In light of these intuitive relations, we are interested by how much the price limits improve under Incomplete Information. To make the analysis more accessible, we refer to the numerical example given in Table
3 which is based on the example case as presented in Sects.
3.2 and
3.7. The
r values represent increasing degrees of risk sensitivity, and the values of the margin
m are chosen such that they vary around the costs
\(c=100\). The numerical example refers to the situation where it is known that the DM is risk averse, because we use the same example in Sect.
6.3 to analyze the effect of this information on the performance of the price limits. See Appendix
4 for a corresponding numerical example without risk aversion.
Table 3
Performance of price limits under risk aversion
147 | 0 | 0.01 | 99.8 | 99.6 | 0.3 | 0.341 | 99.1 | 0.5 | 0.273 | 98.4 | 0.6 |
0.10 | 98.3 | 96.5 | 2.7 | 0.348 | 92.8 | 5.5 | 87.1 | 7.0 |
0.25 | 96.7 | 93.2 | 5.8 | 0.360 | 87.8 | 14.0 | 77.9 | 18.0 |
0.50 | 95.9 | 91.8 | 8.2 | 0.379 | 87.9 | 22.6 | 77.3 | 29.6 |
100 | 0.01 | 100.0 | 98.6 | 1.3 | 0.341 | 98.6 | 1.4 | 97.8 | 2.2 |
0.10 | 99.7 | 90.4 | 9.0 | 0.349 | 89.7 | 12.2 | 83.2 | 17.8 |
0.25 | 99.6 | 87.9 | 12.2 | 0.366 | 86.4 | 21.2 | 76.0 | 28.5 |
200 | 0.01 | 99.8 | 98.6 | 1.8 | 0.273 | 98.6 | 1.9 | 98.6 | 1.9 |
0.10 | 98.4 | 90.4 | 10.6 | 0.279 | 89.7 | 14.9 | 89.3 | 14.5 |
0.25 | 96.2 | 87.5 | 14.0 | 0.295 | 86.1 | 25.2 | 84.7 | 24.0 |
0.50 | 88.1 | 84.4 | 15.6 | 0.344 | 84.4 | 18.5 | 77.3 | 29.6 |
Average | 97.5 | 92.6 | 7.4 | 0.334 | 91.0 | 12.5 | 0.273 | 86.1 | 15.8 |
258 | 0 | 0.01 | 99.6 | 99.2 | 0.4 | 0.240 | 97.9 | 0.7 | 0.152 | 95.6 | 0.9 |
0.10 | 96.9 | 93.3 | 4.2 | 0.249 | 84.9 | 9.1 | 71.6 | 11.5 |
0.25 | 94.0 | 87.6 | 9.4 | 0.266 | 77.1 | 25.0 | 59.9 | 32.6 |
0.50 | 92.9 | 86.0 | 14.2 | 0.296 | 79.6 | 40.1 | 65.2 | 53.7 |
100 | 0.01 | 99.9 | 97.3 | 2.6 | 0.240 | 97.2 | 2.8 | 94.9 | 5.1 |
0.10 | 99.1 | 83.0 | 14.3 | 0.251 | 81.0 | 22.4 | 67.6 | 32.3 |
0.25 | 98.9 | 79.5 | 20.7 | 0.276 | 76.4 | 37.6 | 60.3 | 50.0 |
200 | 0.01 | 99.8 | 97.1 | 4.0 | 0.152 | 97.1 | 4.7 | 97.1 | 4.7 |
0.10 | 97.7 | 84.0 | 16.6 | 0.159 | 82.0 | 27.7 | 81.4 | 27.1 |
0.25 | 94.3 | 80.5 | 26.7 | 0.178 | 78.8 | 48.9 | 77.2 | 47.8 |
0.50 | 80.0 | 74.3 | 25.7 | 0.243 | 74.3 | 34.6 | 65.2 | 53.7 |
Average | 95.7 | 87.4 | 12.6 | 0.232 | 84.2 | 23.0 | 0.152 | 76.0 | 29.0 |
519 | 0 | 0.01 | 99.1 | 97.8 | 0.7 | 0.089 | 91.6 | 1.1 | 0.030 | 76.6 | 1.6 |
0.10 | 92.7 | 83.4 | 6.7 | 0.098 | 59.5 | 20.0 | 32.0 | 30.9 |
0.25 | 87.2 | 73.3 | 15.8 | 0.115 | 53.1 | 51.8 | 33.6 | 71.1 |
0.50 | 86.2 | 74.0 | 26.5 | 0.149 | 64.9 | 72.8 | 53.0 | 88.7 |
100 | 0.01 | 99.5 | 91.7 | 6.4 | 0.089 | 90.8 | 8.9 | 75.9 | 20.5 |
0.10 | 96.6 | 65.8 | 20.1 | 0.099 | 56.5 | 46.4 | 30.9 | 62.0 |
0.25 | 96.0 | 63.8 | 33.9 | 0.126 | 56.5 | 68.8 | 39.7 | 84.9 |
200 | 0.01 | 99.8 | 89.7 | 10.9 | 0.030 | 87.6 | 19.8 | 87.6 | 19.8 |
0.10 | 97.7 | 71.1 | 25.3 | 0.034 | 65.3 | 67.7 | 64.4 | 67.2 |
0.25 | 92.4 | 71.6 | 49.7 | 0.044 | 69.9 | 86.7 | 69.0 | 86.5 |
