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Making Luck Minisode 46: Money Decks Part 2

In this episode, Wandering Winder discusses "Card Evaluation for Money Decks".

In another thread before this one was posted, arflutter said:

[Edited to remove erroneous formula]

This comment is actually on the follow-up money minisode that was published this week, but since there isn't a page for that one at the moment (I realise Adam has his hands full!) I thought I'd post it here. It's about the formula WW uses for calculating money density in decks that have cards that draw other cards (Peddler, Lab and Smithy being the main examples he mentions).

WW starts with pure money decks and simply counts the total money and total number of cards, dividing one by the other to get the money density. No arguments there. In this second episode he then takes a trickier example of adding a Peddler to a deck that previously had 13 cards and 13 money. WW’s assertion is that he now effectively has 14 money in 13 cards because the Peddler doesn’t count as a card. That’s very nearly correct, but the problem is that when you draw five cards to start your turn only 1 card in every 14 is a Peddler, not 1 in 13. If you start your turn with 5 cards drawn randomly from your 14 card deck then the probability of getting the extra dollar that turn is only 5/14. So the average money produced in total on your turn is 5 + 5/14, meaning the effective money density is only 15/14 (exactly the same as if you’d added a Silver) rather than 14/13.

Another way to look at it is that the $1 on the Peddler isn't worth quite as much as each $1 on the other cards, because to get the $1 from Peddler you need to draw it in your 5 card starting hand, whereas to get the money from any other card you can EITHER draw it in your starting hand OR draw it with the Peddler.

In this case the difference in average money is minute ($0.027 per turn), but if you add more Peddlers or other cards such as Labs that draw more than one card then the error in the WW formula becomes more significant. Still, the WW formula is very simple and if you’re trying to keep track of the numbers in your head while playing a real game then this is a pretty good approximation if you’re playing a moneyish deck with just a few Actions.

One more point: cards that draw cards miss more shuffles than ones that don’t, so adding a card like Peddler to a money deck will usually be a little worse than what a formula for “effective money density” would imply.

 

He's completely right, and I'm wrong.

 

What peddler actually gives you fits with what I think is the other way I said you could look at things, the per-hand way: it gives you value equal to your money-density-per-card-in-deck-besides-it, plus $1. And Lab gives you money-density-per-card-in-deck-besides-it times 2. As you get more of these things it becomes harder and harder to tell exactly what these numbers are, because they recurse in on themselves. But the rule of thumb still generally holds, that both become better than silver once your money density exceeds 1, and past that point, lab becomes better than peddler as well (though at that point, the difference is minimal).

 

The other point about thinness vs thickness of deck also still holds - you see ALL of your cards more often the more of these you have as compared to non-drawers. The point about these cards missing the shuffle more often than other cards is true, but only for certain deck sizes. (And not big or small deck size, but modular arithmetic deck size). So it's overall true, but quite minor. (e.g. for a peddler to miss where something else wouldn't, you need to have a deck size exactly divisible by 5 (gets a bit more complicated the more you add), and in this case you can sometimes choose not to play the peddler (which is usually bad, but sometimes very good)). The choice of being able to trigger the shuffle or not is also a small effect but very very roughly cancels this effect out I think? Probably the missed shuffle chance/pain is still higher, but it's incredibly minute, and should almost never be the deciding factor on one of these vs a non-drawer. (Durations et al are another story).

Some time ago, I did some basic analysis of shuffle skipping.  The basic conclusion was that in a deck full of stop cards, if you average out the modular arithmetic, then on average 2 cards skip each shuffle.  If you add a single peddler, it's about 50% more likely to skip the shuffle.  But there's also a second-order effect, because cards won't ever skip two shuffles in a row.

It's rather complicated calculation for an effect that's so small.

So, I'd like to point out a thing.  arflutter mentions two effects, which I'll call the starting-hand effect, and the shuffle-skipping effect.  The starting hand effect says that the value of the peddler is 5/N, where N is the total number of cards in your deck (and not just the number of stop cards).  The shuffle-skipping effect says that the value of the peddler is reduced by the increased chance that it will skip shuffles.

So, the thing I'd like to point out, is that the starting-hand effect, and shuffle-skipping effect, are really two ways to think about the same thing--and that the starting-hand effect is the wrong way.

The starting-hand effect is an accurate way of thinking about it, in the case where you shuffle your deck every turn before drawing cards.  On the other hand, suppose that we had another deck that never gets shuffled.  It's just an infinitely large deck, consisting of a repeating pattern of 13 coppers followed by a peddler.  The probability of drawing a peddler each turn is 5/13, not 5/14.  What we have in reality, is a deck that gets shuffled only some of the time, and a probability that's somewhere between 5/13 and 5/14.

Quite right! There’s more to this than I thought.

My logic was that since the drawing cards obviously miss more shuffles, the situation where your full deck is shuffled before you draw your five cards is better than average as far as finding your drawing cards is concerned, since here they haven’t missed the shuffle. That seems intuitively obvious but turns out to be wrong.

I think in practice 1 Peddler plus n Coppers is still generally closer to (n+1)/n rather than n/(n-1), but I now admit that the former is a lower bound rather than an upper bound. I’m still working on a more accurate formula, but I think we can safely say that WW is right: the differences are tiny compared with the more important factotrs of thin v thick deck and the way the value of the card will change as you add additional cards. Assuming your deck is likely to improve further over the rest of the game (usually the case unless you’re greening or being heavily junked), the Peddler will generally hold its value better than a treasure that gives a similar improvement to instantaneous money density. WW mentioned in the episode the fact that it’s rarely right to buy Copper (over nothing) at the start of the game even though at that stage it improves your money density, because for most of the game that Cooper will drag your money density down. Peddler, Lab etc. don’t have this problem, and this is a much more important consideration than the tiny differences between different formulae for calculating money density.