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Author Topic: Choosing the correct number of terminal actions  (Read 32091 times)

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Epoch

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Re: Choosing the correct number of terminal actions
« Reply #25 on: September 06, 2011, 12:56:57 pm »
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If the only action card present was one of (Bureaucrat, Militia, Thief, Witch, Woodcutter, Workshop), what is the best strategy?

I'll make an argument for "with Woodcutter as the only Action card, there is no particular reason to ever buy it."

Logic:  Woodcutter is different from a silver in only two ways:

1.  In which it collides with another Action.
2.  In which it enables you to use an additional buy.

Obviously, #1 is a way in which it is inferior to Silver.  So when does #2 overwhelm it?

Well, not in a BMU-like deck, I think, ever.  The only things you want to buy there cost $3, $6, and $8, right?

1.  If you have less than $6, then, you can't use two buys since the least amount of money you could spend on two cards is $6.

2.  But if you get $6 or $7, you'd prefer 1 Gold to 2 Silvers, so you don't want +buy there, either.

3.  So what about $9, where you could get 1 Gold + 1 Silver?  But you'd want to get a Province there.

So the only time you'd use your +buy is if you get $11, for 1 Province + 1 Silver.  Which, honestly, this deck is not likely to ever generate.

There's no harm in buying 1 Woodcutter, of course, and sure, maybe it enables that rare case.  But +buy is so un-useful in this deck that I don't think that you could ever make the case for buying 2 Woodcutters, no matter how low the (non-zero) chance of collision is.
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Rabid

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Re: Choosing the correct number of terminal actions
« Reply #26 on: September 06, 2011, 01:08:56 pm »
+1

Estate + Estate with $4?
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Epoch

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Re: Choosing the correct number of terminal actions
« Reply #27 on: September 06, 2011, 01:54:13 pm »
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Estate + Estate with $4?

...Yeah, okay, valid, as is Duchy + Estate with $7 in the late game.  But I still doubt that any chance of collision is worth the possibilities of using an additional +buy.
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AdamH

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Re: Choosing the correct number of terminal actions
« Reply #28 on: September 06, 2011, 03:08:11 pm »
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All right, at least I have an idea about Woodcutter -- I can run the %collision analysis to figure out whether or not your claim
"But I still doubt that any chance of collision is worth the possibilities of using an additional +buy." is true.

The same question still remains for the following terminal actions:
Bureaucrat
Militia
Thief
Witch
Workshop

I would think for each of the attacks, that "BMU + X attack cards" would be the best strategy. The more I think about it, the premise doesn't make much sense for Workshop, since it can only buy other action cards and not treasure or victory cards. I think at this point it makes sense for me to just run the same analysis I'm doing with Smithy on each of these cards and see what happens.

I've tried running a "stress test" with Smithy. The test involved the presence of a third player who played Militia almost every turn after turn 5. I got some results, and they sort of make sense, but I'm not happy with the way it was set up so maybe I can get some help...

Setup: 3-player game, with player 1 being the "Militia Spammer". This guy buys a Militia if he can, then after that he basically does BM. He ends up quickly buying all 10 Militias and plays one pretty much every turn.

The other two players are the ones really competing, since "Militia Spammer" rarely buys any VP cards, let alone provinces, though it does happen from time to time.

I should add that if you're only allowed to buy two Smithys, the best place to do so is when n = 16. It's worth noticing that the %collision value is exactly the same here as it was without the Militia Spammer present! This is because of the reduced hand size -- %collision = 33%.

The analysis goes as before, and I quickly find out that buying two Smithys as soon as possible is the best thing to do when you can buy more Smithys. The real question is when to buy the third Smithy. The %collision for buying the second Smithy is effectively 36%-40%, but this is really a minimum, since there is really no way to buy it with a higher %collision in this circumstance.

Using a similar analysis as before, I discovered the "optimal" time to buy the third Smithy was when n = 20. Attempts to make a 4-Smithy strategy didn't work out, since you either have to buy your 4th Smithy so early that it hurts you, or you never end up buying it because you're buying green cards.

In trying to compute the %collision for this third Smithy, my earlier method doesn't quite measure up. I have three different numbers I came up with: a lower bound, my approximation (2.5*the 2-Smithy %coll), and an upper bound. Those values in this case were 58%, 65%, 71%, which suggests that the %collision is around 65% here.

Remember, though, that the number we're interested in here is the difference between the old and new %collision values. With 2 Smithys already in the deck, our %collision was 26%, so the difference is around 39% -- roughly the difference from when we bought the second Smithy!

Problems: This is a 3-player game. There are 12 provinces to buy and effectively only 2 people buying them. Is there any way around this using the simulator, while still having a Militia Spammer present?

