#### 5.3. Decision Making under Ambiguity

As we already discussed, in each experiment, corresponding to different Italian elections, each subject reported her/his minimum selling price for nine binary lotteries, each lottery paying a given amount of money if a centre-right party/coalition of parties gained a stated share of votes in a forthcoming election. This generated seven different space partitions for complementary lotteries covering the full space of events (100% votes; see

Table 1 and

Appendix A). We also asked subjects to report the corresponding judgement of probability for the same lotteries.

Assuming linearity, as in the case of risk, decision weights under ambiguity can be computed as w(p) = z/x. In other words, the ratio between the minimum selling price and the money outcome of the lottery provides the subject’s decision weight associated with the uncertain event. As stated in

Section 5.2, given that our subjects exhibited probability sophistication under risk, the mere existence of non additivity in probabilities (i.e., sum of probabilities different from 1) can be seen as evidence of source preference according to Hypothesis 1.

We computed

tertiary additivity (

TA) (corresponding here to partitions: A + B + C + D, A + B + H, A + E + D, C + D + I) and

binary complementary (

BC, corresponding here to partitions: A + G, F + D, H + I) to investigate whether subjects showed

ambiguity preference or

ambiguity aversion, as defined and explained in

Section 3. In fact, the sum of decision weights for complementary events, and similarly, the sum of judgemental probabilities provides a measure of reaction to ambiguity under CPT.

Table 2 below reports the median value across subjects of the sum of decision weights and judgmental probabilities for each additivity test. In all the experiments, the median sum of decision weights exceeded unity for TA. These results support Hypothesis 1 of source preference.

Table 2 also shows that the median sums generally increased in the number of space partitions. As far as BC is concerned, we can observe that it is very close to one for decision weights, while it is almost equal to one for judgemental probabilities

4.

Table A3 in

Appendix A reports the proportion of subjects violating each additivity test: these data also support the hypothesis of source preference. In fact, a large proportion of subjects exhibited preference for ambiguity (i.e., a sum greater than unity).

As previously argued, likelihood insensitivity is an important feature of the weighting function and a well-documented dimension of reaction to ambiguity (see

Section 2 and

Section 3): It implies a tendency to overweight low likelihoods and underweight high likelihoods, while being less sensitive to intermediate likelihood variations. In addition, we can use this dimension to test the hypothesis of source preference. In particular, as long as likelihood insensitivity can be captured using the concept of bounded subadditivity (BA, Tversky and Fox, 1995 [

34] (p. 272) and

Section 3 above), a different BA between decision weights under risk and under ambiguity can be interpreted as evidence of ambiguity reaction. Given that our subjects exhibited probability sophistication under risk, the mere existence of BA for the ambiguous lotteries can be taken as evidence of reaction to ambiguity in our experiments. For this reason, we computed lower subadditivity (LSA) and upper subadditivity (USA) tests, using the space partitioning of the lotteries corresponding to the experimental scenarios

5.

Table 3 shows the median values of the individual medians across the various tests—denoted wlsa, wusa, jlsa, jusa. BA is satisfied, for example, when wlsa ≥ 0, wusa ≥ 0, whereas expected utility holds when wlsa = wusa = 0.

Table 3 also reports a measure of global sensitivity, sw = 1-wlsa-wusa, computed by using the first three indexes of LSA and USA: the smaller sw, the higher likelihood insensitivity; if sw = 1, SEUT holds. A similar analysis applies to judgmental probabilities. From

Table 3, it is apparent that the median values of the individual medians were all significantly greater than zero (

p < 0.01)

6. We can take these results as evidence of the existence of BA in the experimental data. Moreover, it turns out that the global sensitivity indexes for judgmental probabilities were higher than the corresponding indexes for decision weights in Milan 1, Milan 2 and Milan 3, supporting Hypothesis 1. The opposite was true for Alessandria.

This means that decision weights exhibited more BA than judgmental probabilities in Milan 1, Milan 2 and Milan 3, consistently with the Tversky and Fox (1995) [

34] hypothesis that decision making follows a two-stage process, thereby judgmental probabilities are firstly evaluated and then transformed into decision weights (see also Wakker, 2004 [

40]). Moreover, whereas in Milan 2, LSA had less impact than USA for decision weights, the opposite result was true for Alessandria, Milan 1 and Milan 3. Hence, the overweighting of low-likelihood events (the

possibility effect) was stronger than the underweighting of high-likelihoods events (the

certainty effect) in these latter experiments

7.

The results of this section are interpreted as evidence of source preference for ambiguity in our framework, supporting Hypothesis 1. First, experimental subjects showed risk neutrality and preference for ambiguity. Second, violation of additivity was stronger when using decision weights as long as the median value of the sum of decision weights was greater than the corresponding value for the sum of judgmental probabilities: This was partly confirmed by the sign test for equal medians and the

t-test for equal means (see

Supplementary Materials). Strong ambiguity preference was also detected when considering likelihood insensitivity.

