Optimal confidence threshold for an uncertainty response ($p>(s+w)/(r+w)$)

Under linear expected utility, let a correct answer yield $r$, an error yield $-w$, and declining yield $s$, with $r+w>0$. Answering has expected payoff $pr-(1-p)w$, so it is preferred exactly when $p>(s+w)/(r+w)$. Equality gives indifference. Different utility, time costs or beliefs change this criterion; behaviour across differently rewarded tasks cannot be compared as if the threshold were fixed.