DM is either of the EU or the RDEU agent. All models share an identical assumption that the DM makes a choice depending on the evaluation of his/her true preference. Let V t (p i , x i ) and V t (q i , x i ) be the utility of the Option A and of the Option B respectively, referring to the EU or the RDEU in every problem t. Thus the DM's calculation is A t ≺ B t and it is determined by the parameters in the EU (the r) and in the RDEU (the r and the g).

The random utility model (RUM)
We assume the parameters in the DM's preference function are constant and the DM evaluates the utility with noise. By this, the stochastic variation ε is added into V t (A t , B t ) and the DM's choice becomes The DM, however, may make a mistake in expressing his or her preference, therefore we involve a tremble parameter (ω) to capture the DM's mistake. Hence the DM chooses an optimal choice in every problem t following V t (A t , B t )+ε with a probability of (1 − ω) and mistakenly chooses a non-optimal choice with a probability of ω. The tremble parameter takes any values of 0 ≤ ω ≤ 1.
Let the y t = 1 if the DM chooses Option A and y t = −1 if the DM chooses Option B in a choice problem t. The likelihood contribution in every problem t is: where Φ (.) is the cumulative distribution function (cdf ) of the normal distribution with parameters mean µ and precision s = 1 /σ. In summary, we have six variations for the RUM from the utility function and the probability weighting function specifications. The variations within the EU specification have estimates of r, s, and ω; and the variations within the RDEU specification have estimates of r, s, g, and ω.

The random preference model (RPM)
This model starts by assuming that a parameter, either r or g depending on the preference function used, is random, and that, once it is drawn from a distribution. Unlike the RUM, the utilities in this model are evaluated without noise. Given this, we will have two specifications of this model according to which parameter is random. 7 The first specification assumes that the r is random and it follows a normal distribution -that the g is constant across all problems. So the DM has his or her mean of the r at µ and standard deviation σ when making the decision in every choice problem. In every V t (A t , B t ), there will be an r * that indicates the indifference between two options. This obviously happens when V t (A t , B t ) = 0. The implication is that if r > r * then the DM chooses the riskier option (Option A), vice versa. We use notations r * 1 and r * 2 to distinguish the r * from the EU and the RDEU specification (for any variations of the utility functions and of the probability weighting functions).
The second specification assumes that the g is random and it follows a lognormal distribution as it always positive -that the r is constant across all problems. 8 By this, we assume that the DM has his or her mean of the g at ln (M ) and standard deviation ln (Σ) when making the decision in every choice problem. Given this, there exists a g * when the DM is indifferent between Option A and Option B as the RDEU values for both options are equal -V t (A t , B t ) = 0. The implication of this specification is that every preference to the Option A will always have the g > g * , vice versa. it should be noted that this model only works under the RDEU with the Power weighting function. The RDEU specification with the Quiggin weighting function provides no solution in finding the g * for any given r.
We then turn to the econometric specification. We use the maximum likelihood technique to proceed with the estimation. Let the y t = 1 if the DM chooses Option A and y t = −1 if the DM chooses Option B in every choice problem t. Within the first specification, the contribution to the likelihood of an observation r * in every problem is: where Φ (.) is the cdf of the normal distribution with parameters mean µ and precision s = 1 /σ. Note crucially that the g enters the likelihood in Equation 6 through r * 2 from the specification of the weighted probability function as defined in the previous section. Meanwhile, within the second specification, the likelihood contribution of an observation g * in every problem is: Pr (y t | ln (M ) , ln (Σ)) = π y t ln (M ) − ln (g * ) ln (Σ) ; y t ∈ {1, −1} where π (.) is the cdf of the lognormal distribution with parameters ln (M ) and ln (Σ). However, we report the non-logarithmic value of M and S = 1 /Σ instead in each model; the S is the precision which is the same to that in the random preference model on r. In summary, we have eight variations from the utility function and the probability weighting function specifications.

