Propensity to risk and the prospect theory

1. Introduction

Nowadays, there is a growing number of research, mostly in behavioural economics, in which impact of psychological factors is emphasised. In behavioural economics is stated that people make many systematic errors in judgements and use certain heuristics while making decisions (heuristics and biases paradigm). People also make mistakes while evaluating past and future utility, which is contradictory with the assumption about utility maximization. With respect to social behaviours, not only selfishness, but also rule of reciprocity seems to be important [Wilkinson 2008], [Rabin 1996].

In context of these studies the question arises, how to measure intensity of psychological features and their impact on economic events. It is connected with more general problem: how to apply the very interesting findings of behavioural economics to real phenomena analysis? In the following monographs [Hozer, Doszyń 2004], [Doszyń 2008], [Doszyń 2013] conception of propensity was proposed. Generally, in these monographs problems connected with measuring propensities were discussed. Also analysis of human propensities’ impact on economic occurrences as well as a proposal of econometric tools enabling identification of this impact were presented.

In this article measures of propensity to risk will be used in context of the prospect theory. With respect to this measures value function will be specified. The hypothesis will be verified if proposed measures of propensity to risk are consistent with conclusions that stem from the prospect theory.

2. Two groups of propensity theories

In the philosophy of science there are many varieties of propensity theories which generally could be divided into two groups: 1) theories in which propensity is understood as a result of all conditions that generate events (K.R. Popper), 2) theories in which propensity is an internal characteristic of an object (C. Peirce).

The first group consist theories in which propensity is treated as a characteristic of a whole situation. Therefore propensity is a tendency of a certain situation. Such propensity depends both on objective and subjective (psychological) factors. Human (psychological) propensities are just one of the factors influencing propensity of a whole situation. This group of theories are mostly based on K. Popper’s works [Popper 1959], [Popper 1990], [Gillies 2000]. The propensity interpretation of probability proposed by K. Popper refers to the tendencies inherent in given situations. In this sense, the propensity is understood as a relative frequency, which results from the interaction of various kind of causes. If business processes are analysed, the propensity (by Karl Popper) stems from the impact of both objective and subjective causes. The propensity inherent in a given situation is the result of the combined impact of all factors.

C. Peirce, on the other hand, claimed that propensities are characteristic of objects. If person is the object, the propensity stems from personality characteristics, and therefore could be identified with psychological factors. In this view propensity describes internal (psychological) structures of people. Propensities are understood as factors describing psychological aspects of human behaviour that make probabilities of certain events higher. Propensities are therefore generalized psychological causes of events.

To sum up, generally propensities could be analysed in two ways: as a propensity of a situation or as a propensity of an object (person). The propensity of object (internal, psychological propensity) has psychological nature and is influencing the propensity inherent in the given situation. Propensity of a situation is therefore characterised by all relevant conditions that generate stable frequency of events. In economics mostly that kind of propensities are analysed. It is probably due to fact that it is not easy to identify internal (psychological) propensities by means of quantitative methods. Statistical and econometric methods that could be useful in such cases are described in [Hozer, Doszyń 2004], [Doszyń 2008], [Doszyń 2013].

Generally, to analyse internal propensity to risk, the assumption should be made, that risky decision is an effect of the propensity to risk. In the prospect theory, propensity to risk is understood as a tendency that depends both on external circumstances and psychological traits. In this theory such factors are influencing behaviours as prospects, level of gains and losses, probability of events, etc.  

3. Measures of propensities

In economics, propensities are usually identified with marginal or average propensities (to consume, save, invest, etc.). These considerations have their origins in a famous book The General Theory of Employment, Interest and Money, in which J.M. Keynes treated propensities as a functional dependencies between certain variables that depend both on objective and subjective factors. According to J.M. Keynes, propensities depend on psychological features, but also on all other important factors. Propensities make connections between variables stable. This is similar to K. Popper propensity theory but there is one difference. K. Popper understood propensity as a certain frequency of events and J.M. Keynes – as a functional dependency between variables (consumption and income, savings and income, investments and income, etc.).

So how to define propensity? Propensity might be defined as a “slope of posture” towards something (or somebody) that makes probability of certain event higher [Hozer, Doszyń 2004]. This definition is consisted with this theories, in which propensity is treated as an internal characteristic of the object. K. Popper, on the other hand, defined propensity as a biased possibility that characterise given situation [Popper 1990].

               Intensity of propensity could be measured by means of frequency and trigonometric methods [Hozer, Doszyń 2004]. Therefore, in the frequency measure, propensity is calculated as a share of cases in which propensity appears in all possible cases. Propensity might be also presented in degrees, by means of trigonometric measure, where propensity is defined as a specified angle [Hozer, Doszyń 2004]. Tangent of this angle could be obtained on the basis of the formula in which frequency measure of propensity is used. The higher propensity, the higher “slope” (and the lower angle). The trigonometric method could be useful while presenting the strength of propensity in a graphical way.           

Both presented measures might be used for both types of propensity theories. If propensity describe the whole situation, these measures describe propensity that stems from all conditions. If propensity is an effect of psychological factors, proposed measures characterise internal tendencies of people.

