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Subject: An Analysis of Early Game Strategy rss

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Matthew Calhoun
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A proper understanding of early game strategy is absolutely essential for optimal dominion play, for the obvious reason that your decisions of in the first rounds dictate the development of your deck for the rest of the game; not to mention that it is at this stage of the game when most players make their first mistakes.

In order to make good decisions in the early stages of the game, you have to first correctly analyze the board and figure out what the best deck type on that board is. To quote from my previous post,

The true secret of dominion strategy is familiarity with all of the possible strategies which exist amongst the current set of available cards, and to know the relative power of the different strategies against each other

So before you gain your first card, you have to know what the best strategy is for the board. Early game strategy is all about deciding what the correct order is to gain and trash cards, based on the assumption that you already know what type of deck you are going for. Each specific deck type requires a specific early game strategy to be properly built, and the priorities of what to gain early can be very different for different deck types.

A great way to think about a generalized game is to picture the set of all possible decisions made by the players as forming a tree (wiki: game theory). Dominion can be thought about as a tree, but the neat thing is that the players progress along the branches of this tree randomly based on the draws they receive throughout the game; the players can influence the probabilities of advancing along various branches of the tree, but ultimately there is a random factor which is ever present. Early game strategy is about controlling these probabilities to give yourself the best possible chance to go down a winning branch. It is possible to strongly control these probabilities in the early, mid, and late stages of the game (that's what a specific deck type is all about), but you have no chance of getting to the end stages of the game (VP race) quickly enough if you don't build your deck correctly in the early stages.

Let's consider the probability tree for the first six turns of a general game of dominion. The first branching takes place before the game even begins; a player can either be dealt a 3/4 or 2/5 money sequence on the first and second turns (you are ~16% to be dealt 2/5). If you gain one card on turn one and one card on turn two, then by the third and fourth turn the branching of the probability tree has become very large. You can either draw both cards you gained in the same hand (this will happen ~30% of the time, so you can see how important it is to plan you first and second turn buys in case you do draw them both in the same hand), or one of the gained cards in one hand and the other in the next, or one of them in one hand and not get the other on the next/previous turn, and about 1.5% of the time you will not get either or them on turn 3 or 4; and for each of those possibilities there is a distribution of possible turn 3/4 money draws to go along with it.

The cards you gain on the first and second turn serve a dual function in the early game, on the one hand, they must enable you to have the best possible turns 3/4 (this can be by a combination of gaining, trashing, and drawing cards), and on the other hand, they must be useful enough to be played twice by the sixth turn. (The cards you gain on the third and fourth turn will only be played at most once by the end of turn 6, whereas cards you gain on the first and second turns will be played at most twice by the end of turn 6.)

On turns 5/6 the branching of the probability tree has become truly enormous; assuming for the sake of argument that a player gains one card per turn, and does not trash or draw, then there are 4 gained cards in a 14 card deck at the beginning of turn 5, and there are many possibilities for turn 5/6 draws with combination's of those 4 cards and corresponding money sequences. Drawing any 2 of these 4 gained cards in the same hand on turns 5/6 is likely to happen; therefore its important that of all the possible two card combination's of the first four cards you gain, no two of them if drawn together could not both be played. By only gaining cards that all work well together if drawn in the same hand, you are avoiding opening yourself to branches of the probability tree where you have really bad turns.

By gaining cards with a similar function, you increase the probability of going down a particular branch. An important fact to know is that if you take two 2 money cards on the first two turns, there is ~90% chance of drawing 5 money on turns 3/4, and ~40% of drawing 6 money by turns 3/4. Since it is often correct to take two 2 money cards on the first two turns, a knowledge of the subsequent probabilities for drawing different money sequences on turns 3/4 is useful, because it allows you to plan your third turn buy with a knowledge of the probability of various fourth turn draws. This problem was solved by Ian Kelly in this thread:

The value of trashing and drawing action cards is not only that they give you on average better draws, but also that they accelerate the rate at which cards you gain can be played. This can be seen from the following table, which is made assuming a player gains one card per turn and does not trash or draw:

End of turn#, # of cards in deck before drawing, # of cards in discard pile
1, 5, 6
2, 0, 12
Reshuffle, 2 gained cards in the deck
3, 7, 6
4, 2, 12
Reshuffle, 4 gained cards in the deck
5, 9, 6
6, 4, 12
Resfhuffle, 6 gained cards in the deck
7, 11, 6
8, 6, 12
9, 1, 18
Reshuffle, 9 gained cards in the deck

As you can see, if by turn 8 you have trashed or drawn two or more cards, then you will reshuffle one turn earlier (giving you access to cards gained on turns 7/8 one turn earlier), also you see that turns 7/8/9 happen with three turns in a row with no reshuffle, whereas every turn before this the deck is reshuffled every two turns. A major difference between the early game and mid game is the number of turns in between reshuffles; in the early game you need to not only gain cards which will all work well together later in three and four turn sequences with no reshuffles, but also to take advantage of draw and trash cards to increase the quality of the cards you gain and the rate at which you get to play them (by reshuffling at a higher frequency), and therfore to open yourself up to branches of the probability tree in which certain lucky draws will give you an advantage over an equally skilled opponent. At the same time as doing all of this, you want to focus on gaining the cards which form the foundation of the deck strategy you are pursuing on the given board. The key is to gain the first several major and minor cards for your strategy in the best possible order by analyzing the various possible branches of the probability tree and choosing the order of buys/trashes for which you have a good plan for each of the subsequent possible draw sequences.

Finally, attack cards play an important role in the early game because disrupting your opponents in the early stages can hurt them for the entire game. Using militia to force your opponent to discard a non estate card, or giving them a curse with swindler/witch is one of the only ways you have of controlling your opponents path down the probability tree. If you use attack cards properly in the early game and your opponent does not, you often have a big advantage. Cards like swindler, militia, and witch are all very powerful because of the useful side ability which allows you to not only play the attack but also have excellent turns.

Please feel free to post any comments and/or contributions.
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