# Supermodularity And Service Bundling

This will be a bit of a wonky and short post with a longer and less technical one to follow some time soon.  Google has just announced a coming update to their privacy policy which will essentially make it possible for Google to integrate all the information it has about a user across its many different services.  This comes at the same time as the revelation that Larry Page apparently explicitly stated the goal of building “a single unified, ‘beautiful’ product across everything.”

While one can come up with many possible verbal explanations for why Google might want to go this direction, there is some powerful math that lies at the heart of it: supermodularity.  Here is the definition:

A function

$f\colon R^k \to R$

is supermodular if

$f(x \lor y) + f(x \land y) \geq f(x) + f(y)$

for all xy $\isin$ Rk, where x $\vee$ y denotes the componentwise maximum and x $\wedge$ y the componentwise minimum of x and y.

If a production function is supermodular then x and y are strongly complementary.  If you want to read the bible on this consult Don Topkis “Supermodularity and Complementarity.”

A firm such as Google for which the production function relies almost exclusively on information (yes, there are servers and people as well) will exhibit super modularity almost by definition.  Why?  Because if X and Y are different information vectors, then as long as they carry some joint signal, the inequality will be met as you can always choose to discard additional information (meaning you always have access to the component wise minimum).  In plain English: if you have access to both the search history (X) and the social graph (Y) of a user, you can always “do better” than two separate services that only have access to one of these respectively.