### 2×2 Grids

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·c is convex_{e}(x) |
by definition |

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

Haha, that last one is just awesome, Dave (even though the Sudoku is over-determined)!