Sydney Afriat arrived in Purdue in the late 60’s with a Bentley in tow. Mort Kamien described him as having walked out of the pages of an Ian Flemming novel. Why he brought the Bentley was a puzzle, as there were no qualified mechanics as far as the eye could see. In Indiana, that is a long way. Afriat would take his Bentley on long drives only to be interrupted by mechanical difficulties that necessitated the Bentley being towed to wait for parts or specialized help.

I came upon Afriat when I learnt about the problem of rationalizability.  One has a model of choice and a collection of observations about what an agent selected. Can one rationalize the observed choices by the given model of choice? In Afriat’s seminal paper on the subject, the observations consisted of price-quantity pairs for a vector of goods and a budget. The goal was to determine if the observed choices were consistent with an agent maximizing a concave utility function subject to the budget constraint. Afriat’s paper has prompted many other papers asking the same question for different models of choice. There is an aspect of these papers, including Afriat’s, that I find puzzling.

To illustrate, consider rationalizing expected utility (Eran Shmaya suggested that expected consumption’ might be more accurate). Let ${S = \{1,2 \ldots, n\}}$ be the set of possible states. We are given a sequence of observations ${\{x^{i},p^{i}\}_{i=1}^{m}}$ and a single budget ${b}$. Here ${x^i_j}$ represents consumption in state ${j}$ and ${p^i_j}$ is the unit price of consumption in state ${j}$ in observation ${i}$. We want to know if there is a probability distribution over states, ${v=(v_{1},...,v_{n})}$, such that each ${(x^i, p^i)}$ maximizes expected utility. In other words, ${(x^i, p^i)}$ solves

$\displaystyle \max \sum_{j=1}^{n}v_{j}x^i_{j}$

subject to

$\displaystyle \sum_{j=1}^{n}p^i_{j}x_{j}\leq b$

$\displaystyle x^i_{j}\geq 0\,\,\forall j \in S$

The solution to the above program is obvious. Identify the variable with the largest objective coefficient to constraint ratio and make it as large as possible. It is immediate that a collection of observations ${\{x^{i},p^{i}\}_{i=1}^{m}}$ can be rationalized by a suitable set ${\{v_{j}\} _{j=1}^{n}}$ of non-zero and nonnegative ${v_{j}}$‘s if the following system has a feasible solution:

$\displaystyle \frac{v_{r}}{p^i_r}\geq \frac{v_{j}}{p^i_{j}} \,\,\forall j, \,\, x^i_r> 0$

$\displaystyle \sum_{j \in S}v_{j}=1$

$\displaystyle v_{j}\geq 0\,\,\forall j \in S$

This completes the task as formulated by Afriat. A system of inequalities has been identified, that if feasible means the given observations can be rationalized. How hard is this to do in other cases? As long as the model of choice involves optimization and the optimization problem is well behaved in that first order conditions, say, suffice to characterize optimality, its a homework exercise. One can do this all day, thanks to Afriat; concave, additively separable concave, etc. etc.

Interestingly, no rationalizability paper stops at the point of identifying the inequalities. Even Afriat’s paper goes a step farther and proceeds to characterize’ when the observations can be rationalized. But, feasibility of the inequalities themselves is just such a characterization. What more is needed?

Perhaps, the characterization involving inequalities lacks `interpretation’. Or, if the given system for a set of observations was infeasible, we may be interested in the obstacle to feasibility. Afriat’s paper gave a characterization in terms of the strong axiom of revealed preference, i.e., an absence of cycles of certain kinds. But that is precisely the Farkas alternative to the system of inequalities identified in Afriat. The absence of cycles condition follows from the fact that the initial set of inequalities is associated with the problem of finding a shortest path (see the chapter on rationalizability in my mechanism design book). Let me illustrate with the example above. It is equivalent to finding a non-negative and non trivial solution to

$\displaystyle \frac{v_{r}}{v_j}\geq \frac{p^i_{r}}{p^i_{j}} \,\,\forall j, \,\, x^i_r> 0$

Take logs:

$\displaystyle \ln{v_r} - \ln{v_j} \geq \ln{\frac{p^i_{r}}{p^i_{j}}} \,\,\forall j, \,\, x^i_r> 0$

This is exactly the dual to the problem of finding a shortest path in a suitable network (I believe that Afriat has a paper, that I’ve not found, which focuses on systems of the form $b_{rs} >$ $x_s - x_r$ ).The cycle characterization would involve products of terms like ${\frac{p^i_{r}}{p^i_{j}}}$ being less than 1 (or greater than 1 depending on convention). So, what would this add?