Alice and Bob are fruit-pickers at an orange orchard.
Alice can pick 6 baskets of oranges in one hour. In contrast, Bob can pick 7 baskets of oranges in the same period of time. However, if Alice and Bob work together, then they can pick a total of 15 baskets in one hour. As part of a team:
- Alice is now picking 7.5 baskets per hour, a most impressive increase in productivity of 25%.
- Bob is now picking 7.5 baskets per hour as well, a modest increase in productivity of approximately 7%.
Both Alice and Bob benefit if they work together (though Alice benefits more). We thus have an example of synergy. To use a cliché: “the whole is greater than the sum of its parts”. If Alice and Bob work alone, they can only pick 13 baskets per hour in total, but if they work together they can pick 15 baskets per hour. Everyone is happy.
Let , let denote the power set of , and let denote the set of nonnegative real numbers. We introduce a productivity function , enumeratively defined as follows
where because the productivity of the “empty team” is zero. Since we have that
we conclude that we have synergy. Note that the existence of a synergistic or synergetic effect is a property of the productivity function . We could attempt to study such property in a more general setting.
We now introduce a definition
Definition: Given a finite set and a function , if the following conditions are satisfied
- for all sets such that
Using the superadditivity property recursively, one can conclude that
for every . For example, if , then we have that the measure of is
Frankly, I have (accidentally) opened a can of worms. I started writing this post thinking about synergy and productivity, and I am now drowning in Measure Theory! As it turns out, the union of all my knowledge of Measure Theory is a set of measure zero ;-) Hence, I will abruptly finish this post with a passage from Wang & Klir :
Observe that superadditive measures are capable of expressing a cooperative action or synergy between sets in terms of the measured property, while subadditive measures are capable of expressing inhibitory effects or incompatibility between sets in terms of the measured property. Additive measures, on the other hands, are not able to express either of these interactive effects. They are applicable only to situations in which there is no interaction between sets as far as the measured property is concerned.
I may return to this topic if I happen to have any interesting ideas.
 Zhenyuan Wang, George J. Klir, Generalized Measure Theory, Springer, 2009.