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Subject: Impact of Purple Cards on Game Length rss

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David J
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Previous articles have expressed dissatisfaction at the game length of Citadels, with many finding it somewhat too long. Many users on this forum house-rule the game in an attempt to shorten the playing time, most commonly either by reducing the number of districts triggering the end-of-game condition from eight to seven, and/or replacing the Warlord with the Diplomat, both of which are within the scope of the printed rules. This article explores an alternate way to alter the playing time: by paying attention to how the purple cards impact game length. In this “variant” article, I analyze the purple cards in terms of whether they tend to slow the game down or speed it up (i.e., bring the game to its end-of-game condition later or sooner). The ultimate aim of this article is to help players decide which purple cards to include in or exclude from the district deck in order to construct a “faster” (or even, if they choose, a “slower”) district deck – i.e., a deck that provides for a shorter (or longer) overall game.

This article assumes that you are using “The Dark City” expansion, which is now included in the standard game. It is this expansion that allows the swapping in and out of bonus purple cards with basic purple cards. In the below discussion, card names include a star (*) when they are part of the expansion (and have a star on the card).

Rule clarification:

The game comes with 12 basic purple cards and 14 “bonus” or expansion-set purple cards. The printed rules state: “Before the game begins, players may agree to add 2-3 additional purple district cards to the District Deck from the 14 available bonus district cards. If players wish to use more than 2-3 of the bonus district cards, they should remove one existing purple district card for each additional bonus district card used.” In essence, this rule may be stated more simply as: “you now have 26 purple cards to choose from; choose 12-15 of them to use in the district deck.” Of course, since this is a variant, there is nothing in theory to prevent you from going over or under the 12-15 purple card limit... however, it turns out that the below analysis will also show that in order to either minimize or maximize game length, you’re going to want to have about that number of purple cards anyway. We now proceed to determine which 12-15 purple cards are the ones you want.

Five Factors:

This analysis proceeds from the assumption that certain factors in Citadels independently shorten/lengthen the time until the advent of the end-of-game condition (i.e., a player building their last district). [Feedback on whether this assumption is warranted is welcome in the comments section.] These factors may be found in house rules, or in the printed text on purple cards. They are:

1. reducing/increasing the number of districts needed to trigger the end-of-game condition. (An example of this factor is house-ruling the number from 8 to 7.)

2. frustrating/facilitating the destruction of districts (An example of this factor is replacing the Warlord with the Diplomat; this “frustrates” (indeed, completely prevents) the destruction of any districts.)

3. facilitating/frustrating the building of districts

4. unlimiting/delimiting player actions

5. enhancing/reducing player options

It turns out that, for five of these factors, each of the purple cards exhibits no more than one of these five factors. (I.e., all of the purple cards that affect game length [not all do] may be sorted into one of these five categories.) For this reason, and also because these factors are straightforward, these five characteristics are relatively easy to analyze.

Two additional factors:

6. the building cost of the card

The five previous factors deal with the printed text of the purple card; this factor takes into account its building cost. Factor #6 is more difficult to analyze than the preceding five factors for two reasons.

The first reason is that a cheap purple district *may* slow the game down, but also *will* speed it up. It may slow it down because it would be a natural Warlord target; thus its very inclusion may “facilitate” the destruction of districts, which is Factor #2. On the other hand, the same card should speed up the game, by lowering the average building cost of the deck. The “average building cost” of the deck is simply the average of all of the building costs of all the cards in the deck. A deck with a higher average building cost should lengthen the play time, for two reasons: First and primarily, because a player must wait longer to gain the gold necessary to build the average district, and second, by increasing the number of district cards deemed unbuildable (due to the opportunity cost required to build them); these unbuildable cards become dead cards that represent a loss of resources, and/or require him to expend a turn selecting a character (i.e., the Magician) that allows him to exchange them for better cards.

So much for the first reason this factor is hard to analyze: cheap purple districts should both slow the game down and speed it up. However, if the Warlord is replaced by the Diplomat, then this uncertainty is eliminated: cheap purple district cards will only speed the game up (and expensive purple district cards will only slow the game down, with or without the Warlord), simply by decreasing/increasing the average building cost.

