## Race — How much to value Draw +1 explore?

I’ve metioned in various places that I’m fond of the Expedition force for it’s “Draw +1 card” during explore phase. How good is this?

Mathematically, this is Hypergeometric distribution (I think). This tells you how likely you’ll get X “good” cards if you draw Y cards (out of a deck of known size and with a specific number of good cards). You can look at the numbers in excel by saying “=HYPGEOMDIST(nGood, nDraw, deckGood, deckSize)”

Note that for us, the “Deck” is all unseen cards (including unseen discards, goods, etc). The real question is “What is a good card?” That’s tough. Early on, most of the deck counts. As you specialize, I think this number shrinks. I’m making a number of assumptions … I’m looking at trends, not exact numbers.

Let’s start assuming that 60% of the deck is good. For simplicity, 100 card deck, 60 good cards. (That assumes we’ve seen a few cards built, and spent a few cards).

Let’s assume someone else explores. If we don’t the expedition force, we get crap 16% of the time. With it, that drops to 6%. Now, if the deck is really good (all great) or really bad, then drawing two or three cards won’t matter. But if the deck is only 10% good (10 out of 100), then the extra card drops garbage from 81% shot to a 73% shot. (8 times out of 100, the +1 draw pays off). For a ‘moderate’ deck, the +1 pays around 15% of the time.

What if we play explore +1/+1. Now we’re drawing 4 keep 2 (instead of 3 keep 2 without the expedition force). Now the odds of getting a terrible draw (nothing good) are pretty minor in a rich deck, but you draw two “good cards” an extra 15-20 times out of 100.

Unsurprisingly, where the +1 pays off least is when you are hunting for the last card in a sparse deck. Drawing 7 and 8 are fairly similar.

So, how good is it? Well, if you assume 5 explores after you get the +1 (whether done by you or others), you stand a expect a good card once more. (Which is to say that some games it won’t help, some games will gift you twice or more, but once is about right). Five explores seems excessive (and you can’t count any you Explore +5), so if you drop it to 3 explores, then the odds drop. (Given all the assumptions, I’m just going to pull a number out of the air and say it helps roughly every other game).

I’m still partial to Expedition Force, but I suspect Space Marines is a better early build (To be fair, it costs more, so it should be better).

[I’ve now split Race posts into their own category, rather than make a gigantic chain of posts].

**Update:** One point I missed … by simply categorizing cards as “Good” or “Bad” I do miss incremental gains. For example, if there are 20 good cards, there may be 7 “Great” and 13 “Goods” and the +1 may let me improve my “Good” to a “Great.” But we’re getting above my mathematical paygrade. [The Dean of Engineering at my undergrad school always said … “The next step is dating a mathematician, who solves this for us.” Good advice.] This is the point where I *could* write a simulation, but I think I’ll pass.

Never knew about the HYPGEOMDIST function before. That would have saved me a lot of time on ***** calculations.

I did more work along these lines and my conclusions are similar to Brian’s. I assumed a two player game and the following distribution:

Early Game = 50 good out of 92 unseen

Mid Game = 20 good out of 70 unseen

Late Game = 5 good out of 55 unseen

Then I assumed two early game explores, two mid games, and one late game (both players alternate picking explore…and yes, that’s quite a few explores). Then I checked extra cards based on how many of these explores during the game you got the +1 benefit for.

If all 5 then 0.53 extra

If 4 then .39

If 3 then .31

If 2 then .16

If 1 then .05

So even if you get it out on the first card it’s only worth an extra half a good card. That’s not that valuable at all. Especially given that seeing 8 instead of 7 in the late game (where the VP swings are the largest) is only worth a benefit once per 20 games (given my 5/55 assumption)!

Overall, considering that a 1 cost card that simply gave you an extra card one time in the early game (like a weak public works) would have a similar ‘good card’ benefit, while also giving you an extra card, I don’t think the explore power of Expedition Force should be valued much at all. Play it for the military if you are going to play it.

***** = a card game played with 52 cards in which some are dealt face down, some face up, and wagers are placed as to who has the better hand. Also a blacklisted word in Brian’s comment filter.

LouJanuary 13, 2008 at 11:55 am

I’ve banned the word poker?

BrianJanuary 13, 2008 at 2:53 pm

For all I know, I may have … (and it may be that my account supercedes that policy. I get comment spam and have banned IPs and a few keywords from time to time).

BrianJanuary 13, 2008 at 2:54 pm

For all I know, I may have … (and it may be that my account supercedes that policy. I get comment spam and have banned IPs and a few keywords from time to time).

BrianJanuary 13, 2008 at 2:54 pm

I’ll try again…

*****

Yup, blacklisted.

LouJanuary 13, 2008 at 7:37 pm

One final (minor) point is that multiple explore powers are much better. If you have a research labs (for example) then the +1 becomes pretty good (I think, I should run more numbers). But I was looking at the isolated case.

BrianJanuary 13, 2008 at 8:59 pm

Actually, going from 1 extra card in the early game, to 2 extra cards in mid-late, only increments the advantage from .53 to .76. If you figure you add SETI for a net +3 for the last two explores you are up to 0.86. Basically, it’s a pretty darn small feature no matter what.

The extra card from SETI and Research Lab is something else entirely.

LouJanuary 13, 2008 at 11:30 pm

Actually, going from 1 extra card in the early game, to 2 extra cards in mid-late, only increments the advantage from .53 to .76. If you figure you add SETI for a net +3 for the last two explores you are up to 0.86. Basically, it’s a pretty darn small feature no matter what.

The extra card from SETI and Research Lab is something else entirely.

LouJanuary 13, 2008 at 11:30 pm

Interesting… FWIW, our experience is that having +2 draws on Explores seems to help a lot more than +1 dfraw. But, this is in part due to receiving 4 choose 1 on other people’s explores as well as 5 choose 2 on your own.

I personally love getting Star Nomad Lair and Expeditionary Force in my opening hand with EE, for a variety of reasons including this one…

Tom_LehmannJanuary 16, 2008 at 8:05 am

Interesting… FWIW, our experience is that having +2 draws on Explores seems to help a lot more than +1 dfraw. But, this is in part due to receiving 4 choose 1 on other people’s explores as well as 5 choose 2 on your own.

I personally love getting Star Nomad Lair and Expeditionary Force in my opening hand with EE, for a variety of reasons including this one…

Tom_LehmannJanuary 16, 2008 at 8:05 am

<blockquote>

The Dean of Engineering at my undergrad school always said … “The next step is dating a mathematician, who solves this for us.”

</blockquote>

On a completely unrelated note, this summarizes perfectly the maddening attitude engineers had towards mathematics that I encountered when <i>I</i> got my undergraduate degree at an engineering school. The professors openly considered math to be an annoying black box that would give them hot formulas that they needed, but the derivations and even the underlying assumptions of these formulas was beneath them (“Too ivory tower”). Naturally, the students picked up on this vibe and emulated it. This attitude amazed me. If my work depended on something, I might not try to master it, but I certainly wouldn’t disparage those who did.

Larry LevyJanuary 16, 2008 at 11:54 am

This brings me to a point that I have been pondering about for quite a bit of time – Reasearch Labs. Am I alone in thinking that this card improves with more players? I dont think I have ever dropped it in 2 player, which makes up about 99% of my games.

zzdromanJanuary 21, 2008 at 7:17 am