Ambiguous Correlation Research Appendix A

A A.1 Appendix Design of earlier experiments (2013 and 2014) We conducted two earlier experiments (in 2013 and 2014) to study the hypothesized behavior of One vs Two. The two experiments had a similar main treatment to the experiment reported in the body of the paper, although in the 2014 experiment we changed the way we handled symmetry (which became a redundant assumption due to the design of the 2015 experiment) and we improved the explanation of the incentive system. More importantly, in the 2014 experiment we added control treatments. In describing the design we will focus on the 2014 experiment as it relates to the 2015 experiment reported in the body of the paper, and note any di¤erences from the 2013 experiment. A.1.1 Subjects, stakes and number of balls Subjects were recruited from UBC’s Vancouver School of Economics subject pool using ORSEE (Greiner, 2015). We made sure that no subject participated in more than a single treatment during all iterations of the experiment (2013-2015). In addition to a $10 participation fee, they could earn up to $51 ($101 in the 2013 experiment) during the one hour experiment. That is, the absolute value of the stakes in the earlier experiments was twice (2014) and four times (2013) higher than in the 2015 experiment. However, the marginal incentive was constant (at $1) through all the experiments. This implied that the proportional premium required to exhibit strict preference is higher in the later experiments than in the earlier one. In other words, the strict ambiguity sensitivity revealed in the later experiments is “stronger” (in relative terms) than in our earlier experiments. A minor di¤erence between the 2013 experiment and later experiments is that in the …rst experiment each urn contained 10 balls, while in later experiments we used only 2 balls. A.1.2 Choice problems Choice problems were organized in pairs which allowed us to infer strict ranking from choices by slightly varying the prizes. Compared to the pairwise choice design used in 2015, there were no pairs with equal payments (which allows us to infer weak preference). Instead, lack of revealed strict preference 32 (choosing the bets with the higher payment) is interpreted as indi¤erence. The design establishes an upper (lower) bound on the indi¤erence (strict preference) class. Complete instructions may be found in Appendix B.2. In the 2014 experiment, subjects were presented with the following choice problems.26 One vs Di¤ erent: Choose between a bet on the color of the ball drawn from a single urn and a bet that pays if the balls drawn from the two urns have di¤erent colors. One vs Same: Choose between a bet on the color of the ball drawn from a single urn and a bet that pays if the balls drawn from the two urns have the same color. Same vs Di¤ erent: Choose between a bet that pays if the balls drawn from the two urns have the same color and a bet that pays if they have di¤erent colors. Color symmetry: Choose between bets on the two possible colors of the ball drawn from a single urn. Standard Ellsberg: Choose between a bet on the color of the ball drawn from one of the two ambiguous urns and a bet on the color of a ball drawn from a third urn containing one red ball and one blue ball (a risky urn as in Ellsberg). A.1.3 Control treatments (2014) The 2014 experiment included two control treatments that were designed to investigate our hypothesis that the preference for One over Two indicates subjects’ambiguity concerning the relation between the two urns. An alternative hypothesis is that a bet on the ball drawn from a single urn is preferred because it is viewed as “simple,”while a bet that depends on two balls drawn from the two urns is more “complex.” In order to test this hypothesis, two urns – each with one red and one black ball, were presented to subjects. There is no ambiguity in this environment, and the probabilities of winning 26 The 2013 experiment included two additional choice problems which are described in Appendix B.1.2, but did not include the color symmetry choice problems which were included in the 2014 experiment. 