What can change in the rods?
Every time I go to fish a Magikarp, I get the choice between 3 rods... that are seemingly all the same.
What is that choice? What is the variance between the 3 rods?
Solution 1:
As far as we know at this point, all three rods are identical from the user's point of view. The game may roll three different Magikarp, or it may simply roll one; given you know nothing to differentiate between the three those would be identical from a probabilistic point of view.
I've done a little research, but not statistics-level yet, to see if the choice seems to have any correlation to the type of bonus you get; in my research it doesn't seem to. I believe the three rods are a meaningless choice.
Solution 2:
The rods are only possibly related to the magikarp you will catch. This can be proven because you can manipulate the game to give you different magkiarp after the rod has been chosen.
When you are standing on the dock fishing you will see a dialog box that has beads of sweat in it. It will flash on screen 1 to 3 times. This indicates the rarity of the magikarp you will catch. If, for example, you only get 1 dialog box pop up (which would indicate a tier 1 magikarp), you could immediately close the game and reopen it. The game will prompt you to pick a rod (you can pick the same rod you did last time) and you will go back to the dock. Once at the dock you can now receive a different rarity of magikarp.
Somewhere, either at the point when you pick your rod or when you are standing on the dock, the game determines what tier of magikarp you will catch. It is unknown if whether the rod has influence on this or if it is purely influenced by the scene at the dock.
One might speculate that the rods are linked to a tier and that when you reload the game it causes them to reroll their tiers. Another option is that the rods are an empty choice and that only the scene determines the magikarp.
So to answer the question, there is a possibility that the rod will determine the rarity of the magikarp you catch.