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On Running Double or Triples of Reusable Economy Cards

Discuss the latest article from Dan looking at how many copies of economy cards you should be running.


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I really like that you used math and determined that the expected value of getting Pro Contacts earlier due to extra copy is 4 credits.

A question:
I believe you showed that the extra copy of Pro Contacts resulted in getting it an average of 4 draws earlier, thus resulting in a pure gain of $4. (Not needing to naturally draw those 4 times).

Drawing a Sure Gamble doesnt give you $4. It gives you the ability to spend a click for $4. So actually HAVING $4 is still better than a Sure Gamble. This makes me think that the bad is actually even HIGHER.

On the other hand, if you have tutor effects, you are way more justified running less copies, if those tutors are cheap. For example, the new card Hostage from Opening Moves will tutor for Kati Jones, or other things. This makes it a lot more viable I think to include less Kati, and some Hostage.

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After I wrote it, I forgot that actually playing the Sure Gamble is a gross gain of 4 credits, but a net gain of only 3 credits so it’s also better than SG. Yeah, there aren’t really single use econ cards that beat 4 additional credits in hand, so you guys were probably right that cutting 1 PC isn’t optimal. It’s weird, but I’m delighted to disprove myself :smile:

Re: Hostage. Yeah, every time I write an article it’s immediately made obsolete by the next data pack. I remember arguing that Medium needed to be included in any Shaper deck, then RDI and Indexing happened a few weeks later.

Edit: That being said, I really like the way Dirty Laundry works in this deck. It opens up a slot in your hand to use PC, and allows you to take a valuable action. I’ve always been happy drawing a DL to plow through an Ice Wall with an SMC’d Parasite.


Its a virtue to be able to recognize when you are wrong, and change a belief. :slight_smile:

I was similarly wrong by initially hating Professional Contacts. If I hadnt been willing to be wrong, I couldnt have made the Professional Kate deck.

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Nice article! Well written and well argued but I think there are some other points to consider which, while not easy to model, make it clearer that two copies is correct.

  1. You have assumed that the card that replaces the third copy will be broadly equivalent, i.e. an economy card. But in a lot of cases it could be right to sacrifice the $4 expected value in order to use the card slot for more utility (perhaps a silver bullet vs. a certain deck archetype). The expectation value is now harder to calculate, it’s $4 lost every game vs. an occasional game-saving/winning card.
  2. You’re assuming that the runner is spending clicks to draw the cards, but actually there are a wealth of deck-thinning tricks which means the the concentration of the desired card potentially increases between draws. This doesn’t apply to all decks of course, but a lot of these devices are in-faction for Shaper so it wouldn’t be unexpected in the same deck as Professional Contacts. For instance, Test-Run and SMC install a card straight from the deck; Replicator and Rabbit Hole pull/install additional cards; Exile draws for free (which could become huge with Pawn in the new set).
  3. Your assumption that runners won’t draw at a rate of 2 cards per turn (unless using Wyldside) is also flawed, I think. Firstly Wyldside is a thing and, combined with Personal Workshop, it’s entirely reasonable that the runner could draw and install cards at that rate. Secondly, the discard pile is increasingly becoming a resource so there’s every reason a player might play a Quality Time and intentionally discard some cards to recycle later on. Freelance Coding Contract presumably synergises with this (or will in the future once there’s a bit more support for recursion).

The other thing that’s hard to measure is what it costs you when the dead card shows up and you’d rather it was something else.

The icing on the cake though is that they spoiled Hostage yesterday :wink:

Update to my previous post:

The Wyldside argument doesn’t really apply to Pro Contacts (since you wouldn’t really play both) but it’s still ok for Kati Jones etc.

I’ve also noticed that the article calculates the probability of drawing the desired card from a 45 card deck, i.e. before the opening hand is drawn. Therefore your chance to draw the card is actually better than stated because of the mulligan. You’re effectively quoting 30% and 21% for having the desired card in your opening hand (i.e. within top 5 draws) for 3 and 2 copies respectively. But if your mulligan strategy is that you will keep almost any hand with the desired card (and mulligan anything without) then you only have 70 or 79% chance of encountering the odds that are in your analysis.

