## The Penultimate Province Rule

The most important — and the most dangerous — Province in the supply is the second-to-last one.  Many games have been lost by buying this Province instead of a Duchy.  Yes, you read that right.  A Province instead of a Duchy can often be a mistake.

How is this possible?  Let’s take a look at the simplest (and most common) scenario involving this rule: a 2-player game between Alice and Bob (Alice being Player 1) that’s coming down to the Province split.  Assume (for now) that there are no +Buy cards, and no Remodel/Workshop/etc. cards that would allow a player to gain multiple Victory cards per turn.

The Provinces are 3-3 thus far.  On Alice’s turn, she gets a nightmarish draw and is only able to purchase a single Estate.  Bob jumps for joy: he’s drawn \$8 and can buy the second-to-last Province to go up by 5.  But doing so would be a critical mistake.  It allows Alice to end the game with a win by buying the last Province on her turn.

Instead, Bob’s correct move is to take a 2VP lead by buying a Duchy.  Now Alice is in the same position as Bob was last turn: forced to buy a Duchy instead of possibly letting Bob split the Provinces for the come-from-behind victory.

Note that this rule didn’t apply to Alice on her Estate turn.  In a tied single-buy game, when you are able to buy the second-to-last Province, you always should, because you are then almost guaranteed not to lose.  In order for your opponent to win, he’d have to counter with an unanswered Duchy, and then the last Province; even if you don’t grab the other Province, the odds that you can’t match your adversary one-for-one with the lesser Victory cards are very low.

Note also that if Alice had gotten a Duchy on her Estate turn instead, meaning Bob can only tie Alice with his Duchy, then he should buy the Province instead.  Intuitively, this can be rationalized by the fact that if Bob chooses to tie Alice, he is at a significant disadvantage: in tie games where Alice goes first, Bob normally only can win if he can get both of the last Provinces, since it’s unlikely that he can take a lead going into a 4-4 split.  In other words, he’s a little more likely to draw, but much less likely to win; he may as well take advantage of his \$8 and hope Alice can’t finish him off.

To summarize, the Penultimate Province Rule states:

If you are trailing, do not buy the second-to-last Province if you can instead purchase lesser Victory card(s) to take the lead.

Naturally, like all rules, the PPR is subject to exceptions.  Understanding these is key to successful 2p endgame play.

First, note that in practice the PPR operates slightly differently for Alice and Bob (P1 and P2).  On Alice’s turn, if she buys the second-to-last Province when both players are tied, she is essentially inviting a draw.  But if Bob buys the second-to-last Province in a tie game, he is actually close to forcing a win: Alice cannot buy the last Province for a draw because she will lose on turns.  It’s an extremely powerful move, and as Alice you should seek to avoid such a situation (perhaps via +Buy cards, as below).

Second, the PPR also depends on your relative deck strengths.  Against an opponent whose deck has more future potential than yours, you should not avoid the penultimate Province and prolong the game, but rather take advantage of the full power of your deck now.  For instance, if you have been Salvaging and Remodeling your Golds into Provinces, and your opponent is running a Hoard/Vault/Trade Route deck, then avoiding the second-to-last Province dooms you to a battle of attrition you cannot possibly win.

Third, as briefly analyzed above, when the only chance you have of winning (or drawing) is to claim both of the last two Provinces, you should absolutely just go for broke.  This includes situations where you can only tie your opponent by avoiding the penultimate Province.  Here, you must take the gamble that your opponent won’t win the game next turn, since if your opponent can buy her fifth Province she will almost certainly win anyway.  This commonly occurs when the Provinces have split 4-2, or when your opponent has such a substantial Duchy/Nobles/VP token lead that you will inevitably lose on a 4-4 split and must go for the 5-3 instead.

There is one exception to this “exception”: in Colony games, it is not necessarily true that you will always lose if your opponent gets 5 Colonies, since it is sometimes possible to win by establishing a major advantage in Provinces and/or Duchies.  For instance, in this game, on turn 20 I draw \$11, but the Colonies have split 4-2 in favor of my opponent and the Provinces are 2-2 for each of us.  So I choose to buy a Province/Trade Route instead.  He (correctly) takes the second-to-last Colony, and on my turn 21 I stock up on Provinces/Duchies/Estates, such that I now trail by 9 points with one Colony left.  In the next two turns, we each get two Duchies, and on the final turn I take the last Colony for the win.  It was incredibly unlikely that I could pull it off (though I was helped by the fact that my deck had untapped future potential — as described above — and Trade Route), but my victory was even less likely if I had gone for both Colonies: the chance I could get both was low, and even if I did, he could buy Provinces in the meantime to win even if I got to the 4-4 Colony split.