But this brings us to the second reason why this factor is hard to analyze, which is that the building cost must always be weighed against the text of the card, which is going to fall into one of the five previous categories. Stated another way, this sixth factor is not independent of the other five, like the first five were independent of each other. E.g., a purple building with a low building cost might have an effect that lengthens the game according to another criterion. More common is the converse: there are many 6-cost purple districts whose printed effects shorten the play time according to one of the five previous criteria. In order to analyze the true effect of the building cost criterion, then, we would need to analyze the previous five in a comparative fashion, and therefore in a more mathematically precise way than I, at least, am able. In this article I therefore merely allude to this sixth, building cost factor in a limited way.

Finally, let us briefly mention one final factor that is not unique to Citadels:

7. Increasing Player Cogitation Time.

This is another factor that, while certainly real, is difficult to analyze. Stated simply if vaguely, a purple district card that introduces significant complexity will increase player cogitation time significantly. (Since all purple cards amend the basic rules, by definition none of them decrease player cogitation time.) The obvious question is, what counts as “significant” complexity? In this article I merely allude to this factor in a few cases, but do not base my argument on it. If someone else can make a case that some purple cards introduce significantly more complexity than others, then I say that’s another article waiting to be written.


Analysis of Individual Cards

I begin with the purple cards that affect the game length directly, sorted into the 5 categories:


1. Cards explicitly affecting the maximum number of districts.

a. Speed up:

Bell Tower*: “When you place the Bell Tower in your city, you may announce that the game will end after the round in which a player first places his seventh district. You may do this even if the Bell Tower is your seventh district. If the Bell Tower is later destroyed, the game end conditions return to normal.”
[Note: if the player holding the card does not benefit from ending the game early (because they need to catch up first), then this card becomes a dead card and, as such, threatens to slow the game down instead. However, I think this is about as likely to happen as drawing a duplicate card that would also become a dead card, so we can treat this card simply as only speeding up the game.]

b. Slow down:

None.


2. Cards affecting destruction of districts. Cards that frustrate destruction speed the game up, while cards that facilitate destruction slow the game down.

a. Speed up:

Great Wall: “The cost for the Warlord to destroy any of your districts is increased by one gold.”

Keep: “The Keep cannot be destroyed by the Warlord.”

b. Slow down:

Armory*: “During your turn, you may destroy the Armory in order to destroy any other district card of your choice in another player’s city.”


3. Cards affecting building. Those facilitating building speed the game up.

a. Speed up:

Factory*: “Your cost of building other purple district cards is reduced by one. This does not affect the Warlord’s cost for destroying the card.”
[Note: Like the Wishing Well*, below, this card may incentivize its owner to build purple cards, which have higher than average building costs. But, unlike the Wishing Well*, since they are doing so at reduced cost, the increase in the average building cost is lessened. I assume the increase in game length due to factor #6 is negligible in this case, and is overridden by the main affect of the card which is to facilitate building, and so does not even warrant a “may” slow down game (as I indicate for the Wishing Well*).]

b. Slow down:

None.


4. Cards affecting player actions. Adding or restoring actions to players speeds the game up, while preventing players from acting slows the game down.

a. Speed up:

Hospital*: “Even if you are assassinated, you may take an action during your turn (but may not build a district card or use your character’s power).

b. Slow down:

Ball Room*: “When you have the Crown, all other players must say “Thanks, your Excellency,” after you have called his character. If a player forgets to address you in this way, he loses the turn.”
[Note: Does anyone really use this card anyway? shake]


5. Cards affecting players’ options. The lion’s share of the purple district cards fall into this category. The cards that speed up the game in this category either provide additional resources or allow conversion of one kind of resource into another; this added flexibility allows the player to get closer to her end-of-game condition. The one card in this category that slows down the game removes its owner’s resources by converting them into victory points.

a. Speed up:

Graveyard: “When the Warlord destroys a district, you may pay one gold to take the destroyed district into your hand. You may not do this if you are the Warlord.”
[Note: this card could also be interpreted as frustrating destruction (by acting as a deterrent), which speeds up the game according to a different reason, #2.]

Laboratory: “Once during your turn, you may discard a district card from your hand and receive one gold from the bank.”

Library: “If you choose to draw cards when you take an action, you keep both of the cards you have drawn.”

Lighthouse*: “When you place the Lighthouse in your city, you may look through the District Deck, choose one card and add it to your hand. Shuffle the deck afterwards.”

