What's the algorithm for converting a scale of values into the statements about the world (i.e. a particular individual's preferences at a particular point in time) contained in that scale of values? What do they say?
In Human Action, chapter 7, Mises describes scales of values for homogeneous goods, and gives an example:
"A man owns five units of commodity a and three units of commodity b. He attaches to the units of a the rank-orders 1, 2, 4, 7, and 8, to the units of b the rank-orders 3, 5, and 6. This means: If he must choose between two units of a and two units of b, he will prefer to lose two units of a rather than two units of b. But if he must choose between three units of a and two units of b, he will prefer to lose two units of b rather than three units of a."
I interpret this as follows: count how many units of a and b you have, say n and m. Look at the nth number in the list-of-rank-orders for a and the mth for b, and take the lowest of the two. The larger this number, the happier you are.
It seems to match the example: losing 2*a makes position = min(4, 6) = 4 and losing 2*b makes position = min(8, 3) = 3, so the man prefers losing 2*a. On the other hand, losing 3*a makes position = min(2, 6) = 2, which he prefers less than losing 2*b.
In other words, this specific scales says that (3a, 3b) beats (5a, 1b) beats (2a, 3b). Those preferences seem to me to be compatible with both "(4a, 0b) beats (3a, 1b)" and its opposite.
Either the scale says something about which of (4a, 0b) and (3a, 1b) beats the other or it doesn't. If it does, doesn't that mean that no scale of values can express the other set of preferences? And if it doesn't, does that not mean that scales of values are unable to distinguish between two distinct sets of preferences?
Also, consider an example with three goods: if I have an a, I want b over c; if not, the opposite. In other words, I prefer (1a, 1b, 0c) to (1a, 0b, 1c) and I want (0a, 0b, 1c) more than I want (0a, 1b, 0c). Is there a scale of values which expresses (perhaps among others) these two preferences? If so, how would such a scale look?
Please help me understand scales of values