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Author Topic: Simulating Challenge - Harem  (Read 5764 times)

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WanderingWinder

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Simulating Challenge - Harem
« on: August 08, 2011, 06:08:57 pm »
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I have a project going on to produce the best-for-the-simulator single-card strategies out of every individual card in Geronimoo's simulator (well, all the nontrivial ones. Obviously I'm not doing throne room, and gardens isn't very interesting by itself, etc.). The way I'm measuring this is either in comparison to any other strategy using only that card (the best one should win more than it loses against every other iteration) or by how big of a margin it beats Big Money by. Now one of the cards that's giving me fits is Harem. There are so many different ways of programming it, it's really difficult to compare them all. So I'd like to know what you guys can come up with. For the record, here's the big money I'm using (it's not totally optimized probably, as you could tweak the endgames real slightly, but it's pretty close given the options we have now).

Code: [Select]
<player name="Big Money">
   <buy name="Province">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Silver"/>
</player>

Davio

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Re: Simulating Challenge - Harem
« Reply #1 on: August 09, 2011, 03:43:06 am »
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This very simple Harem bot beats your BM about ~57% of the time:

Code: [Select]
<player name="BM Harem">
   <buy name="Province"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="3.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="1.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Silver"/>
</player>

It buys Harems after its first Gold and defers from buying Duchies a little longer than your bot.
It's in par with Theory's note about Harems that you probably need them sooner than you think you do.
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Geronimoo

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Re: Simulating Challenge - Harem
« Reply #2 on: August 09, 2011, 04:19:44 am »
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53-42 against Davio's and 64-32 against BM:
Code: [Select]
<player name="BM Harem (Geronimoo)">
   <buy name="Province">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="3.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Silver"/>
</player>

So you need Harems even sooner! After 1 Gold actually.
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DG

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Re: Simulating Challenge - Harem
« Reply #3 on: August 09, 2011, 08:42:03 am »
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Remember to put the harem purchase ahead of the estate purchase!
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DsnowMan

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Re: Simulating Challenge - Harem
« Reply #4 on: August 09, 2011, 10:39:36 am »
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Davio's is buying a Harem after 2 golds (and buys estates before harem's...)
Looks like 1 gold is way to go, but more testing?!
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Superdad

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Re: Simulating Challenge - Harem
« Reply #5 on: August 09, 2011, 10:53:47 am »
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I think this is a perfect case of Paper Rock Scissors. Just from a high-level perspective I don't see how it can be any different.

I have a feeling that we may start out at Case A, then develop Case B that beats Case A. Then Case C that beats Case B. Then we conclude that Case C is optimal, but infact, Case A crushes Case C.

That's my prediction with Harem.

I think that's going to happen in any situation where choosing between two cards (in this case, it's Gold vs Harem) is a sacrifice of buying power for current VP lead.

Then again, it may also be the case that the game will end early enough that the Harem-buying deck can end the game before it is punished for reducing it's buying power for the earlier VP lead. i.e. perhaps we'd see a more interesting Paper/Rock/Scissors playing out if there were 10 provinces in a 2-player game instead of 8.
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #6 on: August 09, 2011, 11:19:09 am »
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Superdad, what do you think A, B, and C are here?

Superdad

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Re: Simulating Challenge - Harem
« Reply #7 on: August 09, 2011, 11:44:35 am »
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Varying strategies for timing of when to hop into Harem. I.e. varying levels of VP curve/Buying Curve trade-offs.

For example, just for argument sake:

Case A: Buy 2 golds then buy harems with $6.
Case B: Buy 1 gold then buy harems with $6.
Case C: Buy harems with $6 always.


Now, without a doubt, the buying power curves and Vp per turn curves of these three cases are different.

Case A has a more exponential-shaped buying power curve and VP curve.
Case C has a more linear buying power curve, and VP point curve.
Case B is in the middle.

The winner in these games will be determined by the length of the game.

What I'm trying to articulate here is that without a doubt, trading buying power for current VP is what is going on here, and if you graphed these two curves, the linear curve starts out ahead, but the exponential curve surpasses it. When? I'm not sure. It may pass it too late, and thus never win (in province games).

So maybe for province games, there may not exist paper-rock scissors, maybe it does. But, if paper rock scissors doesn't exist for Province games, simply due to the length of the game being short enough that perhaps Case C always wins. But I'm fairly certain that Paper/Rock/Scissors will exist for Colony games.


« Last Edit: August 09, 2011, 11:53:23 am by Superdad »
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timchen

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Re: Simulating Challenge - Harem
« Reply #8 on: August 09, 2011, 01:21:31 pm »
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Not necessary true. With a fixed number of turns this is certainly untrue. The question is whether the ending condition creates this "Nash equilibrium": suppose the strategies are fixed in the start, independent of what the opponent is doing.

