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[faust]

We all love new expansions. Unfortunately, more expansions do not improve the game; they make it worse. At least this is the case if you are playing full random. I have been holding this thought for a while, and I think it's an interesting one to share.

In order to explain why expansions make Dominion worse, we are going to start out with a simple proposition: Games with +Buy in them are on average more interesting than games without +Buy. You might not agree with this; just note that the choice of +Buy is arbitrary - I could make the argument that will follow with any other special kind of card, like Villages, or terminal draw, or cantrips - take your pick.

Part 1: Base-only bliss

So to start out, let us strip all the expansions away until only the base game is left. The (second edition) Base set has 26 kingdom card, of which 3 are +Buy. If we choose random kindoms, what are the chances of ending up with a subpar game  (which for us means a game without +Buy)?

The chance that the first card isn't a +Buy card is 23/26. For the second card, assuming the first is already picked and not a +Buy, we have one fewer non-Buy-card in the pool, so the chances to get another non-Buy-card are 22/25 - a lower probability than for the first card. Continuing this for the second to tenth card, we can compute the chances of getting a kingdom without +Buy: It is the product of the chances for all the individual cards. Eventually, we get the result 14/65, or roughly 21.5%

So when Dominion came out, you could expect that a bit over every fifth game would not include +Buy.

Part 2: Infinite variety

Now as a hypothetical, let us imagine the other extreme - a world in which so many expansions have come out that there are effectively infinitely many kingdom cards. Let's also assume that the ratio of cards with +Buy has been kept the same since the Base set - this is roughly true, as Donald X. tries to keep expansions balanced with repect to +Buy cards, Villages, and so on. So each of our infinite number of kingdom cards has a 3/26 chance of offering +Buy.

Now we can do the same probability computation. As before, the chance of the first card not having +Buy is 23/26. But now, after picking it, we haven't reduced the number of options to choose next at all - there are infinitely many of the after all. So the second card will still have a 23/26 chance of not having +Buy. Same for each subsequent card. If we calculate the overall chance of not having any +Buy in the kingdom it is (23/26)^10, or 29.3%.

When we compare this to the situation in the base set, we note that the chance of a "bad game" significantly went up. Adding the huge number of expansions made our average game worse, regardless of the quality of the individual expansion.

Part 3: Shortcomings of the model

There are of course certain gaps in this analysis. First of all, there can never really be inifinitely many cards. As of writing of this article, there are 354 different Kingdom piles, of which 47 have +Buy (if my counting is accurate). This is a slightly higher density of +Buy than in the original Base set. Still, the current chance to get a +Buy in the Kingdom is, using the computation from Part 1, 23.6%, so even though the +Buy density increased, the chance to get a +Buy in your game actually decreased.

There are shortcomings in only considering the Kingdom piles. I have not considered sideways cards (Events, Projects, Ways) that can also provide +Buy. They change the picture for the better as they provide more total slots in which you might find +Buy - increasing the total size of the Kingdom from 10 to up to 12 - but that can only ever be a temporary solution, unless you keep increasing the stuff that is part of the Kingdom.

Finally, how you find a "bad" kingdom is more complicated than saying there's no +Buy. Some Kingdoms without +Buy can be interesting specifically because you have to play around that. But I would argue that most aren't. You could take a measure that would be more universally agreed to be bad, like Kingdom with only terminal cards. That would change around the numbers, but lead to the same general conclusion. You might also try to account for multiple factors - if there's no +Buy and no Villages, that makes things even worse. But I don't think this takes away from the overall point.

Part 4: Ways to fix this

So how can you stop the inflation of bad kingdoms? One solution available is "don't do full random". Another is "don't play with too many expansions at once". These are fine for IRL groups, but in the competitive online scene they get problematic. Another approach is similar to what I mentioned earlier: Increase the size of the kingdom. This was done, in a way, with the addition of sideways cards, but has not been done since. It would probably not be a good idea to add another card type just to increase size, but you could simply add more kingdom cards.

In the current setting, even just adding an 11th card would drop the likelihood of a no-Buy-kingdom from 23.6% to 20.4% - below the chance of that happening in a base-only game.

My proposal would be something like "add an extra Kingdom card for every 10 expansions that you play with". That would help counteract the chances of dud kingdoms increasing.

[AJD]

I mean, of course, by the same token you could say that, for instance, games with Horses are more interesting than games without them. In that case, you're stuck at 100% uninteresting kingdoms until Menagerie comes out, and then Menagerie drastically increases the rate of interesting kingdoms. Therefore new expansions make Dominion better.

[Donald X.]

So how can you stop the inflation of bad kingdoms? One solution available is "don't do full random". Another is "don't play with too many expansions at once". These are fine for IRL groups, but in the competitive online scene they get problematic. Another approach is similar to what I mentioned earlier: Increase the size of the kingdom. This was done, in a way, with the addition of sideways cards, but has not been done since. It would probably not be a good idea to add another card type just to increase size, but you could simply add more kingdom cards.

