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:

1. it increases the error in my posterior estimate when $$\theta=0$$;
2. 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.

1. See here for my discussion of the case when $$\theta$$ and $$s$$ are normally distributed. ↩︎