I think the problem is all these strategies can be more or less described by a single tuning parameter. In this case, imagine a region in parameter space for each strategy which it beats. at the two extreme ends the strategies are lousy and all strategy in between beats them. Consider moving the strategy from one end. The region it cannot beat is somewhere in the middle; as we continuously move toward the middle, that region can move, shrink, or expand. But it cannot move backwards. Since the procedure is continuous, that region cannot jump. In the end, what we have is some optimum strategy in the middle.

Pretty near the middle it is possible for 3 very similar strategy to beat each other, but that is due to the intrinsic discrete nature of the strategy. Not exactly similar to the rock-paper-scissors scenario.
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #9 on: August 09, 2011, 01:42:17 pm »
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I tried your suggestion C in several different permutations, and no matter what I do, it always gets crushed like at least 2-1 by B, which is what it's supposed to beat, right?

Here's the best thing I've been able to come up with so far:
Code: [Select]
<player name="Harem">
   <buy name="Province">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="getTotalMoney"/>
         <operator type="greaterOrEqualThan" />
         <right type="constant" attribute="11.0"/>
         <extra_operation type="multiplyWith" attribute="2.0" />
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="5.0"/>
      </condition>
   </buy>
   <buy name="Silver"/>
</player>

Razzishi

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Re: Simulating Challenge - Harem
« Reply #10 on: August 09, 2011, 10:23:01 pm »
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I tried your suggestion C in several different permutations, and no matter what I do, it always gets crushed like at least 2-1 by B, which is what it's supposed to beat, right?

I'm pretty sure those were examples, and not meant as actual strategies that he had tried and was reporting his results for.
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #11 on: August 09, 2011, 10:42:02 pm »
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Well, if you only read his second post, it sorta seems that way. But this line from his first post
Quote from: Superdad
I think this is a perfect case of Paper Rock Scissors. Just from a high-level perspective I don't see how it can be any different.
seems to imply he actually thinks this is the case here...

HiveMindEmulator

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Re: Simulating Challenge - Harem
« Reply #12 on: August 11, 2011, 03:09:55 pm »
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I think harem is a hard beast to handle with the simulator, since purchase of harems over golds probably could depend on not only how many golds you have, but the score and the count or even parity of remaining provinces. It's infinitely worse than gold when it leads to a 5-3 province split and better if you can still manage 4-4. I suspect you should get duchies earlier in first position and later in second position.

That said, I have a simple strategy that beats geronimoo's 50-46:
Code: [Select]
<player name="BM into harem">
   <buy name="Province"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerThan" />
         <right type="constant" attribute="8.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="5.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Silver"/>
</player>
It buys harems after 2 golds OR after someone buys a province and prefers duchy to gold at 5 provinces left and duchy to harem at 2 provinces left. I haven't really optimized this, but the idea is that the condition for buying harem should be multi-fold.
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #13 on: August 11, 2011, 03:50:00 pm »
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HiveMindEmulator, your bot loses to mine 48/47 after 50,000 games simulated. I wonder if you could get it better with some fine-tuning?

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Re: Simulating Challenge - Harem
« Reply #14 on: August 11, 2011, 05:04:27 pm »
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^I didn't realize you had already posted a strategy with multiple harem-purchase criteria. You can (marginally) improve yours by moving up the money-based harem purchase criterion.
Code: [Select]
<player name="Harem+">
   <buy name="Province">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="getTotalMoney"/>
         <operator type="greaterOrEqualThan" />
         <right type="constant" attribute="11.0"/>
         <extra_operation type="multiplyWith" attribute="2.0" />
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="5.0"/>
      </condition>
   </buy>
   <buy name="Silver"/>
</player>
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #15 on: August 11, 2011, 05:29:55 pm »
0

^I didn't realize you had already posted a strategy with multiple harem-purchase criteria. You can (marginally) improve yours by moving up the money-based harem purchase criterion.
Code: [Select]
<player name="Harem+">
   <buy name="Province">
      <condition>
         <left type="countCardsInDeck" attribute="Gold"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="0.0"/>
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="getTotalMoney"/>
         <operator type="greaterOrEqualThan" />
         <right type="constant" attribute="11.0"/>
         <extra_operation type="multiplyWith" attribute="2.0" />
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="5.0"/>
      </condition>
   </buy>
   <buy name="Silver"/>
</player>
40,000 games in, I'm seeing 47-47-6. Also, it shouldn't make any difference whatsoever, logically speaking. Neither of the two lines between the harem buy conditions should ever come into play when you have $6+ and harems left in the supply, as in order for their few-enough-provinces condition to be met, the harem buy above them would have kicked in.
On the other hand, switching the buy condition on province away from needing a gold to needing at least 23 money in the deck improves the bot by a bit over 1%.
Code: [Select]
<player name="Harem">
   <buy name="Province">
      <condition>
         <left type="getTotalMoney"/>
         <operator type="greaterThan" />
         <right type="constant" attribute="11.0"/>
         <extra_operation type="multiplyWith" attribute="2.0" />
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Harem">
      <condition>
         <left type="getTotalMoney"/>
         <operator type="greaterOrEqualThan" />
         <right type="constant" attribute="11.0"/>
         <extra_operation type="multiplyWith" attribute="2.0" />
      </condition>
   </buy>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="5.0"/>
      </condition>
   </buy>
   <buy name="Silver"/>
</player>

