Transitivity, Completeness and Reflexivity

Completeness

It is assumed that individuals must have a preference relationship between any two sets of goods; either we must be able to say that they weakly prefer A to B, or that they weakly prefer B to A, or both (indifference). If we are told that Dave strictly prefers larger chocolate bars to smaller ones, this gives us enough information to completely define Dave’s preferences over the entire space of chocolate bars. If Susie says that she always prefers the bigger and darker chocolate bar, we do not have enough information to define a preference relationship across the entire space of chocolate bars – if one bar is darker and smaller than another, but lighter and bigger than a third, which is preferred to which?

Transitivity:

Transitivity refers to the property of preference relationships that if one bundle (bundle A) is preferred to another (bundle B), and that bundle is preferred to a third (bundle C), then the first bundle must be preferred to the third.

The relationship between the first and the third bundles will be governed by the strongest preference relationship in the set; if A is strictly preferred to B and B is weakly preferred to C (at least as preferred as C), A is strictly preferred to C. If A is weakly preferred to B and B is strictly preferred to C, A is strictly preferred to C.

Given that the measurement of cardinal utility is beyond the realms of practicality (its measurement is theoretically possible by making use of nuclear imaging such as a Functional Magnetic Resonance Imager), we are left able only to perform ordinal analysis. If all we are able to do is rank preferences, it is important that we are able to compare these ranks.
Reflexivity

Relationships are reflexive if they can be applied when both sides of the relationship are the same – i.e. I am at least as old as myself  (I am in fact exactly as old as myself, but the statement is not incorrect, merely imprecise). Weak preference relationships are reflexive; a bundle of goods can be said to be weakly preferred to itself, but not strictly preferred to itself (in fact, it can be more accurately said to be exactly as preferred as itself).

Completeness

It is assumed that individuals must have a preference relationship between any two sets of goods; either we must be able to say that they weakly prefer A to B, or that they weakly prefer B to A, or both (indifference).