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.

Thinking aloud here:

Say I'm an agent that wants to increase u, but not "too strongly" (this whole thing is about how to formalize "too strongly"). Couldn't I have a way of estimating how much other agents who don't care about u might still care about what I do, and minimize that? i.e. avoid anything that would make other agents want to model my working as something more than "wants to increase u".

(back in agent-designer shoes) So we could create a "moderate increaser" agent, give it a utility function u and inform it of other agents trying to increase v, w, x, y, and somehow have it avoid any strategies that would involve "decision theory interaction" with those other agents; i.e. threats, retaliation, trade ... maybe something like "those agents should behave as if you didn't exist".

I generally think of satisficing being a local property of a utility function. I may want to maximize my utility, but locally speaking I am indifferent among many, many things, so I satisfice in respect to those things.

I think that looking at it in this way might be more productive.

Aside from that,

Would not effectively aid M(u), a u-maximiser. Would not effectively resist M(-u), a u-minimizer.

It is likely to do the former up to a point, and would certainly do the latter after a point. As Vaniver noted, you haven't specified that point.

Would not effectively resist M(-u), a u-minimizer.

I'm not sure how that's supposed to work. S(u) won't do much as long as the desirability threshold is obtained, but if M(-u) comes along and makes this difficult, S(u) would use everything it has to stop M(-u). Are you using something beyond desirability threshold? Something where S(u) stops not when the solution is good enough, but when it gets difficult to improve?

The general problem seems difficult. There are plenty of ways to model satisficing behavior.

Suppose we are given a utility function u. If we have some information about the maximum, here are some possibilities:

• The satisficer follows some maximization algorithm and terminates once some threshold is attained (say, 90% of max(u)).

• The satisficer uses a probabilistic algorithm that terminates after a certain threshold is probably attained.

• The satisficer has some threshold, but continues searching as long as it's "cheap" to do so, where "cheap" depends on some auxiliary cost function. It could, for example, measure the number of steps since attaining the best known value thus far or it could measure the rate of improvement and stop once that dips too low.

If we don't know much about the maximum, we can't very well have a predefined threshold, but there some strategies:

• The satisficer follows some algorithm for maximizing u and terminates after, say, N steps and chooses the best value found. Or one can add some adaptive behavior and say that it terminates if there has been no improvement in a certain time interval.

• In a constrained problem where searching is expensive, the satisficer accepts the first solution satisfying the constraints.

What the above have in common is that the satisficer is basically carrying out some optimization algorithm with alternative stopping conditions. As such, it may exhibit some of the drawbacks one finds with optimizers. Maybe there's a model that's not built on an optimizer that I haven't thought of.

This contradict the intuitive picture that many people have of a satisficer, which is an agent that does the minimum of effort to reach its goal, and doesn't mess up the outside world more than it has to. And if it can't accomplish the goals without messing up the outside world, it would be content not to.

There's a tacit assumption here that messing with the outside world involves effort. Without it, you can get all kinds of unexpected behavior. If the satisficer is fine with "make 10 paperclips", but the difference between that and "make 10 paperclips and kill all humans" is small, the satisficer may well go with whatever showed up first in its search algorithm.

That's just a relic of mathematics though. If you're trying to find the maximum of, say, the size of the projection of some 10-dimensional set on a particular 2-dimensional plane, there may be a lot of room in those extra dimensions that doesn't impact the actual value to be maximized. Trying to find something merely close to the max would seem to exacerbate this issue.