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[DG]

General Principles

Without either gains or extra buys a deck is likely to be quite one paced and will take a number of turns to buy cards for income then more turns to buy cards for victory points. This is seen most clearly with alternate (kingdom) victory point cards since it takes far too many turns to buy the vicory cards and also buy any other cards needed to boost their scores.

Trashing cards like copper out of a deck takes time and can be a waste of time if good cards cannot be put into the deck quickly enough to replace them, usually with buys and gains. The faster that you can put cards into your deck with +buys and gains, the more turns you can spend trashing down to a controlled deck beforehand. The reverse can also be true: the more compact your deck becomes, the more often you will be able to play cards for buys and gains (although you still need income to make use of those buys).

Buys and gains obviously accumulate many cards and suit big deck strategies for kingoms cards like philosopher's stone. Some curse ridden decks will never be repaired and will be improved by accumulating basic treasures such as silver and sometimes even coppers. Action based becks (engines) usually need to accumulate many action cards in order to function. Many key cards in a kingom, such as fool's gold, might work better in large quanitites and it is important to accumulate them faster than your opponents. Buys and gains can deliver the quantity of key cards and can often deliver them rapidly, however time spent preparing the buys and gains could be time spent buying the key cards. This can be particularly important in multiplayer games or when using expensive actions such altar or university.

End Game

There may be no point taking many turns to slowly build a strong deck if you cannot earn a high reward. If an opponent gains a early lead it might be hard to overtake them just buying one vp card per turn. Buys and gains allow you to take more turns to build a strong deck on the assumption that it will be able to score more heavily with more victory cards in later turns. This can certainly be worthwhile if a deck can greatly improve each turn before greening or risks drawing badly as soon as victory cards are addded.

With extra buys you can earn the capability to end the game early by emptying 3 piles when you are ahead. Throne room, procession, and king's court can be strong in these situations too if they can conjure extra buys or gains. It can be difficult to force a three pile ending if you can only buy one card each turn, assuming the opponents don't make it easy for you. This leads onto the topic of defensive play.

The simplest vp purchasing strategy is to buy the highest victory card available on your turn. Another strategy however is to score vp while denying your opponent the chance to win on their next turn. Since the game only ends when the province pile or three piles are empty, you simply leave enough cards in the piles so that so your opponent is unable to empty them (and win) next turn. The most well known example of this is the penultimate province rule (PPR), which relies on the opponent being unable to win until the penultimate province is bought, but as soon as they do that they offer you a chance to win with the last province providing the scores are close. If you have extra buys and gains available you are in a better position to purchase vp defensively, such as buying two duchies instead of the penultimate province or perhaps switching to alternate vp if it is in the kingdom.

Furthermore, if you can buy or gain extra vp cards on your turn it is more difficult for an opponent to play defensively since you can potentially score more on your turn. Similarly if you have many buys and gains it becomes more difficult for opponents to prevent you from emptying piles. Expert players will often try to control the endgame by having more buys and gains than their opponents, and often create situations where they can end the game but opponents cannot. 

Buys

In principle, buys are much more simple to use than gains even though they can require decision making during play on how to spend the coins. Generally you want extra buys with large income hands. As soon as +buys are in your deck each coin generated is more likely to be useful. It isn't wasteful to have +buys when you don't need them as long as you have  enough buys when you do need them. That's all obvious but it is easy to lose sight of income when crafting a fancy deck. It's also possible to push too hard for income in the early game and belatedly spend a lot of coins for a cheap +buy card in the late game.

For very high income hands, +buys can make sure that all income can be used. A bank is a prime example of card that is best used with +buys, otherwise any extra income it provides above a gold might go to waste. More generally +buys offer the choice to buy multiple cheap cards rather than one expensive card. One expensive treasure is usually better than two cheaper basic treasures, so this more often applies to victory cards, action cards, and key kingdom cards. If kingdom cards are cheap and at different costs then there will me more choice how to use +buys. Extra buys also offer flexibility when dealing with potion cost cards and hands that miss key price points, such as 7 coin hands. A large income hand with a potion often needs a buy for the potion cost card and a buy for the remaining coins.

When you are wanting to build a deck through high spending and extra buys it can be important to keep a focus on income cards. Dominion already needs judgement on the value of a trashing, gaining, or attacks compared to income cards. When a deck has +buys, each coin is more likely to be useful and should be valued slightly more.

Buys naturally support the accumulation of action cards to build an engine deck. Extra buys suit some unique kingdom cards that benefit from buying coppers and curses but I won't discuss them much here. Cards like goons, counting house, gardens, and trader all have their own play style that does not fit any general advice. Black Market has many unique tricks as well. Cost reducers such as bridge, highway, quarry, and peddlers work very well with extra buys and the more that you can combine these reducers with +buys the better the results, even producing 'megaturns' when enough are put together. Coin tokens and +buys both provide flexibility on how coins are spent and it may depend on the kingdom whether they do much more together than either would do alone. Extra buys do however allow the hoarding of coin tokens for big spending turns later in the game.

