First things first, Seprix, I'm not writing this because I think there's anything wrong with what you wrote. Just because I'm bored and I don't let things like "someone else already did it" stop me from making needless work :3
1700 words, no formatting because I'm on my phone at the moment. Edit: now with bold titles!
Pearl Diver is among the most basic examples of a cantrip in Dominion. Other than Page, no other cantrip does less on-play than Pearl Diver. All Pearl Diver does is check the bottom card of your deck, which you can move to the top. The effect does not stack well, since if you keep the bottom card on the bottom of the deck, future Pearl Divers just check that same card repeatedly unitl you reshuffle. So it really is a cantrip with a bonus so weak it is negligible in most cases.
Pearl Diver is an excellent illustration of opportunity cost in Dominion. It is a card that, once in your deck, usually does not hurt the deck. It might even save a good card from missing the shuffle once in awhile. But with limited economy, gains, or buys, getting Pearl Diver when you could have gotten a different, better card is often difficult to justify.
Playing Pearl Diver is simple enough, so it is best to focus on when it is worth gaining Pearl Diver in the first place.
When Should I Gain Pearl Diver?
To answer this question, first consider the attributes of Pearl Diver.
- It is named Pearl Diver
- It is an Action card.
- It costs $2.
- It gives +1 Card and +1 Action.
- It does a bottom-of-deck inspection.
- It comes from an Action Supply Pile.
The short version is, Pearl Diver is worth it if you need cards with one or more of these attributes in your deck and a better card cannot be gained instead. Below are some examples of when each trait is relevant. They are by their nature edge cases, but they should give you a good idea of what to notice when considering Pearl Diver.
When You Need A Differently Named Card
This is kind of a gimme, because it can justify the gaining of basically any card, but Pearl Diver is a potential unique card that is easy to incorporate into almost any deck (terminal draw BM excluded). Horn of Plenty and Fairgrounds are two powerful cards that care about uniques.
Hunting Party is also relevant. In certain decks you may run out of unique cards for it to sift for, causing excess HP plays to only draw a single card. If you add cantrips, you can have your excess HPs dive for the cantrip, then play all copies of the cantrip, then play HP again to fish for another copy of that cantrip. This ensures your handsize continues to increase with each HP play.
When You Need An Action Card
Certain cards really like your deck to have a lot of Action cards in them. Scrying Pool draws through Actions and stops at stop cards. Graverobber and Advance trash Action cards for benefit. Vineyard craves numerous cheap Action cards. Throne Room needs to collide with SOMETHING in order to not be a dead card. Gaining Pearl Diver always increases the proportion of your deck that is Action cards, and always increases the number of Action cards in your deck. By being a cantrip, Pearl Diver shouldn't hurt your deck too much versus your other choices for cheap Actions in that particular Kingdom, especially if it is one of the only nonterminals (more on that later).
Depending on how you gain it, you may be able to remove a stop card while doing so, for example by Remodeling a Copper or Curse into Pearl Diver. Ironworks is a card gainer that needs to gain an Action card in order for it to be nonterminal. It's plausible the choice for Ironworks is between Pearl Diver or not being played at all (if you can't afford deck space for another Ironworks). Or maybe you just have an extra $2 and 1 Buy at the end of a turn, and it's the best Action card you can justify.
Procession and Death Cart both are trash for benefit cards that require you to trash Actions, so Pearl Diver is a good card to feed these with if you can get it easily.
When You Need A Card That Costs $2
Pearl Diver costs $2, and all sorts of things in Dominion care about that. Trash for benefit cards are the big winner here. There's the Remodel (or Governor) case, where you may need something that costs up to $2 more than a Copper or Curse. In many cases, the cantrip Pearl Diver is the card you should get.
More importantly though, Pearl Diver can itself be trashed for benefit. Salvager can use it as fuel to give you a little money (and more importantly a +Buy). Remodel can turn it into a $4 cost card once your Estates are gone. Remake can turn it into at worst a Silver. Expand and Graverobber can trash it and turn it into any card costing up to $5, an elite tier of Action cards.
The most notable synergy is with Procession. Procession can play Pearl Diver twice, giving +2 Cards +2 Actions, and then gain an Action card costing $3. Pearl Diver could be the start of a powerful Procession chain, as well as a source of +Actions for an engine lubricated by Procession.