Risk was generated by lotteries based on chance devices. Uncertainty was generated by lotteries based on future election outcomes. Therefore, it is possible that experimental subjects showed a preference for betting on uncertain events for which they had emotional involvement and/or comparatively superior knowledge than on affect-poor risky events based on luck. Moreover, this result may depend on (positive) emotion and knowledge providing extra utility to subjects. The next sub-section will investigate these hypotheses.

#### 5.4. Emotion, Knowledge and Decision Making under Ambiguity

This sub-section will investigate the hypothesis that emotion and knowledge had an effect on ambiguity attitudes (see Hypothesis 2 in

Section 3). More specifically, this sub-section will investigate the role of emotion and knowledge in generating (i) preference for ambiguity, (ii) stronger violation of the additivity tests when using decision weights rather than judgmental probabilities, and (iii) effects on the elevation and the shape of weighting function and judgemental probabilities

8.

As argued above, we can take the sum of decision weights and the sum of judgmental probabilities as a measure of reaction to ambiguity. For given subject i = 1 … N and state partition of events t = A + B + C + D, A + B + H, A + E + D, C + D + I, A + G, F + D, H + I, we covered the 0–100% election outcome space. For example, considering partition A + B + C + D, the sum of decision weights for subject i is DW

_{iABCD} = W

_{i}(A) + W

_{i}(B) + W

_{i}(C) + W

_{i}(D), while the sum of judgmental probabilities is: JP

_{iABCD} = J

_{i}(A) + J

_{i}(B) + J

_{i}(C) + J

_{i}(D). Assuming separability between decision weights and judgemental probabilities, we can take their difference, Y

_{it} = DW

_{it} − JP

_{it}, as a measure of reaction to ambiguity that depends on a subject’s willingness to bet. Finally, Y

_{it} was regressed on the subjects’ individual characteristics, controlling for space partitioning under a linear specification:

Emotions and Knowledge are the subject’s self-reports (or actual questionnaire scores, when applicable). Individual is a vector of subject’s characteristics

9. Space partitioning is a matrix of dummy variables for different partitions

10; and ε

_{it} is an additive normally distributed random term.

Table 4 below presents the results of random-effects Generalised Least Squares estimates controlling for individual unobserved heterogeneity

11. The findings show that being either more confident in Milan 1 (i.e., showing higher self-assessed knowledge, see columns (a) and (b)) or more competent in Milan 2 (i.e., showing higher knowledge of politics, see columns (e) and (f)) was associated with a higher willingness to bet (

p-values < 0.06 or less). This result suggests us that a superior knowledge of the decision context in these experiments enhanced the willingness to bet, consistently with Heath and Tversky (1991) [

10] competence hypothesis. However, in Milan 3, controlling for other students’ characteristics (i.e., being graduate and Italian) that were specific to this experimental session only, it turned out that superior knowledge was associated with a lower willingness to bet (

p-values < 0.07; see columns (g) and (h) of

Table 4). A possible explanation for this finding is that a superior knowledge of Italian politics had a stronger direct effect on judgemental probabilities (i.e., Kilka and Weber’s 2001 [

14] finding) than on the willingness to bet, thus lowering the gap between decision weights and judgemental probabilities in Milan 3. Finally, note that subjects in Alessandria showed no knowledge effects (but see

Table 5 below).

Emotions, however, were not correlated with the willingness to bet but marginally in Milan 2 (

p-value 0.091) for negative emotions

12. A possible interpretation for this finding is that, when knowledge matters, people disregard their emotions (Schwarz, 2012 [

29]). Alternatively, this finding may signal the absence of an emotion-based elevation effect, but not necessarily of a curvature effect on the decision weighting function.

To test for the effects of emotion and knowledge on LSA and USA, hence on the elevation and shape of the weighting functions and judgemental probabilities, we further ran the linear model:

I

_{ih} is the test for either LSA or USA

13, where is i = 1 … N is the individual subject and h is the specific test;

**Individual** is a vector of individual controls (see footnote 9); Test is a matrix of dummy variables that depends on the space partitions

14; finally, υ

_{it} is an additive normally distributed random term.

Table 5 and

Table 6 below show the results of random-effect GLS estimates that use robust standard errors clustered by subjects.

Table 5 refers to Milan 1 (columns (a)–(d)) and Alessandria (columns (e)–(h)). These findings show that, in both experiments, positive emotions and a superior knowledge of the decision context were positively correlated with the overweighting of low likelihoods for decision weights (columns (a) and (e) for WLSA,

p-values < 0.05), while no correlation was detected with high likelihoods (columns (c), (g) for WUSA) in both experiments. Stronger negative emotions were associated with less overweighting of low likelihoods for decision weights in Milan 1 (column (b) for WLSA,

p < 0.1). For Milan 1, these results are consistent with the hypothesis that negative emotions induce a more neutral attitude to uncertainty. No significant emotion and knowledge effects were found for judgemental probabilities (see columns (d) and (h))

15. These results suggest the presence of an asymmetric curvature effect of emotion and knowledge on the decision weighting function, enhancing the willingness to bet, but at low likelihoods only (i.e., increasing lower subadditivity), supporting our Hypothesis 2.

