# Are Preference Relations Robust to Aggregation

This is more a question of putting the theory into practice, or at least clarifying the theory in practice. The question has to do with the application of preference relation axioms and aggregation. Take the transitivity relation for example;

$\forall (x,y,z) \in X,$ if $x \succ y \wedge y \succ z \iff x \succ z$

Suppose I have a set of data on the preferences of $n$ consumers $C_1,C_2,...,C_n$ over products $a, b,$ and $c$ such that;

$C_1$ prefers $a \succ b \succ c$

$C_2$ prefers $a \succ c \succ b$

$C_3$ prefers $c \succ b \succ a$

and so on until consumer $n$, each with a respective WTP for each product.

So the question is, if the data were aggregated over $n$ consumers we would achieve an aggregate WTP for whichever preference dominates. Lets say that the aggregate data reveals $a \succsim b \succsim c$. Does this result violate transitivity and completeness since the individual consumer $C_3$ prefers $c \succ b \succ a$. Do individual and aggregate preferences have to be consistent.

• Depending on the specific aggregation mechanism you consider, the aggregated preference may or may not satisfy transitivity. However, I'm not clear what you mean by the aggregated preference satisfying transitivity at the individual preference level. On the other hand, completeness will usually be satisfied at the aggregate level, provided that each individual preference is complete. Dec 29 '15 at 23:45
• Aggregating individual preferences to a social preference is in the realm of social choice theory, which studies the properties of various aggregation mechanisms and their associated social preferences (aka "social welfare functionals"). Dec 29 '15 at 23:55

I presume you mean to take the total (or average) willingness to pay for each alternative to be the social willingness to pay. $WTP_{soc}(a)=WTP_{1}(a)+WTP_{2}(a)+WTP_{3}(a)$ Then $WTP_{soc}$ is a representation of a social preference relation $a\succsim b \Leftrightarrow WTP_{soc}(a)\geq WTP_{soc}(b)$. This preference is transitive and complete. Individual and aggregate preferences are consistent in the sense that they satisfy the Pareto criterion and the independence axiom of Arrow's impossibility theorem. The way we have tricked the impossibility theorem is that we have taken an ordinal preference relation and turned it into a cardinal concept (WTP).