Sunday, 1 February 2015

Revealed preference, WARP, SARP and GARP

The basic idea behind revealed preference is incredibly simple: we try to infer something useful about a person's preferences by observing the choices they make. The topic, however, confuses many a student and academic alike, particularly when we get on to WARP, SARP and GARP. So, let us see if we can make some sense of it all.
          In trying to explain revealed preference I want to draw on a study by James Andreoni and John Miller published in Econometrica. They look at people's willingness to share money with another person. Specifically subjects were given questions like: 
Q1. Divide 60 tokens: Hold _____ at $1 each and Pass _____ at $1 each. 
In this case there were 60 tokens to split and each token was worth $1. So, for example, if they held 40 tokens and passed 20 then they would get $40 and the other person $20. Consider another question:
Q2. Divide 40 tokens: Hold _____ at $1 each and Pass ______ at $3 each.
In this case each token given to the other person was worth more than that kept. So, for example, if the subject held 20 tokens and passed 20 they would get $20 and the other person $60. The figure below plots the feasible set for each of the two questions.


           Suppose a subject called James chooses A when asked Q1 and B when asked Q2. In other words he passes all 60 tokens in question 1 and holds all 40 tokens in question 2. Is that 'sensible'? When answering Q1 James could have chosen B; he could have given $0 to the other person and kept $40 for himself. Instead he chose A; he chose to give $60 to the other person and keep $0 for himself. We say that A is directly revealed preferred to B. But, then look at question 2. In this case James could have chosen A but chose B. So B is directly revealed preferred to A. This seems inconsistent; how can A be preferred to B and B preferred to A. It is a violation of the Weak Axiom of Revealed Preference (WARP).
           Why would James or anyone else violate WARP? There are three basic reasons: (1) James does not have consistent preferences. Which is another way of saying he behaves irrationally. (2) He is indifferent between A and B. I'll come back to this later. (3) There are menu or framing effects. To illustrate how menu effects work it is easier to consider a slight variation on Q1 and Q1. Specifically, consider questions:

Q3. Would you choose: A. $0 for yourself and $60 for the other person, or B. $40 for yourself and $0 for the other person.

Suppose James chooses B. So, B is directly revealed preferred to A. Now consider an alternative question:

Q4. Would you choose: A. $0 for yourself and $60 for the other person, B. $40 for yourself and $0 for the other person, or C. $0 for yourself and $50 for the other person.

Clearly option C is a nonsense option because A beats C. But, the presence of C may sway James to choose A. This would be a violation of WARP caused by an attraction effect: option A looks best because it is clearly better than C. There is nothing inherently inconsistent, therefore, in some violations of WARP.
          Let's move on now to the Strong Axiom of Revealed Preference (SARP). Suppose, as before, that James chooses B when asked question 3. So, B is directly revealed preferred to A. Suppose also that when asked he would choose A over C. So, A is directly revealed preferred to C. Given that B 'beats' A and A 'beats' C we would say that B is indirectly revealed preferred to C. Suppose, however, that when asked James choses C over B. Then we have a violation of SARP.
          Note that there is nothing in the story of the previous paragraph that violates WARP. That, however, is because we only consider pairwise comparisons. If we ask James to choose between A, B and C then, whatever his answer, we would get a violation of WARP. More generally, if we elicit James' choices over all the possible combinations of options then WARP and SARP essentially become equivalent: if SARP is violated then so must be WARP and vice versa. The setting of Andreoni and Miller is like this. In order to observe a violation of SARP without a violation of WARP it must be that we only observe choices for some possible combinations of options. That, however, is not unlikely in applied settings. Preference reversals being one example.
         WARP and SARP get most of the attention. They have, however, a basic 'flaw'. Suppose James is indifferent between A and B. Or, to give a slightly more mundane example suppose that when he goes to the canteen for lunch, James is indifferent between ham sandwiches at $2 or cheese sandwiches at $1.50. If we observe James sometimes buying ham sandwiches and sometimes cheese we would say that his preferences violate WARP and SARP. But, there is nothing inconsistent in his preferences or choices. To get out of this problem we need the Generalized Axiom of Revealed Preference (GARP).
          To sometimes choose A and sometimes choose B, or to sometimes choose ham and sometimes choose cheese, is not a violation of GARP. What is? Suppose we observe James sometimes choosing ham and sometimes cheese. Then the price of a cheese sandwich increases from $1.50 to $1.75. If James is observed buying a cheese sandwich after the price rise we say that cheese is strictly directly revealed preferred to ham. This is a violation of GARP. The rise in price should have broken James' indifference between ham and cheese but it did not.
         In order to fit this into the setting of Andreoni and Miller we need a slight reframing of James' choice. Let us consider how much he is willing to give to the other person. With question 1 it costs James $60 to give $60 to the other person and reach point A. With question 2, how much would it cost James to get to point A? It would only cost $40. (Inefficient though it is he would hold 0 tokens and pass 20 tokens, losing 20 tokens in the process.) We said that James would say A when asked question 1. Given that it is 'cheaper' to say A when asked question 2 it would be a violation of GARP for James to now choose B. If he did so it would be the case that B is strictly directly revealed preferred to A.  
         Let me finish by briefly commenting on why WARP, SARP and GARP matter. If GARP is satisfied then preferences are 'well behaved' and we can model choice using a utility function. Given that utility maximization drives most of economics this is clearly crucial! WARP and SARP are less relevant because of the indifference issue. For example, preferences could satisfy GARP and not WARP or SARP.
          That just leaves the question of whether preferences satisfy GARP? There is abundant evidence of violations of GARP. So, in a strict sense people's preferences don't satisfy GARP; something many behavioural economists emphasize. But, to say that preferences don't satisfy GARP is different to saying we cannot make meaningful predictions by assuming they do.

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