So the correct calculation (for this primitive mulligan rule) for the number of draws is:
P(opening hand) * 0 + P(not opening hand) * original quoted numbers

If you run these numbers it works out to a weighted average of 5 and 9 draws required (for 3 and 2 copies respectively), which is actually still the same difference in clicks. I thought the gap could widen by a card or two which would improve the expected value of 3 copies, but it turns out not to.

I think this tells us that you save yourself 4 draws (clicks) by trying to mulligan for the card you want, but the number of copies in the deck doesn’t change the relative EV of the two deck constructions so you don’t harm yourself.

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Ah that’s true, I neglected to include mulligans. I think I’ll edit the article a bit to include some points made in this thread. To respond to your points:

  1. On silver bullets. That’s correct, however the point of the article was looking solely at econ cards. If you’re on the fence about Plascrete vs a third PC, well that’s entirely meta- and preference-dependent. If you’re on the fence about Dirty Laundry vs a third PC, well that we can model.

  2. Deck thinning will not be relevant until you get into the 15-20 cards left range, because a thinner deck improves the likelihood to draw PC both in the 2- and 3-copy versions, and we would only be looking at the difference in expected draw time between the two. Any benefits a 2-copy PC deck would get from thinning would also be experienced in a 3-copy PC deck. The difference between the double and triple cases are marginal. So with 3 copies and 30 cards left, you’d go from 3/30 to 3/29 with every deck thinning, but with 2 copies left you go from 2/30 to 2/29. That is to say, both versions benefit from the thinning by about the same amount, so we can leave out this consideration. I’d also venture a guess and say you’ll probably take a pass on installing PC once you’re 20-30 cards into your deck :smile:

  3. I think there are definitely some edge cases where a runner will be aggressively drawing more than 2 cards/turn in the early game (in this case we’re talking about 2 vs 3 KJ, drawing prior to seeing KJ). However, unless you’re getting Katana’d or dumping your hand to Freelance, I don’t see myself ever drawing more than 2 cards/turn on average. I’m not talking about 2 cards in one turn – I’m talking about the average number of draws/turn in the first 5 turns or so of the game. Yes, if you have Wyldside or some strange scenario, you should adjust accordingly, but I’d guess I draw between 1 and 1.5 cards/turn.

I think the conclusion at KJ can be intuited if you think back to any game where you get her in your opening hand. Just think about how much work she does for you over the course of 4-5 turns and it’s obvious that’s pretty much better than any other econ card you could ask for there.

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One thing I realized after Alex’s posts is that single-use economy cards are actually better in a PC deck than in other decks. This is because (hold the tomatoes) drawing the card really is part of the cost, since the opportunity cost is taking a credit. For most decks, drawing a card means you’re giving up the opportunity to take a credit, but in a PC deck that’s not true anymore.

Once PC is in play, you can basically add 1 credit to SG, DL, and Daily Casts in your analysis. So therefore you should run those. But they’re only better if you have PC in play, so you should run more of those. So… ahghsdhhghhh, confusing :smile:

The tables in this article are accurate but the conclusion is not. You need to factor in the number of cards you’ll be drawing to get the expected value of the extra Pro Contacts. The equation is going to look something like (sum of (percent chance of drawing your first Pro Contacts at a given point * number of cards left you expect to draw)). I’m not sure I believe my equation but if it’s correct it looks like you need to plan to draw 35 cards before the third pro contacts is worth $4.

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I was going to write up a paragraph explaining why the waiting time can be used, but decided to just math it out and see for myself what the difference in EV was.

So you can compute the EV of PC by adding together the following:
The probability of drawing PC in your opening hand * (value of PC for the remaining draws - 5 - 1)
The probability of not having drawn PC by draw X * drawing PC at the Xth draw * (value of PC for the remaining draws - 5 - 1)

And you would sum this over all draw points where installing and using PC has positive value (i.e. you will draw enough over the remaining game to make up for the install). Also it’s - 5 - 1 for the 5-cost install and the click to install.