33 when betting on One, Same and Di¤erent are all equal to 0.5. However, the calculation of probabilities in the latter two bets is more involved and requires that the subject multiply probabilities correctly (reduction of compound objective lotteries - ROCL). There exists strong experimental evidence (Halevy 2007; Abdellaoui, Klibano¤ and Placido 2015; Chew, Miao and Zhong forthcoming; Dean and Ortoleva 2016; Gillen, Snowberg and Yariv 2016) that violation of reduction is frequent in the general population, so choosing One over Two might be a manifestation of violation of ROCL. If the frequency of indi¤erence in One vs Two is signi…cantly higher in the risk control than in the two-urns ambiguity treatment, we conclude that the behavior of One over Two re‡ects something beyond the violation of reduction. We implemented also a second control for two reasons, the …rst being that indi¤erence in the risk control above is extremely fragile. Secondly, we wanted to examine whether our interpretation of the preference One over Two can be supported also when the environment is ambiguous. As noted above, there exists empirical evidence that relates ambiguity aversion and violations of ROCL, and such violations in turn may be responsible for a preference to bet on one urn. In this control, two draws were made with replacement from a single ambiguous urn containing two balls (with unknown composition). If a subject exhibits R1 Same in this environment, it cannot be because of ambiguity about how the urns are related since the two draws are made from the same (ambiguous) urn and hence from the identical composition.27 However, our interpretation of One over T wo would be supported if the frequency with which subjects exhibit the ranking R1 Same is signi…cantly higher in the two-urns treatment than in this single-urn control. A.1.4 Symmetry in colors The 2013-2014 experimental designs relied on color symmetry (Ri Bi for i = 1; 2) to identify the behavior in One vs T wo, since we asked subjects to pick a single color to bet on. As a result, a Bayesian subject who holds nonsymmetric beliefs may strictly prefer One over Same and Dif f . Though we believe that symmetry in colors is a very natural assumption in the current setting, we took two alternative approaches to handling the issue empirically. In the 2013 experiment, before subjects made any payo¤-relevant choices, they were presented with four pairs of bets. The symmetry between urns 27 Note that in this environment, R1 Dif f is easily rationalized, for example, by a Bayesian model. Thus we focus on the the choice between R1 and Same. 34 and between colors in the experiment implied indi¤erence within each pair. Following the presentation of the four pairs, each subject was asked, in a non-incentivized question, whether she agreed with the indi¤erences. Since symmetry serves as an identifying restriction to evaluate various models in light of the data, we thought it important to clarify this assumption to subjects. Their response (agree/disagree) indicates their understanding of the symmetry encoded in the experiment. We hypothesized, based on other studies, that most of the subjects will hold symmetric beliefs anyway. However, we cannot exclude the possibility that this form of communication manipulated some subjects with non-symmetric beliefs into symmetry.28 Regardless, we believe that this is a desirable outcome since otherwise one would interpret behavior in light of an identifying assumption (symmetry in colors) that may not hold. In the 2014 experiment we adopted the more conservative approach of asking subjects to make pairwise choices between bets on the two colors, thus identifying possible non-symmetry directly and eliminating any potential contamination of the other choice problems. A.1.5 Incentive structure and hedging As in the 2015, payment was determined by randomly choosing one choice problem before subjects made their choices. Also, each subject was asked to choose at the beginning of the experiment an urn (1 or 2) to bet on. An important distinction from the 2015 experiment is that subjects did not bet on both colors: the subject chose a color (red or black) to bet on in the bets that involve a single urn. As a result, there is no risk that choices consistent with One Two actually re‡ect hedging rather than preference for a bet on bias to a bet on relation. The empirical question whether subjects hedged in spite of our implementation is discussed further below in Section A.2.4. A.2 More Results (2013-2015) A total of 80 subjects participated in 4 sessions of the 2013 experiment. Subjects were paid a total of CA$4,851 (an average of just over $60 per subject). Out of the 80 subjects, a total of 24 were removed from the analysis 28 One may argue that the “instruction” of symmetry may have a¤ected choices made elsewhere. However, we believe that such a scenario is highly improbable and we could not …nd evidence to that e¤ect in the 2014 experiment. 35 reported immediately below. In particular, 18 subjects were removed due to choices violating either monotonicity or transitivity, and 4 were removed for disagreeing with symmetry over colors and urns (see below).29 This leaves 56 subjects. We consider this retention rate to be high when taking into account the strong consistency (transitivity and monotonicity) that we imposed. We attribute this rate to the high stakes (more than $100) employed in the experiment, which provided subjects su¢ cient incentive to minimize arbitrariness and to consider their choices seriously. 