Observatory: “If you choose to draw cards when you take an action, you draw three cards, keep one of your choice, and put the other two on the bottom of the deck.”

Park*: “If you have no cards in your hand at the end of your turn, you may draw two cards from the District Deck.”

Poor House*: “If you have no gold at the end of your turn, receive one gold from the bank. Gold placed on your district cards does not count as your gold for this purpose.”

Quarry: “When building, you may play a district card already found in your city. You may only have one such duplicate district in your city at any one time.”

School of Magic: “For purposes of income, the School is considered to be of the color of your choice...”

Smithy: “Once during your turn, you may pay two gold to draw three district cards.”

Throne Room*: “Every time the Crown switches players, you receive one gold from the bank.”

b. Slow down:

Museum*: “On your turn, you may place one district card from your hand face down under the Museum. At the end of the game, you score one extra point for every card under the Museum.”
[Note: in addition to this reason, the Museum* might further increase game time by increasing player cogitation time; see the note under Map Room*, below.]


0. [zero]
I now consider those remaining purple cards that don’t fall under any of the categories 1-5. We’ll call this group the “zero category.”

Dragon Gate and University: “This district costs 6 to build, but is worth 8 points at the end of the game.”
[Note: The only possible effect on game length is that these *may* slow the game down due to increasing the average build cost (factor #6).]

Haunted City: “For purposes of victory points, the Haunted City is considered to be of the color of your choice. You cannot use this ability if you built it during the last round of the game.”
[Note: This card is quite plausibly a Warlord target: it is the only purple card to be so cheap (cost=2), it will usually be played prior to the last round, and would typically be played to grant the player 3 VP by providing the fifth color. Therefore, its presence *may* facilitate the destruction of a district (itself), and thus *may* slow the game down according to factor #2.]

Imperial Treasury*: “At the end of the game, you score one point for each gold in your possession. Gold placed on your district cards do not count toward this total.”
[Note: see Map Room* immediately below.]

Map Room*: “At the end of your turn, you score one point for each card in your hand.”
[Note: both the Imperial Treasury and the Map Room convert other, constantly-changing resources into victory points, and thus would seem to make an endgame move for either their owners or other players more difficult to calculate. This could possibly increase player cogitation time (factor #7), and therefore, it could be argued that these cards *may* slow the game down according to this factor, which is beyond the scope of this article.]

Wishing Well*: “At the end of the game, you score one point for every other purple district in your city.”
[Note: Since in theory this card incentivizes its owner to build more purple buildings, which on average have higher build costs than non-purple buildings, this card *may* actually slow the game down according to factor #6.]

Analysis of the “0 category” cards: It turns out that the purpose of all of these cards is to alter the victory point scoring only after the game has ended. None of them speed up the game according to factors 1-6, but all of them *may* slow the game down according to one of these factors. At the end of this analysis I make a distinction between those cards that *may* slow the game down and those that *do*. If my suggestions that any of these six cards may slow the game down are not persuasive, you can simply treat those cards as having no affect on game length. As I will argue in the final analysis, if you are trying to create a faster deck, this won’t matter either way, as there are plenty of certain “fast” cards to choose from before dipping into these six, whose effects are less certain.

Summary of categories 0-5:

Sorted first by effect on game length, then by build cost, then alphabetically:

Cards that speed the game up:
3 Keep
3 Lighthouse*
5 Bell Tower*
5 Graveyard
5 Laboratory
5 Observatory
5 Poor House*
5 Quarry
5 Smithy
6 Factory*
6 Great Wall
6 Hospital*
6 Library
6 Park*
6 School of Magic
6 Throne Room*

Cards that do not speed up the game, but may slow it down:
2 Haunted City
4 Imperial Treasury*
5 Map Room*
5 Wishing Well*
6 Dragon Gate
6 University

Cards that slow the game down:
3 Armory*
4 Museum*
6 Ball Room*


Conclusion: Constructing the Fast Deck.