Superdad

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Re: Simulating Challenge - Harem
« Reply #16 on: August 12, 2011, 03:12:49 pm »
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Actually, it was a busy day at work and I think I was a bit mixed up. I was picturing something high-level, where a different strategy would lose to another strategy (and each option would lose to another one).

Infact, the only thing that matters here is the game-length - that is the only true variable. Once this is decided, only one discrete answer should win.

High-level you can visualize each deck graphed, with the Y-axis being current VP points, X axis is game-turns. Imagine three cases, resulting in different trade-offs for buying power vs Current VP. The three graphs are:

1) Root, ex Y = squareroot of X
2) Linear, say Y = x + 0.000001
3) Square, say Y = X^2

Before X = 1, The Case A wins.
After X = 1, Case C wins
Right on, or very near X = 1, Case B wins.

However, generally there is only 1 winner.


Perhaps the paper/rock/scissors effect only happens when the decks can interact with eachother. And since there is no interaction here, it is actually impossible for paper/rock/scissors to occur?



I actually think the thing I was thinking of is that the results of this simulation will only be valid for the assumptions made in the case to be studied. I.e. Province vs Colony games (as that is really the only variable in terms of game-length parameters).
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #17 on: August 12, 2011, 04:10:09 pm »
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No, I totally agree that there are lots of instances for rock-paper-scissors, at least on broad strategies, to occur. I just didn't really think they would here. Also, I think in many of the RPS scenarios, there's probably some kind of equilibrium which requires you to be as flexible as possible for as long as possible

chwhite

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Re: Simulating Challenge - Harem
« Reply #18 on: August 12, 2011, 04:31:07 pm »
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Imagine the following board:

Ambassador
Horse Traders
Gardens
Vault  (Envoy/Smithy would work here too)

and some other stuff.  Ambassador should beat Vault-Big Money; Vault-Big Money should beat Horse Traders + Gardens, but Horse Traders + Gardens should beat Ambassador.

That's the clearest example of rock-paper-scissors I can think of off the top of my head.
« Last Edit: August 12, 2011, 04:33:29 pm by chwhite »
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ARTjoMS

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Re: Simulating Challenge - Harem
« Reply #19 on: August 12, 2011, 06:47:16 pm »
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Ambassador should beat Vault-Big Money
No, it shouldn't.
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Fangz

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Re: Simulating Challenge - Harem
« Reply #20 on: August 14, 2011, 12:46:39 pm »
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Yeah I think Ambassador might beat smithy BM or something but vault is specifically strong against deck junking - everytime you draw the vault with 2 silvers or a gold, that's a province, and every time you don't, that's a gold, and I don't think ambassador can really stop that.
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chwhite

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Re: Simulating Challenge - Harem
« Reply #21 on: August 15, 2011, 01:31:21 pm »
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Yeah I think Ambassador might beat smithy BM or something but vault is specifically strong against deck junking - everytime you draw the vault with 2 silvers or a gold, that's a province, and every time you don't, that's a gold, and I don't think ambassador can really stop that.

Fair enough, Ambassador by itself is probably too slow for Vault (though not Envoy/Smithy).  However, I suspect Ambassador with support (villages to play multiple Ambassadors and start passing Curses; some other source of virtual money that likes a thin deck, such as Peddler or Conspirator) would probably overpower even Vault.
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HiveMindEmulator

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Re: Simulating Challenge - Harem
« Reply #22 on: August 15, 2011, 05:22:19 pm »
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We're well off the topic here, but I don't think ambassador is good against money-heavy strategies in general (even smithy big money) unless there are alternative sources of VPs to allow you to really drag the game out. Handing out coppers is simply not disruptive enough to those types of strategies as they race to provinces, compared to the amount you slow yourself down spending turns to trash 1-2 cards at a time. In a battle of action chains, ambassador usually beats any other opening, but money strategies are precisely what ambassador openings are bad against.
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fp

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Re: Simulating Challenge - Harem
« Reply #23 on: August 22, 2011, 04:28:07 am »
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...
Quote
Case A has a more exponential-shaped buying power curve and VP curve.
Case C has a more linear buying power curve, and VP point curve.
Case B is in the middle.