The one variable here is the expected number of draws you plan to make throughout the game. If you can win the game with your opening hand, PC is useless. If you’re going to draw through your whole deck PC gains value. So we have to do a little sensitivity analysis to see what kind of values we get.

What I found is (drumroll, please) I’m lazy and could’ve done a better job. I used the expected waiting time because this would be approximately correct in a world where the probability of having drawn PC as you go through your deck increases linearly. This felt sufficient, and I figured it’d be pretty close, but apparently I was a bit off.

Here’s what I found:
If you expect to draw down to 5 cards left, the EV of the first copy of PC you draw is 24.6 credits with 3 copies, vs 20.5 credits with 2 copies.

If you expect to draw down to 10 cards left, the EV of the first copy of PC you draw is 19.9 credits with 3 copies, vs 16.4 with 2 copies.

If you expect to draw down to 20 cards left, the EV of the first copy of PC you draw is 10.7 credits with 3 copies, vs 8.1 with 2 copies.

So yeah, that would’ve been a much more complete article if I had done it this way. The approximation I used overestimates the value a bit – I think most games I’ve played with this deck take me down to about 15 cards remaining, so a third PC would net you about 3 credits in hand. That’s the equivalent of replacing it with a Sure Gamble (you’re netting +4 but taking a click to play it). I guess in the end, same same.


I’m getting slightly different numbers than you are. With mulligan I get 23.9 vs 20.4, 19.1 vs 16, and 9.9 vs 7.8. Without a mulligan, I see 21.7 vs 18.1, 17 vs 13.9, and 8.1 vs 6.2. Your formula is the same one I’m using though. So by my numbers you need to draw down to 10 to make it equiv to sure gamble, by yours to 15.

Oh duh, I was using a 49 card deck (doy, doing this during the trading day seems like a bad idea).

I get roughly your numbers using a 45 card deck, to within .1 credit or so which is whatever.

So maybe it is slightly better to just run 2. Interesting stuff =0) Now I don’t feel so bad about running doubles. It’s good to finally get a grip on more accurate numbers, though. Thanks for pointing this out!

Running 3 hostages and 1 or maybe 2-offs of your key resources is also pretty attractive. Depending on how much you draw through your deck, you get most of your extra click back from hostage because you’re thinning the deck, so it ends up costing you $1 and a partial click, which is good value with the flexibility. Also lets you use PC in criminal without paying exorbitant influence.

Also this teaches you to mulligan for PC. Even at the most pessimistic that’s a better EV boost than you’d see if you mulliganed for sure gamble, although it won’t give you the early game rush effect that sure gamble does.

I think you’re maths is a little out here, as what you’re actually interested in is the expected number of cards required to get your first PC. You’ve assumed it occurs when the probability of having drawn a PC is 50%, but this isn’t accurate. Unfortunately, the distribution of “number of cards to first success” when the underlying distribution is hypergeometric is not a common one, and I couldn’t find an expression for the mean (either on the net or from first principles). If our underlying distribution was binomial, the distribution of interest would be the geometric distribution.

Thankfully, finding the mean for small discrete distributions is fairly straightforward, and we’re only interested in a couple of cases, so it’s fairly easy to calculate in excel by finding all possible values and the probability of each. The expected number of cards you’ll need to draw with 3 PC is 11.5, vs 15.33 with 2 PC. So your conclusion is almost spot on, even if your calculations are slightly off :slight_smile:

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Yeah, I think the conclusion is basically: run 3 if you like, or run 2 if you like. The third copy is like another econ card, so if you need econ and you like PC run 3, otherwise you’re fine without it. It’s like if someone ran Easy Mark over a Hedge Fund - it’s not really a big deal, and probably won’t be a deciding factor in your game.

