The loTTery-panel Task for bi-dimensional parameTer-free eliciTaTion of risk aTTiTudes

In this paper, we propose a simple task for eliciting attitudes toward risky choice, the Sabater-Grande and Georgantzís (SGG) lottery-panel task, which consists in a series of lotteries constructed to compensate riskier options with higher risk-return trade-offs. Using Principal Component Analysis technique, we show that the SGG lotterypanel task is capable of capturing two dimensions of individual risky decision making: subjects’ average willingness to choose risky projects and their sensitivity towards variations in the return to risk. We report results from a large dataset obtained from the implementation of the SGG lottery-panel task and discuss regularities and the desirability of its bi-dimensionality both for describing behaviour under uncertainty and explaining behaviour in other contexts. keywords Decision-making; Lotteries; Psychometric tests; Risk aversion. resumeN En este trabajo proponemos una tarea sencilla que permite obtener la actitud frente a la toma de riesgo monetario, y que llamaremos tarea Sabater-Grande y Georgantzís (SGG) de riesgo. Esta tarea consiste en una serie de loterías construidas para compensar las opciones de mayor riesgo con un mayor retorno. Utilizando la técnica de componentes principales, encontramos que la tarea SGG es capaz de capturar dos dimensiones de la toma de decisiones individuales: por un lado, la voluntad promedio de los sujetos de elegir proyectos arriesgados y, por otro, su sensibilidad hacia las variaciones en el retorno por riesgo. Presentamos los resultados de una gran muestra de datos obtenidos a partir de la implementación de la tarea SGG, y discutimos las regularidades y la conveniencia de su bidimensionalidad tanto para describir el comportamiento en condiciones de incertidumbre como para explicar el comportamiento humano en otros contextos. PAlAbrAs clAve Aversión al riesgo; Loterías; Tests psicométricos; Toma de decisiones. revistA iNterNAcioNAl de socioloGíA (ris) Special Issue on Behavioral and Experimental Economics Vol. 70, extra 1, 53-72, marzo 2012 ISSN: 0034-9712; eISSN: 1988-429X DOI 10.3989/ris.2011.07.1A 54 • A. GARCÍA-GALLEGO, N.GEORGANTZÍS, A. JARAMILLO and M. PARRAVANO


Introduction
Testing for interdependence across different aspects of behavior requires jointly studying stated or observed attitudes which are informative on the corresponding individual attributes. 1  Beyond the question of what explains what in such studies, the search of associations among decisions in different tasks is a main motivator for experimentalists.A systematic rejection of such associations would confine experimental results to the specific setting in which they were obtained, undermining the practical relevance of our research outside the lab.
In order to produce reliable tests, psychologists invest a substantial amount of effort in (i) developing the task and proposing it to the scientific community, (ii) standardizing the format and applying it among large populations, (iii) generating result distributions by subject category, (iv) identifying successful tasks as reliable approximations of an idiosyncratic factor, and (v) identifying contexts in which behavior correlates with performance in a given task.This process is parallel and significantly synergic to the very important endeavor of producing correct theories on the measured aspect itself.However, metaphorically speaking, looking for appropriate tasks in the absence of a perfect theory is like the practice in medicine of establishing clinical protocols for the cure of a disease even before the disease is fully understood.This paper is inspired by the surprising observation that the process with which the existing tests of risk attitudes in economic domains are chosen and used totally ignores stages (ii) and (iii) above, while (i), (iv) and (v) are rarely performed in an intentional and systematic way.Economists usually aim at testing theories, rather than at relating risk attitudes with behavior in other contexts.Even the need for external risk measurements is often not recognized by some economists 2 , often explaining the effect of risk preferences on observed behavior by theoretically deriving the sufficient conditions for this effect to emerge, thus explaining fact Y by its sufficient (but not necessary) condition X .
The remaining part of the paper is structured as follows: Section 2 reviews economic theories of risky decision making and comments on some devices used to elicit risk attitudes as an external explanatory factor of behavior in other contexts.Section 3 reports results obtained from the application of the lottery-panel test by Sabater-Grande and Georgantzís (2002), SGG.
1 For example, when studying the effects of psychometric intelligence on complex decisions, psychologists correlate scores in, say, Raven (1976)'s Advanced Progressive Matrices (APM), and performance in complex microworlds, like NEWFIRE or COLDSTORE.On this, Rigas, Carling and Brehmer (2002) note that performance in APM and each one of these complex tasks correlate because they provide different measurements of intelligence. 2Some famous examples of inferring risk attitudes without using an external risk elicitation task are Cox and Oaxaca (1996), inferring risk attitudes from bidding in private value auctions, Goeree, Holt and Palfrey (1999) whose data are from laboratory matching pennies games and Campo, Guerre, Perrigne and Vuong (2002) on real timber auctions.Section 4 concludes.In a longer working paper, we provide more information on the design of the test, as well as instructions for subjects and the experimenter.3