I'm still not thrilled with the way I calculate %collision with 3 terminal actions present. I think my approximation is OK but I will have to crunch some numbers to convince myself that it really is.

Anyways, it's a preliminary result: The baseline for Smithy is a %collision of 33%, which can go up to 40% in extreme circumstances.
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Epoch

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Re: Choosing the correct number of terminal actions
« Reply #29 on: September 06, 2011, 03:47:50 pm »
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The more I think about it, the premise doesn't make much sense for Workshop, since it can only buy other action cards and not treasure or victory cards.

That's not true.  Workshop can gain any card costing $4 or less.  You can use it to gain Silvers and Estates in an "only Workshop" board.  And other Workshops, of course.  And Coppers, if for some reason you wanted to.
« Last Edit: September 06, 2011, 03:53:58 pm by Epoch »
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AdamH

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Re: Choosing the correct number of terminal actions
« Reply #30 on: September 06, 2011, 11:29:59 pm »
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You are correct. And now I have no idea what the optimal strategy for workshop boards is... anyone?
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Jimmmmm

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Re: Choosing the correct number of terminal actions
« Reply #31 on: September 06, 2011, 11:35:46 pm »
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And Coppers, if for some reason you wanted to.

And Curses.
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AdamH

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Re: Choosing the correct number of terminal actions
« Reply #32 on: September 08, 2011, 10:53:48 am »
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I've updated the original post to reflect some of the discussion that's gone on so far. I've also added a concise version of all of the results to that post as well.

I ran a few more simulations on a couple of other terminal actions (Woodcutter and Council Room). The results are on the initial post, but I'll go into a little more detail on the process for Council Room here.

Testing circumstances were similar to the Smithy tests, but with a few more scenarios. It turns out that when your opponent is also using Council Room, your hand size is large enough that your %collision doesn't get low enough to justify buying a second CR. The actual numbers I got were put up against the "BMU + 3 Smithys : n=20" strategy. Win rates were actually fairly competitive between these two strategies, but CR proved to be a better buy when they were both up against the Militia Spammer.

I'll continue running some simulations, but I'm still looking for a few things:
- more Workshop strategies (or is BMU + Workshop the best one?)
- some more ideas for stress tests on some of the attack cards. The idea is to find a situation where playing the terminal action more often outweighs the risk of collision.
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DStu

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Re: Choosing the correct number of terminal actions
« Reply #33 on: September 08, 2011, 10:58:47 am »
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The best Workshop strategy should be Workshop->Gardens. Should be implemented in the simulator.
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AdamH

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Re: Choosing the correct number of terminal actions
« Reply #34 on: September 08, 2011, 11:10:16 am »
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All right, I suppose the best we'll do is a baseline of BMU + workshop (since Workshop -> Gardens depends on the presence of another card) and then Workshop->Gardens for the stress test...

The more I think about this, the less Workshop benefits from knowing about %collision. If I understand Workshop->Gardens correctly, %collision is completely a non-factor in that strategy, and BMU + Workshop just doesn't seem like a viable strategy. I guess it will be interesting to see if the simulations prove me wrong...
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Geronimoo

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Re: Choosing the correct number of terminal actions
« Reply #35 on: September 08, 2011, 11:21:06 am »
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Collision is a problem for the Gardens/Workshop, but that strategy needs to empty the Workshops to win, so that factor is more important than the occasional collision. If you toss in a few Fishing Villages the strategy empties 3 piles 2 turns faster...

Here's a sample:

Code: [Select]
<player name="COMBO - Workshop/Gardens + FV">
   <buy name="Gardens">
      <condition>
         <left type="countCardsInDeck" attribute="Workshop"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="8.0"/>
      </condition>
   </buy>
   <buy name="Fishing_Village">
      <condition>
         <left type="countCardsInDeck" attribute="Fishing_Village"/>
         <operator type="smallerThan" />
         <right type="countCardsInDeck" attribute="Workshop"/>
      </condition>
      <condition>
         <left type="countCardsInDeck" attribute="Fishing_Village"/>
         <operator type="smallerThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Workshop"/>
   <buy name="Estate"/>
   <buy name="Silver"/>
   <buy name="Copper"/>
</player>
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Epoch

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Re: Choosing the correct number of terminal actions
« Reply #36 on: September 08, 2011, 12:55:47 pm »
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The more I think about this, the less Workshop benefits from knowing about %collision. If I understand Workshop->Gardens correctly, %collision is completely a non-factor in that strategy, and BMU + Workshop just doesn't seem like a viable strategy. I guess it will be interesting to see if the simulations prove me wrong...

