Dominion strategy guide

Dominion strategy guide

While there are many different ways to play Dominion I will go over the idea of an expected value strategy. The idea behind this strategy is that in order to win the game you must buy a greater value of victory cards. This is completed by having enough money to beat the other person to the end goal of having more victory cards. As a result we are looking to maximize the expected value of each hand, in order for us to be able to purchase, near the end of the game, the highest amount of victory cards.

      At the beginning of the game we have seven coppers and three estates; this leads to an expected value of 3.5 money per hand. Eliminating one estate gives us an expected value of 3.88 money per turn, while increasing our deck to 11 cards and purchasing a silver gives us an expected value 4.09 money per hand. Under this framework we can begin to see how it is possible to evaluate various decisions based on their merit in regards to increasing our expected value per hand. While this is an all right way to evaluate non-action cards, action cards have to be given a fairly different evaluation even though their net effect is the same. Any card that does not contribute, i.e. is it says +1 action and +1 card, can be taken out of the denominator. That being said it does not account for cards that allow us to perform more actions. In order to figure out how much an action is worth, we can figure out the expected value of terminal cards and discount back the value of the action based on what the action produces when an action card is played. For example, let's say in hand we have a militia, a village, a smithy and two estates and our expected value for a single card equals 1.1. Based on this idea, we can effectively evaluate the smithy as worth 3.3. We can also valuate the militia at 2. If these are the only two terminal action cards in our deck, then the average terminal action is worth 2.6 ((3.3+2)/2).

     The next problem to consider is if the terminal action is worth X and it draws cards how do we evaluate it for the possibility of drawing another action card which cannot be played. In this case, we simply subtract all of the values from terminal action cards remaining in the deck from the deck's total value and then proceed to divide by the number of cards remaining in the deck. In this way we can effectively evaluate the possibility of drawing a card which cannot be played and compute the value of the cards drawn given the possibility that a card may be drawn which cannot be played. (23 money in cards remaining in the deck, 9 from terminal action cards and 14 from money. If there are 14 cards remaining in the deck, then we subtract out 9 from the terminal action cards to get to 14. Then we divide 14 by 14 to get expected value of 1 money per card drawn.)

     The next level in evaluating this method is when to switch the expected value of the deck. At the beginning the game we start with 10 cards and let's say our first two turns we buy a silver and a money lender. Let’s say on our first turn we bought a silver. Obviously we cannot add in the value of the silver to the remaining cards in our deck. In this regard, the deck shifts from one expected value to the next every time the discard pile is shuffled into our draw deck. Over the course the game we can see the expected value shift from at the beginning, a value of .7 money per card to .916 (11 total value (7 copper, 1 silver and money lender (worth 2 because of the necessity of trashing a copper to produce 3) equals 11 which is divided by 12 as we purchased two cards) and up and up from there. Expected value is therefore a system that has systematic regime change. While the expected value does not change instantaneously it does drastically change every time the discard pile shuffled back into the draw deck. This brings up an interesting factor where we can see that when action cards draw as part of their text, they effectively accelerate our deck towards the next regime change. Say that current value of the deck is 18 with 15 cards in the deck. After reshuffling we can expect to find a total value of the deck at 27 with 18 cards. Before reshuffling, the deck has an expected value of 1.2. After reshuffling the deck has an expected value of 1.5. The difference between the two values is .3. We then divide the difference by the number of cards in the deck (.3/15) to come up with a value of .02 which does not look like much. However when evaluated on a relative basis instead of absolute basis, we discover that it is somewhat more promising. For example, adding silver to a deck of 15 cards the total value of 18 increases the deck to a total value of 20 with 16 cards. The change in expected value for the deck before the silver and after the silver changes from 1.2 to 1.25 (18/15 and 20/16). This means that the expected value of buying a silver is .05. The expected value of drawing a card is .02 in regards to deck regime change. This effectively means that drawing three cards is better than buying silver when regarded on a holistic level. In this manner it can sometimes be seen that terminal action cards that draw, when there are no additional actions remaining, can still be better than a terminal action card that provides a specific value. This concept is called cycling or deck acceleration. We would like to get to the cards that we just purchased as quickly as possible and sometimes cycling just to get to the deck is highly beneficial, especially if the cards purchased with the current draw deck are much better than the cards already in the deck.

     Another factor that we have to consider in regards to dominion is that all players are fighting for a limited resource, the victory point cards and more specifically the provinces and or colonies. However since players are not immediately able to purchase the victory point cards, each player must build the deck in a way that the deck will be able to purchase the high cost victory point cards. In effect this means that we are in a race to purchase most of the province cards. However this does not mean we should be able to purchase a province every turn. Most likely if we are able to purchase a province every turn we have decided to purchase victory cards too late and our opponent will have an equal amount if not majority of victory point cards. This means that our deck must be able to burst at certain points in order to gain the victory point cards before our deck is sustainably able to do so. This is particularly interesting because the distribution of the amount of money we can expect to receive on any given term is not a normal distribution. Since we cannot have negative value money cards that we must play, our hand will initially take on a right skewed distribution after several turns of buying positive value cards:

                   _

      .2         /  \_

      .15     /        \_

      .1      /             \___

      .05  /                       \

             2 3 4 5 6 7 8 9 10 :Hand Value

    When we begin to see this type of distribution we know that it is time to start purchasing victory cards. The purchase of the victory cards will shift our distribution so that it more closely resembles a normal distribution. When not able to purchase victory cards while in the stage, the goal of our deck must be to increase the average value of the distribution and especially increase the probability of obtaining a hand value that lies in the right tail of the distribution. In effect, we wish to increase the likelihood that our deck will give us high hand values. While this is by no means a complete guide, it serves as a baseline for making logical, numeric decisions in the world of dome.

JP