Theories and tests of risk attitudes
An early explanation of why subjects do not evaluate risky choices by their mathematical expectation is attributed to the Expected Utility Theory (EUT) by von Neuman and Morgestern (1944).According to the theory, when comparing a lottery . The preference for less risky projects is then explained by a negative second derivative of

) (x U
, implying a decreasing marginal utility from money, a condition often used as synonymous to risk aversion.Despite its survival as the main paradigm in economics as observed by Rabin and Thaler (2001), the EUT was proved to be an incorrect descriptive model since Allais ' (1953) paradox, emerging when subjects are faced to alternative lottery pairs with same probability/reward ratios.According to (1), such lotteries should be ranked in the same way, whereas people systematically change their choice in favor of the certain payoff when this becomes part of the feasible set.Kahneman and Tversky (1979) proposed an alternative model, Prospect Theory (PT), assuming that people implicitly use non linear weights ) ( p w to evaluate probabilities.Therefore, in our example, 1 L would be strongly preferred to 2 L , if: (2) PT accommodates Allais' paradox, whereas it reduces to EUT for p p w = ) ( . Tversky and Kahneman (1992) assumed later a power utility function defined separately over gains and losses: . So a and b are risk aversion parameters, and λ is the coefficient of loss aversion.This new version, called Cumulative Prospect Theory (CPT), defines probability weighting over the cumulative probability distributions, offering an explanation of risk-loving behavior for payoffs below their reference point (losses), while exhibiting risk-averse behavior for rewards above their reference point (gains).The form of the probability weighting function proposed by Tversky and Kahneman (1992)  .Therefore, in its simplest formulation, CPT explains risk attitudes using a minimum of four parameters, a , b , λ and γ.Our overview does not pretend to narrate the history of economic theories of decision making. 4We simply want to stress the fact that the evolution of these theories achieves the aim of accommodating phenomena which invalidated earlier theories by the use of more degrees of freedom.
Contrary to this evolution of theories towards more complete and complex descriptions of human behavior in risky environments, all tests currently used are fundamentally unidimensional, despite their creation in the post-PT era.This does not mean that all studies of behavior under uncertainty have ignored the multi-dimensional approach dictated by modern theories.In fact, a fruitful line of research has specifically designed and analyzed data obtaining parameters for utility and probability weighting functions. 5However, in order to produce ready-to-use data, the elicitation of risk attitudes as an explanatory factor of behavior in another context should not depend on the parameterization or even the theory used. 6  A measure of risk aversion is obtained in recent economic studies by the use of the Holt and Laury (2002) HL procedure.Although the task was not, initially, proposed as an external riskrelated task to explain behavior in other contexts, it has served this purpose in several occasions. 7Due to its uni-dimensionality, costlessly allowing a one-to-one mapping of choices on specific utility parameters, the test entails a possible loss of information due to underspecification of risk attitudes, which is also likely to reduce its power to explain behavior in other contexts.This is also true for the whole set of alternative procedures used by economists to elicit risk attitudes. 8The task elicits one individual datum from each block of 10 binary choices, designed to obtain the switching point from a less risky to a more risky alternative.
Furthermore, the nonlinearity of responses to probabilities has even been confirmed at the level of neural responses by Hsu, Krajbich, Zhao and Camerer (2009), and, for aversive outcomes, by Berns, Capra, Chappelow, Moore and Noussair (2008), while it is rejected in a study of neural signals reflecting reward uncertainty reported by Schultz et al. (2008). 6Mapping choices on parameters of utility and probability weighting functions is further complicated by Harrison and Rutström's (2009) observation that we may even have to switch between theories in order to account for the heterogeneity observed. 7It has been used to explain behavior in strategic games (Goeree, Holt and Palfrey, 2003), agricultural economics (Lusk and Coble, 2005), risky settings outside the lab (Harrison, List and Towe, 2007), and setups relating risk attitudes and discounting (Andersen, Harrison, Lau and Rutström, 2008). 8A variety of alternatives to HL, adopted by Wakker and Deneffe (1996), Bleichrodt and Pinto (2000), Abdellaoui (2000) and Abdellaoui, Bleichrodt and Paraschiv (2007), use the trade-off method based on a series of binary choices between lotteries aiming at separating between attitudes toward consequences and attitudes toward probabilities.A second approach, adopted by Hey and Orme (1994), Camerer and Ho (1994), Carbone and Hey (2000) and Stott (2006) uses a large number of independent binary choices between lotteries to estimate risk attitudes.Both sets of procedures are specific to the EUT and are even more time-consuming and cognitively demanding for the subjects than the more frequently used HL procedure.This causes a practical problem since some choices do not satisfy the "single-switching" condition.Posterior applications have opted for different solutions to this problem, leading to a variety of alternative implementations which, together with the plethora of designs aimed at identifying other biases 9 of the set up, have created an -undesirable, for our purposesplethora of non comparable datasets.Contrary to the problem of non comparability among small data sets, several studies 10 use hypothetical simple questions among large and even international samples, which however have not been used to explain behavior in other contexts.
A broadly used test among psychologists is Zukerman's (1978) Sensation Seeking Scale (SSS) with which our test exhibits some correlation 11 .The test is structured as a YES-NO questionnaire on attitudes towards risky activities under four subscales separating subject's riskiness in different domains, none of which is strictly speaking financial.The economic domain is used in the Iowa Gambling Task (IGT), introduced by Bechara, Damasio, Damasio and Anderson (1994).The task was originally aimed at measuring a subject's difficulty to identify the most profitable deck, from which he or she should, thereafter, extract all cards.Using the task as an external risk attitude elicitation device implies significant loss of control, because it mixes risk preferences with a subject's learning ability (a "slow" learner can be confused with a risk loving subject or one with low levels of loss aversion) and it does not fully account for different learning histories.For space reasons, we will not review other tests occasionally used to elicit risk attitudes as an explanatory factor of behavior in other contexts.Rather, we will risk a generalization.All existing tasks suffer from either lack of systematic replication in a stable format generating statistics with large comparable datasets, or they are insufficiently justified as measures of risk attitudes isolated from other parallel phenomena.Furthermore, they are all uni-dimensional.

