Contested-Pool Mathematics and When to Abandon a Three-Star Chase

A three-star chase either finishes or it doesn't, and by the mid game you can usually tell which — by counting. Every copy has a location: your board, your bench, an opponent's board or bench, or the pool. The pool is the only place you can roll from, and there is a specific round where continuing to roll for the last copies becomes negative expected value. Finding that round is the skill.

The copy count

Copies in the pool per unit, by rarity: Common 27, Uncommon 22, Rare 18, Epic 14, Ultra 10. A 3★ needs nine total. So for a Common carry, 9 of 27 must reach you; for a Rare, 9 of 18 — half the entire supply; for an Epic, 9 of 14, which basically requires that nobody else touches it.

Counting your outs

Scout and account for every copy:

• You hold — on board + bench.
• Opponents hold — count copies on every other board and bench. Two players each with a 2★ of your carry is 6 copies gone.
• Pool — total supply minus everything held. These are your outs.

If you need 4 more copies and only 5 remain in the pool, you need an ~80% capture rate against everyone else's rolls — not happening unless they're all off it. If 12 remain, you're fine. The number that matters is outs remaining ÷ copies you still need; below about 2, the chase is in trouble; below 1 it's mathematically dead.

When outs return

Held copies aren't gone forever:

• A player who dies dumps their whole board and bench back into the pool.
• A player who pivots off your carry sells their copies back.
• A player 3-stars it and stops buying — no new copies leave the pool from them, and if they later sell the 3★ it returns as multiple copies.

So a chase that looks dead can revive. If a contesting player is at 15 health and dropping, their copies are about to be yours — it can be correct to hold gold, field a 2★ version for a round or two, and roll hard the moment they die.

The abandon decision

Abandon the 3★ chase when all of these are true:

• Outs ÷ copies-needed is below ~1.5 and no held copies are likely to return soon.
• The gold to complete it (at your hit rate) exceeds your surplus above the interest cap, i.e. finishing means breaking economy.
• You're taking real damage now that a 2★ board plus levels/items could reduce.

When you abandon, the 2★ version of the carry plus a level, a second synergy, or a completed item is often only slightly weaker than the 3★ would have been — and you have it now instead of in six rounds. Take the 2★ line, redirect the gold, and stop looking at the shop for that unit.