I want to build a Relentless Rats deck, but I don't know if i use a monoB with Marrow-Gnawer or a BUG with Damia, Sage of Stone. Why do you think someone should pick the monoB version? Why it's better?

Maizena, I like my mono-black because it gives me focus. It allows me to get the maximum effectiveness out of devotion cards like Nykthos, Shrine to Nyx and Gray Merchant of Asphodel.
It does let me concentrate on just one color to ramp with Caged Sun and Crypt Ghast, and a harder hit from Nightmare Incursion, Corrupt, Crypt Rats, and Mutilate.
Additionally, Marrow-Gnawer gives you access to fear for all your rats, and infinite rat tokens.

It does, however, add vulnerabilities to pro-black, and to nasty things like Iona.
If those became a more consistent threat in my meta, there are plenty of colorless removals, both targeting and wipe, that I could introduce.

Ultimately, it's up to you what you like. I can say for sure that making a mono-color mana base is MUCH easier on the wallet... $$$

I'm sold! Congrats on the deck!

Btw, what about Cabal Coffers?

Like others before me your rat deck has inspired me to build my own... +1 from me. Thanks.

I run Marrow-gnawer, with a very similar list, and the all-star of my deck is Altar of the Brood, being a 1 mana cost win condition in the deck, Lightning Greaves is a strait upgrade to swiftfoot boots. Just a few suggestions from a similar deck

This deck is awesome, I may even build it myself. Although, I would make a few tweeks... probably add Door of Destinies and Patriarch's Bidding. Thanks for posting this build. +1 from me.

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Soul Foundry would work wonders in this deck.

please let me know if I missed something, but is there a way to prevent blue players from just banishing all relentless rats from your deck? or do you just have to pray your friends aren't jerks