The SGG lottery-panel test
The SGG lottery-panel task was originally used to study risk preferences parallel to cooperation/competition in prisoner's dilemma games.Riskier subjects were found to be more cooperative.The task consists of four different panels, like those in Figure 1, every one of which contains ten different lotteries.In each lottery, subjects can win a payoff ) (x with a probability ) ( p and otherwise nothing.2006) on the embedding bias induced by the fact that subjects tend to change their switching point when some extreme alternatives of binary choice are removed. 10See Wang, Rieger and Hens (2010) and Weber and Hsee (1998, 1999).  1This is based on small sample reported in Georgantzís, Genius, García-Gallego and Sabater-Grande (2003) in which only results from the first panel of the SGG test exhibited a weak correlation (-0.248) with SSS on the expected direction: more sensation seeking, riskier choices.
Subjects choose (marking the preferred lottery as in the example of Figure 1) one of the ten lotteries from each panel.In the implementation of the task with real money, only one of these four panels, selected randomly at the end of the session, is used to determine a subject's earnings in the experiment.The range of winning probabilities in all panels is the same (from 1 to 0.1 in steps of 0.1).The payoff associated to each lottery's winning probability is constructed using the rule: . The parameter j c is a constant amount of money which is fixed for this dataset to 1€.The parameter premium, which generates an increase in the lotteries' expected values as we move from safer to riskier options within the same panel.All the panels begin with a sure amount of 1€, which is increased as winning probabilities are decreased, resulting in increments of expected values as we move from left to right within each panel.These increments are larger as we move from panel 1 to panel 4.This structure implies that more risk-averse subjects choose lotteries closer to the left of a panel. 12All risk neutral and risk loving subjects should choose the lotteries at the far right extreme of the panels.
Considering the fact that with 4 choices the researcher obtains 4 different observations (as opposed to 10 choices for 1 observation in HL) per individual subject, we can easily see that the test parsimoniously produces a panel rather than a single column of data.By definition, this corresponds to a multi-dimensional description of individual attitudes towards risk.

