The linked paper seems to associate a cost \alpha for guessing too high vs. too low, and then gets into some analysis for how to also model the cost of the game going forwards (i.e., now that you’ve made this one guess, what’s the expected cost of continuing to play the game?)
I’ll have to think more about how to model this, but it looks hard to me! In particular, the paper poses this as an open problem to find \gamma, and whether the functional form is the correct one, I think?