Principles for front of pack scoring systems

Several front-of-pack or supermarket-shelf nutrition labeling programs use scoring systems to rate food products.

Examples include the Hannaford supermarket chain's Guiding Stars system ...
... and the NuVal system used by some other supermarket chains.
The computation of the scores seems fairly arbitrary to me. It might help to start with some basic mathematical principles. Here are three principles that I think a food scoring system should satisfy.

1) The score should not depend on the serving size. A major limitation of all such scoring systems is that our health really depends on how much of each food we eat, and in what combination. Unfortunately, the scores necessarily apply to each food as it sits on the supermarket shelf, not as we consume it. Because the creators of the score have no knowledge of how much we eat, the score should be treated as a description of the density of good and bad nutrients per unit of weight or food energy. The score should not depend on the serving size convention used or the number of servings in a package. I have seen some scoring systems that appear at first glance to be independent of the serving size, but on closer inspection have quirky limits on total daily nutrients per serving that contribute to the score.

2) The score should rate mixed foods in a consistent way. Take the example of a ham and cheese sandwich. Suppose the sandwich is, by weight or by calories, 40% bread, 30% meat, and 30% cheese. The score for a packaged ham sandwich in the supermarket should be the same as a weighted average of the scores for bread, meat, and cheese purchased separately:

SandwichScore = 40%*BreadScore+30%*HamScore+30%*CheeseScore.

Without this principle, the scoring system will be biased in favor of or against manufactured mixed foods in place of separate ingredients. I think a good scoring system would have no such prejudice.

3) The score should rate each good and bad nutrient independently. Suppose adding 20% more salt to a high-fiber food reduces the score by 20%. How much should adding 20% more salt change the score for a low-fiber food? Many of us would say the score should again change by 20%. Unfortunately, existing scoring systems may have strange interactions across the good and bad nutrients, so that the effect of one nutrient on the score depends on the value of other nutrients in a way that the authors probably did not intend.

Currently, the scoring systems seem to me to have an ad hoc quality that makes it difficult to take the quantitative scores very seriously. Broadly speaking, a scoring system may seem to work correctly in giving healthy foods a good score, and unhealthy foods a bad score, but consumers understood those broad outlines of their food options already even without a scoring system. The whole purpose of a scoring system is to add quantitative rigor to the information provided. They may have a long way to go.

In other recent reading on front-of-pack labeling, see the recent series by Timothy Lytton at the Fooducate blog. In general, Lytton suggests that FDA should not get into the business of developing its own front-of-pack system, but instead should just enforce existing rules against making implicit healthy claims for foods that fail to meet FDA's definition of "healthy." One exception to Lytton's hands-off recommendation is that he feels there may be a need for stronger regulation of the complex scoring systems.