Paying for the truth

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 $s\left(x\right)\in \{0,1\}$ with precision $$Pr\left(s\right(x)=\theta )=\frac{1+x}{2}.$$ 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 $s\left(x\right)$ and my prior ${\theta }_0$ to form a posterior estimate $$\hat{\theta }\left(s\right(x\left)\right)=Pr(\theta =1|s\left(x\right))$$ via Bayes’ rule. I care about the mean squared error $$MSE\left(x\right)=E\left[{(\theta -\hat{\theta }(s\left(x\right)\left)\right)}^2\right]$$ of my posterior estimate, where $E$ is the expectation operator taken with respect to the joint distribution of $\theta$ and $s\left(x\right)$ 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 $x^{\ast }$ that minimizes $$f\left(x\right)=MSE\left(x\right)+cx.$$ The chart below plots my objective $f\left(x\right)$ 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 $c^{\ast }$ be the threshold value of $c$ at which I stop paying for information: the “choke price” of truth. How does $c^{\ast }$ 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 $c^{\ast }$. The second effect makes me think I need less information, decreasing $c^{\ast }$. 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 $$WMSE\left(x\right)=E\left[W\right(\theta )\cdot {(\theta -\hat{\theta }(s\left(x\right)\left)\right)}^2],$$ where the weighting function $$W\left(\theta \right)=\{\begin{array}{cc}1& \text{if}\theta =1\\ w& \text{if}\theta =0\end{array}$$ has $w\ge 1$. Increasing $w$ nudges my optimal posterior estimate towards zero because I want to avoid being “confidently wrong” when $\theta =0$. Since $WMSE\left(x\right)$ is concave in $x$, I still optimally pay for all the truth or none of it. But now the choke price $c^{\ast }$ at which I stop paying for the truth depends on my prior ${\theta }_0$ and the error weight $w$.

The chart below shows that $c^{\ast }$ 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.