Weakened-Weak Axiom of Revealed Preference and Consistency in Decision Making

[Somdeb Lahiri]

The following occurred to me while working in a different context and I request you to let me know whether the "money pump argument" at the end is valid or not.

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In what follows there are no cardinality restrictions on sets from which alternatives are chosen. They may be finite or infinite.

The Weak Axiom of Revealed Preference (WARP) says that if y is available when x is chosen, then it should not be the case that when y is chosen, x is available but not chosen.

Fifteen years ago (in 2008), Lars Ehlers and Yves Sprumont suggested a property called Weakened-WARP.

Weakened-WARP (W-WARP) says that if y is available and not chosen when x is chosen, then it should not be the case that when y is chosen, x is available but not chosen.

Clearly WARP implies W-WARP but the converse is not true as the following example shows.

Example: Let X = {x,y,z} and the choice function C on Y(X) satisfies C({x,y}) = {x}, C({y,z}) = {y}, C({z,x}) = {z} and C({x,y,z}) = {x,y,z}.

It is easy to see that C satisfies W-WARP but violates WARP.

When C is “singleton valued” then WARP is equivalent to W-WARP..

Now given any choice function (possibly multi-valued) there is the "natural" menu-dependent strict preference relation associated with it which for every menu (feasible set) of options A and x,y belonging to A says that x is strictly preferred to y at A "if and only if" x is chosen at A but y is (available but) not chosen at A.

Thus, Weakened WARP says that there does not exist two menus of options A and B, such that x is strictly preferred to y at A and y is strictly preferred to x at B.

Now suppose a decision-maker (DM) violates W-WARP. 

Thus, there exists two menus of options A and B, such that x is strictly preferred by the DM to y at A and y is strictly preferred by the DM to x at B.

Suppose that the DM values money (as an instrument with which he/she can derive satisfaction from the consumption of goods and services). 

This is a very crucial assumption for the argument that follows.

Consider the following thought experiment, where the DM starts off with x at A. Now, if he is given the option of choosing y from B in lieu of x which is also in B, then he will be willing to pay a strictly positive amount of money, say 'a', to get y in exchange of x since at B, the DM strictly prefers y to x. If after he has received y at B against a payment of 'a', he is given the option of choosing x from A in lieu of y which is also in A, then he will be willing to pay a strictly positive amount of money, say 'b', to get x in exchange of y since at A, the DM strictly prefers x to y.

In this way, by alternating between x from A and y from B, the entire wealth of the DM can be “pumped out” leaving the DM totally bankrupt. 

Thus, if not going bankrupt in a thought experiment based on a “money pump argument” is considered to be a minimal requirement of consistency in decision making, then the choice function of the DM should satisfy W-WARP.

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What I am not sure about is the validity of the argument and conclusion, the entirety of which is marked in "brown", and at the same time I can't see why and where the argument is wrong. Inputs from you would be most welcome.

Apologies for the "longish" email.

Thanks and regards.

Somdeb. 

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Somdeb Lahiri
https://sites.google.com/view/somdeblahiri

https://independent.academia.edu/LahiriSomdeb 

I believe in non-violence (= absence of violence).

Lecture Notes: Demand, Demand Curves and Consumer's Surplus

[somdeb...@gmail.com]

The published version, the only thing preciously new about which is the quote due to Gurudev (spiritual guide or murshid and not a salaried teacher) Rabindranath Thakur.

https://drive.google.com/file/d/1Em1qvKUxRgASDNsAEsM4wRqekAwbUEsp/view?usp=sharing

Thanks.

Somdeb.

[somdeb...@gmail.com]

I came across a paper on "revealed preference" in Acta Oeconomica, late evening this Monday, while my work was in progress on the paper available at the following link:

https://drive.google.com/file/d/1B9HL-SlsKpda9CEqJ2Riho80SR58bPHB/view

To me it appears that there is a difference in the representation/understanding of "revealed preference", particularly in the context of hyperbolic discounting, which justified my reasons for writing this paper.  I was aware of the use of hyperbolic discounting as an example to illustrate "time inconsistency" in the context of dynamic programming and not as an example to illustrate the violation of Weakened-WARP. Hence, it did take me a while to write a couple of paragraphs about the compatibility of hyperbolic discounting with Weakened-WARP. 

Your comments on the document available at the above link would be most welcome.

Thanks.

Somdeb.