Table 6 below (columns (a)–(d)) reports the estimates of Equation (2) for Milan 2. It turns out that a superior actual knowledge was associated with higher overweighting of low likelihoods (low subadditivity) (no correlation between knowledge and underweighting of high likelihoods was detected: column (c); the results for JPUSA are not reported), for both decision weights (column (a) and (b),

p-value < 0.01) and judgemental probabilities (column (d),

p-value < 0.05). This latter result suggests a direct, but asymmetric competence effect on the degree of belief. Relative to Milan 1 and Alessandria, emotions did not affect BA in Milan 2.

The picture emerging from Milan 3, however, is different.

Table 6 (columns (e) and (f)) illustrates.

Table 6 shows that stronger positive emotions were associated both with lower overweighting of low likelihoods (WLSA,

p-value < 0.1), and with higher underweighting of high likelihoods (WUSA,

p-value < 0.01) (lower but also upper subadditivity). A possible interpretation of this result is that subjects who felt more positive about the election victory of the centre-right coalition in this election also were fearful of an electoral defeat, irrespective of the likelihood associated with each share of votes, which gave them disutility.

Alternatively, subjects who felt more positive about the victory engaged in some sort of “magical thinking” (Heath and Tversky, 1991 [

10] (p. 8)). Here, this “illusion of control” might imply the belief that, by not being too optimistic in pricing lotteries, one could exercise some control over the outcome before the election by not “bringing bad luck” to the preferred party. However, note that stronger negative emotions were associated with lower underweighting of high likelihoods (WUSA, column (g)), as expected.

The positive correlation between actual knowledge of the decision context and underweighting of high likelihoods (

Table 6 columns (f) and (g) for WUSA,

p-value < 0.01) and between actual knowledge and overweighting of low likelihoods (column (h) for JPLSA,

p-value < 0.05; no effect was detected on JPUSA, not reported) is more puzzling. Whereas the latter effect is consistent with the hypothesis that a superior knowledge boosts the degree of belief at low likelihoods (see also

Table 6, column (h)), the former effect is not easily explained. A possible interpretation is that, in this sample, being more knowledgeable generated disutility from predicting the correct result, especially when the likelihood of an election victory was high. Finally, please note that Italian subjects were associated with a more rational assessment of election probabilities (which was significant only at low likelihoods, column (h),

p-value < 0.05), in line with what one would expect.

The different results on the effect of emotion and knowledge on the curvature and elevation of the weighting function and judgemental probabilities could be better interpreted by considering the different political and electoral contexts. In 2001 (Milan 1), the centre-right House of Freedoms coalition won the general elections and did very well in Milan. Political scientists consider this election as a watershed: for the first time in the Italian Republic’s history, the majoritarian electoral system produced an alternation in government between the incumbent centre-left coalition and the incoming centre-right coalition (Newell and Bull, 2002 [

48]). By 2004, PM Berlusconi’s

Forza Italia was less popular and performed badly in the European elections, as one would expect if citizens use European elections both to express their dissatisfaction with governing parties and to signal their sincere political preferences (especially considering the proportional voting system that is in place for European Elections, Hix and Marsh, 2007 [

49]).

In 2013, the political and electoral context was radically different. First, and most importantly, the new anti-establishment Five Stars Movement (5SM), completely disconnected from the bi-polar centre-right vs. centre-left division characterising the Italian political space since 1994, entered the political arena. In other words, as long as the 5SM was not in the event space before 2012, starting with this election, the assessment and emotional involvement associated with centre-right coalition parties may have changed drastically, especially among younger voters (i.e., 18–24 years old), who strongly supported 5SM in both the 2013 and 2018 general elections

16. Actually, in 2013, the centre-right coalition lost both popularity and the elections (Newell, 2013 [

50]); although it regained votes and popularity in 2018, when the election resulted in a hung parliament (Chiaramonte, 2018 [

51]). However, this better performance was due exclusively to

Lega Salvini-premier, which—despite being a component of the centre-right coalition—was able to preserve its anti-establishment features and attract the younger voters. This context may partly explain why, in the experimental data, there is a clear trend towards more negative emotional involvement (and lower emotional variability) for the centre-right coalition in Milan 2 (2013) and Milan 3 (2018) compared to Milan 1 (2001) and Alessandria (2004) (see

Table A6 in

Appendix A). This is particularly true for Milan 3. In fact, although one would expect that emotional involvement could be lower in Milan 3, both because of the presence of non-Italian subjects and because the questionnaire was in English, the experimental data show more extreme emotional reactions, with a clear prevalence of negative involvement.

Second, the voting system for the general election was changed twice, shortly before voting. In 2013, the new electoral law was basically proportional with thresholds in the House of Commons (where 18–24 years old have the right to vote) and proportional with a majoritarian bonus at the regional level in the Senate (where younger people cannot vote). In 2018, a new electoral law introduced a mixed first-past-the-post/proportional voting system with blocked lists.

The presence of both the 5SM in the event space and the more complex mixed electoral voting systems may have contributed to a change in preferences and an increase in uncertainty, which may partly explain the different effects of emotion and knowledge on the decision weighting function in the 2018 Milan 3 experimental session.