A Simple Utility Maximization Problem?
Consider a rational agent with a utility function defined by \(U(x,t,\ell) = x - \sqrt{t} + \ln(\ell)\), where \(x \sim Ber(p)\) is a Bernoulli random variable representing the existence of the page that you’re looking for with success probability \(p = P(x = 1 \mid t) = \frac{t}{24}\), \(t\) is the time spent searching for this page, and \(\ell\) is the time spent on other pages on this website. Given a time constraint of \(t + \ell = 24\), the agent maximizes their expected utility, specifically:
\[\begin{aligned} E(U) & = U(1,t,\ell) \cdot P(x = 1|t) + U(0, t, \ell) \cdot P(x = 0|t) \\ & = \frac{t}{24} - \sqrt{t} + \ln(\ell) \\ & = \frac{t}{24} - \sqrt{t} + \ln(24 - t) \end{aligned}\]The agent notices that the marginal utility from an increase in the time \(t\) spent searching for the webpage is strictly negative, specifically:
\[\frac{\partial E(U)}{\partial t} = \frac{1}{24} - \frac{1}{2\sqrt{t}} - \frac{1}{24 - t} \leq 0 \quad \forall \, t \in [0,24]\]As such, the agent selects an optimal \(t^* = 0\), and maximizes their utility with the vector \((x^*, t^*, \ell^*) = (0, 0, 24)\). In other words, they allocate all their time endowment to other pages instead of searching for this specific page – I would suggest doing the same here. How about going back to the home page?