In conducting research, drawing 2-by-2 grids has some sort of magical power over me. One typical example occurs on page 4 of a paper by me and my colleague Deepc; it looks something like:
|approximation ratio||Packing problems||Covering problems|
|k constraints per variable||Θ(k)||Θ(log k)|
|k variables per constraint||n||k|
For me, it was useful in developing a line of research since three of the entries had been investigated before, and we were able to investigate the fourth one (top left), as well as fix a bug in another (bottom right). As a reader I also like it since I can immediately perform one of several comparisons with little effort. Moreover, when I go back to study the paper, I find this the easiest way to refresh my memory.
This happened again today to me while I was studying routing games in algorithmic game theory, for teaching later this term. This time, I was finally able to internalize a group of concepts which previously I could never fit together the right way:
|characterizations||Nash Equilibrium||Social Optimum|
|local optimum||by definition||by a potential function|
|global optimum||by first-order conditions, if x·ce(x) is convex||
As of today, I understand why some theorems assume this convexity and some don’t; and I can give a more intuitive explanation for my class.
To illustrate my love for all things 2-by-2, here is a diagram.
Filed under: diagrams, math | 1 Comment