A special case of this principle is when both you and your opponent are capable of gaining 2 Provinces at once.  For example, with 4 Provinces left, if you are slightly behind, you might hesitate about buying the double Province if you believe your opponent is going to double Province for the win.  This is especially pertinent in Tactician games.

Fifth, the PPR becomes considerably more complex with Remodel, Expand, Forge, Salvager, or any trash-for-benefit card.  Similar to Tactician games, you have to watch out for double-Province gains: Remodel can turn a \$11 hand into two Provinces, and it’s much easier to draw \$11 + Remodel than \$16.  But more importantly, it’s even easier to draw \$8 + Salvager + a Province.  If, in our first example, Bob blindly follows the PPR to take the lead with a Duchy, Alice can simply buy one of the Provinces and use one of her Provinces in hand to trash Bob’s next Province out of the supply (metaphorically).  In other words, just because there are two Provinces left doesn’t necessarily mean you’ll get a shot at one of them!

Finally, the PPR is difficult to implement in games that involve substantial amounts of VP chips.  The ability to gain an extra 4-10VPs while buying the last Province places such games in the same category as +Buy games, except most +Buy games don’t feature players able to buy a Province + 3 Duchies at once.

The real takeaway from all of this is that endgame play is not simply about buying the biggest green card you can.  There’s usually a very limited number of ways the Victory cards can split; good 2p endgame play depends on being able to manipulate the game state such that the most likely possible endings are in your favor.

Sample games

A canonical example of the Penultimate Province Rule.  On Turn 13, I am down by 2VP with the Provinces split 3-3.  With \$8 in hand, I go for a Duchy and Estate rather than a Province, and pull out the victory as a result.

An example of when buying the second-to-last Province is OK, since when I buy it (on turn 13) I hold the lead.  My opponent is forced to “Duchy-dance” with me for a while, which is inevitably unsuccessful.

An example of when I need to buy the second-to-last Province, because my only hope of winning or drawing is to get both of the last Provinces.  I already have a substantial lesser Victory lead so that when I do get to 4-4 Provinces, his lesser VP buys in the meantime aren’t enough to overtake me.

An example of when P2 errs in buying the second-to-last Province in a tie game: although it is a very strong play in single-buy games (because P1 cannot end in a draw without losing on turns), it is suicide in multi-buy games if P1 is able to buy something else along with the final Province.

This entry was posted in Articles. Bookmark the permalink.

### 26 Responses to The Penultimate Province Rule

1. tlloyd says:

Excellent article! Well done. I’ve been feeling for some time now that the key for me to take my game up a notch is mastering the art of manipulating the end-game.

2. tlloyd says:

“if Alice had gotten a Duchy on her Estate turn instead, meaning Bob can only tie Alice with his Duchy, then he should buy the Province instead. Intuitively, this can be rationalized by the fact that if Bob chooses to tie Alice, he is at a significant disadvantage”

Here’s how I understand this: in the circumstance where Bob can only tie by buying a Duchy, if on her next turn Alice has the \$8 for a Province, she will buy one regardless of what Bob does. If Bob buys a the tying Duchy, Alice will buy the Penultimate Province without fear, since she seemingly can’t lose (she’s up a province with only one province left). If Bob buys the Penultimate Province for the lead, the Alice will buy the final Province, recovering the lead and winning the game. Therefore, the only way Bob wins is if Alice doesn’t have \$8, and if she doesn’t the best thing he can do is take the lead with the Penultimate Province.

My only problem with this analysis is that Alice is assumed to be indifferent to winning and tying (at least she would very much rather tie than risk losing), but we’re telling Bob that he should risk losing in order to avoid the very likely tie. Or do we assume that Bob won’t be able to keep up in the coming turns, since he has already seen more of his money go by than Alice?

3. tlloyd says:

Okay, another question/comment:

“if Bob buys the second-to-last Province in a tie game, he is actually close to forcing a win: Alice cannot buy the last Province for a draw because she will lose on turns. . . . as Alice you should seek to avoid such a situation”

Does this actually suggest a related rule that, as P1, you should not buy the third-to-last Province if doing so evens the score?

• timchen1017 says:

In an ideal world of Dominion without shuffling and with fixed income, yes.

However, the key to this aspect of strategy (and basically why the penultimate Province is so different) is that the draw is random. Without knowing whether whoever will miss a province buy, the best strategy is just to buy whatever you can. It is only when the game is getting close to the end AND that you can expect your opponent to get to Provinces with some consistency that you have to worry about this.

• rrenaud says:

Anyone want to compute the optimal strategy assuming different province buying probabilities for players when 2 are left?

