In a previous post, I showed that if the truth doesn’t matter then I’m better off being an ideologue with ideological friends. I discussed the trade-off between (i) experiencing reality and (ii) experiencing what my friends experience. Truth-seeking made sense only when the benefit of (i) exceeded the cost of forgoing (ii). This post discusses another cost of truth-seeking: having to pay—financially, cognitively, or emotionally—for information.
One way to model that cost is as follows.1
Suppose the truth is determined by a random variable \(\theta\in\{0,1\}\).
I learn about \(\theta\) by observing a signal \(s(x)\in\{0,1\}\) with precision
$$\Pr(s(x)=\theta)=\frac{1+x}{2}.$$
The parameter \(x\in[0,1]\) determines the signal’s quality.
If \(x=1\) then the signal is fully informative; if \(x=0\) then it is uninformative.
My prior estimate \(\theta_0\in[0.5,1]\) of \(\theta\) is based on no information; it reflects my ideology.
I use the realization of \(s(x)\) and my prior \(\theta_0\) to form a posterior estimate
$$\hat\theta(s(x))=\Pr\left(\theta=1\,\vert\,s(x)\right)$$
via Bayes’ rule.
I care about the mean squared error
$$\newcommand{\E}{\mathrm{E}} \newcommand{\MSE}{\mathrm{MSE}} \MSE(x)=\E\left[\left(\theta-\hat\theta(s(x))\right)^2\right]$$
of my posterior estimate, where \(\E\) is the expectation operator taken with respect to the joint distribution of \(\theta\) and \(s(x)\) given my prior \(\theta_0\).
But I also care about the cost \(cx\) I endure from observing a signal of quality \(x\).
This cost reflects the resources I use to seek the information and process it (e.g., money, time, and mental energy).
I choose the quality \(x^*\) that minimizes
$$f(x)=\MSE(x)+cx.$$
The chart below plots my objective \(f(x)\) against \(x\) when I have prior \(\theta_0\in\{0.5,0.7,0.9\}\) and face marginal cost \(c\in\{0,0.1,0.2,0.3\}\).
Since \(f\) is concave in \(x\), it has (constrained) local minima at \(x=0\) and \(x=1\).
My choice between these minima depends on the value of \(c\).
If it’s small then information is cheap and I “buy” as much as I can.
If it’s large then information is expensive and I don’t buy any.
But there’s no middle ground: I seek all the truth or none of it.
Let \(c^*\) be the threshold value of \(c\) at which I stop paying for information: the “choke price” of truth.
How does \(c^*\) depend on my prior \(\theta_0\)?
Intuitively, increasing \(\theta_0\) has two competing effects:
- it increases the error in my posterior estimate when
\(\theta=0\); - it increases my confidence that
\(\theta=1\).
The first effect makes me want more information, increasing \(c^*\).
The second effect makes me think I need less information, decreasing \(c^*\).
The chart below shows that the second effect dominates.
The more ideological I am about the value of \(\theta\), the cheaper the truth must be for me to seek it.
If I’m a pure ideologue (i.e., \(\theta_0=1\)) then I won’t seek the truth even if it’s free.
One reason the first effect might dominate is if I care about errors when \(\theta=0\) more than when \(\theta=1\).
For example, if \(\theta\) indicates whether it will be sunny then I’d rather bring an umbrella I don’t use than be caught wearing flip-flops in the rain.
I can capture that asymmetry by replacing the MSE component of my objective with a weighted version
$$\newcommand{\WMSE}{\mathrm{WMSE}} \WMSE(x)=\E\left[W(\theta)\cdot\left(\theta-\hat\theta(s(x))\right)^2\right],$$
where the weighting function
$$W(\theta)=\begin{cases} 1 & \text{if}\ \theta=1 \\ w & \text{if}\ \theta=0 \end{cases}$$
has \(w\ge1\).
Increasing \(w\) nudges my optimal posterior estimate towards zero because I want to avoid being “confidently wrong” when \(\theta=0\).
Since \(\WMSE(x)\) is concave in \(x\), I still optimally pay for all the truth or none of it.
But now the choke price \(c^*\) at which I stop paying for the truth depends on my prior \(\theta_0\) and the error weight \(w\).
The chart below shows that \(c^*\) is non-monotonic in \(\theta_0\) when \(w\) is large.
This is due to the two competing effects described above.
The first effect dominates when \(w\) is large and my prior is low.
In that case, it’s really bad to be wrong and I’m not confident I’ll be right.
Whereas the second effect dominates when \(w\) is large and my prior is high.
In that case, I’m so confident I’ll be right that I don’t care what happens if I’m wrong.
This example raises a philosophical question: what does it mean for the estimate to be “wrong?” For example, suppose I thought there was a 30% chance of rain. If it rained, was I wrong? What if I thought there was a 5% chance? A 95% chance? Where should I draw the line? On those questions, I recommend Michael Lewis’ discussion with Nate Silver about 17 minutes into this podcast episode.