It is arguable that the two most common ways of reducing the playing time are not as effective as they might seem at first. The 7-max-district rule makes it harder to complete the 5-color bonus, so a player may well refrain from building a 7th district if the only one he may build is the wrong color and he needs those bonus points to win. The Diplomat has more to consider in acting than the Warlord, such as 5-color bonus and income benefits accruing to him after he acquires the district, and he must consider this for two buildings, not just one, all of which may cause the Diplomat player to have a higher player cogitation time than the Warlord player. In addition, the inclusion of the Diplomat may further slow down the game because the Graveyard, a card that speeds up the game according to the above analysis, may not be used when the Diplomat is in play. But even if these are unprovable assertions, these and other methods for shortening play time may be used in conjunction with the method outlined in this article. Furthermore, for those who think the basic rules (8 district max; use the Warlord) otherwise work very well as is (for various reasons), then it is the selection of the purple cards for the deck that is the best method of shortening the length of play.

Thus, in what follows I offer some closing thoughts on building a faster district deck.

Assuming we follow the printed rule about the number of purple cards (quoted and restated near the beginning), this means constructing a deck that has between 12 (the basic number) and 15 (12+3) purple cards. Keeping in mind that the basic deck of 12 comes with two Keeps, the first list above containing cards that speed the game up consists of 17 cards, more than enough to choose 12-15 from. Minimally, then, we can say that only those cards on that list of 17 should be chosen. We might further argue that within this list, we should favor the lower cost cards, for reason #6, but, again, it is harder to make a general rule here, as this depends upon the relative weights of the effects within this group of cards.

But we might further debate whether, if we want the fastest district deck, the printed rule should be followed. If all 17 of the purple cards in the first group speed up the game, then for the fastest game, shouldn’t we add all 17 of them? That is a reasonable assumption, although at least one argument might work against this conclusion: A deck with all 17 cards will increase average building cost over a deck with 12-15 cards, if the latter are selected from the group of 17 according to lowest building cost. However, we can at least state that we should not dip into either of the other two groups, since these “may” and “will” slow the game down; thus, at most the first 17 cards should be used – not too far off the stated rules which stipulate no more than 15 purple cards.

This is about as far as I can take this, but any comments that further the conversation are welcome.
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John
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Interesting analysis. A couple of comments:

Lighthouse*: “When you place the Lighthouse in your city, you may look through the District Deck, choose one card and add it to your hand. Shuffle the deck afterwards.”

I would expect this to slow down the game unless the player building it was experienced enough to know what to look for (and have a back up plan) due to the time it could take an inexperienced and/or indecisive player to chose a card.

Dragon Gate / University - expensive and certainly worth building if you draw them (since they score more points than the gold cost) so may slow down the game as people try to save up for them.
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Daniel Blumentritt
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IMO #7 is the most important factor. My guess is 80% of people who say Citadels takes too long are having some AP issues in their group, and I say this because I've played the game in several groups like this. 5-10 seconds less of "which character do I choose" decision time per instance can take 10-15 minutes off the total play time.

Quote:
Note: if the player holding the card does not benefit from ending the game early (because they need to catch up first), then this card becomes a dead card and, as such, threatens to slow the game down instead


It can speed the game up if a player who is tired of playing uses it to end the game early just to get it over with. Had that happen once. Nobody was satisfied by that move except for her - including the winner.
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David J
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zabdiel wrote:
Interesting analysis. A couple of comments:

Lighthouse*: “When you place the Lighthouse in your city, you may look through the District Deck, choose one card and add it to your hand. Shuffle the deck afterwards.”

I would expect this to slow down the game unless the player building it was experienced enough to know what to look for (and have a back up plan) due to the time it could take an inexperienced and/or indecisive player to chose a card.


That is a good point. It would then be another instance of the hard-to-assess Factor #7, increased player cogitation time. The possible slow-down due to #7 needs to be weighed against what I argue is a certain speed-up, the effect of the card once it is played. This effect I categorize as #5, adding a player resource, in this case, a card in hand (and specifically, a good card). I assume this will speed up the game, since they now have something in hand worth building.

Something I didn't say explicitly but perhaps should have, is that if a player reaches his end-of-game condition one turn early, that speeds up the game by an entire round. Whereas playing the Lighthouse only slows the game down for that one turn -- it's a one-time effect.
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David J
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zabdiel wrote:

Dragon Gate / University - expensive and certainly worth building if you draw them (since they score more points than the gold cost) so may slow down the game as people try to save up for them.


My first reaction was that I already covered that. But you're making me realize that since Factor #6 is the ONLY factor impacting those two cards, then that factor is not "difficult to assess" for them; these are instances where Factor #6 functions independently. Thus I should recategorize these from "may" slow down the game to "will" slow down the game.

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