Exploring Game Theory (Using Catan)
There is something incredibly satisfying about Game Theory. First off, much as the name implies, it can apply to games – tabletop, video, and the kinds of games we play with one another in everyday life. Secondly, it is such a simple mathematical theory that, when explored, most of the readers of this article will probably go “oooh yeah, that makes sense” and never play Catan in the same way again. It is so simple, so effective, and so geeky it makes the very nerdcore of my brain quiver with joy. You see, at its simplest, Game Theory is about winning games.
Over the past couple of years, I have had to explore Game Theory for one reason or another. From a professional perspective, it has had some use in my job. From a personal perspective, it is just such a cool theory to know that I have actively spent time looking at how it can be applied to tabletop and video games.
At this point, I feel like I should say a disclaimer. I don’t set out to wreck any games for anyone, so please use all the strategies I post with a pinch of salt. They won’t always work, and part of the fun is not always winning. It’s trying different things each time. Not to mention, we’ve all played games enough to know that the laws of mathematics don’t always hold up in such short bursts of probability and with so many variables. Or, to be more precise, they may hold up but the strategies do not. Now that has been said, let’s talk about Game Theory.
What is Game Theory?
The definition of Game Theory is not sexy to read about, so let’s just rip the band-aid off. Wikipedia defines Game Theory as:
“The study of mathematical models of conflict and cooperation between intelligent rational decision-makers.”
Mmm, mathematical models. Yum. Let’s try and put it in a way that is far nicer to understand: Game Theory is the theory of using mathematics to win games and win at life. After all, it was Galileo who said: “Mathematics is the language in which God has written the universe.” Religion or no, mathematics (“maths” or “math”, depending on your point of origin) is the backbone for a lot of things.
To put very basic Game Theory into context, let’s use an example. Here is a very simplified version of Catan.
What is a Working Example of Game Theory?
Okay, yes, it’s an impossible setup. I get that but imagine it isn’t (a Catan player just fainted somewhere, after fanning themselves and saying “sacrilege!”). Instead, imagine this is what you see and this is all that matters. Another player (blue) has placed on the wheaty six. The question is where, with you as the next player, do you place your first settlement and road?
The question is not an easy one. There are a few options. Logic dictated that a six is fairly likely in Catan (although what we said earlier about probability comes into its own) so going on one of those two spaces makes sense. The brick untouched, and a fairly decent resource to have early on in the game (although wheat is used more for building later on) so there is an interesting idea there.
Ultimately, you may decide to go for brick, or you may decide to go for grain. Most players would agree that the desert space is probably a bad move, but other than that does it really matter?
So does it matter? Well, yes, in Game Theory it does, and the above placement is the wrong one to do. You may be asking: aren’t all placements equal (obviously depending on the number), especially that early on in the game?
The answer, according to Game Theory, is no. The best placement is here –
So why is that the best placement? The answer is actually remarkably simple, although it does involve using Photoshop, so this is going to be an interesting experience for me (I have the design capability of a small teaspoon).
Firstly, it blocks off the wheat. The blue player cannot just circle the wheat and monopolise off it. Secondly, and more interestingly, that is the better position from a mathematical perspective.
The whole reasoning for this, within the world of Catan, is the viability of the longest road. If we look at the first placement from afar we can see the following split of the board.
What we can see is the blue player starts off with 2/5 of the board. The orange player starts off with 1/5 of the board. This is their starting real estate with the other player struggling to advance into their real estate space should they choose to. The space between them is 2/5 wide (I’m going by rows of hexagons), which they can both lay claim to. This space they split evenly. This results in the orange player starting with 2/5, with the blue starting with 3/5. Thus the blue player has the advantage, with more space to go for the longest road should they want it.
On the flip side of the coin, if the two players start beside each other, the board looks like this:
Without wanting to use Photoshop again, what we can now see is the blue player still has 2/5, but now so does the orange player. They are 2/5 apiece. They share 1/5. This gives them each control over 2.5/5 of the board – or half the board each. This, for those interested in the technical terminology, is a (kind of) version of a Nash Equilibrium.
Now, obviously, this isn’t how Catan works, so thank you for bearing with me, however, it is a good way of exploring Game Theory. With three players the principles are the same, the only thing that changes is each player is looking for at least 1/3 of the board. In four players, each player wants at least 1/4. You can see the benefit though. Game Theory can even up, and even win, a game, right from the outset (and not just with Catan but with other games too).
There is an everyday application to this, outside of gaming. Have you ever wondered why you see no coffee shops for miles, and then suddenly you come across a Starbucks, Costa, Nero, and other coffee shops within a few hundred yards of each other? The concept is exactly the same. Chain coffee shops do better business when they are closer to other chain coffee shops because they monopolise on more real estate. It makes perfect sense when you think about it. Think of it as keeping your friends close, but your enemies closer.
This whole idea is called Hotelling’s Model of Spacial Competition and is a basic concept in Game Theory. It is just one way in which data can be used to create a fantastic, mathematically apt, outcome to give you a benefit over competitors or competition. There is a fantastic video about this I have embedded below (by TedEd):
I would recommend you give it a watch.
Obviously, this is just scratching the surface of game theory; however, I would be interested to know what you think. Are you excited by mathematics or do you rather just play a game?
‘Are you excited by mathematics or do you rather just play a game?’ is an interesting question… assuming maths can help you win, then this is really questioning whether one plays to win or plays for fun I guess!
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As someone who ended up doing very little mathematics in high school, it surprises me how interesting I find it in regards to gaming. Hearing of Nash Equilibriums brings me back to my early university days where I was trying to be an economist.
I think game theory is really interesting overall, and I look forward to reading more about it on your blog. I almost feel as though the Model for Spacial Competition has some parallels with the Two Prisoner Dilemma, except that the outcomes aren’t worse for the people involved, but for the customers.
Want to blow your mind – dealing with Kim Jong-un using game theory.