Most importantly, all the benefits from buys do not come in isolation. For example you might be able to build a deck quite strongly before deciding to pursue a gardens strategy, buying multiple gardens in one turn, adding coppers to boost the scoring, and using buys to force a three pile ending. For an engine builder, not only do extra buys allow more engine components to be added each turn but they can allow an engine to deliver bigger vp turns at the end. This means an engine can be built up over more turns with more resilience, power, and a great deal of end game control.

[DG]

Gains

When new Dominion players sit down to play the first recommended kingdom, someone will see that the workshop immediately gives you a card worth four coins without even having to use a +buy. Silver only gives 2 coins and needs to be used with a buy. That player usually buys workshops and then loses the game as they don't know to use them. The power from gaining cards is there but it needs to be harnessed correctly with good play in the right kingdoms.

How gaining works can vary greatly depending upon the kingdom card used. The main difference from +buys is that gaining cards does not usually rely on income and can work with little or no conventional economy. This can be useful if you feel that maintaining an economy with basic income cards, such as copper and silver, is going to be bad for your deck. However gaining a card often provides no income and therefore no initial benefit to a purchasing strategy that needs to buy key cards at higher costs. A deck that is gaining cards can often work well if there are low cost kingdom cards available for low income hands. From a comparison of upgrade and junk dealer, we can see that +1 coin from the junk dealer suits a purchasing strategy (with high cost cards or using +buys) better than upgrade.

The second difference is speed at which you gain cards. It usually takes time/turns to buy a workshop, say, and then play it to gain a village when it would have been possible to buy the village directly. However you can keep gaining cards every time you play the workshop so you usually want to play it as frequently as possible. Rapid cycling decks can certainly suit card gaining. Furthermore, gained cards are available in the discard pile in the same turn that they are gained, vastly improving the pace of decks that can use them.

The third difference is the restriction of the cards you can gain. If you cannot work with this restriction then your deck will suffer. If you are restricted from gaining the important kingdom cards then all the effort put into gaining unimportant cards is likely to stop you from buying the important cards. Even when you can gain key kingdom cards you might want to consider what options remain once the pile runs out. Some common restrictions are

Basic Cards - Gaining basic cards, such as silver, can suit any sort of deck but the basic cards themselves often suit big treasure decks more than complex engines. Gaining cards can sometimes lock you into strategies since you don't have the same freedom in gaining cards as you do when buying cards, however this is particularly true when you can only ever gain a named basic card.

Cost Restriction - Card cost is a frequent restriction on the cards you can gain. Bridges and highways can ease the restriction but you need to be watchful for strange outcomes, such playing a forge after highways. In general though, if important kingdom cards are outside the cost limit then you need to consider carefully whether gaining less important cards is worth the effort. Haggler, horn of plenty, and stonemason can have variable price points based on your played cards but it is still prudent to consider what cards can be gained at a range of outcomes.

Hand Restriction - More complex card gaining can have additional restrictions based on cards played/trashed/discarded from hand. With the wrong cards in hand it will be impossible to gain the required cards, such as a mine with no treasures in hand or a tournament without a province. The simplest remedy is to increase the hand size or the quality of the cards drawn so that you have the maximum choice of cards to gain. This is quite different from a buying strategy that will typically take income from any source.

Trash and Gain - Some actions allow you to gain cards at a cost related to trashed cards. These require planning to ensure that:
(1) there are good cards in the kingdom at the right cost to replace the trashed cards (or nothing at all if you just want to trash)
(2) cards added to the deck have potential to be trashed for valuable cards later in the game (usually victory cards).
If we take remodel as an example, we might want good kingdom cards at cost 2 to replace copper, good kingdom cards at cost 3-4 to replace estates, and cards at cost 6-7 in the deck to trash for provinces later. Cards like develop and procession need useful kingdom cards at a sequence of card costs and this can make them difficult to use well throughout the game. Despite the severe restrictions, trashing bad cards to gain a good cards can be be a big step towards restructuring a deck.
 
Opponent's Deck - Thieves, Noble Brigands, Smugglers, Jesters, Possession, and even Rogues can gain cards based on your opponent's deck. Although it sounds obvious to make sure that your opponent's deck will supply useful cards it is very easy to get that wrong, especially when your opponent is working against you. Gaining too many terminal actions is the common problem however it is also possible to gain unhelpful treasures. The opponent may also be pursuing a different vp strategy meaning that you can't gain the cards you need to boost your own score.

Often card gaining has an interaction with another copy of the played card and this will be available in any kingdom. For instance you can gain ironworks with ironworks, remodel an estate into a remodel, or upgrade an upgrade into gold. It is no coincidence that farmland adds a card (itself) to the deck with the potential to be trashed later to gain a valuable card (province) with another farmland.

Some cards provide benefit by trashing other cards but if you want to play them repeatedly you can run out of cards to trash. Buying new cards, just to be trashed as fuel, can often be expensive and provide little net benefit. Gaining cards as fuel can be efficient and productive.