In the current setting, even just adding an 11th card would drop the likelihood of a no-Buy-kingdom from 23.6% to 20.4% - below the chance of that happening in a base-only game.

My proposal would be something like "add an extra Kingdom card for every 10 expansions that you play with". That would help counteract the chances of dud kingdoms increasing.

The fact that pure random would mean a lower % of "important stuff that I nevertheless don't want in every game" as time went on was not invisible. My actual solution was to slightly up the % of these things in the later expansions. The landscapes also help.

[JW]

The fact that pure random would mean a lower % of "important stuff that I nevertheless don't want in every game" as time went on was not invisible. My actual solution was to slightly up the % of these things in the later expansions. The landscapes also help.

I find that the greater prevalence of +Actions in recent expansions is a substantial benefit even when playing games with only one or two expansions. In addition, the reduction in the percentage of cards in newer sets that are duds has made full random Dominion more interesting. The revisions to the base set and Intrigue were also great in that regard. Hoping for Seaside and Prosperity revisions next!

[faust]

So how can you stop the inflation of bad kingdoms? One solution available is "don't do full random". Another is "don't play with too many expansions at once". These are fine for IRL groups, but in the competitive online scene they get problematic. Another approach is similar to what I mentioned earlier: Increase the size of the kingdom. This was done, in a way, with the addition of sideways cards, but has not been done since. It would probably not be a good idea to add another card type just to increase size, but you could simply add more kingdom cards.

In the current setting, even just adding an 11th card would drop the likelihood of a no-Buy-kingdom from 23.6% to 20.4% - below the chance of that happening in a base-only game.

My proposal would be something like "add an extra Kingdom card for every 10 expansions that you play with". That would help counteract the chances of dud kingdoms increasing.

The fact that pure random would mean a lower % of "important stuff that I nevertheless don't want in every game" as time went on was not invisible. My actual solution was to slightly up the % of these things in the later expansions. The landscapes also help.

Interesting! I sort of figured that you would be more focused towards making the game best for people who might only have 2 or 3 expansions, and increasing the % would end up distorting those kind of games slightly.

This article originated form a feeling that there were more "boring" games online recently; obviously that's not exactly hard data. It probably also factors in that I've been playing the game for years and maybe don't get excited as easily anymore.

[schadd]

not exactly what you're talking about but i've definitely enjoyed the game a little more with each new expansion

[Robz888]

When I saw this topic, I actually thought you were going to argue something close to the reverse—each expansion increases the complexity dramatically and thus increases the odds that you are going to have build some kind of engine deck, whereas you can still win with some cute, simpler money strategy a decent percentage of the time if you stick to base game plus an expansion or two. I don't personally agree that that version of Dominion is better, but I could at least see that argument: It's a matter of personal preference, and some people might prefer it. I occasionally miss the days when winning games like that was somewhat more common. (Obviously, it still happens, but at high-skill levels of play, it's rarer, I think.)

I have a very hard time believing that adding expansions raises the odds that a random kingdom is going to be lacking some key component.  Perhaps this is mathematically true in the event that there were infinite cards, but as you note, there are not. Note that cards don't just do one thing, and cards do different things depending on what other cards are there. I.e., Sheepdog is not really a draw card on its own, but if you have non-terminal mid-turn gaining, then suddenly it is. The odds of having mid-turn non-terminal gaining have certainly increased as we've added expansions. If Sheepdog was dropped into Base, you would need it to be present with Workshop or Artisan and a village to make it draw productively for you.

I would argue that if you're restricting yourself to fewer expansions, you increase the need for every kingdom to have +buy and +cards and +actions in order to be interesting (again, if by interesting you mean vaguely engine-y; interesting is a really subjective thing). With many more expansions, it doesn't matter so much if one of those components is strictly absent, because the kingdom will have some other exciting thing that you can do, like gain a bunch of VP chips, or acquire cards and points from landscapes, or whatever.

[Cuzz]

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

[Awaclus]

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

It might be an interesting coincidence to roll an all-village kingdom, but it's certainly not an interesting game.

[xyz123]

Is +buy this example really just trying to say boards that only allow you to gain 1 card per turn? If so, then I would agree. Those games are less interesting.

There are plenty of example though where with gainers, remodellers, etc, you can have a board where multi-card gains per turn are possible. Sometimes the workaround to a lack of +buy makes these games more interesting.

The presence of gainers or remodellers alone though doesn't mean that a board without +buy will be interesting though. They need to be able to combine together, with the right support if necessary, to be able to reliably gain copies of the key cards when needed. Without that, for certain key cards you might be limited to using your single buy per turn to gaining them.

