Paying for precision

Suppose my payoff \(u(a,\mu)\equiv-(a-\mu)^2\) from taking an action \(a\in\mathbb{R}\) depends on an unknown state \(\mu\in\mathbb{R}\).¹ I can learn about \(\mu\) by collecting data \(X=\{x_1,x_2,\ldots,x_n\}\), where the observations \(x_i\) are iid normally distributed with mean \(\mu\) and variance \(\sigma^2\):² $$x_i\mid \mu\sim N(\mu,\sigma^2).$$ I use these data, my prior belief $$\mu\sim N(\mu_0,\sigma_0^2),$$ and Bayes’ rule to form a posterior belief $$\mu\mid X\sim N\left(\frac{\tau_0}{\tau_0+n\tau}\mu_0+\frac{n\tau}{\tau_0+n\tau}\bar{x},\frac{1}{\tau_0+n\tau}\right),$$ where \(\tau_0\equiv1/\sigma_0^2\) is the precision of my prior, \(\tau\equiv1/\sigma^2\) is the precision of the \(x_i\), and $$\bar{x}\equiv\frac{1}{n}\sum_{i=1}^nx_i$$ is their arithmetic mean. Then my expected payoff from taking action \(a\) equals $$\DeclareMathOperator{\E}{E} \DeclareMathOperator{\Var}{Var} \E[u(a,\mu)\mid X]=-(a-\E[\mu\mid X])^2-\Var(\mu\mid X).$$ I maximize this payoff by choosing \(a^*\equiv\E[\mu\mid X]\). This yields expected payoff $$\E[u(a^*,\mu)\mid X_n]=-\frac{1}{\tau_0+n\tau},$$ which is increasing in \(n\). Intuitively, collecting more data makes me more informed and makes my optimal action more likely to be “correct.” But data are costly: I have to pay \(\kappa n\tau\) to collect \(n\) observations, where \(\kappa>0\) captures the marginal cost of information.³ I choose \(n\) to maximize my total payoff $$\begin{align*} U(n) &\equiv \E[u(a^*,\mu)\mid X]-\kappa n\tau, \end{align*}$$ which has maximizer $$n^*=\max\left\{0,\frac{1}{\tau}\left(\frac{1}{\sqrt\kappa}-\tau_0\right)\right\}.$$ If \(1\le\sqrt\kappa\tau_0\) then \(n^*=0\) because the cost of collecting any data isn’t worth the variance reduction they deliver. Whereas if \(1>\sqrt\kappa\tau_0\) then \(n^*\) is strictly positive and gives me total payoff $$U(n^*)=-2\sqrt\kappa+\kappa\tau_0.$$ Both \(n^*\) and \(U(n^*)\) are decreasing in \(\kappa\). Intuitively, making the data more expensive makes me want to collect less, leaving me less informed and worse off. In contrast, making my prior more precise (i.e., increasing \(\tau_0\)) makes me want to collect less data but leaves me better off. This is because being well-informed means I can pay for less data and still be well-informed.

Curiously, making the \(x_i\) more precise (i.e., increasing \(\tau\)) makes me want to collect more data but does not change my welfare. This is because the cost \(\kappa\tau\) of each observation \(x_i\) scales with its precision. This cost exactly offsets the value of the information gained, leaving my total payoff \(U(n^*)\) unchanged.