If you actually model Dominion with a differential equation and solve said differential equation, buying curves (meaning the amount of $ you have, on average to spend that turn) are, at best t^{2/3}- which is not only not exponential and sublinear, but concave down.

Quote
The winner in these games will be determined by the length of the game.

This is, with no doubt, the key to unlocking Harem's power.

Quote
What I'm trying to articulate here is that without a doubt, trading buying power for current VP is what is going on here, and if you graphed these two curves, the linear curve starts out ahead, but the exponential curve surpasses it. When? I'm not sure. It may pass it too late, and thus never win (in province games).

So maybe for province games, there may not exist paper-rock scissors, maybe it does. But, if paper rock scissors doesn't exist for Province games, simply due to the length of the game being short enough that perhaps Case C always wins. But I'm fairly certain that Paper/Rock/Scissors will exist for Colony games.

I believe that one of the fundamental flaws in many player's analysis is that they are assuming that the optimal Harem/Gold buying ratio exists. And I am not so sure one does. As superdad points out the power of a Harem strategy will depend upon the length of the game, but a large part of that "length" is determined by the parody of the piles.

In my longer analysis, which I will post once I fix all the mathematical errors (which, as mentioned above includes differential equations), it turns out that, on an average basis Harem performs equally with Gold. Specifically, given a fixed amount of time, if you were to draw an "average" hand every turn, then the Gold and Harem strategies can yield the same number of points in that period of time. The key, however, is that Dominion is not a fixed time game, and that a player buying a Harem (instead of a Gold, say), is lengthing the game. A Gold player can then lengthen the game buy buying a Gold (instead of a first Province). In turn, this gives the Gold player more power buy more provinces later in the game. If the Gold player buys 5 provinces, with high likely hood, that would be enough to overcome the Harem player's bonus. In particular, lengthening the game by buying treasures over Provinces may not benefit either player in general, and may be determined by the exact situation- including the "oddness" and "evenness" of each pile. Specifically, I think Harem is way too chaotic than it seems- especially if both players may have strategies regarding the Harem.
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DG

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Re: Simulating Challenge - Harem
« Reply #24 on: August 22, 2011, 09:28:53 am »
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Quote
if you were to draw an "average" hand every turn, then the Gold and Harem strategies can yield the same number of points in that period of time.

An important feature of gold is the added variance it gives to draws. Assuming that many provinces are bought with an average draw below 8 each turn, in those situations the variation from gold will provide more province buying hands. Buying a province and estate is always better than buying two duchies, but it's especially important when considering the last province in the supply since it gives the option to close a winning position.
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WanderingWinder

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Re: Simulating Challenge - Harem
« Reply #25 on: August 22, 2011, 09:52:26 am »
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fp: I'm skeptical, but I am eagerly looking forward to your analysis and will withhold comment until I see it.

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Re: Simulating Challenge - Harem
« Reply #26 on: August 25, 2011, 02:16:16 am »
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WW, I don't want to disappoint you, but I have been sort of stuck.

What the analysis says, pretty much, is that Harems versus Golds yield the same result. However, this is caused by a few nonsensical assumptions:

1) You always draw an "average" hand.
2) Cards you buy mix instantaneously
3) The game is continuous (specifically, you can use "half-a-turn" to buy "half-a-province", etc.)
4) The PPP does not apply.

2 is a unharmful assumption since no strategy benefits from it. However, clearly 1, 3, and 4 are assumptions that greatly hurt the Harem strategy for three reasons:

1) You never draw a "random" hand. Buying extra Harems for a slightly longer period of time induces less variance in the deck whereas buying Golds for a shorter amount of time induces more variance.
3) The continuousness of the game allows a Gold player to use "one-third-of-a-turn" to buy "one-third-of-a-province" which would make up one Harem. Specifically, the all or nothing affair makes catching up harder
4) The Harem player has an advantage if going into the PPP phase of the game by the nature that he or she "starts" with more points.

Things that I will consider, but they will take me a lot of time:

1) How to model the variance of a hand (or shuffle) and how that affects what one can buy. Specifically, having an average above $1.6 does not guarantee the ability to buy a province, and nor does $1.8 or $2.0. However, the higher, the more likely you are able to buy a province... based upon the variance of your deck.

3) Using a differential equation (e.g., continuous model), makes things easier. Using a discrete recurrence relation might be better, but they are often much more painful to solve and manipulate.

4) I don't know. I think the next simulation challenge should be encoding a successful PPP. The PPP ultimately decides many games, that will ultimately be the key for a good simulator.
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