Abstract

 

 

The introduction of a concave value-function is supposed to stimulate the slightest risk-taking behavior. We posited that people conduct themselves as Kahneman and Tversky described within their paper Prospect Theory: an analysis of decision under risk, published in Econometrica in March 1979. They have been proven to be statistically risk-averse in the gain domain and risk-seekers for the losses. Thanks to an experiment that has been set up in order to replicate Prospect Theory for comparison and so as to extend conclusions, regarding behaviors when risk is at stake, we have been able to modify the hypothesis of the utility-function. Through the Binary Lottery Procedure (BLP), we manipulated this function to change the basic assumption of risk-neutrality when responding to this BLP. Participants completed four tasks involving monetary gains, points’ gains, monetary losses and points’ losses and finally preferences information. After having analyzed results, subjects behaved in a way that was not expected since they even took more risk when they could win points, whereas they were supposed to be risk-averse in the BLP because of the concave value-function. Comparison between Money Treatments (MT) and BLP suggests that people may have a cognitive bias in their perception of monetary gains and non-monetary gains.

 

 

Introduction

 

 

In everyday life, choices have to be made with their part of uncertainty and/or risk. Dilemma emerges when there are risks, but also when the result can be of the greatest value. Take for example the relatively sure investment in treasury bills, but not that much remunerating, or the riskier investment in shares on the stock exchange market, but so much more rewarding if the share rate turns to your advantage. Basically, if two investments have the same expected return, the one with the lower risk will be preferred. The counterpart: the riskier investment needs to have a higher expected return in order to give an incentive for a risk-averse investor to select it. What is the process involved in decision making that triggers risk taking even if information and consequences are not enough well-measured, or known? Why most of the people behave like Kahneman and Tversky described: risk-averse in gains and risk-takers in losses? Is it still true when you modify their utility-function and you intend to inject a risk-averse behavior as a basic assumption thanks to the concavity of this function?

When they published their paper in Econometrica in 1979, Daniel Kahneman and Amos Tversky introduced Prospect Theory as an analysis of decision under risk compared to a choice between prospects or gambles. They infer through their experimental researches that people are risk averse in the gain domain and they have a risk-taking behavior when losses are at stake. The aim of this research project is to implement a new condition in the value-function of the subject, regarding Kahneman and Tversky lottery-choice experiment. The control of the value-function by the binary lottery procedure will consider a linear function (control #1); a concave function (control #2) and an S-shape function (control #3). All along this paper, assumptions and results will focus on the introduction of the concave function (control #2). In the wake of Prospect Theory’s behavioral description, the goal is to analyze what is due to behavior inducing and what is due to outcome weighting, as far as people’s decision making is concerned, and after having established the shape of the concave value-function.

 

It is stated that an event is uncertain when it is just a possibility among others and the final outcome is not yet known. Uncertainty affects economic behavior in the sense that people have to make decisions right now, not knowing, or partially knowing, what will happen in the future. Uncertainty can embody two forms. First, the risk when one has an imperfect knowledge of the outcome but has a perfect knowledge of the possibilities. And second, the proper uncertainty where one has both an imperfect knowledge of the outcomes and on their distribution. This notion of uncertainty enables to consider irrationality in the wake of decision making in economics. On this precise point, psychology plays an important part: Kahneman and Tversky’s Prospect Theory brought some psychological factors into economics. This theory enabled to unshackle from the theoretical basis of expected utility theory, that is to say, to emancipate from what is supposed to happen for decisions under uncertainty (purely theoretical); and to lead the way to a more descriptive approach of what actually happens when those decisions (under risk) are at stake. With prospect theory, decisions are not taken in the wake of pure rationality anymore (Montier, 2002). In order to test how people behave when they have been induced a concave utility-function which is supposed to provide them a risk-averse behavior, subjects’ preferences were controlled using the binary lottery procedure developed first by Roth and Malouf (1979) and developed later by Berg, Dickhaut and O’Brien (1986); Prasnikar (2000) and Berg, Dickhaut and Rietz (2006). This procedure will be explained in the development of the overview of the study and also recalled in the results section.

 

 

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Method:

 

Participants:

 

The overall experiment has been passed by 123 subjects. They were divided into groups of 20 or 21 people so there were 6 groups in the end. Groups 1 and 2 participated in the experiment with the linear utility-function hypothesis. Groups 3 and 4 had the concave utility-function hypothesis, which is the case treated in this thesis. Finally, groups 5 and 6 had an S-shaped utility function (convex in the loss domain and concave in the gain domain).