I’ll tell you what, PC in a game vs Jinteki is pretty sweet though :smile: I played a game tonight that ended with 3 cards left in my stack. I could just PC for money over the entire game and never had to use a single other econ card. This left a lot of net damage fodder to call bluffs with.

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Ok, so I realised what I posted above is probably not the best method of analysis, given the circumstances of the game.

Given that you start with 5 cards in your hand regardless, it doesn’t make sense to weight the analysis in favour of 3-of due to the fact that you’re more likely to get it as the first rather than the fifth card in your starting hand, as you have no choice about drawing the first 5 cards.

So I re-did my analysis with a starting 5 card hand as given, and looked at your expected number of draws (not including your starting hand) in order to find the first PC.

3 PC, no mulligan: 7.14 draws
2 PC, no mulligan: 10.77 draws

If you mulligan for a PC:

3 PC, mulligan: 4.97
2 PC, mulligan: 8.48

So the extra PC giving you 3.5 extra creds on average seems fairly robust, regardless of how you model it. FWIW, on the basis of this I would always play 3 PC over 2 PC + an economy card, as you may not even see the additional economy card over the course of the game.

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Yeah, one thing that’s underscored when you do the math in Excel is how much value an early PC gives you (in the 20-30 credit range).

I know it sounds weird, but a lot of time I find Kati to be excessive when I have PC going. Like the only time I ever click Kati is when I’d click for a credit – I just click her instead. A lot of times I’ll be hitting other econ cards as I draw through the deck so that running/installing has better value than clicking Kati (especially against cheap opponents who don’t pressure the runner).

I’d like one more solid Connection to Hostage for and then I think triple Hostage with 1-of Connections would be pretty amazing. I don’t think Masanori is it, but rather something that stacks with non-draw actions like running or installing.

You’re right, but not really for the reasons that you are implying. You seem to be suggesting that PC is worth more credits if you get it earlier. You’re not wrong but the claim here (made by several different models) is that actually that economic edge isn’t that big and it boils down to personal preference.

You may be right in your assessment of how much money PC generates over the course of a game, but we’re only interested in the difference in value between the second and third copies. Of the “20-30” credits you’re claiming, many of them you’d get anyway with a mid-game PC, so we’re only interested in the gain that’s made by having the third copy - which we seem to be fairly agreed upon now.

You get PC (on average) 3.5 cards sooner if you have an extra copy. Given that you only play PC if you’re intending to carry on drawing, those 3.5 cards translate directly to $3.5 that you would have had if you had a third PC. So that’s the end of the story as its economic value goes.

Where I think you have a point, but didn’t really back it up, is that there is additional value in getting it slightly earlier because of the extra play opportunities and tempo it can give you. Essentially you have a “3.5 card window” in the game with only 2 copies of PC where drawing and gaining money are still separate actions. So, for instance, if you run on click 1, leaving yourself broke, and hit a snare you can’t recover cards and clear the tag. Whereas with PC you’re clawing back the damage and gaining $2 to clear the tag in the same turn, which potentially takes you out of range now of a storm of Neural EMPs. The other thing about this “window” is that if you need a small amount of money to pay for an installation or to afford a crucial run, you’re losing the additional value of a card when you gain $1.

So really it’s not about the $ value missed by clicking for single cards, but about the card value lost by clicking for single credits in this window! That’s much harder to quantify, because you could potentially sit in the window for a long time and might click for $1 a bunch of times before drawing through the 3.5 cards.

Obviously you can’t actually know when you’re in “the window”, because it’s an average integrated over all deck layouts. But essentially, whenever you’re clicking for $1 before your PC shows up, you might be in the window if you only have two copies.

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Oh yeah, I totally agree. Sorry I don’t think it was clear when I replied, I was responding to @creedofhubris’s comment about Hostage. Hostaging for an early PC can result in a ton of value, that’s all.

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I know it sounds weird, but a lot of time I find Kati to be excessive when I have PC going. Like the only time I ever click Kati is when I’d click for a credit – I just click her instead.

First world problems!