0.50 | 65.8 | 59.2 | 40.8 | 0.092 | 59.2 | 68.8 | 53.0 | 88.7 |
Average | 92.1 | 76.5 | 21.5 | 0.088 | 68.7 | 46.6 | 0.030 | 56.0 | 56.6 |
For Version 1 of Incomplete Information
I2, we proceed as described in Sect.
5.5 when determining the level of the allowed violation,
\(\underline{\epsilon }_{D, a}\), by eliminating the effect of the price, while
\(\underline{\epsilon }_{D, a}\) still depends on
m and
p. The
\(\underline{\epsilon }_{D, a}\) values are thus driven by both the DM’s preferences and the decision problem, i.e., by the risk sensitivity
r as well as by
m and
p. Version 2 goes one step further and eliminates the influence of
m and
p in addition to that of the price
\(\pi \). This is done by extending the considered profit range to
\([-99, 200]\), so the same value of
\(\underline{\epsilon }_{D, a}\) can be used for all decision problems satisfying
\(c=100\),
\(m \in [0, 200]\), and
\(p \in [0.01, 0.5]\) and thus for all problems included in Table
3.
5 As a consequence, the differences in the
\(\underline{\epsilon }_{D, a}\) values given for Version 2 are not driven by the decision problem anymore, but only by the DM’s risk sensitivity. The specific values of
r are chosen such that the
\(\underline{\epsilon }_{D, a}\) values under Version 2 of Incomplete Information
I2 match with Levy et al (
2010, Table 7).
6
Here, we focus on the
\(\Pi \) columns, while the
\(\Gamma \) columns are discussed in Sect.
6.3.
\(\Pi _C\),
\(\Pi _{{{I}_{1} ,a}}\), and
\(\Pi _{{{I}_{2} ,a}}\) give the reduction of the range enclosed by the upper and the lower price limit relative to No Information without the information of risk aversion, i.e.,
\(\Pi _C \equiv 1 \!-\! \frac{\underline{\pi }_C - \overline{\pi }_C}{\underline{\pi }_N - \overline{\pi }_N}\),
\(\Pi _{{{I}_{1} ,a}} \equiv 1 - \frac{{\underline{\pi } _{{{I}_{1} ,a}} - \overline{\pi }_{{{I}_{1} ,a}} }}{{\underline{\pi } _{N} - \overline{\pi }_{N} }}\), and
\( \Pi _{{{I}_{2} ,a}} \equiv 1 - \frac{{\underline{\pi } _{{{I}_{2} ,a}} - \overline{\pi }_{{{I}_{2} ,a}} }}{{\underline{\pi } _{N} - \overline{\pi }_{N} }} \). This performance measure is not defined if the best price limits under No Information are the same. This happens when
\(c = m\) in conjunction with
\(p = 0.5\), so this setting is omitted.
In addition to a confirmation of Lemma
4, we make the following observations from Table
3.
For the intermediate risk sensitivity, i.e., \(r = 258\), the price limits are narrowed on average by at least 76%, which makes up for more than 79% of the maximum performance given by the performance under Complete Information. Overall, the improvements vary between 59.2 and 99.6% for Incomplete Information I1 and between 30.9 and 99.1% for Incomplete Information I2.
To make this observation, fix the values of m and p, concentrate on one of the Incomplete Information scenarios, and then decrease the value of r from 519 via 258 to 147. For example, under Incomplete Information \(I_{2}\) (Version 2) with \(m = 100\) and \(p = 0.1\), this yields performances of 30.9, 67.6, and 83.2%.
