Q:

Why exactly did Cartman let Kyle in on his voting scandal in Obama Wins?

South Park season 16 finale episode "Obama Wins" starts off with Cartman calling Kyle over to look at the boxes of stolen ballots- to which of course, Kyle vows to not let Cartman get away with it.  

Cartman had planned to get Kyle to try and stop him- but how exactly did this contribute to his plan? 
His rival even ends up ultimately ruining his scheme in the end. So what exactly was the point of this, and how did it benefit his scheme?

A:

Cartman was letting Kyle to get external help only to get him fooled. Similar to Cow boy and the Tiger story.
From what happens in the episode, we can assume that Cartman had two plans to execute. 

Call a friend and ask them to help him in the plan. Take a promise not to spill the secret to anyone.
If they don't agree with him, let them go call the police or other external help. Then he hides all the ballots. He convinces them that he is innocent. That is what he did in the case in Kyle when he brought cops for raiding his home. By doing this, no one would suspect that he is involved in the voting fraud.

But the plan backfired because Kyle also informed it to his friends where he finds out that Butters was also part of Cartman's plan in hiding the ballots. This was not expected by Cartman. It is highly possible Cartman took a promise from Butters not to spill this secret to anyone. He asked Kyle too but Kyle didn't promise. When Cartman visits Butters in the hospital,  

Butters: Please, Eric! I tried to be quiet, I swear!
Cartman: Shhhh, it's okay, Butters.
  Nobody's going to find the election ballots.
  I have them hidden away, somewhere nobody would ever look.
  A place in town people barely even know exists.
Obama Wins episode Script

