## Armies and Roads (Or Maybe Not)

I’ve mentioned the *longest road* and *largest army* before on here, and undoubtedly anyone visiting this blog knows what they are – 2 points for owning the longest trading route of roads (and/or ships if we’re playing the Seafarers expansion) as well as 2 points for having the largest army of knights (from Development Cards). A couple of questions jump to mind:

- What is the main role of each of these outside of the 2 points they give?
- How useful are these two extra ways of gathering points?

The main role of road building is to develop settlements. The rules of Settlers state that two roads need to separate every connected settlement. Roads are necessary for any development past the original two settlements of each player. We can then say that roads are really the most fundamental way of expanding spatially (no roads are necessary to upgrade a settlement to a city). This is an inherit part of our overall strategy, as it is next to impossible to win with only the original settlement spaces (I suppose that one *could* do it by upgrading both to cities, obtaining the *largest army*, and purchasing their remaining 4 points in development cards, but that’s quite the challenge).

The main role of playing knights is to move the robber. According to the normal distribution of rolls (graph and explanation can be found here), the robber moves once in every 6 rolls (7’s are rolled 16.67% of the time). So, intervening in the natural movement of the robber is by no means a permanent fix. If I find myself plagued with the robber on my most productive hex, I can play a knight in order to move the robber, but he will most likely be back in around 6 rolls. Control over the robber also gives us some time to control the market price, as we have already talked about.

Taking this into consideration, I would claim that the road building process is more relevant to good strategy than the process of buying and playing knights. Does this mean that the *largest army* is worth less than the* longest road*? This brings us back to the second question raised above.

When talking about the utility of the *longest road* or *largest army*, we are given a very clear value of worth: 2 points. In a regular Settlers game, this is 20% of the total points needed to win (2 out of the total of 10 needed). This holds quite a bit of weight then. The absolute value of the *longest road* and the *largest army* are equal (we gain 2 points for each), but the relative value may not be.

Relative value takes more than just the points awarded into consideration. For example, since I have claimed that road building is more valuable than playing knights, I could in turn claim that the *longest road* has a higher relative value than the *largest army.* Another factor to consider is the availability of each. The *longest road* is attained by stringing together 5 or more roads and or ships. Assuming we start with our original road from the beginning of the game, we need to build 4 roads. This is equivalent to 4 wood resource cards and 4 brick resource cards, totaling 8 resource cards. To obtain the *largest army*, we need to buy 3 knight cards. This is equivalent to 3 sheep, 3 wheat, and 3 ore, totaling 9 cards. This is not much difference, especially when trying to compare through shifting value of each resource. But, when we look at the resources spent in retaining the *longest road* and the *largest army*, we can find some differences. If someone steals the *longest road*, it takes at least 2 roads to take it back – 1 road to tie, and another to overtake the opponent. This is an equivalent of 4 resource cards. Using the same logic for retaining the *largest army*, we see that we need to spend 6 resource cards to buy the 2 knights necessary. Also, as discussed in Utility of Development Cards, we are not certain of drawing a knight. In fact, we have about a 50% chance of drawing a knight. So, on average, we need to spend DOUBLE to obtain and retain the *largest army*. This is the most obvious disadvantage to trying to hold the *largest army*.

So, we can say that the *longest road* holds a lot more relative value than the *largest army*. Statistics reinforce this idea:

The winner holds the *longest road* about **48-53% of the time.**

The winner holds the *largest army* about **25-30% of the time.**

This is a much more accurate description of value than simply saying they are both worth the 2 points we receive from them.

Howdy, great post. One thing you forgot is that each time you play a knight you move the robber and then take a resource from an opponent. So yes it costs 3 resources for a knight, you do eventually get a resource back meaning only 2 resources were spent. The same as a road.

I recently looked into the value of development cards from a statistical point of view. You see, it’s not accurate to simply say you have a 50% chance of drawing a knight every time. Drawing development cards follows a hypergeometric distribution. Don’t worry about the name. All this means is that as development cards are drawn, the odds of drawing a specific card change. Excel has this formula. I calculated the percentage of acquiring 3 knights for largest army based off of how many cards one draws:

1 card: 0% obviously

2 cards: 0%

3 cards: 15.826% that you will have drawn three knights

4 cards: 39.57% that you will have drawn either three or four knights

5 cards: 62.17% that you will have drawn either three, four, or five knights

6 cards: 79.13% that you will have drawn either three, four, five, or six knights

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So on and so forth.

My point then boils down to this, if one wants a reasonable assurance of acquiring largest army, then one needs to purchase >=6 (arguably 5) development cards. Broken down: 6 Ore, 6 Wheat, 6 Sheep. 18 resources will give you a reasonable guarantee of 2 victory points. I probably shouldn’t use the word “guarantee” seeing as the largest army can be challenged. However, in my experience, I rarely see a largest army race between players.

Here’s the hidden gem though. Buying six development cards will get you a really good chance of obtaining 2 VP via largest army. What about the other cards that you draw that aren’t knights? I ran a similar analysis on victory point cards. If you buy six development cards, there is a 43.77% chance that one of those cards will be a victory point, and a 27.36% chance that two of those cards will be victory points.

In conclusion, those 18 resource cards that you spend to get largest army, will likely result in not only the two victory points from largest army, but also in 1-2 more victory point cards, or a total of 3-4 victory points. Of course, this doesn’t account for the benefit from on of those six cards likely being a road building, monopoly, or year of plenty card, or for the resources you get back from robbing people with the knight card.

I used to be the player that never touched development cards. I also am the player that usually loses with 8-9 victory points. I am considering experimenting with development cards more to see if that will help elevate my game.