87 subjects participated in the main treatment of the 2014 experiment, of which 75 had no monotonicity or transitivity violations. The risk control included 47 subjects, of which 43 had no monotonicity violations. The single-urn control included 42 subjects, of which 37 had no monotonicity or transitivity violations. Subjects were paid a total of CA$6,734 (an average of about $38.26 per subject). All the results reported are robust to employing a less strict retention criteria as reported in Appendix A.2.5. A.2.1 Control treatments (2014) In the risk control, 23 out of 43 subjects (53.5%) did not exhibit strict preference in choosing between One vs Same and One vs Dif f . In the two-urn treatment we …nd that 28 out of 78 subjects (35.9%) exhibited this choice pattern in this two choice problems. The di¤erence is signi…cant at 5% (p-value of one-sided Z test is 0.0330 ). The …nding that almost one half of subjects in the risk control did not reduce objective compound lotteries is consistent with the experimental evidence noted above. The fact that the frequency of indi¤erence is signi…cantly lower in the two-urn treatment indicates that choices in One vs T wo are a¤ected by more than the “simplicity” of a bet that depends on a single draw. In the single-urn control, 8 out of 37 subjects (21.6%) with monotone and transitive choices strictly preferred One to Same, while in the two-urn 29 We removed 11 for choosing in at least one pair of questions two lotteries with prizes of $100, and 7 due, for example, to choices revealing that Same R1 Dif f based on pairwise choice between R1 and the other two bets but choosing consistently with Same Dif f in the direct pairwise choice. Two more subjects in the …rst session were caught cheating and their choices were excluded from the analysis (one of them had nontransitive choices, and would have been removed in any case). 30 A one-sided Fisher exact test, which is very conservative in comparing two binomial samples, yields a p-value of 0.046. 36 Ellsbergian Ambiguity Attitude 2013 # % strictly averse (nonPS) 37 66.1 strictly seeking (nonPS) 4 7.1 neutral (PS) 15 26.8 Total 56 100 2014 # % 29 52.7 0 0 26 47.3 55 100 Table A.1: Ellsbergian ambiguity attitude 2013-2014 treatment 34 out of 75 subjects (45.3%) with monotone and transitive choices exhibited this pattern. This di¤erence is signi…cant at 5% (p-value of onesided Z test is 0.00731 ). In summary, the two control treatments indicate that although a bet on one urn is simpler and also may be attractive relative to the bets Same and Dif f for subjects who do not reduce compound objective lotteries, it’s appeal is signi…cantly strengthened when the relation between the compositions of the urns is ambiguous. A.2.2 Probabilistic sophistication We now discuss the distribution of choices in the two behaviors–Ellsberg’s two-urn classic problem and One vs T wo–that tested probabilistic sophistication. We restrict to subjects who did not exhibit strict color preference.32 Although strict color preference does not necessarily imply non-symmetric beliefs, (even some subjects in the risk control exhibited this behavior), we decided to err on the side of caution and excluded such subjects when calculating adherence to probabilistic sophistication. Results including the other subjects are reported in Appendix A.2.5. Ellsbergian ambiguity (2013-2014) Out of 111 subjects, almost 60% (66 subjects) exhibited strict Ellsbergian ambiguity aversion when asked to choose between betting on a red draw from the risky urn (with probability 0.5 of winning) and betting on a chosen 31 p-value of one-sided Fisher exact test is 0.012. The fraction of subjects who exhibited strict color preference in the single ambiguous urn control is similar (9 out of 39 subjects) and is lower in the risk control (6 out of 44 subjects). 32 37 One vs Two 2013 # % One Two (nonPS) 23 41.1 Two One (nonPS) 4 7.1 One Two 20 35.7 non-symmetric beliefs (PS) 9 16.1 Total 56 100 2014 # % 18 32.7 6 10.9 26 47.3 5 9.1 55 100 Table A.2: One vs Two: 2013-2014 color from one of the urns with unknown composition. Almost 37% (41 subjects) chose in a way that does not reveal ambiguity aversion or seeking, and the remaining 4 subjects exhibited ambiguity seeking. That is, about 63% of the subjects were not probabilistically sophisticated in a standard Ellsberg experiment. This proportion is consistent with existing studies that use certainty equivalent elicitation or choice data.33 One versus Two (2013-2014) There were 41 subjects (37% of 111 subjects) who exhibited One Two, and 10 subjects who exhibited Two One (7 of them ambiguity averse in Ellsberg). These 51 subjects (46% of 111) violated probabilistic sophistication.34 The choices of the remaining 60 subjects (54% of the 111 subjects) can be rationalized by probabilistic beliefs. 