2×2 Grids

09Mar11

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

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.

 

The bottom-right entry is in fact a book, “The Power of the 2 x 2 Matrix: Using 2×2 Thinking to Solve Business Problems and Make Better Decisions.” Yours for only $50 on Amazon!
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One Response to “2×2 Grids”

  1. 1 Daniel

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


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