When one was asked whether he prefered either an apple to a banana, or a banana to an apple, he told that he chose an apple. After that the same person was asked whether he prefered either a banana to a peach, or a peach to a banana, he told that he prefered a banana. Can we infer something about his preferences? We know that first he had chosen an apple over a banana, and then chose a banana over a peach. We might be able to assume that this person like an apple more than a peach. This is called the Revealed Preference Relations.

Since the preferences of human are abtract and unobservable, economists proposed the so called Revealed Preference Relations to explain the behavior of human. We observed what choices are made by a person and then made the conclusion on his preferences from what we observed. However, we expected something special with someone to be called a rational person. One will be called a rational person if he satisfies the "Weak Axiom of Reveal Preferences" (WARP).

I will use the same example to explain WARP. Suppose that I have three kind of fruits as above. Suppose that I first had an apple and a banana available for you and required you to choose only one of them. You said you chose an

__apple__. "OK" said I. Then I did not give an apple to you but instead asked you again that this time I have one more peach as the third choice and now which one of them you prefered. This time you told me that among those three I prefer a__banana__the most. "OK. I know that you are not a reasonable person!!! and I won’t give you anything" said I. You may said "What wrong!! I chose what I prefered the most". Why can I conclude that you are not rational?What I will tell you is

__you are not satisfy WARP__becuase when at first apple and banana are available, you chose apple and next when BOTH APPLE and BANANA available again and a peach presented as the third alternative, but this time why did you choose a banana instead of an apple?To explain mathematically, let me give you the definition of WARP first

**A person is said to satisfy WARP if when a,b available, he chose a over b. Later if a,b available again, he must not choose b over a.**

**The implication of WARP is that the preference should be independent of the sets of available choices.**

Let a=apple, b=banana, p=peach. In addition let

**X**be a set of all available alternatives, and**C**(·) be the "choice rule" on available alternatives. In this example we have**X**={**a**,**b**,**p**} and**B1**,**B2**are subsets of**X**where**B1**={**a**,**b**} and**B2**={**a**,**b**,**p**}. Observe that__are in both__**a**,**b****B1**and**B2**. First you had**C**(**B1**)=**C**({**a**,**b**})={**a**} ie you choose an apple. Next you had**C**(**B2**)=**C**({**a**,**b**,**c**})={**b**} which contradicts to WARP. To satisfy WARP, you have to choose either**a**or**p**.Also, WARP can not apply to me. This time I have three alternatives at Cornell,

**X**={**reading**,**enjoying**,**F in Q Exam**}. When only reading and enoying are presented, certainly enjoying seem more appeal to me, ie**C**({**reading**,**enjoying**}) = {**enjoying**}. Later when all of three alternatives are presented, I must choose reading over enjoying to avoid failing in the Q Exam ie**C**({**reading**,**enjoying**,**F in Q Exam**}) = {**reading**}. Am I not a rational person?????
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I every studied weak axiom and reviewed preference when I was sophomore.I forgot all of them. This make me miss Economist so much.Have a great success na krab!