As far as groups 3 and 4 are concerned (i.e. the concave groups), there were 41 people on the overall. The computer room has 21 computers available at the same time; that is why the experiment for the concave hypothesis, inter alia, has been divided in two sessions. Group 3 is composed of 15 males and 5 females; and group 4 is composed of 9 males and 11 females. So the concave session has been released by 24 males and 16 females.

Eventually, data from 41 subjects (16 female, ages 18-28 years, M = 22.7, SD = 3.4 and 24 male, ages 19-31 years, M = 22.7, SD = 3.5) were included in analyses. Informed consent was obtained for each subject according to a protocol approved by Paris School of Economics, and each received monetary compensation for their participation.

 

Task design:

 

The experiment has been coded with the software z-tree, after having asked for a free license to the University of Zürich. Each one of this thesis research has participated to program the experiment.

Participants had to complete the experiment which is divided into 4 parts and 40 questions on the overall.

 

Part 1 (Questions 1 to 9) involves monetary gains. People have to choose between several prospects that are presented, and that were taken from Kahneman & Tversky’s prospect theory, just by modifying the monetary outcome, since the experiment design displays euros when Kahneman & Tversky’s problems refer to Israeli currency. But, proportions of the eventualities and probabilities are consistent with prospect theory.

 

Part 2 (Questions 13 to 30) is a Binary Lottery Procedure (BLP) involving gains. In this part, people have to make a choice between several lotteries involving points’ gains. These points will be later transformed into a probability to win a fixed price of 20€. In this part, the concave utility function is supposed to induce people’s choices since each point is associated to a proper probability to win. The transformation curve that has been used follows the Arrow-Pratt formula u(c) = 1-e^-ac. By comparing a = 0.11/ 0.1/ 0.09/ 0.08/ 0.07/ 0.06/ 0.05/ 0.04/ 0.03/ 0.02/ 0.01, it appeared that = 0.05 provides the most appropriate curve. Eventually, the value-function that has been used is f(x) = 100(1-e^-0,05x). Participants were provided the transformation curve associated to their group (concave group here) and a points’ transformation table.

 

 

Part 3 (Questions 28 to 36) is composed of 3 lotteries of monetary losses (Q10, Q11, Q12) and 6 BLP involving points losses. On the opposite of part 2, earning many points increases the chance to lose a fixed price of 20€, according once again to the transformation curve provided.

 

Part 4 (Questions 37 to 40) is composed of 4 questions of evaluation to study the WTP (Willingness to Pay) and WTA (Willingness to Accept) and preference reversal. This part is not linked to the outcome of participants.

Eventually, in the first 3 parts, subjects will have made choices and at the end of the experiment, one for each part is randomly selected for payment.

 

Manipulation of risk preferences:

 

Subjects were given “Fiche A”, “Fiche B” or “Fiche C” at the beginning of the experiment, according to their group belonging. In the concave utility curve group, participants were told to observe “Fiche B” with the concave curve and the points’ transformation table before answering the BLP. They acknowledged that they had to consider differently points, money and probabilities. The concave condition is supposed to encourage people to take less risk since for instance, between 15 and 100 points earned, they have more than 50% of chance to win the fixed price of 20€. Indeed, within this condition, at question 25 (Q28 in the design), subjects have to choose between A: (50 points, 50%; 0, 50%) and B: (25 points, 100%). Thanks to the risk manipulation, it is expected that subjects choose in majority prospect B, because 25 points = 71.35% and 50 points = 91.79%, which represent a difference of 20% to win but people are supposed to take less risk and should choose the sure prospect where they win 25 points (50% difference between the two prospects).

 

Data acquisition:

 

Data was recorded in excels files directly from participants’ computers using z-tree. Each file covers one session so on the overall there were 6 sessions so 6 excel files to gather. These files contain the number of the session, the identification of the participant – numerical ID – and his/her answer to each question; and eventually his/her monetary reward, according to his/her career in the experiment.

 

Data analysis:

 

Data was reorganized into pivot tables in Excel in order to have a list of subjects’ answers inside each question (20 or 21 subjects per question). Eventually, data was merged and imported in Stata v.12 in order to perform every data analysis that seemed relevant to interpret. These analyses include descriptive statistics, t-tests, and summarizes. As far as t-tests are concerned, a result to a lottery with a rate of 80% of subjects choosing a specific prospect is significant if the p-value is lower than the threshold that has been used (p < 0.1; p < 0.05 and p < 0.01).

 

 

 

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