A large dataset
Since its first implementation, the SGG test has been used in several occasions producing various small experimental datasets. 13Here, we report results from a large dataset 14 (N=785) 12 In terms of EUT, García-Gallego, Georgantzís, Navarro-Martínez and Sabater-Grande (forth.)observe that a subject with constant relative risk aversion (CRRA), as implied in the utility function r x x U r − = − 1 ) ( 1 makes choices which associate higher risk aversion parameters r to safer choices in each panel.Furthermore, for a given risk aversion parameter, weakly monotonic transitions towards riskier choices are predicted as we move from panel 1 to panel 4. 13 Brañas-Garza, Guillén and López del Paso (2008) have shown that choices in the test do not correlate with subjects' mathematical skills.García-Gallego, Georgantzis, Martínez Navarro and Sabater-Grande (2010) warn us that repeated implementation without any intermediate treatment generates regression to the mean phenomena.Implementation by Brañas-Garza, Georgantzís and Guillén (2007) in a gambler anonymous session among pathological gamblers and their spouses captures an unprecedented riskaverse behavior by the latter.Earlier, Georgantzís et al. ( 2003) had studied the effect on choices of knowing expected utility theory and hypothetical vs. real monetary rewards.obtained under comparable conditions, paying special attention to the bi-dimensional nature of decision making and its implications for the explanation of behavior in other contexts.Figure 2 depicts the frequency of choices when all data from all panels are pooled together.Given the variation in prizes and payment methods, this image corresponds to what could be seen as a randomized experiment over the probability space.The peak on the certain payoff captures a certainty effect.A peak on the other extreme (p=0.1) as well as a valley on p=0.9 are both compatible with over-(under-) weighting of small (large) probabilities predicted in PT.Strong attraction of choices towards the "center" (p=0.5) may be the result of subjects' familiarity with the p=½ probability or simply because of an embedding bias similar to that reported by Bosch-Domènech and Silvestre (2006) on HL.No matter what causes this attraction to the center, this property favors close-to-normal distributions of the resulting variable, making it appropriate for simple OLS regressions.

FIGURE 2 HERE
In Figure 3 we present the same dataset broken down by panel, gender and reward method (hypothetical, N=384; real money, N=401).Males are less risk-averse than females.However, males and females behave in more different ways when playing hypothetical lotteries than real ones.Actually, with real rewards, mean choice varies significantly across genders only in panel 3 and 4 (2.7 and 3.9 percentage points at 5% and 1% confidence level, respectively).Responsiveness to risk-premium increases, captured by choice variation across panels, is similar for males and females.Specifically, when faced with hypothetical payoffs, both males and females make less risk-averse choices, the higher the reward, while, counterintuitively15 , when playing with real payoffs, riskier choices are observed in panels with lower risk-returns.

FIGURE 3 HERE
We have argued that it should be a main concern for experimentalists and decision theorists whether a subject's decision under one condition meaningfully relates to behavior under another condition.

FIGURE 4 HERE
Figures 4 and 5 present an aspect of behavior which is missed by other tests.Each graph presents the joint density of individual choices across panel pairs.Each color represents a percentage, i.e. the proportion of subjects whose choice combinations in each panel pair correspond to that specific chart label.Higher risk aversion in one panel predicts a higher risk aversion in another and, at the same time, reactions to the variation of risk returns across different panels seem to be rather moderate.

FIGURE 5 HERE
As expected, reactions are more visible across more "distant panels", showing that a bigger shock is necessary to guarantee a change of choices.This within-subject pattern reproduces in a more reliable way what we have already observed, namely, that the use of real rewards makes subjects to switch to safer options in the presence of higher returns to risk.