The frequency measure could be used to calculate propensity to risk if we have two outcomes: risky one and safe (safer) one. That kind of decision problems were analysed by D. Kahneman and A. Tversky in their famous article about the prospect theory [Kahneman, Tversky 1979]. If there are many outcomes, propensity to risk could be estimated by means of measure constructed on the basis of the game against nature [Doszyń 2008]. Decision problems with many outcomes were analysed by D. Kahneman and A. Tversky in the cumulative prospect theory [Kahneman, Tversky 1992].

Output matrix in the game against nature should be constructed in such way that strategies are ordered with respect to propensity to risk. To each strategy different level of propensity to risk is assigned. According to Hurwicz criterion, given strategy is an optimum for given propensity. The higher propensity to risk, the higher dispersion of the results in a given strategy. Output matrix should be specified exactly in this way. Propensity to risk might be then estimated as a weighted average, because we know how many respondents choose each strategy.

4. Prospect theory and propensity to risk

The first version of the prospect theory was described in the famous article Prospect Theory: An Analysis of Decision Under Risk, published in 1979 by D. Kahneman and A. Tversky in Econometrica. It is now accepted that the model proposed in this article has, however, some limitations. Primarily, it can be only applied to gambles with two nonzero outcomes. In 1992 D. Kahneman and A. Tversky published a new version of the model, which is called cumulative prospect theory. There is a lot of literature on this topics. A brief description is presented for example in [Barberis 2013]. Nowadays cumulative prospect theory is usually implemented. The most important findings are presented in [Kahneman, Tversky 1992].

In the cumulative prospect theory gambles with many outcomes are considered [Barberis 2013]. Under expected utility theory, every individual make decision on the basis of the utility levels weighted by objective probabilities of events. In case of cumulative prospect theory each decision is based on values of changes (with respect to reference point) weighted by the so called decision weights [Kahneman, Tversky 1992], [Barberis 2013]. Therefore, in the cumulative prospect theory not state of wealth is important but changes of wealth (gains, losses) with comparison to settled reference point. 

Usually four elements of prospect theory are emphasised [Barberis 2013]: a) reference dependence, b) loss aversion, c) diminishing sensitivity, d) probability weighting.

As it was mentioned, the prospect theory states that people derive utility not from absolute levels of wealth but from gains and losses, with relation to the reference point. This is why value function (with changes) and not utility function (with wealth states) is considered. General explanation for this assumption, known as reference dependence, was proposed by D. Kahneman and A. Tversky. After many experiments they noted that our perception works in such way, that we are rather more sensitive to changes of attributes than to absolute magnitudes.

Another very important feature is that the value function captures loss aversion, so people are much more sensitive to losses than to gains of the same level. Loss aversion makes the value function much steeper in the region of losses. The value function is also concave in the region of gains and convex in the region of losses. This is known as a diminishing sensitivity and it implies that, for example, a change of gain (or loss) from $100 to $200 has a higher utility impact than change of gain (or loss) from $1,000 to $1,100. The concavity in case of gains makes that people are risk averse over moderate probability of gains. In losses regions people tend to be more risk seeking (over moderate probability).

The last feature of the prospect theory is that people don’t weight outcomes by using objective probabilities, but rather by decision weights. The decision weights are obtained by means of a weighting function, where objective probability is taken as an argument. The weighting function was proposed by A. Tversky and D. Kahneman [Kahneman, Tversky 1992]. This function has many interesting features. At first, it is consisted with empirical data [Kahneman, Tversky 1992]. It has only one parameter and encompasses both concave and convex regions. The general conclusion is that the weighting function overestimates low probabilities and underestimates high probabilities. People usually have tendency to overweight unlikely extreme outcomes. D. Kahneman and A. Tversky found that it is consistent with the fact that people like both lotteries and insurance. In case of lotteries, value of winning $1000 with probability 0,001 is higher than sure gain of $1. But at the same time, people prefer to pay $1 than to lose $1000 with probability equal to 0,001.

               When people evaluate uncertainty, two natural boundaries are considered: certainty and impossibility. It is known that an increase of 0.05 in the probability of winning has stronger impact when it changes the probability from .95 to 1.0 or from 0 to 0.05. When the probability of winning changes from 0.40 to 0.45 or from 0.7 to 0.75 the influence is much weaker. The diminishing sensitivity is a reason why weighting function is concave near 0 and convex near 1. It is however worth noticing that the function is not well-behaving near the endpoints. If we have a very small probabilities, they can be both overestimated or neglected.

It is also worth noticing that the transformed probabilities do not represent erroneous beliefs, but they are exactly decision weights. So someone who is offered a 0.01 chance of winning, knows exactly what it means to have a 0.01 probability of occurrence.

