"I have two related questions on Choice.
I know that we can satisfy WARP but have nevertheless a violation of GARP. My question is if we can have a situation where WARP is violated, but GARP is satisfied?
Secondly, from the definition of GARP it is always spoken of a bundle being revealed preferred to another bundle through a chain or ""sequence"" of revealed preferences. My question is, if this defined ""sequence"" can consist of only two observations, so that we have actually a direct revealed preference after all? In other words, does ""revealed preferred"" include the case of ""direct revealed preferred""?"
The quick answers are, to the first question, no, and to the second question, yes. In other words, GARP implies WARP and the chain can have only two elements, which is WARP directly.
A longer response would include making a comparison between revealed preference and the usual assumptions made about preference. These are about a preference relation, R. xRy is then read as, x is preferred to or indifferent to y. So from R one can also define P and I by:
- xPy if xRy and not yRx.
- xIy if xRy and yRx.
There are three "logical" assumption about preference orderings.
R is complete. For every x and y in the Consumption set (the set of possible consumption bundles) either xRy or yRx. This means comparisons can be made between any two consumption bundles. Note that neither P nor I are complete.
R is reflexive. For every x in the Consumption set xRx.
R is transitive. For every x, y, and z in the Consumption set, if xRy and yRz then xRz.
These properties allow one to define a choice, provided the choice set is closed. In this case if the choice set is C then x in C is a choice (a maximal element under R) if xRy for all y in C. Note that there is no greatest number less than 100. If you posit it is 99, then 99.9 is greater and you can always add an additional 9 to the right. So for a choice to exist, the choice set must be closed. 100 is the greatest number less than or equal to 100. The choice set being closed means it includes its boundary.
To these logical assumptions, one usually adds an economic assumption - monotonicity or more is preferred to less. This assumption rules about satiation points as well as "thick indifference curves," The upshot of this assumption is that when the choice problem is given by a budget set, the choice will always be on the budget line, never inside the line.
A further assumption that is frequently made is that preferences are Convex, which gives indifference curves their usual shape. This is done do when the Budget environment changes in a small way, the choice also changes at most in a small way. Or, if you prefer, the demands are continuous function of the budget environment.
Now, with all this machinery, what does WARP get you? In this way of thinking, WARP is equivalent to completeness, reflexivity, and monotonicity. You need GARP to bring in transitivity. There is also something called SARP, the strong axiom. It is WARP plus the assumption that preferences are strictly convex, so the choice is always unique.
I hope that helps.