Results and analyses
We estimate the parameters using a pooled subject rather than taking individually. We shall begin the discussion of which the two models is best fitted to explain the data. In total, we have fourteen variations from two models -the RUM and the RPM. We go on in answering this question by comparing the corrected log-likelihood. This is due to the difference in the number of parameters in each model and can be of a formal comparison of our models.
We use three alternatives of measuring the goodness-of-fit for the corrected log-likelihood: (i) Akaike information criterion (AIC); (ii) Bayesian information criterion (BIC); and (iii) Hannan-Quinn information criterion (HQC). 9 Appendix 2 presents the model selection according to the corrected log-likelihood. It ranks the best-fitted specification according to each measure of the goodness-of-fit in all models. It suggests the RUM specified with RDEU Quiggin using CARA utility function to be the best fit the data according to the AIC, the BIC, and the HQC. The RPM on r, however, varies only slightly with the RUM according to the BIC measure. A few number of problems could be the main reason in which the subjects had a constant preference across problems. Results in Appendix 2 also show that the use of the RDEU as a core theory is better to model our stochastic story in this paper in comparison to the EU.
We delve deeper the investigation on the subjects' risk aversion (Appendices 3 and 4).
Most of the variations suggest that the subjects are risk-averse; with variations within the RPM show that the subjects have a strong tendency to be risk-averse. Note that this estimate is unique for all subjects. Secondly, we estimate the probability weighting parameter in all variants using the RDEU, in which we specify with the Power and the Quiggin functions (Appendix 5). All variants using the Power function suggest a convex probability weighting function, and all variants using the Quiggin function suggest an inverted S -shaped probability weighting function. Both exhibit an identical behaviour implication where the subjects over-weigh the small probabilities and under-weigh the large probabilities. Lastly, we estimate the tremble within all variants in the RUM which appears to be relatively high. One plausible explanation for this is the subjects had difficulties in understanding the nature of the problems during the experiment.

Conclusions
This paper discusses mainly on the modelling the preference under two stochastic theories: the RUM and the RPM. The results give us a clear message that the RUM can well explain the subjects' behavior in Pradiptyo et al.'s (2015) experiment. Given this, the subjects were rather having a constant preference across problems. The results of this study also show that the RDEU is well-applicable, rather than that of the EU, to model the preference in our stochastic stories.
9 AIC = 2k − 2 ln (LL), BIC = k ln (n) − 2 ln (LL), HQC = 2k ln (ln (n)); where k is the number of estimated parameters, n is the number of observations and LL is the maximised log-likelihood. AIC provides a simple approach and is widely used in practice among analysis of complex data but may not perform well if there are too many parameters, whereas BIC and HQC try to reduce its potential bias by imposing a more stringent penalty on the number of parameters than that of AIC (Haggag 2014).
The advantage of our paper is that it allows us to characterise and to identify the source of noise in the subjects' preference. This is our primary contribution. We apply the model to capture the stochastic process since we initially hypothesised that the data would be noisy, particularly due to the characteristic of the subjects. Another point to note is that since the subjects are trader, it may be worth investigation to have comparison between occupations and, perhaps, between educational backgrounds. An additional extension may be to expand the choice problems in the experiment to enable analysis of the decision under uncertainty.

EXPERIMENT INSTRUCTIONS
Welcome and thank you for your participation in this experiment. You will take a participation in a decision making under risk experiment. These instructions will help you to understand this experiment. You will have a chance to win some money in cash by the end of this experiment depending solely on your answer. Before you go on to the main experiment, you are asked to listen to the PowerPoint instruction. It will appear on the big screen upfront. Please read and listen to these two types of instruction for you to understand this experiment. There will be a practice session after PowerPoint instruction has been presented.
At the end of the experiment, you will be asked to fill the personal information form. We will keep your personal information and it will be only used in this experiment. You are also asked to turn-off your mobile phones and to not making any forms of communication with others, unless allowed by the experimenters. Do not hesitate to raise your hand if you have any questions. Either the experimenters or helpers will come to you to answer your questions.

The Experiment
You will be presented 20 pairwise-choices, all in the same type. On each problem you will have to choose one of two options that you think you prefer to it. Your choice will have no impact to others but yourself. You can finish all problems in this session anytime you wish to. There is no time limit for you to complete all problems.
A picture below gives you an example of a problem in this session.
Translation -Problem 1. Which one of these two options (Option A and Option B) that you will choose according to your preference? Option A will give you a 90% chance to win IDR5,000 or a 10% chance to win IDR10,000. Option B will give you a 95% chance to win IDR5,000 or a 5% chance win IDR10,000. "Next" button.
If you choose an Option A you will have a 90% chance to win IDR5,000 or a 10% chance to win IDR10,000. If you choose an Option B you will have a 95% chance to win IDR5,000 or a 5% chance win IDR10,000. You have to click the "Next" button and you will be directed to this page: Translation -Confirmation for Problem 1. You have chosen Option A. Is this your true preference? If yes please click the "Save and Continue" button (the right one), otherwise click the "Back" button (the left one).
The picture above is the confirmation screen. If you think you are sure with your answer you should click the "Save and Continue" button. Otherwise click the "Back" button (the left one) to modify your answer. Once the software has saved your answer you cannot modify your answer.

Appendix 7. Examples of screenshot in the decision making under risk session (in Indonesian)
Translation -Question 1: Which one of these two options that you prefer to choose?
Once a subject has his or her choice by clicking one of the two options then this screen below will appear: Translation -Confirmation of Question 1. You have chosen "A". Is this your true answer? Please click "Record and Continue" button if you think you are sure with your answer otherwise click "Back".