Preface: I wrote an earlier article about this, but from the outset I wanted to run some simulations and return with a cleaner article. Here it is! I have an associated GitHub repository but it's attached to my real name so I'll only share over pm.Despite the great variation between different games of Dominion, there is one strategy concept that is particularly enduring: the dichotomy between decks that draw (here called "Engines"), and decks that do not (here called "Big Money"). In this article, I will not give strategy advice, but instead try to understand the dichotomy through the lens of physics. I will show that the line between engines and big money is a phase transition, much like the transitions between water, liquid and gas.
Understanding phase transitionsMost people are familiar with the phase transitions of water. At a certain temperature, water will freeze; at another temperature, water will boil. The phase of water also depends on pressure. We can draw a phase diagram which shows the state of water as a function of temperature and pressure.
Although water is the most familiar example, phase transitions occur everywhere in physics, and in pure math as well.
It is particularly informative to consider the
bond percolation model. Suppose that we have a network of pipes, with hubs arranged in a 2-dimension grid. Some adjacent hubs are connected by pipe, and others are not. If the pipes are arranged randomly, what is the probability that water will be able to flow from point A to point B, in the limit as A and B are infinitely far apart?
We have two possible scenarios:
- The hubs are broken up into a series of islands, each inaccessible from the others. Some islands might be very large, but they are still finite in size. As the distance between A and B goes to infinity, there is zero probability that they are connected on the same island.
- There is at least one infinitely large network of hubs. Although there will still be some disconnected islands, there is now a nonzero probability that neither A nor B is on an island, and they are connected to each other.
If you increase the density of pipes, there will be a sudden transition from case 1 to case 2. This is a phase transition. Although the model is based on an infinite grid of pipes, a phase transition still occurs even in a finite grid--it just won't be quite as sudden.
Phase transitions in DominionIn Dominion, we are not trying to traverse a network of pipes, we are trying to traverse our deck by drawing cards. As with the network of pipes, we will consider the limit as the deck is infinitely large.
The simplest deck that we can consider is a deck with just Coppers and Labs. In this deck, we can tune one parameter, the fraction of Labs (L). As a function of L, what is the expected payoff of a single turn?
Here's what the simulations show. For L < 1/2, the expected payoff is finite. For L > 1/2, the payoff is infinite (i.e. it gets larger and larger the longer the simulation runs). L = 1/2 is the phase transition.
At first, it would seem you can't do any better than infinity. However, what you can do is increase the probability of drawing infinitely on any given turn. I refer to this probability as
Reliability.
Interestingly, at exactly L = 1/2, the expected payoff is infinite, but the reliability is zero. However, this only applies to an infinite deck. In a finite deck, the phase transition is not as sharp, and the payoff is capped by the total payoff in our deck. The phase transition also occurs slightly earlier because we draw 5 cards for free.
The Village/Smithy Phase DiagramNow we'll consider an infinite deck with Villages, Smithies, and Copper. Now we have two tuning parameters, the fraction of Villages (V) and fraction of Smithies (S)--and the fraction of Copper is just 1-S-V. With two parameters, we can draw a two-dimensional phase diagram, just like the phase diagram of water.
For some values of S and V, the expected payoff is finite, which is shown with one color scale. For other values of S and V, there is a probability of infinite payoff; in these cases I show the reliability with a different color scale. The diagram is shaped like a triangle because the total number of Villages and Smithies must be less than the number of cards in the deck.
The diagram also shows that if you add a few Smithies, you can improve your deck, even in absence of any Villages. This is the "Big Money plus terminal draw" strategy. However, this strategy is not associated with any phase transition, and is fundamentally distinct from the Village/Smithy engine.
ConclusionsThis analysis shows how Engine and Big Money strategies are fundamentally different. It also shows why, if you go for an engine, you usually want to go all the way. Moving towards the engine phase transition has some opportunity cost, and sometimes, the opportunity cost is worth it to see that big increase in payoff that occurs at the phase transition. Sometimes, the opportunity cost is not worth it. Rarely is it best to move towards the phase transition, only to stop short.
This analysis applies to a wide variety of Engines in Dominion, but it is also interesting to consider exceptions. In sufficiently small decks, phase transitions become meaningless. There is no phase transition in draw-to-X decks, such as the one enabled by Minion, and City Quarter evades analysis as well. Some sifters actually generate multiple phase transitions, the first phase transition allowing you to draw through your deck, and the second allowing you to draw the cards you discarded.
Although a simple physics-based analysis cannot teach you how to play Dominion well, I hope that it has helped you appreciate some of its contours.
