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Dominion is getting worse with each expansion

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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.:

--- Quote from: faust on August 01, 2021, 03:30:51 pm ---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.
--- End quote ---
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:

--- Quote from: Donald X. on August 02, 2021, 11:52:05 am ---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.
--- End quote ---

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:

--- Quote from: Donald X. on August 02, 2021, 11:52:05 am ---
--- Quote from: faust on August 01, 2021, 03:30:51 pm ---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.
--- End quote ---
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.

--- End quote ---
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.