ICE Preference Axioms
My most recent ICE publication [Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptability. Health Services and Outcomes Research Methodology 2008; 8: 31-56] started out as a much shorter paper with a different title: ICE Preference Maps: An Axiomatic Basis.
The (x, y) notation for (effectiveness, cost) differences between two competing treatments used in these axioms assumes [as explained on my ICE notation and terminology page] that both of the x and y coordinates are expressed in the same units (either both in cost units or else both in effectiveness units.) In other words, the problem has been placed into its canonical (standardized) form where the Shadow Price of Health, Lambda, can be visualized as having been forced to equal 1.
When first examining the following table, it may be helpful to note that the linear preference map of Net Benefit Analysis, NB(x, y) = x - y, clearly satisfies all four of these axioms. The first two axioms represent basic concepts that appear to have been well known and widely accepted before my ICE publications; the importance of the final two axioms was essentially unrecognized before my work.
The Four Axioms
of ICE Preference
Direction of Preference:
P(x, y) = 0 when x = y,
P(x, y) > 0 when x > y, and
P(x, y) < 0 when x < y.
P(x, y) > = P(xo, yo) for
all x > = xo and y < = yo.
P(x, y) = -P(-x, -y)
P(x, y) = P(-y, -x)
= -P(y, x)
The following are my (intuitive) motivations for these four axioms...
 ICE Indifference and Direction of Preference
When x = y, society receives exactly the difference in effectiveness for which it pays, no more and no less. Therefore, there is no compelling reason to prefer one treatment relative to the other, i.e., P(x, y) = 0. If the new treatment is more effective, it is commensurably more costly. If the new treatment is less costly, it is commensurably less effective.
When x > y, the difference in effectiveness of the new treatment compared to the standard exceeds the cost difference between the treatments. Society thus receives a level of incremental effectiveness worth more than its incremental cost. Therefore, the new treatment is preferred over the standard, i.e. P(x, y) > 0.
When x < y, the difference in cost is larger than the difference in effectiveness. Society thus receives a level of incremental effectiveness worth less than its incremental cost. The standard treatment is then preferred over the new treatment, i.e. P(x, y) < 0.
This first axiom is by far the most restrictive of the four. It dictates an infinite, linear interface of standardized slope 1 separating positive from negative ICE preferences. In original units, this is the line DC/DE = l with slope determined by the shadow price of health, which ICE analysts may wish to deliberately vary to perform sensitivity analyses that turn out to be anything but subtle.
 ICE Cartesian Monotonicity
A fundamental property of all sensible preference maps is that P(x, y) is > or = P(x0, y0) for all x > or = x0 and all y < or = y0. If the effectiveness of a new treatment is increased at the same time its ultimate cost is decreased, preference for that new product over a fixed standard treatment certainly cannot decrease. Remember that we are assuming that x has been defined so that larger (more positive) values of standardized effectiveness are more favorable to the treatment currently called new. Similarly, y must be defined so that smaller (more negative) values of standardized cost are more favorable to the treatment currently called new.
 ICE Re-labeling
One meaning of P(x, y) = -P(-x, -y) is that, when reversing treatment labels (new and standard) on a single pair of treatments, the direction of preference is reversed while the strength of preference is preserved. The implications of this axiom are broader in the sense that this same preference equality must also hold when a fixed new treatment is either preferred or not preferred to a fixed standard by a fixed, specified amount. This axiom imposes a form of fairness or even-handedness upon head-to-head ICE treatment comparisons.
 ICE Symmetry and Anti-Symmetry
Axiom 4 can be expressed in two equivalent ways. Starting with the third axiom plus either form of the fourth axiom, the other form of the fourth axiom follows immediately by simple algebra. For example, the re-labeling property, P(x, y) = -P(-x, -y), can be combined with the symmetry property, P(x, y) = P(-y, -x), to yield P(x, y) = -P(y, x), which is the anti-symmetry property.
The ICE symmetry axiom requires that preferences for any pair of outcomes symmetrically located relative to the upper-left to lower-right standardized diagonal, x = -y, of the ICE plane must be identical, P(x, y) = P(-y, -x). In other words, for any (x, y) outcome, the alternative outcome of (-y, -x) must yield the exact same strength of preference in the same direction (new over standard or vice versa).
Suppose now that (xo, yo) is any fixed point within the NE quadrant, 0 < xo and 0 < yo . As a result, (-yo, -xo) is then a fixed point within the SW quadrant. Denoting the standardized ICE ratio (slope) corresponding to (xo, yo) by so = yo/xo > 0, ICE symmetry implies that the standardized slope corresponding to the outcome pair (x, y) = (-yo, -xo) is s = y/x = -xo/-yo = +1/so > 0. In other words, one immediate implication of this axiom is that all (xo, yo) and (-yo, -xo) pairings with equivalent preferences have standardized ICE ratios, s = y/x, that are numerical reciprocals or inverses.
This inverse relationship has considerable intuitive appeal. Within the NE quadrant, s = y/x is a positive “loss over gain” ratio; the positive numerator represents an undesirable additional cost (loss) while the positive denominator represents a desirable increase in effectiveness (gain.) Meanwhile, within the SW quadrant, s = y/x is a positive “gain over loss” ratio; the negative numerator represents a desirable cost reduction (gain) while the negative denominator represents an undesirable reduction in effectiveness (loss.)
In other words, numerically small and positive standardized ICE ratios are desirable within the NE quadrant where they represent loss/gain ratios, while numerically large and positive standardized ICE ratios are desirable within the SW quadrant where they represent gain/loss ratios. By assuring that outcomes within the NE and SW quadrants that yield equivalent preferences also yield standardized ICE ratios that are numerical reciprocals (yo/xo and -xo/-yo= xo/yo), the ICE symmetry axiom simply formalizes basic intuition. (This highly cogent motivation for ICE symmetry is based upon the "give/get" versus "get/give" arguments of Ken Buckingham in his unpublished paper on ALICE curves defined using a "frontier" formed by ICE rays with reciprocal slopes.)
Next, note that the linear preference map, NB(x, y) = x – y, possesses a purely optional property that is much stronger and more restrictive than P(x, y) = P(–y, –x). Linear NB preference is constant everywhere on the straight line passing through the points (x, y) and (–y, –x). Again, when one’s preference map is linear, preference is assumed constant on all straight lines (x – y = constant) that are parallel to the lower-left to upper-right diagonal (x = y) of the ICE plane.
Finally, note that ICE preference symmetry property does impose an additional restriction besides reciprocal ICE ratios, y/x and –x/–y = x/y, for the corresponding "matched" outcome pairings. Namely, all such pairs clearly also have the same ICE radius,
In its own right, the anti-symmetry requirement that P(x, y) = -P(y, x) is quite intuitive. It requires symmetry in strength of preferences about the x = y diagonal. However, this property is called anti-symmetry here because the direction of preferences is reversed on the two different sides of the x = y diagonal. After all, when pairs of outcomes of the form (x, y) and (y, x) are not on the x = y diagonal, they are symmetrically located relative to this x = y diagonal.