For an explanation, realize that a weaker risk sensitivity means less pronounced curvatures of the risk-averse utility function \(u_{2}\) and thereby a higher level of the largest allowed violation \(\epsilon_{D} (\pi )\) and of its approximation \(\underline{\epsilon}_{{D,a}} \). The violations \(v_{a} (F_{\pi } ,S)\) and \(v_{a} (S,F_{\pi } )\), in turn, are monotonic functions of the price \(\pi\): the first violation decreases with \(\pi\), while the latter increases. Therefore, the increases in \(\epsilon_{D} (\pi )\) and \(\underline{\epsilon}_{{D,a}} \) translate into tighter and thus better performing price limits.
From the table, the increasing effect of decreasing the risk sensitivity on the largest allowed violation can be observed only for
\(\underline{\epsilon}_{{D,a}} \) because the function
\(\epsilon_{D} (\pi )\) is not shown. To see the effect for
\(\epsilon_{D} (\pi )\), we first observe that the largest allowed violation for the relevant utility functions
\(D = \{ u_{1} ,u_{2} \}\) is driven by utility function
\(u_{2}\), i.e.,
\(\epsilon_{D} (\pi ) = \epsilon_{{u_{2} }}\), since
\(u_{1}\) reflects risk neutrality which implies
\(\epsilon_{{u_{1} }} = 0.5\). For
\(u_{2}\), the largest allowed violation according to (
4) is
\(\epsilon_{{u_{2} }} = \rho /(1 + \rho)\) with
\(\rho = u_{2}^{\prime } (\overline{x})/u_{2}^{\prime } (\underline{x} )\) as the ratio of the smallest to the largest slope of the utility function over the support
\([\underline{x} ,\overline{x}]\). Decreasing the risk sensitivity
r makes the positive slope ratio
\(\rho\) increase, so that the largest allowed violation
\(\epsilon_{{u_{2} }}\) also increases. For
\(\underline{x} < 0 < \overline{x}\), for example,
\(\rho\) amounts to
\( \exp [r \cdot (23\underline{x} - 22\overline{x}) \cdot 10^{{ - 6}} ] \), and its derivative with respect to
r, i.e.,
\(\rho \cdot (23\underline{x} - 22\overline{x}) \cdot 10^{{ - 6}}\), is negative.
Observation
2 can be generalized in two ways.
The first part of the lemma means that Observation
2 does not depend on the specific choices
\(F_{\pi }\) and
S. This is due to assumption (
1) which implies that the violation measures are monotonic in the price. The second part generalizes the form and number of exponential utility functions. With respect to the form, the direction of the effect of the risk sensitivity on the largest allowed violation described above for the specific utility function
\(u_{2}\) is the same for any utility function of the same type. For
\(\underline{x} < x_{0} < \overline{x} \), for example,
\(\rho\) amounts to
\(\exp [r \cdot (c_{2} \underline{x} - c_{1} \overline{x})] \) and its derivative with respect to
r to
\(\rho \cdot (c_{2} \underline{x} - c_{1} \overline{x}), \) which is still negative. With respect to the number of exponential utility functions, we realize that increasing the allowed violation for any relevant utility function, i.e., increasing
\(\epsilon_{u}\) according to (
4), increases the allowed violation for all relevant utility functions, i.e., increases
\(\epsilon_{D}\) according to (
5).
Take \(m = 100\) and \( p = 0.1 \) as an example. For high risk sensitivity, i.e., \(r = 519\), performance drops by 9.3 percentage points when switching from Incomplete Information I1 to Incomplete Information I2 (Version 1). The corresponding performance loss for \(r = 258\) is smaller, namely 2.0 points, and that for \(r = 147\) only amounts to 0.7 points. The performance losses when switching from Incomplete Information I1 to Incomplete Information I2 (Version 2) show the same pattern.
The observation is driven by the dependence of the largest allowed violation on the price: the weaker the risk sensitivity, the less does \(\epsilon_{D} (\pi )\) react to changes in \(\pi\) and the better does its approximation in the form of the constant \(\underline{\epsilon}_{{D,a}}\) become.
