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I was telling my friend how after 172 sit n gos, my ROI is 20%.

He also plays sit n gos but on a different site. He was given a free 50$ bonus for signing up. His bank roll is now 95$ or something. He says his bank roll is infinity.

I keep telling him that that's flawed logic because the money is yours, each time you make a buy in, it's an "investment" and the ratio between money invested and profit is the ROI

He keeps saying that on a whole, his ROI is infinity because he didn't make a deposit. Then I said by his logic, I made a 50$ deposit and now my bank roll is 815$ so my ROI is 1500%.

Please clarify this so I can tear into my friend.

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Heh, sounds like a dumb argument. His return on the $50 taht was given to him is 90%. The $50 became his when the poker site gave it to him. Same as if a stranger gives me $1 to buy a lottery ticket and I win $2. My ROI is 100%. I'm keeping the $1 the stranger gave me. If I gave it back, I would have an undefined ROI. I assume he is planning on keeping the $50 the poker site gave him?

He is arguing he made a $0 out of pocket investment. Therefore, his ROI on his out of pocket investment of $0 is undefined. It's a silly argument. He can argue about ROIs when he has more than a double digit bankroll.