2 subjects exhibited Same R1 Dif f , and 12 subjects exhibited Dif f R1 Same (11 of the latter exhibited Ellsbergian ambiguity aversion). Combining both the Ellsberg and One vs Two choice problems, we …nd that only 27 out of the 111 subjects made choices that are consistent with probabilistic beliefs. That is, more than 75% were not probabilistically sophisticated in at least one of the choices. 33 Note that 41 is an upper bound on the number of ambiguity neutral subjects–the proportion of ambiguity neutral subjects may be even smaller since the increment of $1 used in the experiment to detect strict preference may have been too big for some subjects. 34 Remember that we restricted the sample to subjects who agreed with the symmetry statements in the …rst experiment and were consistent with color symmetry in the second experiment. 38 Ellsbergian One vs Two Total Ambiguity nonPS Symmetric PS non-symmetric PS nonPS 20 20 23 63 PS 15 40 11 66 Total 35 60 34 129 Fisher exact test p-value=0.003 Table A.3: Ellsbergian ambiguity and One vs Two: Pairwise Choices (2015) A.2.3 Relation between attitude to sources of ambiguity 2015 Tables A.3 and A.4 summarize the association between attitude to Ellsbergian ambiguity and choices made in One vs Two in the two elicitation methods. Each cell counts the number of subjects who exhibit the pro…le of behaviors. The rows in both tables correspond to probabilistically sophisticated behavior in the standard Ellsberg experiment, where the top row counts the subjects who are either strictly Ellsbergian ambiguity averse or seeking (nonPS), while the second row counts all the rest (PS). The columns correspond to probabilistic sophistication in One vs Two. The leftmost column corresponds to subjects who are not probabilistically sophisticated (nonPS, exhibiting either One Two or Two One). The other columns include subjects that made choices consistent with probabilistic sophistication. The middle column in Table A.3 includes only subjects who did not exhibit any strict preference in One vs Two, while the right column includes probabilistically sophisticated subjects who chose at least one bet with a lower prize (and hence revealed strict preference) in One vs Two. Fisher exact test for association reveals a tight association between behaviors in the two problems, but it is important to note that it is not that failure of probabilistic sophistication in one problem is associated with similar behavior in the other problem (this e¤ect is not signi…cant here); rather, probabilistically sophisticated behavior in the Ellsberg problem is associated with symmetric probabilistically sophisticated behavior in the One vs Two. In other words: subjects who always chose the bet with the higher prize in one problem tended to do the same in the other problem as well. Table A.4 presents the association based on choice list elicitation. Since 39 Ellsbergian One vs Two Total Ambiguity nonPS PS nonPS 28 12 40 PS 15 19 34 Total 43 31 74 Fisher exact test p-value=0.034 Table A.4: Ellsbergian ambiguity and One vs Two from Choice Lists (2015) Ellsbergian One vs Two Total Ambiguity nonPS Symmetric PS non-symmetric PS nonPS 37 20 13 70 PS 14 26 1 41 Total 51 46 14 111 Fisher exact test p-value<0.00011 1 p-values of Fisher exact tests in the 2013 and 2014 experiments (separately) are 0.008 and 0.113, respectively. Table A.5: Ellsbergian ambiguity and One vs Two in 2013-2014 experiments the tendency to make probabilistically sophisticated non-symmetric choices in One vs Two is much lower, we can combine the two probabilistically sophisticated columns. Here we …nd that failure of probabilistic sophistication in Ellsberg is signi…cantly associated with (but distinct from) violation of probabilistic sophistication in One vs Two. 2013-2014 As in 2015, when distinguishing between forms of probabilistic sophistication in One vs Two the association is signi…cant, but when pooling the two forms of probabilistic sophisticated behavior the association is less strong (one sided p-value=0.043). When pooling all 240 subjects (2013-2015) who participated in the pairwise choice experiment, the association between probabilistic sophisticated behavior in the two domains is signi…cant (one side p-value = 0.008), but the behaviors are far from being identical. 