All the end game benefits discussed at the introduction can be achieved with gains. Extreme engine decks can often draw more smoothly if they can gain victory cards without the need to buy them with lots of treasures. A deck that fills with green cards might see income drop but still be able to gain cards rather than buy them. It usually requires planning however to ensure that you can gain the right victory cards in the end game. For example, if you want to play a throne room on a remodel to trash two gold for provinces, you need to ensure that those cards are not only all in your deck but also drawn together in hand. It is not the same as putting together a coin income from any assortment of cards in your deck.

Gaining cards to empty piles can work very well. Some foresight is needed however to make sure any restrictions do not prevent the gaining of important cards from low piles. The endgame strategy needs to fit the capability of your deck. Cards that can gain copies of themselves can potentially empty their own pile in one turn, if enough gained cards can be both drawn from the discard pile and played the same turn (example rats).

At the end of the buys article I suggested that the benefits of +buys were available throughout the game and could help at any time. This is not generally true for gaining cards and it can make a big difference in the end game when any restriction on gaining cards will hurt the most. For example, you can only smuggle a victory card if your opponent gained one last turn, if it costs less than six coins, and there are still copies of that specific card in the supply. You might be able to smuggle a treasure or action but that might be no help if it will never be played. It can certainly be harder to play defensively since the restrictions on gaining cards do not offer the flexibility of coins and buys. Even if you realize that upgrading a coppersmith to a duchy will put you a losing position there may be no alternative way to score with those cards.

[DG]

These articles cover a very big subject and I know that it can't cover everything and can't offer enough depth to the subject matter. There still seemed to be a big gap in the Dominion regarding buys and gains so I wrote the article anyway. This is a not about exception cases, such as smuggling border villages to gain duchies, so please keep feedback to the general principals of buys and gains.

[DStu]

+3

The second difference is speed at which you gain cards. It usually takes time/turns to buy a workshop, say, and then play it to gain a village when it would have been possible to buy the village directly. However you can keep gaining cards every time you play the workshop so you usually want to play it as frequently as possible. Rapid cycling decks can certainly suit card gaining. Furthermore, gained cards are available in the discard pile in the same turn that they are gained, vastly improving the pace of decks that can use them.

I would explicitly state that this means that they can, in the right situations, be played in the turn they are gained.  I was stuck a second on this formulation, as being in the discard pile is not really the best place to be...

[shark_bait]

Understanding extra buys/gains differs in regards to what stage of the game you are in.  I think you mention these in the article but I think it is good to highlight these changes.

Early game, extra buys/gains can help win crucial splits.  Having an extra buy/gain can allow for winning a Fool's Gold, Garden or Peddler split.

Mid-game your extra buys/gains are most useful for deck development.  In engine building the ability to gain an extra Village, Smithy or Conspirator goes a long ways for putting your deck above your opponents.

End-game your extra buys/gains (assuming you have more than your opponent) give you control over the 3-pile ending.  Your ability to do so puts your opponent on the back foot as they must always respond to the capabilities of you deck.

Understanding extra buys/gains can really go a long ways in improving your skill as a Dominion player.  You can win so many games that should have been lost by specifically understanding mid-game/end-game concepts of buys/gains wherein your opponent lacks that understanding.

[jomini]

A couple of points:
1. Gains can give you a lot of flexibility. Remodel can allow you to flip silvers into festivals (e.g. if your opponent starts going Noble Brigand) or just keep adding Hamlets from copper. You can much more quickly retool your deck with gainers (as a general rule) than just by swapping buy strategy. The best example of this is when your opponent draws something out of the BM deck that is game changing, generally gaining will let you pick up the reactions/attacks/etc. to adapt quicker.

2. Game ending gains can grow exponentially. One gainer (e.g. Hop) can become two, which can become four the next turn and eight the turn after that. This sort of explosion can force you into a lot of defensive greening (which in turn can destroy your deck).

3. Outpost can work quick well as gainer even in decks where you can't build a reliable engine. Figure on getting a $2 or Silver without much draw. When strong it effectively doubles your gain/buy limit and doubles your thinning rate (e.g. Pool/scheme decks).

4. Graverobber may be able to gain "from the opponent"; particularly if coupled with Knights or the like.

5. Burning points off the total available is quite strong with TfB gainers. Remodel Province -> Province is a classic way to control end game. This doesn't gain you any points, but lowers your 50%+1 victory threshold by 3. This is also viable for three piling - duchy -> duchy or component -> component (or even province -> component) expand the ways to control end game.

6. Timing the buying of weak gains/+buy is very key. Do not be afraid of spending $5 or more on a cheap +buy. It is often more important to thin/build draw early than to get a weak +buy; buy your +buy/gain when it makes sense in the build - even if you have to forgo things like early lucky gold/province. I'd love to hear more about this, because it is a very tricky dance when you are deciding between engine reliability and faster component gain.