Emotional decision-makers and anomalous attitudes towards information

Abstract

We use a simple version of the Psychological Expected Utility Model (Caplin and Leahy, Q J Econ 116:55–80, 2001) to analyze the optimal choice of information accuracy by an individual who is concerned with anticipatory feeling. The individual faces the following trade-off: on the one hand information may lead to emotional costs, on the other the higher the information accuracy, the higher the efficiency of decision-making. We completely and explicitly characterize how anticipatory utility depends on information accuracy, and study the optimal amount of information acquisition. We obtain simple and explicit conditions under which the individual prefers no-information or partial information gathering. We show that anomalous attitudes towards information can be more frequent and articulated than previously thought.

Appendix

1.1 Proof of Remark 4

• (i) From Eq. Eq. 3.6 it is easy to check that, when w ₁ − w ₂ < 1, $f_{2}^{\prime }(\cdot )$ is negative for $q\in \left[ \tfrac{1}{2},1\right]$.

• (ii) This is obvious from Eqs. 3.6 and 3.4.

• (iii) When w ₁ − w ₂ ≥ 1, $f_{2}^{\prime }(\cdot )<0$ for $q=\tfrac{1}{2}$ and $f_{2}^{\prime }(\cdot )>0$ for q = 1. Thus, starting from $q=\tfrac{1}{2},$ f ₂(·) is first decreasing and then increasing in q. From Eq. 3.6 it is easy to see that the higher w ₁ − w ₂, the shorter the subinterval of $\left[ \tfrac{1}{2},1\right]$ where f ₂(·) is decreasing in q.

1.2 Proof of Lemma 1

• (i) When $q=\frac{1}{2}$, it is easy to see that:

  $$ \left. \frac{\partial ^{2}U\big(q;w_{1},w_{2}\big)}{\partial q^{2}}\right\vert _{q= \frac{1}{2}}=2u^{\prime }\left( f\left(\frac{1}{2};w_{1},w_{2}\right)\right) +u^{\prime \prime }\left( f\left(\frac{1}{2};w_{1},w_{2}\right)\right) $$

  where $u_{1}^{\prime }\left( \cdot \right) =u_{2}^{\prime }\left( \cdot \right) =u^{\prime }\left( \cdot \right) ,$ thus:

  $$ \left. \frac{\partial ^{2}U\big(q;w_{1},w_{2}\big)}{\partial q^{2}}\right\vert _{q= \frac{1}{2}}>0\Leftrightarrow R_{u}(f)<2f\left(\frac{1}{2};w_{1},w_{2}\right). \label{conditionq12minimum} $$

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• (ii) This corresponds to the case where $\left. \frac{\partial ^{2}U(q;w_{1},w_{2})}{\partial q^{2}}\right\vert _{q=\frac{1}{2}}<0.$

1.3 Anticipatory utility decreasing in the signal precision for q = 1.

Remark 6

Anticipatory utility is decreasing in the signal precision for q = 1 if and only if:

$$ \frac{u^{\prime }\big( w_{1}\big) }{u^{\prime }\big( w_{2}\big) }< \frac{1-\big( w_{1}-w_{2}\big) }{1+\big( w_{1}-w_{2}\big) } \label{conditionq1decrescente} $$

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Two necessary conditions are w ₁ − w ₂ < 1 and u(·) concave enough.

Proof

When q = 1, it is easy to see that:

$$ \begin{array}{lll}\left. \frac{\partial U\big(q;w_{1},w_{2}\big)}{\partial q}\right\vert _{q=1}=&\frac{1 }{2}\left( w_{1}-w_{2}+\big( w_{1}-w_{2}\big) ^{2}\right) u^{\prime }\big( w_{1}\big) \\ &+\,\frac{1}{2}\left( -\big( w_{1}-w_{2}\big) +\big( w_{1}-w_{2}\big) ^{2}\right) u^{\prime }\big( w_{2}\big) \end{array}$$

Thus, for q = 1, anticipatory utility is decreasing in the signal precision if:

$$ \left( 1+\big( w_{1}-w_{2}\big) \right) u^{\prime }\big( w_{1}\big) -\left( 1-\big( w_{1}-w_{2}\big) \right) u^{\prime }\big( w_{2}\big) <0 \label{inequalityconcavity} $$

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From Eq. 6 condition 5 can be immediately derived. Moreover Eq. 6 can be verified only if w ₁ − w ₂ < 1 and it implies that u(·) must be concave enough. In fact, inequality 5 is verified only if $u_{1}^{\prime }\left( w_{1}\right)$ is sufficiently lower than $u_{2}^{\prime }\left( w_{2}\right).$

1.4 Simulations

Let’s consider the power function $u\left( x\right) =\frac{ x^{1-\gamma }}{1-\gamma },$ with γ re-interpreted as the parameter of constant relative aversion to information.¹⁴ In this paragraph we provide examples of all the possible results described in Table 2.