40 A.2.4 RIS and hedging As discussed earlier, the standard use of RIS in ambiguity experiments (where the selection of the choice problem that counts for payment is performed after choices have been made and balls have been drawn) is open to the criticism that it could provide the subject with an opportunity to hedge the uncertainty using the randomization device employed in the incentive system. This implementation of the RIS induces a choice problem in which the subject faces Anscombe-Aumann acts and thus where she can hedge ambiguity using the randomization device employed in the incentive system by choosing certain combinations of ambiguous bets.35 For this reason we performed the randomization before subjects made their choices and the draws from the urns were made. It is an empirical question whether subjects conformed with this order when evaluating lotteries, and if they did not, whether they hedged. These aspects of our experimental design as well as the empirical question whether there is evidence that subjects hedged in spite of our theoretically incentive compatible implementation are discussed in this Appendix. Hedging might be revealed in di¤erent ways. For example, consider the two choices comprising One vs Two (in 2013-2014), and suppose, for simplicity, that the individual attaches probability 1=2 to payo¤s being dependent on each of these questions. Then, a subject who is ambiguity averse and acts as if the randomization occurs after the balls are drawn, could choose Same over R1 in one question and Dif f over R1 in another, and be left with the same state-independent payo¤ (in expected utility units) that she would obtain from betting on drawing red from Ellsberg’s risky urn. Accordingly, if she is ambiguity averse in the Ellsberg problem, then the preceding combined choices would be preferable to R1 and we would observe Two One. However the latter is observed for only 15 subjects (6.25% of subjects) in the three experiments (2013, 2014 and 2015). Another possibility for hedging that existed only in 2015 was to make 35 The signi…cance we are attaching to when the randomization takes place runs counter to Anscombe and Aumann’s well-known “reversal of order axiom.” However, the descriptive appeal of that axiom has been put into question by evidence that subjects who are ambiguity sensitive usually do not satisfy the reduction of compound lotteries assumption (Halevy 2007; Abdellaoui, Klibano¤ and Pacido 2015; Chew, Miao and Zhong forthcoming; Dean and Ortoleva 2016; Gillen, Snowberg and Yariv 2016). In a more normative vein, Seo (2009) argues for a relaxation of the “reversal of order”axiom and shows how it can be part of an axiomatic characterization of the “smooth ambiguity model.” 41 choices consistent with One Two (choose R1 and B1 ). This possibility was not available to subjects in the 2013-2014 experiments, where they were asked to choose a color to bet on in advance. The fact that the frequency of Ellsbergian ambiguity averse subjects making choices consistent with One Two in 2013-2014 is almost three times higher than in 2015 (28/111 compared with 12/129) suggests that this cannot be a major concern in our experiment. Similarly, in the questions that compare Same to Dif f directly (in 20132014), hedging would imply the absence of a strict preference between Same and Dif f . This follows since if the subject had perceived the problem as if the randomization used for incentives occurred after the balls are drawn from the urns, then choosing the bet with a prize of $51 ($101 in the 2013 experiment) in each of the two questions would leave the subject with a lottery that pays $51 ($101) with probability 1=2 independently of the state, thus completely hedging the ambiguity. However, empirically about 36% of the subjects exhibit a strict preference in these questions,36 and close to half (about 49%) of the subjects who are ambiguity sensitive in the Ellsberg choice problem (and thus may wish to hedge), made choices inconsistent with indi¤erence between Same and Dif f: We conclude that we do not …nd convincing empirical evidence that subjects used the RIS to hedge any source of ambiguity in our experiment. A.2.5 Weaker inclusion criteria (2013-2014) Throughout the paper we excluded subjects who made choices that were not consistent with transitive and monotone preferences in one of the choice problem. In 2013-2014 we also excluded subjects who did not agree with the symmetry statements (2013) or exhibited strict preference to betting on one of the colors.37 The number of subjects excluded decreased as we progressed, since the instructions they received improved. Footnote 9 in the paper details the 24 subjects excluded in 2015, and below we describe the subjects excluded from the analysis of the previous two experiment. In this Appendix we show that their inclusion does not substantially a¤ect our conclusions. As noted above, out of 80 subjects who participated in the 2013 experiment, 24 were removed from the analysis: 11 were removed due to violations of monotonicity or transitivity in at least one pair of questions, 7 due to 36 7 subjects strictly preferred Same, 33 subjects strictly preferred Dif f , and the remaining 71 subjects made choices consistent with indi¤erence between Same and Dif f . 