Anders Gabrielsson
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Uppsala
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Edit: I've corrected the horrendous formatting that I completely missed when I pasted the text from Notepad. Everything should be a lot more legible now.

Inspired by the discussion in the Numerical Take thread I put together a spreadsheet to make some calculations on the odds of drawing a favorable card with the various 6-cost developments.

This is in no way meant to be a complete analysis of these cards; it's merely an analysis of one aspect of the value of these cards.

WARNING: There may be mistakes in my math or in the database, but I think they are not big enough to have more than a very minor impact on the end results. If you find any errors, please let me know!

If you're not interested in theoretical discussions, skip down to The Actual Numbers, though you may want to read Terminology to keep from being confused by strange, well, terminology.

Notes

Why is this interesting?
Knowing how many cards you need to draw to have a good chance of getting a card that's useful for your 6-dev can be useful for choosing your strategy or deciding if you want to play it at all (for the points). If you're already doing a trade side-strategy a 6-dev that has low odds per card of getting something but is otherwise useful becomes more viable than if you're playing a strategy that is low on card-draws. Besides, math is fun.

Methodology
I created a spreadsheet with all relevant information about the cards in the game (including both expansions). I then created a formula for each 6-dev that calculates the VP it gives for each card. This gives me the information I need to calculate the odds of drawing cards giving a certain number of VP (a specific number, more than zero, or any other combination) from the deck.

Then I created a simple table with the combinatorial data and just looked up the various probabilities.

Limitations
All calculations are based on drawing from the full deck minus the particular 6-dev in question. For most of the developments this will be good enough in most situations; however, there are some corner cases that need to be specifically mentioned. Developments that rely on a very few cards to generate their points (I'm looking at you, Galactic Renaissance!) can have their probabilities heavily skewed by the situation for one or two of their cards. This also depends on which those cards are: for Galactic Renaissance, it makes a big difference if you have Artist Colony and Galactict Trendsetters in play and are drawing for Research Labs than if you have Artist Colony and Research Labs and are drawing for Galactic Renaissance. I'll try to mention some of these situations below.

However, such special cases aside, the below calculations should be good enough to evaluate the power of the different cards from a VP perspective.

Also, I have not included calculations on the probabilities of drawing more than one VP-giving card, mostly because it isn't very interesting. The question you want answered in the normal situation is "How big a chance do I have to draw a useful card?" and that is the question I will answer below.

I have left out two of the 6-cost developments: Pan-Galactic Research and Galactic Exchange.

Pan-Galactic Research gains no VP for other cards, and while it does change the VP/CC (see below) for developments it is still outside the scope of this discussion. For Galactic Exchange the probabilities change dramatically depending on which and how many types of worlds you already have in play or in hand, and listing all the 17 (if my off-the-cuff estimation is correct) cases is a bit much, especially since making assumptions about cards already in play requires changes to the basic assumption of drawing from the full deck. I may return to these cards some other time, but for now I'm going to ignore them.

Terminology
I will use percentage notation for all probabilities.

In some cases I will discuss VP in relation to "card cost", where card cost is the number of cards required to get the desired card into play. For cards other than military worlds this will normally be the cost of the card plus one for the card itself. Another interesting figure here is the VP a card gives in relation to its card cost, which I will call VP/CC.

Example: Alien Data Repository has a cost of 7 and gives 6 VP. It's card cost is 8 (to play Alien Data Repository you need eight cards: Alien Data Repository itself and seven cards to discard), making it's VP/CC 6/8 or 0.75.

VP/CC does not take into account the bonus card gained from the Settle action since it is situation specific.

For military worlds this value seems meaningless: provided you have the relevant military the CC is alwas one. However, if you try to include the number of cards required to get to that military you have to make a lot of assumptions about how often certain cards are played etc, making the VP/CC for them measure something completely different than for the other cards. As a rough equivalent you can measure their VP/Def, or VP per point of defense. This is simply the VP the card gives divided by the defense value of the card.

However, for the calculations below I have used VP/CC (with CC=1 for military worlds) for the sake of simplicity. To combine VP/CC and VP/Def I would still need to get into making assumptions about how the game is played that I would rather stay away from at this point. Furthermore, at the point where you are able to play a specific military world, CC=1 is accurate so for a mature tableau it is good enough.

Any changes in VP/CC takes into account discounts gained from the 6-dev itself with the following exceptions:

* Galactic Bankers: The free card is not counted as a discount since you get it even if you don't place a development.

* Rebel Alliance: The discount is ignored, since that bonus is situation specific - you don't use it if you conquer the world.

* Terraforming Guild: The bonus for settling is counted as a discount, but card cost is never counted as being less than one even when this would be the case after the bonus. (For example, settling a military world with Terraforming Guild in play would normally reduce card cost to zero, but for the purpose of these calculations it is still assumed to be one.) This is mainly to avoid having to deal with division by zero.

For developments that give military this has no effect, which is another weakness of setting VP/CC = VP/Def for those cards, but as discussed above doing it any other way will result in other complications.

I have only noted these numbers where I think they are particularly interesting since they are a bit speculative and (despite the length of this secion) not the main thrust of this post.

Now that my academical side has been mostly satisfied, let's get on to the interesting stuff: the actual numbers.

The Actual Numbers

Here I will present each 6-cost development with the probabilities of drawing a VP-giving card depending on the number of cards drawn. Remember, these values are for drawing from the full deck minus the development itself.

All tables show the number of cards drawn in the left column and the probability of drawing at least one VP-giving card in the right column.

