I don't think "satisficer" is a good name for the concept you're describing here. For one thing, I think it's weird to see a satisficer with just a function (I presume that's what u is?) as its input--where's the threshold of acceptability?

I think a well-specified satisficer as traditionally conceived looks like this:

1. It has some function u(x), that it uses to determine the desirability of any solution x.
2. It has some desirability threshold u_0, and terminates when it finds a solution where u(x)>=u_0.
3. It has some proposal sequence (x_t), that it uses to determine what solution to consider in index t.

Thus we can consider some problem (basically, the set X that describes all possible solutions), and throw a satisficer S(u,u_0,(x_t)) at it, and know exactly which solution x will be picked by S. (I think my notation for the proposal sequence is awkward; if there's a finite set of solutions, we can describe it as a permutation of that set, but that implies some restrictions I don't like. I'm using the parentheses so I can differentiate (x_t), the set of all of them, and x_t, the t-th one.)

Some obvious generalizations suggest themselves: the desirability threshold, rather than just looking at u(x_t), could look at some function of u(x_i) and x_i where i ranges from 0 to t. x_t+1 could depend on u(x_i) rather than just t. x_t+1 could have a source of randomness (in which case we now have a distribution over solutions, rather than a single known solution).

We can then talk about other properties. Perhaps we want to describe satisficers that consider the entire solution space X as "complete," or the ones that only consider each solution at most once "nonrepeating." Most importantly for you, though, a "well-behaved" satisficer has that the property that the proposal sequence x_t is arranged in ascending order by some measure n(x_t) of effort and negative externalities. Maintaining the correct level of illumination in a room by adjusting the curtains comes very early in the ordering; launching a satellite to block the sun comes much, much later; launching a mission to destroy the sun comes so late in the ordering it is almost certain it will not be reached.

This property suggests that a well-behaved satisficer is basically solving two optimization problems simultaneously, on u and n. (The way you'd actually write it is min n(x) s.t. u(x)>u_0, x\in X.)

To maintain the computational simplicity that satsificers are useful for, though, we wouldn't want to write it as a minimization problem. This requires us to use a less restrictive version of 'well-behaved,' where the proposal function is 'generally' increasing in effort rather than strictly nondecreasing in effort.