Also, what about some rush games. A gainer, some alt VP, the right support and you might not need to bother about cards that are too expensive for the gainer.

[NoMoreFun]

The joy of dominion for me is engaging with new cards and mechanics and seeing how it fits together with the existing ones. Each new expansion is actually more interesting than the last because there are more interactions to discover.

Also Donald X has gotten better at making sets with more interesting cards and fewer duds.

[infangthief]

Quality vs Novelty

Playing base only might give you the best chances of avoiding a low "quality" board (eg a board without +buy) but it also gives you the best chances of getting a board that feels like you've played it before.

Maybe not the exact same board, but one with enough similarities.

Adding in each new expansion might give a marginal drop in quality probability, but surely gives an increase in novelty.

[LastFootnote]

Quality vs Novelty

Playing base only might give you the best chances of avoiding a low "quality" board (eg a board without +buy) but it also gives you the best chances of getting a board that feels like you've played it before.

Maybe not the exact same board, but one with enough similarities.

Adding in each new expansion might give a marginal drop in quality probability, but surely gives an increase in novelty.

If you want a big shot of both quality and novelty, just play a game with 10 cards from a new expansion.

[Awaclus]

Quality vs Novelty

Playing base only might give you the best chances of avoiding a low "quality" board (eg a board without +buy) but it also gives you the best chances of getting a board that feels like you've played it before.

Maybe not the exact same board, but one with enough similarities.

Adding in each new expansion might give a marginal drop in quality probability, but surely gives an increase in novelty.

If you want a big shot of both quality and novelty, just play a game with 10 cards from a new expansion.

The novelty comes from the larger variety of boards you could possibly get, which makes each board feel less similar to the ones you already played. Restricting yourself to just the newest expansion defeats that purpose.

[LastFootnote]

Quality vs Novelty

Playing base only might give you the best chances of avoiding a low "quality" board (eg a board without +buy) but it also gives you the best chances of getting a board that feels like you've played it before.

Maybe not the exact same board, but one with enough similarities.

Adding in each new expansion might give a marginal drop in quality probability, but surely gives an increase in novelty.

If you want a big shot of both quality and novelty, just play a game with 10 cards from a new expansion.

The novelty comes from the larger variety of boards you could possibly get, which makes each board feel less similar to the ones you already played. Restricting yourself to just the newest expansion defeats that purpose.

I only meant, do it for a game or three, then branch out.

[Holger]

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

I see what you did there https://en.wikipedia.org/wiki/Interesting_number_paradox

But the reasoning doesn't quite work in this case, as there are plenty of kingdoms in which it is impossible to ever empty the supply - specifically, in any kingdom without +buy and without gainers, the 3-pile rule ends the game long before you could empty the supply. Your argument only proves that any kingdom in which you can empty the supply is interesting - but if the uninteresting kingdoms are a subset of the kingdoms in which you cannot empty the supply, there is no ordering and no "minimal number" exists ...

[faust]

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

I see what you did there https://en.wikipedia.org/wiki/Interesting_number_paradox

But the reasoning doesn't quite work in this case, as there are plenty of kingdoms in which it is impossible to ever empty the supply - specifically, in any kingdom without +buy and without gainers, the 3-pile rule ends the game long before you could empty the supply. Your argument only proves that any kingdom in which you can empty the supply is interesting - but if the uninteresting kingdoms are a subset of the kingdoms in which you cannot empty the supply, there is no ordering and no "minimal number" exists ...

It also fails on another level. The set of "minimal uninteresting kingdoms" could be very large; indeed, it is possible that the number of decisions is exactly the same for each uninteresting kingdom. Then being minimal with respect to the ordering gives no additional information, and hence no justification for making the kingdoms interesting.

The interesting number paradox sort of requires a unique minimal element, and that is not given here.

[Cuzz]

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

I see what you did there https://en.wikipedia.org/wiki/Interesting_number_paradox

But the reasoning doesn't quite work in this case, as there are plenty of kingdoms in which it is impossible to ever empty the supply - specifically, in any kingdom without +buy and without gainers, the 3-pile rule ends the game long before you could empty the supply. Your argument only proves that any kingdom in which you can empty the supply is interesting - but if the uninteresting kingdoms are a subset of the kingdoms in which you cannot empty the supply, there is no ordering and no "minimal number" exists ...

Just ignore the 3 pile rule to make it well-defined.

There is no such thing as a "bad" kingdom, and in fact:

Theorem: All kingdoms are interesting.

Proof:

Suppose there exists an uninteresting kingdom.
Order the countably many kingdoms by the minimal number of decisions it takes to empty the supply in a two player game with perfect shuffle luck.
There then exists at least one minimal uninteresting kingdom with respect to this ordering.
But this property makes these kingdoms interesting.
This is a contradiction.