Alien Tech Institute
1 9.5%
2 18.1%
3 26.0%
4 33.2%
5 39.6%
6 45.5%
7 50.9%
8 55.7%
9 60.1%
10 64.1%


Basically, unless you're drawing a huge amount of cards you can't expect to get a card giving you extra VP more than about every other turn.

Free Trade Association
1 12.3%
2 23.1%
3 32.7%
4 41.1%
5 48.5%
6 55.0%
7 60.7%
8 65.8%
9 70.2%
10 74.0%


Galactic Bankers
1 45.3%
2 70.2%
3 83.8%
4 91.3%
5 95.3%
6 97.5%
7 98.7%
8 99.3%
9 99.6%
10 99.8%

Galactic Bankers has the lowest VP/CC effect on its related cards (0.33 compared to the average of 1.02).

Galactic Federation
1 45.3%
2 70.2%
3 83.8%
4 91.3%
5 95.3%
6 97.5%
7 98.7%
8 99.3%
9 99.6%
10 99.8%

The VP/CC change on the affected cards is average, but calculated for all cards it has the second highest VP/CC bonus after Terraforming Guild. This is mainly because reducing the cost of the many cheap developments by one more cards has a big effect on their VP/CC.

Galactic Genome Project
1 10.1%
2 19.2%
3 27.4%
4 34.8%
5 41.5%
6 47.6%
7 53.0%
8 57.9%
9 62.4%
10 66.3%


Galactic Imperium
1 21.2%
2 38.0%
3 51.3%
4 61.9%
5 70.1%
6 76.7%
7 81.8%
8 85.8%
9 89.0%
10 91.4%


Galactic Renaissance
1 2.8%
2 5.5%
3 8.2%
4 10.8%
5 13.3%
6 15.8%
7 18.3%
8 20.6%
9 23.0%
10 25.2%

Unsurprisingly, Galactic Renaissance has very low probabilities for drawing useful cards.

Galactic Survey: SETI
1 62.0%
2 85.7%
3 94.7%
4 98.0%
5 99.3%

Unsurprisingly, with SETI you're virtually guaranteed to draw a useful card even with very limited drawing capability. On the other hand, getting a card worth more than one extra VP is slightly harder than for Alien Tech Institute above (8.4% for one card, 35.8% for five cards).

Imperium Lords
1 26.3%
2 45.7%
3 60.1%
4 70.8%
5 78.6%
6 84.4%
7 88.6%
8 91.7%
9 94.0%
10 95.7%


Imperium Seat
1 11.7%
2 22.1%
3 31.4%
4 39.6%
5 46.8%
6 53.2%
7 58.9%
8 63.9%
9 68.4%
10 72.3%


Merchant Guild
1 16.8%
2 30.8%
3 42.5%
4 52.3%
5 60.5%
6 67.3%
7 73.0%
8 77.7%
9 81.6%
10 84.8%


Mining League
1 12.8%
2 24.1%
3 34.0%
4 42.6%
5 50.1%
6 56.7%
7 62.5%
8 67.5%
9 71.9%
10 75.7%


New Economy
1 33.5%
2 55.9%
3 70.9%
4 80.8%
5 87.4%
6 91.7%
7 94.6%
8 96.5%
9 97.7%
10 98.5%


New Galactic Order
1 10.6%
2 20.2%
3 28.7%
4 36.4%
5 43.3%
6 49.5%
7 55.1%
8 60.0%
9 64.5%
10 68.4%


Pan-Galactic League
1 27.4%
2 47.4%
3 61.9%
4 72.5%
5 80.2%
6 85.8%
7 89.8%
8 92.7%
9 94.8%
10 96.3%


Prospecting Guild
1 57.5%
2 82.1%
3 92.5%
4 96.9%
5 98.7%
6 99.5%
7 99.8%
8 99.9%

The numbers here are virtually identical to SETI in all aspects, including VP/CC changes.

Rebel Alliance
1 23.5%
2 41.5%
3 55.4%
4 66.0%
5 74.2%
6 80.4%
7 85.2%
8 88.8%
9 91.5%
10 93.6%


Terraforming Guild
1 22.9%
2 40.7%
3 54.4%
4 65.0%
5 73.2%
6 79.5%
7 84.4%
8 88.1%
9 91.0%
10 93.1%

Terraforming Guild has the single greatest effect on the VP/CC of all cards together, and is second only to the highly specialized Uplift Code in the VP/CC effect it has on the cards it gives bonuses to.

Trade League
1 12.3%
2 23.1%
3 32.7%
4 41.1%
5 48.5%
6 55.0%
7 60.7%
8 65.8%
9 70.2%
10 74.0%


Uplift Code
1 4.5%
2 8.8%
3 12.9%
4 16.8%
5 20.6%
6 24.3%
7 27.8%
8 31.2%
9 34.4%
10 37.5%

Here even massive card drawing capability won't get you more than one card every few turns. In other words, don't play it for VP unless you already have the cards. On the other hand, the cards you do get VP for get a massive VP/CC boost (2.66 compared to the average of 1.02, the single highest value).
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Sarah Allen
United Kingdom
Colchester
Essex
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Re Uplift Code - the other night I had SIX uplift worlds in my tableau and could I find this card?!!!

I thought three VPs per relevant card with Uplift Code was a bit high, but seeing your numbers sets me straight.
 
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Anders Gabrielsson
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I feel your pain.

Sometimes it's a bit frustrating that getting the right development is a bit random. I guess playing with drafting helps with that, but I haven't tried that yet.
 
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