Regarding the Lab Engine:
When I read the articles I was really confused what 'payoff' and 'reliability' are supposed to be and doubted it could be as simple as
Let x be the average value of a card. The average value of a turn is simply 5*x.
x = 1-D + D*2*x
x = 1+D/(1-2D)
so i took the time to think about it myself. I said that you draw 5 cards, play all labs at once, draw all the cards, play all the Labs from those cards etc until you only draw Copper. To end up with M money you need to play exactly M-5 Labs. The probability that you're able to draw and play exactly l Labs is given by L(l,5), where L is
l is the number of Labs you still need to play, c is the number of cards you're drawing and p
Lab is the probability of drawing a Lab.
If you already played all (M-5) Labs, you need all cards that you draw to be Coppers, otherwise you can draw any number (k) of Labs, but not more than (M-5) and not more than you draw cards. You then play those k Labs, draw your 2k cards and reduce the amount of needed Labs by k.
Prob gives you the probability that the c cards you draw contain exactly k Labs.
I wrote a script to evaluate that formula for me and it gave me this plot:
To my surprise that showed quite nicely what 'reliability' is: The probability to draw an infinite amount of cards or "The white space in my plot (as x approaches ∞)". It's non-trivial that such a probability exists, but it actually makes a lot of sense: For a high density of Labs you almost double the amount of cards you draw each cycle of playing and drawing, so the probability to draw enough Coppers to end it vanishes so quickly that it converges against a value < 1.
I assume with 'payoff' you mean the average amount of money made each turn. In that case it's really confusing to say that payoff is infinite, since the reliability to get infinite money is 0%. I'm not sure how to determine what actually happens at p
Lab =.5
I think it's possible to do a similiar calculation for a Herald engine, but that's probably too difficult for me already. Not sure what you'd gain from it anyway.
Dan is playing this season?
