To reduce the resistance
Here is a great way to look at a problem where all resistances are equivalent:


Intuition: Lets (temporarily) remove the vertical resistors and attempt to analyze the problem. After removing the vertical resistors, lets look at what elements exist at the nodes of the voltage source:


Connected between both nodes are 3 branches in parallel 
Inside of each branch are 2 resistors in series.


Since all three branches are identical, lets look at only a single branch first. Lets call the node in between the two resitors "MiddleNode". We apply voltage division and we find that that voltage at MiddleNode is 0.5*Vsource (because both resistors are identical). 

Note that the MiddleNode voltage is identical on each of the three branches. 

So if we look at the potential difference (voltage drop) between two MiddleNodes, we find that the voltage drop will be 0V. If you have a 0V potential difference between two nodes (two MiddleNodes in our case), no current will flow through an element placed there. Hence, the vertical resistors play no role.
Solution: You have 3 branches in parallel with 2R resistance per branch. 

2R//2R//2R = (2/3)R