4. Jahz says:

Excellent article.
The key point for me in games with +Buy or with cards to gain VP is to really keep better track of these cards in the opponent’s deck, which I’m almost not doing atm.

5. Geronimoo says:

I implemented the Penultimate Province Rule (PPR) into my Simulator and ran a million games between two Big Money Players (no Kingdoms). One of them applied the PPR and the other completely ignored it. Here are the results:

PPR player (45,7% wins)
PPR was applied 24161 times (so 2,5% of all games)
40% wins, 40% losses and 20% ties in those games where PPR was applied

Simple Big Money Player (45,4% wins)
PPR was ignored 30634 times (so 3%)
40% wins, 57% losses and 3% ties in those games where PPR was ignored

This confirms the validity of the rule (at least for the Big Money Strategy). But… ignoring it isn’t a very costly mistake (-0,3%).

• Thisisnotasmile says:

That looks to me like applying PPR won’t really enhance your chances of winning, but ignoring it will most probably cause you to lose. Interesting.

• DG says:

The rule will not apply well to big money decks as big money decks are unlikely to have regular province buying turns at the end of the game.

Theory has suggested a vp buying strategy. It isn’t always applicable and you don’t always have to use it. By using this strategy in the right games at the right times, rather than playing mechanically, you can win a few more games.

6. ecks says:

I just looked at what I’d read and didn’t realize it was so long – that was how well written it was!

Very helpful to have it in a format like this; I can do the math pretty quickly on the spot lategame, but having already read this will make it a lot easier. Thanks!

7. Meej says:

“Alice cannot buy the last Province for a draw because she will lose on turns.

Hmm. I hadn’t actually realized that this was a rule; so it’s only possible to tie the game as Player 2 (or greater)?

• theory says:

Yes. P1 can never end the game in a tie.

• kn1tt3r says:

So if P2 ends the game in a tie, it’s an actual tie? Or is it a P1 victory too?

At home we’re just counting a tie as a tie no matter what, so I’m not familiar with these actual rules…

• theory says:

The rulebook states that in the event of a tie, the player who took the fewest turns wins. Players who took the same number of turns “rejoice in their shared victory”.

• Meej says:

Out of curiosity, is that “real turns” that are counted (especially on isotropic, I suppose), or does taking extra turns with Outpost or Possession count against you?

8. guided says:

It’s good to raise awareness about the idea of avoiding the trap of handing a game-winning opportunity to an opponent when you don’t need to.

I do want to caution players looking to apply this knowledge to be acutely aware that it is not at all uncommon in PPR situations that your best chance to win will actually be to buy the Province and hope your opponent cannot use the opportunity to win. Trading Duchies favors first player and/or whoever’s deck has the most tolerance for more Duchies. This is at least partially covered by the “the PPR also depends on your relative deck strengths” paragraph.

• Epoch says:

Yeah, I’ve been in quite a few situation where I thought, “Well, if I take this province/colony, and my opponent takes the next one next turn, they’ll win. But… they’re winning anyway. They’re increasing their lead. My only real hope is to gamble on my taking both remaining provinces/colonies with an unlucky draw for them next turn.”

That’s not a great situation to be in, but it’s one everyone will be in from time to time.

9. Mick_Swagger says:

Best article I’ve read on this site yet!

10. ycz6 says:

“In a tied single-buy game, when you are able to buy the second-to-last Province, you always should, because you are then almost guaranteed not to lose. In order for your opponent to win, he’d have to counter with an unanswered Duchy, and then the last Province; even if you don’t grab the other Province, the odds that you can’t match your adversary one-for-one with the lesser Victory cards are very low.”

Oh, theory.

http://councilroom.com/game?game_id=game-20110330-094133-66f28631.html

11. Lenoxuss says:

Dominion is nim!

It’s even an anagram for “No, I do nim.”

This presumes you even know how many points are in your deck let alone your opponent’s. If just playing the base game, perhaps, but what about a Gardens game yet still people are buying Provinces? Then there are the other Victory cards – Duchy, Duke, Tunnel, Island, Nobles, Harem, etc. There exists people with great memory who could keep track, but realistically you will not know how many points are in anyone’s deck.

• rrenaud says:

Even before the point counter, most of the good players did know the exact score or close enough to it most of the time in online games.

The pure VP per cards aren’t hard to keep track of at all. Yes, Gardens is a problem, and Duke a bit less so.

In general, the blog tends to assume all of the players are very good players, who will mostly make the right choices. Keeping track of relevant, totally observed information is (an uninteresting) part of playing well.

13. arigaa says:

TIL that the PPR should be ignored if your opponent has a loose deck, and has only drawn gas out of their deck so far.