Theorem on zero tolerance and behavior control

We formally state and prove the fact about zero-tolerance and progressive penalties described in a prior entry.

Definitions. A utility function is any real function. A penalty is a real non-decreasing function p(x) such that p(x) = 0 if x ≤ x_legal, for some fixed x_legal value. The penalty is said to be zero-tolerance if it has the form p₀ + r(x) for x < x_legal, with p₀ > 0 and r(x) non-decreasing. On the other hand, p(x) is progressive if it is continuous.

Theorem. Let u(x) be a continuous unimodal utility function with maximum at x_max > x_legal and p(x) = p₀ + r(x) a zero-tolerance penalty such that u(x_legal) − ε ≥ w(x) = u(x) − p(x) for x > x_legal and some ε > 0. There is then a progressive penalty p'(x) such that u(x_legal) > u(x) − p'(x) and p'(x) ≤ p(x) for x > x_legal.

Proof. Choose some continuous strictly increasing function d(x) on [x_legal, x_max] with d(x_legal) = 0, d(x) ≤ ε in all its domain (for instance, d(x) = ε(x − x_legal)/(x_max − x_legal) ) and define p'(x) on [x_legal, x_max] as

p'(x) = u(x) − u(x_legal) + d(x),

which is clearly non-decreasing given the unimodality of u(x). Then, for any x in (x_legal, x_max] we have:

u(x) − p'(x) = u(x_legal) − d(x) < u(x_legal),

and also:

p'(x) = u(x) − u(x_legal) + d(x) ≤ u(x) − u(x_legal) + ε ≤ p(x),

where the last inequality follows directly from the theorem hypothesis on p(x). To complete our definition of p'(x), extend it for x > x_max as p'(x) = p'(x_max). p'(x) is then clearly a progressive penalty and verifies all the conditions of the theorem.

It is not possible to relax the theorem hypothesis to merely requiring that u(x_legal) > w(x) = u(x) − p(x) for x > x_legal; the figure shows a counterexample:

The zero-tolerance penalty function p(x) tends to u(x_max) − u(x_legal) when x tends to x_max from the left, even though p(x_max) > u(x_max) − u(x_legal). So, for any continuous p'(x) ≤ p(x) we will have p'(x_max) ≤ u(x_max) − u(x_legal), making p'(x) an inefficient penalty for u(x). Anyway, this limit case has no real-life significance.