Finally, Pearl Diver is a card that is cheaper than $3. This makes it immune to all trashing attacks, including Warrior, Knights, and Rogue. This is relevant if you are building an engine based on Teacher or a token event - Pearl Diver is a safe destination for tokens that trashing attacks can't remove from your deck. Even without token cards, a deck full of Peddlers has a lower density of more valuable cards, increasing the chances of a trashing attack miss.
When You Need A Cantrip
Pearl Diver is more or less a vanilla cantrip. It doesn't do much else! So really, this is its biggest selling point, those +1 Card and +1 Action, which make it usually not have an impact on your deck's terminal space or handsize.
Some cards really, really value the presence of cantrips. The biggest example are Throne Room variants. You can Throne Room a Pearl Diver to give +2 cards +2 actions, which increases the available terminal space in a deck. If Pearl Diver is the only cantrip, the extra Action can be invaluable, because otherwise you have to rely on chaining Throne Room / Throne Room / Terminal Draw / Throne Room / Action / Throne Room... for your engine to work. If you miss a chance to play Throne Room on itself, the chain is broken. If you Throne a cantrip, then you can end a Throne Room chain and still play other Actions.
Other Throne Room variants also like cantrips. Royal Carriage engines require cantrips if there is no Village in the kingdom in order to play more than a single unique terminal action. Procession not only gets the extra Actions, but it upgrades Pearl Diver into a card that might actually be good. King's Court is probably the least reliant on Pearl Diver, as it is easier to chain multiple King's Courts than it is to chain multiple Throne Rooms, but Pearl Diver can still be nice in a pinch for KC engines.
Conspirator requires a certain number of Actions to have been played in order to activate, so a Conspirator deck benefits from having vanilla cantrips sprinkled throughout it. Herald and Vassal both play Actions if they are on your deck, so they value a deck with many Actions, especially cantrips. Pearl Diver is good filler for these cards as well.
When You Need A Bottom Of Deck Inspection
This is, oddly enough, both the thing that makes Pearl Diver unique and the least common reason you'll purchase the card. But it does have this benefit. Opening Pearl Diver / something on 5/2 decreases your chances of the $5 card missing the shuffle versus opening something / nothing. This might matter at some point.
The only other situations that come to mind are when you care about the top card of your deck. For example, Mystic likes Pearl Diver because it can guarantee a draw.
When You Need A Card From An Action Supply Pile
Teacher and the +Token Events deserve special mention here, as Pearl Diver is among the cheapest possible targets for these cards / events. Pearl Diver becomes a $2 Laboratory plus with the addition of a +Card token. It can be a half-decent cheap Village with the +Action token, or a $2 Peddler with the +Coin token. The +Buy token basically makes it a boring Market Square variant. These situations are usually fairly obvious.
When To Not Gain Pearl Diver
The most obvious time to not gain Pearl Diver is when a better card can be gained for the same opportunity cost and do the same job as Pearl Diver. If you just need an Action card, and there is another cheap Action card, maybe it benefits your deck more than Pearl Diver. If you just need a cantrip, maybe you can afford a pricier one depending on your gain method, or even cards like Vagrant which are the same price. The bottom of deck inspection is the only unique attribute of Pearl Diver other than its name, and that effect isn't strong enough to usually justify it outside of these other factors.
You also don't want to gain Pearl Diver when your coins, Buys, or gains are best used on other cards. These are sometimes scarce, and Pearl Diver is usually only good when you have exactly $2 and a spare Buy, or you have a gainer you were otherwise not going to use. Lots of times there just isn't the time to gain it versus other things you need more.
One special case when it is worse to get Pearl Diver than it is to get nothing at all is when you play a terminal draw card with no actions left. In this case, like any cantrip, Pearl Diver becomes a stop card instead of a freebie. Big Money should stay away from Pearl Diver for the most part.
The other special case where Pearl Diver is particularly bad is discard attacks. These attacks make all cantrips a little bit worse, since you have to decide between discarding the cantrip or a different card (without knowing the replacement card the cantrip will draw). In the case of more powerful cantrips like Peddler variants, Grand Market, etc. the benefit of playing the cantrip itself justifies their inclusion in a deck vulnerable to these attacks. But Pearl Diver's secondary effect is so minimal, its inclusion in your deck just becomes a liability. Pillage is an exception - there Pearl Diver is helpful because it effectively conceals the information of what card replaces it from your opponent.