However, the diminishing distance between
\(\epsilon_{D} (\pi ) \) and
\(\underline{\epsilon}_{{D,a}}\) is not the only effect to take into account. From the explanation of the previous observation, we know that weaker risk sensitivity also means that the largest allowed violation rises. In view of Propositions
1 and
2, this means that the section of the violation measure that becomes relevant for determining the price limit shifts. If this new section of the violation measure exhibits a lower slope, the shift causes the price limits to diverge and thereby counteracts the converging effect of the converging largest allowed violations. Therefore, Observation
3 does not hold in general.
The performance measures only give the aggregate effect from both price limits. That is why Observation
4 cannot readily be made from Table
3. For
\(m=0\), this problem can be overcome by first referring to the best upper price limit under No Information which equals the indifference price under risk neutrality, i.e.,
\(\overline{\pi }_{N, a} = \pi _n\); see (
6) in Appendix
1. This means that the upper price limit is unaffected by the information about the largest allowed violation. Hence, all performance figures given in Table
3 for
\(m=0\) are solely driven by improvements of the lower price limit rather than the upper price limit.
Finally, we look at the performance gain from the additional information that the DM is risk averse. Due to this information it is valid to use the price limits \(\underline{\pi }_{X, a}\) and \(\overline{\pi }_{X, a}\) with \( X \in \{ N,{I}_{1} ,{I}_{2} \}\) instead of \(\underline{\pi }_X\) and \(\overline{\pi }_X\) with \( X \in \{ N,{I}_{1} ,{I}_{2} \}\). At the same time, this describes the switch from FSD and AFSD to SSD and ASSD.
The types of information carried by the largest allowed violation on the one hand and risk aversion on the other are different. This section helps to disentangle and quantify the effects of both types of information on the performance of the price limits. It raises the awareness for the different kinds and effects of preference information and provides implications for the elicitation of the DM’s preferences. For instance, it suggests testing for risk aversion when the information on the largest allowed violation is weak.
The following result confirms the idea of Lemma
4 that more preference information leads to better price limits. Here, it is the information that the DM is risk averse. The lemma is driven by the hierarchy between AFSD and ASSD.
In Table
3, the corresponding performance gain is measured by
\(\Gamma _X \equiv \frac{\Pi _{X, a} - \Pi _X}{\Pi _{X, a}}\) with
\(X \in \{{I}_{1} ,{I}_{2} \}\). The performance
\(\Pi _X\) is the counterpart of
\(\Pi _{X, a}\) and thus gives the reduction of the range enclosed by the upper and the lower price limit relative to No Information without the restriction to risk aversion, i.e.,
\(\Pi _X \equiv 1 \!-\! \frac{\underline{\pi }_X - \overline{\pi }_X}{\underline{\pi }_N - \overline{\pi }_N}\) with
\( X \in \{{I}_{1} ,{I}_{2} \}\). That is,
\(\Gamma _X \!\cdot \! 100\)% (not percentage points) of the
\(\Pi _{{{I}_{1} ,a}}\) and
\(\Pi _{{{I}_{2} ,a}}\) performances are due to the information of risk aversion. For
\(r=258\),
\(m=100\), and
\(p=0.1\), for instance, 22.4% of the performance of 81% under Incomplete Information
I2 (Version 1) go back to switching from AFSD to ASSD.
This observation can be made from \(\Gamma _{{{I}_{1} }}\) and \(\Gamma _{{{I}_{2} }}\) values exceeding 50%, which is equivalent to \(\Pi _{X, a} \!-\! \Pi _X > \Pi _X\) for \(X \in \{{I}_{1} ,{I}_{2} \}\). We also observe that the weaker the information about the largest violation, the more likely it is. This is addressed by the following observation.
To observe the effect of the DM’s risk sensitivity, stick to one of the Incomplete Information scenarios and compare the
\(\Gamma \) values across the risk sensitivities for given values of
m and
p. This part of Observation
6 relates to Observation
2: the higher the risk sensitivity, the lower is the largest allowed violation
\(\epsilon _D\) and its approximations
\(\underline{\epsilon }_D\) and
\(\underline{\epsilon }_{D, a}\), and the more important is the other piece of information, namely the information of risk aversion.
The effect of weaker information about the largest allowed violation can be seen from Table
3 by realizing that the
\(\Gamma \) values deteriorate when switching from Incomplete Information
I2 (Version 2) via Incomplete Information
I2 (Version 1) to Incomplete Information
I1. However, this effect does not always occur.
The common intuition behind both parts of Observation
6 is that the two pieces of which the incomplete preference information consists, namely the information about the largest allowed violation and the information about risk aversion, are substitutes in terms of their effect on the price limits’ performance.