lederhosen: (Default)
[personal profile] lederhosen
One of the things I like about Settlers/Cities & Knights of Catan is that dealing with randomness is a big part of the strategy (as opposed to things like Fluxx, where you can't really plan ahead past the next turn or so). I'm quite fond of prob-theory, so that works nicely for me.


The obvious application is identifying which building sites are likely to give you a good resource income, but that's pretty easy - especially since the numbers are marked to show how likely they are to come up. The next step is to look at the variance of that income.

Scenario: Your only wheat production is a settlement on a 4, which comes up 3 times in 36 rolls. You're not getting enough wheat to keep your one and only knight activated, so you want to expand. All else being equal, would you rather expand to a new wheat hex that also comes up on 4s, or one that comes up on 11s?

The simple answer is that 11s only come up twice in 36 rolls, so expanding to the 4 will give you better wheat income... on average. But that income will be patchy. Most of the time you'll get none; occasionally you'll get two. If you don't want to be done over by a run of lean years, you'll need to keep a reserve of wheat in your hand, and the more you hold the more risk you have of losing it.

Expanding to the 11 gives you slightly less wheat, but it makes that income more dependable, which may be a Good Thing.

So, is variance always a bad thing? Not at all. There are several cases where a high-variance placement is good:

- If you're trying to build cities, you'll want masses of iron and wheat. Building them up gradually puts you at a high risk of losing your precious resources to other players, or the robber. A high-variance placement gives you a better chance of picking up all your metal (or wheat) in a short burst, letting you spend it before anybody can take it off you. I guess the same consideration applies to clay for city walls, but they're not a big part of my strategy (largely because I try to keep my hand as small as possible.)

- If you're relying on trading, the same consideration applies. When you get pairs of the same resource in one roll, you've got a much better chance of trading them away before you lose them.

- If you're losing, a high-variance strategy is good, because it puts you in a position where good luck could get you ahead. This is also important when you have several opponents - high variance increases your chances of coming first (although it also increases your risk of coming last...)

After variance, the next thing to consider is covariance: what events are likely to happen together?

Scenario: Placing your first settlement, you have to choose between two spots. They're equally good, except that one gives wood on a 5 and clay on a 9; the other gives both wood and clay on 5. Which should you choose?

If you just look at the average production, these two options are equally good - 5s and 9s both come up 4 in 36. But in fact, the second option is MUCH better, because it delivers wood and clay together - and you almost always spend these two resources together, whether it's roads or settlements.* If you take the first option, you need a 5 and a 9 to come up before you can use the proceeds; the second option only needs a 5.

This applies even more to metal and wheat; it takes five of them to build a city, and you really don't want to be holding on to four cards turn after turn. Better to have none for a while, and then get several at once.

Scenario: Your opponent has one city that produces cloth on an 8. You already have a city that gets cloth on a 4. You want to get the metropolis by beating your opponent in cloth production (at the moment you're running neck and neck) so you're planning to build another city. All else being equal, should you take an option that produces cloth on a 6, or one that gives cloth on an 8?

Again, the average production is the same for both. Both options give you the same variance in production. If you take the 6, odds are that you will out-produce your opponent, because on average the 6 matches their 8, and the extra production from your 4 should give you the edge. But averages don't always eventuate; it's still possible to get a run of 8s, and that will really ruin your day.

You're much better off taking the 8, because this locks a lot of the chance in your production in with theirs. On a run of 6s, neither of you get production; on a run of 8s, both of you get some. Eliminating most of the randomness from this competition means that your edge - the extra production on 4s - becomes more important, giving you a better chance of winning. If possible, you want to pick the same 8 as your opponent, to lock production even closer to theirs (by avoiding the situation where a robber blocks your production but not theirs).

Again, if you're losing, you probably want to play the reverse strategy: pick different numbers so a streak of lucky rolls can make a difference. You're still not guaranteed to win, but at least it improves your chances.
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