I see what you did there https://en.wikipedia.org/wiki/Interesting_number_paradox

But the reasoning doesn't quite work in this case, as there are plenty of kingdoms in which it is impossible to ever empty the supply - specifically, in any kingdom without +buy and without gainers, the 3-pile rule ends the game long before you could empty the supply. Your argument only proves that any kingdom in which you can empty the supply is interesting - but if the uninteresting kingdoms are a subset of the kingdoms in which you cannot empty the supply, there is no ordering and no "minimal number" exists ...

It also fails on another level. The set of "minimal uninteresting kingdoms" could be very large; indeed, it is possible that the number of decisions is exactly the same for each uninteresting kingdom. Then being minimal with respect to the ordering gives no additional information, and hence no justification for making the kingdoms interesting.

The interesting number paradox sort of requires a unique minimal element, and that is not given here.

I don't think it's required. I'm claiming that the minimality property makes all such kingdoms interesting. Generically there will be some moderate number of minimal kingdoms and it will be fair to call these interesting, and on the other hand special cases like you mention where e.g. the number of decisions is exactly the same for each uninteresting kingdom are even more interesting.

[chipperMDW]

Unique Kingdom Numbering Lemma: Each kingdom can be assigned a unique natural number.

Proof: Assign a unique prime number to each of the following kingdom elements:

   each kingdom card pile
   each event, landmark, project, and way
   the inclusion of Colony/Platinum
   the inclusion of shelters

For the given kingdom, generate a number that is the product of the primes associated with the elements included in that kingdom.  This number will be uniquely associated with that kingdom by the fundamental theorem of artihmetic.

Optionally, depending on one's definition of what constitutes a kingdom, additional information may be encoded into the number.  For example, the prime associated with the Young Witch kingdom card pile may be raised to the power of the number associated with the pile chosen as bane pile.  A number of other examples have been included in the exercises at the end of the chapter.


Minimal Kingom Corollary: Every non-empty set of kingdoms has an element whose number is minimal for the less-than relation.

[ehunt]

Speaking as an old person, Dominion is more fun now than it was a decade ago, although it's less addictive.

[Cuzz]

Speaking as an old person, Dominion is more fun now than it was a decade ago, although it's less addictive.

By remarkable coincidence, Dominion suddenly became less addictive exactly around the birth of my first child.

[Cuzz]

Speaking as an old person, Dominion is more fun now than it was a decade ago, although it's less addictive.

By remarkable coincidence, Dominion suddenly became less addictive exactly around the birth of my first child.

Mafia remained exactly the same amount of addictive which is why I can never play it again.

[fika monster]

I agree fully with the technical analysis, but i do agree in a more general sense (there are plenty of cards that are boring in isolation). one solution could be that in online, at least one kingdom pile is designated "+buy".

idk how to fix this otherwise. I wish i could dislike/ban more cards online, since there are 349 cards, and i can ban 5 and dislike 5, so thats sortoff banning 7.5 cards. And there are more then 7.5 bad cards in dominion in my opinion.

[dane-m]

Speaking as an old person, Dominion is more fun now than it was a decade ago, although it's less addictive.

I occasionally get to play against someone who will play only with the base set.  I can say with some confidence that if that were all that I could now play with every time, I would find Dominion a lot less addictive than I do.

In recent times I've played plenty of games online with interesting kingdoms, plus several that had 'boring' kingdoms, but even those were interesting in the sense that they required me to think about how they had to be tackled.  I even appreciate the occasional slogging match in which everyone's deck get filled with Curses and Ruins because there's no trashing.  I wouldn't want to be in such games too often, but I think I'd miss them if they never cropped up.  I suppose to some extent I want to have to answer the question "Is this a kingdom for which building an engine is sensible?" before moving on to the question "What's the best engine?"  That might be a reflection on the fact that I'm far from the world's best engine builder!

[dz]

idk how to fix this otherwise. I wish i could dislike/ban more cards online, since there are 349 cards, and i can ban 5 and dislike 5, so thats sortoff banning 7.5 cards. And there are more then 7.5 bad cards in dominion in my opinion.

If banlists were expanded, sure I'd probably use it, but I don't think it's that important or likely to happen. There are a lot of people with either empty or incomplete banlists, which indicates that people don't need larger lists. Well you could also argue that a lot of people don't even know that banlists exist, but I've still seen people around my level (58+) that don't have banlists.

And maybe I'm just less picky than others, but I don't have a giant list of cards that I'm ready to add if banlists get expanded. I have maybe 3 cards in my mind, and would struggle with coming up with more (especially for landscapes).

I think a more likely compromise is: if you aren't fully using your banlist, you can add more cards to your dislike list. There are people who have empty banlists, but a full dislike list. And it seems unfair to force them to ban cards (that they don't want to ban), just so they feel like they're fully using their lists. So if you only have 2 cards banned, you can dislike 8 cards.