• Case (a) Let’s start with the more interesting sub-case: a preference for the partial informative signal. According to Remark 5, let’s take $\gamma <2f(\tfrac{1}{2};w_{1},w_{2}).$ The uninformative signal is a minimum for w ₁ ∈ $[w_{2}+1-\sqrt{ 4w_{2}+1-2\gamma },$ $w_{2}+1+\sqrt{4w_{2}+1-2\gamma }]$ and $\gamma <\frac{1 }{2}+2w_{2}.$ Moreover, for w ₁ − w ₂ < 1 the derivative of anticipatory utility in q = 1 is negative.¹⁵ Figures 1 and 2 show anticipatory utility in the interval $q\in \left[ \tfrac{1}{2},1\right]$ when u(·) is the power function and w ₁ = 0.55, w ₂ = 0.023. It can be seen that for γ = 0.42 (Fig. 1) the uninformative signal dominates the fully informative one, whereas for γ = 0.4 (Fig. 2) the opposite holds. According to intuition, all else being equal, the lower the relative aversion to information, the closer to 1 the optimal precision of the signal. In “The case of total utility” below we present the case of total utility (anticipatory utility plus expected physical utility): an interior solution still exists and it is compatible with higher values of information aversion.

  The second sub-case arises when $q=\tfrac{1}{2}$ corresponds to a global minimum and full information is the optimal choice. This occurs, for example, in the case γ = 0.3, ω ₁ = 0.5 and ω ₂ = 0.2.

• Case (b) The uninformative signal corresponds to a global minimum and full information gathering is the optimal choice. This occurs, for example, when γ = 0.5, ω ₁ = 2.5 and ω ₂ = 0.5 .

• Case (c) In the first sub-case $q=\tfrac{1}{2}$ is a global maximum and no information is the optimal choice. This occurs, for example, when γ = 0.7, ω ₁ = 0.45 and ω ₂ = 0.2. In the second sub-case anticipatory utility is first decreasing and then increasing in information accuracy and either full information or no information are the preferred choices. When γ = 1.2, ω ₁ = 1, ω ₂ = 0.3 full information dominates no information whereas when γ = 1.2, ω ₁ = 0.99, ω ₂ = 0.2, the opposite occurs.

• Case (d) Anticipatory utility exhibits a global minimum for partial information gathering. The preferred choice can be either the fully informative signal or the uninformative one. When γ = 1.1, ω ₁ = 1.3, ω ₂ = 0.05 no information dominates full information whereas when γ = 0.8, ω ₁ = 1.3, ω ₂ = 0.05 the opposite occurs.

1.5 The case of total utility

As we mentioned before, using anticipatory utility only instead of total utility (consisting of anticipatory utility plus expected physical utility) as the DM’s objective involves no loss of generality. To show that, in this paragraph we obtain sufficient conditions such that partial information is the preferred choice (as in Remark 5) when the DM’s well-being is measured by total utility. Total utility is¹⁶:

$$ \begin{array}{lll} \hat{U}\big(q;w_{1},w_{2}\big) &=&\frac{1}{2}u\big( qw_{1}+(1-q)w_{2}-q(1-q)\big(w_{1}-w_{2}\big)^{2}\big) \\ &&+\,\frac{1}{2}u\big( (1-q)w_{1}+qw_{2}-q(1-q)\big(w_{1}-w_{2}\big)^{2}\big) \\ &&+\,\frac{1}{2}\big(w_{1}-w_{2}\big)-q(1-q)\big(w_{1}-w_{2}\big)^{2} \end{array} $$

Since a convex term has been added to the DM’s objective function, Lemma 1 and Remark 6 (the latter in the Appendix) must be slightly modified.

Corollary 1

The uninformative signal is a local minimum of total utility if

$$ R_{u}\left(f_{\frac{1}{2}}\right)<w_{1}+w_{2}-\frac{1}{2}\big( w_{1}-w_{2}\big) ^{2}+\varepsilon \label{inequality1qbis} $$

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where ε ≡ $\frac{2f\big(\tfrac{1}{2};w_{1},w_{2}\big)}{ u^{\prime }\left( f\big(\tfrac{1}{2};w_{1},w_{2}\big)\right) }.$

Note that $f(\tfrac{1}{2};w_{1},w_{2})$ and ε are positive for w ₁ ∈ ($w_{2}+1-\sqrt{4w_{2}+1},$ $w_{2}+1+\sqrt{4w_{2}+1}$). The following corollary is the equivalent of Remark 6 in the case of total utility, that is, it derives the condition such that total utility is decreasing in the signal precision for the fully informative signal.

Corollary 2

Total utility is decreasing in the signal precision for q = 1 if and only if:

$$ \frac{u^{\prime }\big( w_{1}\big) }{u^{\prime }\big( w_{2}\big)}< \frac{1-\big( w_{1}-w_{2}\big)}{1+\big( w_{1}-w_{2}\big) }-\frac{ 2\big( w_{1}-w_{2}\big) }{u^{\prime }\big( w_{2}\big) \left[ 1+\big( w_{1}-w_{2}\big) \right] }. \label{conditionq1decrescentebis} $$

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Finally, the following corollary states sufficient conditions for an internal solution in the case of total utility:

Corollary 3

With total utility, if both the uninformative signal is a minimum of anticipatory utility and anticipatory utility is decreasing in the signal precision for the fully informative signal, that is if conditions 7 and 8 simultaneously hold, a partially informative signal is the DM’s optimal choice.

Note that, with total utility, it is less likely that the objective function is decreasing in q for the fully informative signal, however the condition such that the uninformative signal is a local minimum is less stringent than before.

To summarize, when the DM maximizes total utility and an internal solution exists, the latter is compatible with levels of information aversion higher than the values we observed when anticipatory utility was considered alone. Our simulations show that an internal solution with total utility can be found, for example when γ = 0.56, w ₁ = 0.53 and w ₂ = 0.023.