               D. Kahneman and A. Tversky also proposed a specific value function, with different form in region of gains and losses [Kahneman, Tversky 1992]. In case of the value function the following problem arises: does the propensity to risk affect its parameters? We could assume that parameter present in the value function is not constant and could be different in other groups of people. So how this parameter could be specified? In this article an attempt was made to treat one of the parameter as a propensity to risk, calculated by means of proposed measures of propensity to risk. The general conclusion is that the higher propensity to risk, the more steeper value function. Therefore frequency measure of propensity to risk differentiates values assigned to each gain (loss). On the basis of the conducted analysis it could be state that specification of the value function by means of frequency measure of propensity to risk gives results that don’t deviate from the prospect theory, but this finding should be confirmed in further analysis.         

5. Empirical example

               The main aim of the undertaken research was to use proposed measures of propensity to risk (frequency measure and measure based on game against nature) to specify the value function proposed by D. Kahneman and A. Tversky [Kahneman, Tversky 1992]. Mainly students took part in the survey. 48 respondents were asked to fulfill the questionnaire. Most of the participants were women (75%). Majority of respondents were 21 – 22 years old. Most of men were at least 23 years old.

In the first part, the frequency measure of propensity was applied. In this measure propensity to risk is measured on the nominal scale. In decision problems we have only two outputs. Intensity of propensity to risk is a share of respondents who make risky choices. That kind of problems were analysed by D. Kahneman and A. Tversky in the first version of the prospect theory. The respondents had to answer the question (problem S1), in which expected value of the lottery was the same as a sure gain. It was assumed that people who choose lottery have strong propensity to risk.

               In the next part, propensity to risk was measured by means of measure, where there are many outputs presented in the form of game against nature. That kind of measures could be useful in context of cumulative prospect theory. The respondents had to decide, which strategy to choose in the problem (problem S2), where there were five strategies with different level of propensity to risk. To build that kind of output matrix Hurwicz criterion was applied. If we know how many respondents choose each strategy, we could calculate propensity to risk as a weighted average.

               What seems to be very interesting, propensity to risk calculated for all respondents by means of two measures is the same and equal to 0,354. The general conclusion is that propensity to risk among all respondents is rather low. Also trigonometric measure of propensity was applied. According to sex, in the first problem (S1), in which propensity is measured on nominal scale, propensity was a little higher for women. In the second problem (S2), where propensity is calculated on the basis of output matrix with many strategies, propensity was higher in case of men. These differences are probably due to fact, that only 12 men took part in the research and it is hard to make a valid conclusions. Despite this drawback, obtained results were used to specify the value function proposed in cumulative prospect theory to show how propensity to risk could affect choices under risk and uncertainty.

Concluding remarks

               Many researchers state that it is hard to implement very interesting findings of behavioural economics. It is usually not so obvious how to implement general findings proposed by behavioural economists. In this article conception of propensity was proposed that could be useful in behavioural economics. Definition of propensity, propensity measures and different theories of propensities were discussed, mainly in context of propensity to risk. Two versions of the prospect theory were also considered, the first version [Kahneman, Tversky 1979] and cumulative prospect theory [Kahneman, Tversky 1992].

The main aim of the article was to propose a new possibility of specification of the value function in the prospect theory, where the measures of propensity to risk are used as a parameters. Two measures of propensity to risk were introduced. The first measure is a share of risky choices in all considered cases. This measure is useful if we have two outputs with known probabilities (decisions are made under risk), as in the first version of the prospect theory. The information about propensity to risk is exhibited on nominal scale.

The second measure of propensity is based on game against nature with unknown probabilities (decisions are made under uncertainty). Output matrix is constructed with respect to Hurwicz criterion. It enables identification of intensity of propensity to risk. This measure might be useful in situations, where we have many outputs, such as in cumulative prospect theory. Presented ideas should be verified in futher research to confirm that proposed specification of the value function is consisted with real decisions made under risk and uncertainty.  

Bibliography:

[1] Barberis N.C. (2013), Thirty Years of Prospect Theory in Economics: A Review and Assessment, Journal of Economic Perspectives 27/2013.

[2] Doszyń M. (2008), Statystyczno – ekonometryczna analiza skłonności ludzkich [Statistical and econometric analysis of human propensities], WNUS, Szczecin.

[3] Doszyń M. (2013), Skłonności w ekonomii. Ujęcie ilościowe [Propensities in economics. The quantitative approach], WNUS, Szczecin.

[4] Gillies D. (2000), Varietes of Propensity, British Journal for the Philosophy of Science, 51.

[5] Hozer J., Doszyń M. (2004), Ekonometria skłonności [Econometrics of propensities], PWE, Warszawa.

[6] Kahneman D. (2011), Thinking, Fast and Slow, Penguin Books.

[7] Kahneman D., Tversky A. (1979), Prospect theory: an analysis of decision under risk, Econometrica 47, 2. 

[8] Popper K.R. (1959), The Propensity Interpretation of Probability, British Journal for the Philosophy of Science, 10.

[9] Popper K.R. (1990), A World of Propensities, Thoemmes, Bristol.

[10] Rabin M. (1996), Psychology and economics, Department of Economics, University of California - Berkeley.                    

[11] Tversky A., Kahneman D. (1992), Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, 5(1992).

[12] Wilkinson N. (2008), An Introduction to Behavioural Economics, Palgrave Macmillan.

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