37 This is not a necessary assumption in the 2015 experiment. 42 Ellsbergian Ambiguity Attitude 2013 # % strictly averse (nonPS) 51 68.9 strictly seeking (nonPS) 6 8.1 neutral (PS) 17 23 Total 74 100 2014 # % 45 52.9 3 3.5 37 43.6 85 100 Table A.6: Ellsbergian ambiguity attitude 2013-2014: maximal inclusion 2013 # % One Two (nonPS) 31 40.2 Two One (nonPS) 6 7.8 One Two 25 32.5 non-symmetric beliefs (PS) 15 19.5 Total 77 100 One vs Two 2014 # % 30 38.5 10 12.8 28 35.9 10 12.8 78 100 Table A.7: One vs Two 2013-2014: maximal inclusion cyclic choices between R1 , Same and Dif f , 4 for disagreeing with the symmetry over colors and urns, and two more subjects in the …rst session were caught cheating and their choices were excluded from the analysis. Out of 87 subjects who participated in the main treatment of the 2014 experiment, 75 had no monotonicity or transitivity violations and 20 showed strict color preference. In order to include as many subjects as possible from the 2013 experiment, several relaxations of the inclusion criteria were employed. First, the answer to the non-incentivized question concerning symmetry was ignored. Second, in case of cyclic choices between R1 , Same and Dif f , the direct comparison between Same and Dif f was not taken to invalidate the choices made in One vs Two (which relies on comparing R1 to Same and R1 to Dif f ). Third, if a subject had non-monotone choices in only one pair of questions (assuming transitivity), (s)he was not removed from the analysis. Instead, choices in other questions, together with transitivity, were used in order to 43 extend the preferences to the suspect direct comparison.38;39 We used the …rst two methods for the 2014 experiment as well. The distributions of Ellsbergian ambiguity and One vs Two are presented in Tables A.6 and A.7. The marginal distributions of ambiguity attitude expressed in the standard Ellsberg experiment and in One vs Two are remarkably similar to those for the smaller sample, taking into account that the bigger sample includes subjects with non-symmetric color preference, which may a¤ect the choices that comprised ambiguity attitude (seeking) and One Two. A.3 Details for the source model This appendix provides supporting details for the model in Section 5.2. To de…ne utility precisely, let u1 , u2 and u3 be strictly increasing vNM indices that will apply to the three issues respectively. Let the cdf F describe conditional beliefs about p, generalizing from the uniform distribution in Figure 5.1. Then, for any bet (or act) f over S1 S2 , its utility (in certainty equivalent units) is computed recursively by:40 V2 (f ) = u2 1 U (f ) = u1 1 ( u1 (V2 (f )) + (1 )u1 (W2 (f ))) , Z Z 1 u2 (V3 (f ; p)) dF , W2 (f ) = u2 u2 (W3 (f ; p)) dF p V3 (f ; p) = u3 , p 1 Z u3 (f ) d [(p; 1 p) (p; 1 u3 (f ) d [(p; 1 p) (1 p)] , and S1 S 2 W3 (f ; p) = u3 1 Z p; p)] . S1 S 2 38 This applies to 5 subjects. An extreme example is provided by Subject 315: (s)he did not agree with the suggested symmetry in colors and urns, and her/his choices in R1 vs Same were inconsistent with monotone preferences. However, the choices in R1 vs Dif f and Dif f vs Same were consistent with R1 Dif f and Dif f Same; so this subject was classi…ed as exhibiting One T wo. 39 We omitted only the two subjects who were caught cheating and another subject who made choices inconsistent with monotone preferences in both R1 vs Same and R1 vs Dif f . 40 Symmetry in urns is built into the model, and symmetry in colors is implied if we assume, as we do, that F is suitably symmetric (the compositions (p; 1 p) and (1 p; p) are equally likely). 44 Now specialize to uniform F , = 12 and linear u3 , with u3 (x) = 1 and u3 (0) = 0. Then the utility of the bet R3 on drawing red from the risky Ellsberg urn is given by U (R3 ) = u3 1 1 2 = 12 , and the utility of the bet R1 on drawing red from a single ambiguous urn is Z 1 u2 (p) dF . U (R1 ) = u2 p Thus R3 is preferred if u2 is concave, as noted in the text. One Two is implied if u1 is more concave than u2 : u1 U (Same) = 1 u 2 1 u1 = u1 Moreover, u2 1 32 u2 (1) + 13 u2 21 + 12 u1 u2 1 32 u2 (0) + 13 u2 u2 1 12 32 u2 (1) + 31 u2 12 + 12 23 u2 (0) + 31 u2 12 u2 1 13 u2 (0) + 31 u2 21 + 13 u2 (1) = u1 U (R1 ) . 45 1 2