Principal Component Analysis
It is clear that multidimensional descriptions of risk attitudes require obtaining more than one choice per individual.This is done by the SGG test through the use of the four panels.However we have not shown yet that, first, the additional information obtained significantly improves the description of behavior and, second, that this improvement leads to a higher power of our task to explain behavior in other contexts.
We use Principal Component Analysis (PCA) to construct two synthetic variables (the first two components) capturing 85% of subjects' choice variance.These variables have the following advantages: (1) they are subject to economic interpretation and, (2) since they are by construction orthogonal among each other, they can be used as explanatory variables of the same model.Intuitively, the first component can be interpreted as an arithmetic mean of choices across the four panels given that the loads of each panel in this component are similar and of the same sign.The second component involves a juxtaposition of panels 1 and 2 on one hand and 3 and 4 on the other, which can intuitively be seen as a measure of sensitivity to riskpremium variations.As observed in Table 1, the component is loaded more by the extreme panel 1 (negatively) and 4 (positively) than by choice differences across the adjacent panels, 2 and 3. Intuitively, the first component is increasing in the average probability of the lottery chosen in the four panels and can be seen as a standard measure of risk aversion.The second component can be seen as a measure of a subject's sensitivity to variations in the return to risk in the "counterintuitive" direction of lower risk taking in the presence of higher returns to risk.While this confirms our comments on Figures 4 and 5, it provides a formal motivation for the use of bi-dimensional descriptions of risk attitudes, summarized as individual choice averages and choice variability across contexts (panels).

TABLE 1 HERE FIGURE 6 HERE
Using these two components we reconsider gender and hypothetical/real reward effects.It can be seen on Figure 6 that gender differences are specific to the first component, while they diminish or even vanish in the second component.Therefore, males are less risk averse than females but both genders are similar in terms of their sensitivity to variations in the return to risk.Regarding differences between hypothetical and real rewards, both components are relevant.According to the first component, subjects make safer choices in hypothetical lotteries, while, according to the second component they switch more across panels with real rewards, but opposite to the expected pattern of riskier choices for higher risk-returns.
3.3 Using the SGG test to explain behavior: An example.
García-Gallego, Georgantzís, Pereira and Pernías-Cerrillo ( 2005) conducted experiments on pricing where firms have some captive clients and they also compete for informed consumers using price comparisons on the Internet.During 50 periods, subjects face the dilemma of setting high prices to benefit from captive clients or lower prices to compete for informed consumers too.Parallel to the main experiment controlling for more and less competitive markets and complete or incomplete price indexing (Treatments T1-T4), the SGG risk elicitation task was implemented with hypothetical rewards.
Following the estimates on Table 2 and abstracting from the specifics of the main experiment, 16 we see that risk attitudes provide significant explanatory power for the pricing behavior observed.In fact, both first and second principal components are necessary to identify the effect of risk attitudes on pricing behavior.On one hand, the first component capturing safe choices is associated to more competitive pricing.That is, more risk-averse subjects set lower prices in order to avoid the risk of not having the lowest price indexed by the engine.On the other hand, the second principal component is also associated with lower pricing.This means that subjects, recognizing the increased profitability of riskier choices across panels, also realize that setting higher prices guarantees profits which do not depend on the excessive randomness of the search process.

Conclusions
We have discussed the properties of risk attitudes as captured by the SGG elicitation task.The danger of using unidimensional descriptions of risk attitudes goes beyond the incompatibility with modern economic theories like PT, CPT etc., all of which call for tests with multiple degrees of freedom.Faithfull to this prescription, the contribution of this paper is an empirically and endogenously determined bi-dimensional specification of risk attitudes, sufficient to describe behavior under uncertainty and necessary to explain behavior in other contexts. 16Apart from the expected effect of firm number on prices, the model identifies a decreasing time trend and adoption of higher prices when the firm has not managed to be the cheapest on Internet in the last period.0.0%-2.0%2.0%-4.0%4.0%-6.0%6.0%-8.0%

Figure 3 .
Figure 3. His stograms of sub bjects' probabil lity choices by p panel, implemen ntation conditio ons and gender

Figure 4 .
Figure 4. Subject's choices across panel pairs for hypothetical payoff lotteries.Legend percentage ranges refer to proportion of subjects choosing combinations indicated in each chart label.

Figure 5 .
Figure 5. Subjects' choices across panel pairs for real payoff lotteries.Legend percentage ranges refer to proportion of subjects choosing combinations indicated in each chart label.

Table 1 .
Kern nel density estim mates for first a and second com mponent scores, by gender and d reward method Cumulative percentages of components eigenvalues (top) and loads per component (bottom).

Table 2 .
Random effects GLS regression: Pricing explained by risk attitudes.