Content warning: the following newsletter will only be of interest to those with familiarity with (and enthusiasm for) the board game Settlers of Catan. You cannot say you weren’t warned.
A colleague recently told me about his experience playing Settlers of Catan with some MBAs, and mentioned an ill-fated attempt to establish a commodity futures market. It makes some sense - Catan is, after all, a game about commodities where future abundance is unknown but drawn from a known distribution. Commodities play multiple roles there, both used in production and for trade. There are many rules, and one unwritten but nevertheless important principle:
No one wants your f*cking sheep.
Perhaps the most common market condition is a glut of sheep. Other resources are usually hard to trade for while sheep is always on offer at generous terms. In the extreme case, sheep may be traded in large numbers as a fungible currency. It strikes me that this appears to be a case of Gresham’s Law: bad money drives out good [Wikipedia]. But why?
It turns out that looking at the math of the game’s economy explains why: sheep are a commodity of relative abundance and the low use value. Just as Gresham’s Law predicts, sheep will become the most freely offered resource while more valuable resources are scarce on the market.
Abundance: A Field of Dreams
It is easy to see that sheep will be relatively prevalent in terms of supply. The global distribution of resources is fixed by the number of the tiles. There are four each of wood, sheep, and grain and three each of ore and brick.
We also have random frequency of resources, with those neat little buttons which mark which resources pay out on a given dice roll. Those are randomly assigned, and so it’s easy to figure out the relative distribution of the resources. There will be roughly 4 wood, sheep, and grain generated for every 3 ore and brick. You can do it mathematically but it’s simple enough to do a Monte Carlo simulation to figure out what the average yield per resource will be for a single dice roll:
Every roll of the dice, approximately 0.36 lumber, grain, and sheep will be added into the economy and 0.27 ore and brick. Every game is different, and the numbers per game will be different. So sheep is relatively abundant in general, though not uniquely so. The key to sheep’s lack of desirability isn’t just in it being more prevalent but in the unappealing uses to which it can be put.
Use Value: A Dog In Sheep’s Clothing
The use value of sheep is more difficult to determine, and highly dependent on assumption structures. I make a big one here, which is that the value of a given commodity is judged in terms of victory points (VPs)!1 Points are, after all, how one wins the game. So if a purchase is worth one VP, and requires a single resource, the use value of a resource in that purchase is one VP - if it requires two, it’d be half a VP, and so on.
So it’s easy enough to calculate the value of spending resources on towns and cities:
Roads are a little trickier. But at least in the early part of the game, roads represent an investment towards the coveted 2-point “Longest Road” achievement. So a resource spent there is 1/2 of 1/5 of a 2-point purchase:
Development cards are somewhat of a nightmare, in terms of assumptions. However, they can fit into this structure. There are many different development cards with different values, and so the expected value of a development card can be expressed as the sum of the probability of drawing a card type i times the value of card type i:
Some of these values are easy to figure out. A VP card is worth one VP. The value of a Knight card is similar to a road, as a contribution towards largest army. For the Year of Plenty card granting two resources, it seems reasonable to assume it has the value of the 2 times the highest use value on the board so far (0.4). The Monopoly card is hard to value as it’s dependent on board conditions, but I assume that it is of similar value to Year of Plenty.2 This sum comes out to a meager 0.755 points per development card, comparing poorly to a settlement or city, and thus we see that sheep lacks particularly high-value applications. This is particularly acute if the Largest Army card is already owned, as that sharply cuts the point potential of purchasing a development card.
Sheep is clearly less valuable than both grain and ore, and brick’s relative scarcity will make it more desirable even with comparably low use value. And yet, despite looking roughly comparable to lumber on pure use value and abundance, sheep still finds itself as the (forgive me) black sheep of the resource family.
Gresham’s Revenge
Lumber is similar in overall abundance, only two use cases, and low average value of deployment. Why does lumber generally not find itself in the cursed position of sheep? The answer here is not really about lumber or sheep, but about their complementarity requirements.
The reason why my chart of use value is incomplete is because realizing the use value of any resource is conditional on having other resources. Both lumber and sheep have a low-expected-value use case that is dependent on other rarer resources (roads and development cards respectively). However, using a sheep for a development card requires two other resources, one of which (ore) is relatively rare and both of which have a substantially higher use value in another application. So if we use the table above, we can say that using a grain and an ore for a development card has an opportunity cost of 0.8 (red) against a realized value of .5, whereas using a brick for a road only has an opportunity cost of .25 (green) against a realized value of .2.
The opportunity cost of purchasing a development card is very high! Those ore and grain are really more valuable when spent on a city.
Thus we have arrived at the conclusion every Catan player already knew: no one wants your sheep. The structural glut of sheep is why most games feature players desperately scrambling to offload their sheep in search of more useful resources, and a general unwillingness to accept them. Similar to the Turkish Lira or Argentinian Peso, sheep are available in abundance to anyone willing to take them.3 Ore plays the role of the US dollar in emerging markets, subject to hoarding and often disappearing from common market exchange. The Settlers have landed, and brought Gresham’s Law with them.
An alternative framework is that resources are often spent in search of future resources, which is true, but also it ultimately boils down to getting points.
You can mess with this value assumption a bit but the conclusions remain unchanged as there are only two Monopoly cards. Sorry, dev card bros (assuming such a thing exists).