Well, I suspect that BMU + 1 Workshop might be okay.  You have less chance for a Gold earlier compared to pure BMU, but you'll end up a few Silvers up, and BMU pretty much likes Silver.  But it's obviously in a much lower tier of terminal action for the purposes of enabling BMU than pretty much any card drawer, or a decent attack.
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WanderingWinder

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Re: Choosing the correct number of terminal actions
« Reply #37 on: September 08, 2011, 01:16:25 pm »
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BMU should basically never buy a workshop. The extra silver doesn't do near enough to compensate for the missing tempo.

Epoch

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Re: Choosing the correct number of terminal actions
« Reply #38 on: September 08, 2011, 01:37:00 pm »
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BMU should basically never buy a workshop. The extra silver doesn't do near enough to compensate for the missing tempo.

Yeah, you're right.  I was thinking, "You close the Silver gap by the first reshuffle, and getting a Gold during the first reshuffle is pretty unlikely, so why not?"  But I wasn't counting in that Workshop makes one hand in each future reshuffle considerably worse.  If Workshop, like, self-trashed after a few uses, or had +1 card associated with it, it'd probably be fine, but BMU handily beats BMU + 1 Workshop.
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WanderingWinder

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Re: Choosing the correct number of terminal actions
« Reply #39 on: September 08, 2011, 01:44:42 pm »
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BMU should basically never buy a workshop. The extra silver doesn't do near enough to compensate for the missing tempo.

Yeah, you're right.  I was thinking, "You close the Silver gap by the first reshuffle, and getting a Gold during the first reshuffle is pretty unlikely, so why not?"  But I wasn't counting in that Workshop makes one hand in each future reshuffle considerably worse.  If Workshop, like, self-trashed after a few uses, or had +1 card associated with it, it'd probably be fine, but BMU handily beats BMU + 1 Workshop.
Actually, silver/silver gets a gold on first reshuffle over 42% of the time. Workshop/gold will get it about 8.8% of the time. So the difference is slightly more than 1/3 of games.

zorch

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Re: Choosing the correct number of terminal actions
« Reply #40 on: September 22, 2011, 10:17:12 am »
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Returning to the simulation that shows that the optimal time to buy a Second Smithy is n (decksize) = 22.
In a BM + Smithy game, with no extra buys, that corresponds to turn 18 (you have 22 cards after turn 17.  What this tells me is that you rarely want to buy that second smithy, in practice, since BM+Smithy buys it fourth provine at about turn 14.5 (about n= 19-20).

It's a useful metric -- inasmuch as it is another way to look at BM+single Smithy is better than BM+ 2 Smithy.  By the time that the second Smithy becomes useful, the game should be out of hand in favor of BM + single Smithy.

...and that is where the mathematics come into play, imho.  The "fact" of BMU making 4 provinces in about 17 turns, and BM+Smithy making it in 14.5 are the results of the math (ok... simulations).  It resulted in a rule of thumb.  A guideline.  If you stragey can't outbuy (in total, not just provinces) BM+Smith it is inferior to it.  What is your metric for that?  Well, discounting chips and the other complexities of the game it is that turn 14.5 benchmark.  It fits your intuition, surely, but now there is a number to your pain. 

Also, there are "big money plus Smithy plus curse games".  I assume that we can posit the thesis that in these games, when your deck gets to 22 cards, buying that second smithy will be worthwhile.

The concept of collision percentage is equally useful, if it can be turned into a similar guideline.  I will collide 26% of the time.  OK, so third f the time, I going to have this bad turn -- equivalent to a loss of tempo.  Is the upside worth it?  For which cards, under which types of boards?  The questions are fine.. but it is the decision making framework that comes from the rules of thumb, dervied from the concepts and simulations that add to the game.  Or, at least, that create structures and framworks that people can follow, and use.  The beenfit of this card is (say) 1.5 whatever that means) it will colide 35% of the time, thus I roughly break even by buying it.

I'll make an analogy to the game of Bridge.  It was very popular, even before the great bridge analasts of the day (1920's-30's ish, i fyou include auction bridge) and writers intorduced fairly straightforward (if somewhat flawed) and thus powerful ways to evaluate the playing strength of a bridge hand.  But, by their anaysis, these writers (in this case, Milton Work) introduced the idea of the point count, and this was a guideline -- something that a beginner, with a logical mind could grasp and actually use. 

Prior to that, high level hand evaluation was in the realm of the more creative mind.  Point count created a framewor that people could use to compare their hands strength to baseline expectations, and then act on it.

Mathematical constructs wont answer questions like how to prepare your deck to take advantage of menagerie, but it does answer questions like how fast must I be to beat BMU-Smithy.
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