Sunday, May 21, 2017

Increasing utility

I played a couple of games of Castles of Mad King Ludwig over the past couple days and have been smashing my brain against the approximate value of the utility cards in the game.  For those that haven't played, Castles is a game where you build a castle out of a bunch of differently shaped and sized rooms that you fit together into something wild and crazy.  The rooms have a variety of properties like colour, shape, door number and location, and scoring bonuses or penalties.  Utility cards are cards each player gets in secret and they award points for various things.  You might get 3 bonus points for each size 400 room, for example, or 7 bonus points if you have a room of every colour.

Initially I thought the utility cards were worth about 3 points each.  This is important because you have opportunities to acquire more utility cards throughout the game and it is really useful to have a ballpark for what you can get out of them.  It is theoretically possible to get as much as 28 points out of a single card, but generally the high is around 12 with a minimum of 0.  In my game at Naked Man's place I got 28 points from my four cards and he questioned my estimate of 3 points per card, so I thought I would take a look at what utility cards are usually worth.

I took a castle that I finished a game with on Saturday that was pretty middle of the road and figured out what each utility card would have been worth if I had been awarded it at game end.  The high was 12, five of the cards were worth 0, and the average was 3.04 points per card.  Pretty good so far!

But there is more to it than that.

When you get new utility cards you get dealt two of them and you choose one to keep.  When you take the distribution of cards I got from Saturday's game and choose the best of two of them you get an average of 4.49 points per card instead.  So getting a fresh card on the last turn of the game is likely worth close to that value.  There are two cards that give a lot of points if you have fulfilled a difficult condition and if you happen to have a castle that has already done both of those things your expected value rockets up to 5.54 instead, and you know this before you make a play to get a card.  If you don't want to bother counting up your castle it is likely right to count a new utility card for 5 points, which is rather a large number.  It means that unless any other option has some really powerful benefit utility purchases on the last turn are likely to be top notch.  For certain if you feel like you are in a bad position in the game you should go for utility cards at game end to try to luck your way out.  If you bust and get no points you are still losing, but if you get lucky and hit a 10 or 12 point card it might eke out a victory.

However, that is only the easy part of the calculation.  The hard part is figuring out what a utility card is worth partway through the game.  At the start of the game you get three utility cards and you keep two of them.  These help guide your building by changing your evaluation of the tiles.  My initial impression was that early utility cards would be extra powerful because you can then tailor your builds towards them as the game progresses.  Knowing that I get bonus points for size 400 rooms means I can load up on them, right?

Not exactly.

The problem is that even though I know that I get bonuses for such rooms I often can't capitalize anyway.  Sometimes those rooms just don't come out or the person ahead of you is grabbing them all for unrelated reasons.  Even if you do get them you are usually giving something else up for them.  If I take a 4 point tile because I have a utility card that grants 3 bonus points on it I only really make the full 3 points if I pass up another 4 point tile.  Often I end up passing up a 6 point tile to get 4+3, which is still a benefit, but not nearly as large as the point values from the utility cards make it appear at endgame.  You usually lose something to alter your gameplan to maximize your utility cards.

The other reason that early utility cards aren't great is that your choice isn't terribly useful.  You might get offered a orange bonus card and a green bonus card, but since you haven't seen which tiles come out during the game your choice is often really just a guess.

But when you get a choice at the end of the game it is *much* better.  You can know which will give you more points for sure so you don't end up with a worthless card very often.  Utility cards at game start have higher potential but higher tradeoff and a far greater chance to totally whiff.

It is pretty easy to come up with a useful range for early utility cards even if it would take an enormous amount of data to get a really firm average.  Even if you don't bother to look at the card at all you have a floor of 3 points per card, as noted above.  That means that you definitely do better than that since you can alter your plan slightly to maximize your value there, at least some of the time.  I can't imagine you could exceed 5 points per card though as you would have to double its expected value while giving up only .5 points / card and from experience I can say that doing so isn't practical.  My experience and instinct tells me that the value is somewhere between 4 and 5.

I imagine that the optimal time to get a utility card is in the middle of the game.  By that point you can see whether or not you have a basement set up, which tiles you are already hunting for to finish off your big point rooms, and you know what right hand opponent is angling for so you can avoid that.  You also can see some of the tiles that will come out in the next turn so you can make some good guesses about what will be available to you later.

Combine that information with the flexibility to spend the last five turns of the game picking the tiles that work with your utility card and I feel like the midgame is the ideal place to farm up utility cards.

In any case talking about the ideal point in the game to take the utility card is kind of silly because if they fluctuate in value between 4 and 5 throughout the entire game there isn't much real sense in shifting your internal valuation as time passes.  Simply put, I think the utility cards in your initial hand where you have no information are worth about 4 points, but since you are simply handed those at the start you just take the best of what you have.  Once the game is going and you have a grasp of what is happening I think you should assume that utility cards are worth 5 points, but never forget that their high variability favours buying them when you are losing.

Tomorrow I will write about which utility cards are most valuable and discuss strategies based on which King's Favour pucks come out at the start of the game.


  1. Nice analysis! However, there's one step further to take it. The rooms that give Utility cards are usually 1 point rooms, which means that the actual expected value of the room is 1 + 4 or 5, or 5 or 6. This is low-to-average value for rooms, in my experience. So unless you get lucky with the draw, I don't think Utility rooms are quite as valuable as the main opinion of them is.

    Of course, most groups I play with push Utility rooms to the far left of the chart, assuming they can get 8 - 12K for them. So maybe it's less that Utility rooms are mediocre and more that most players vastly overestimate them.

    1. You definitely need to add the 5 to the value of the utility room - there are 9 of them, 3 each at point values 1, 2, 3. Consequently you can estimate them as being worth 6,7,8 points each. That is decent, but obviously not enormous, as there are plenty of tiles you can just buy without any sort of plan that make that many points when you drop them.

      I agree that paying 10 or 15 for a utility room is bonkers. They aren't worth that unless maybe the King's Favour pucks really shift the point value of them up. I think they are nice solid middle of the pack rooms, something I am happy to scoop up in the 1,2,4 slot and okay with 6, very rarely going to be willing to pay 8+.

      If your groups are all slamming Utility rooms to the 8-15 range... I think they are doing something very wrong. Perhaps I should play in those games for money. :)

      I have another pair of Castles posts lined up, one more about utility cards in particular (strategy based on Kings Favour pucks) and then another one discussing the value of paying more money for more points and the ratio you want to hit.

    2. I pay 8 or 10 for a utility room early and am beyond thrilled to do it. I think you're really underestimating how much you can sculpt a plan around them. Especially in combination with the blue bonus you can force your size buildings to come up and be worth even more extra points.

      I also think they're a fair bit worse at the end of the game, but maybe that's because I don't think you should expect to have an 'average' castle. You want a specialized castle! Maybe towards the global bonuses, maybe towards your personal bonus cards, maybe towards the piles that ran out quickly... But specialized in some way!

      I'm also not convinced there are a plethora of 8 point tiles just lying around? That's a pretty good dungeon tile's worth of points, and those don't go cheap.

    3. The more specialized your castle is the better the utility rooms are at the end. Having more 0s and more 10s when you get to pick the best of 2 is a big advantage, rather than 2s and 8s.

    4. If you're only looking at point ceiling, sure. But I'd rather look at a pair of 2s than a pair of 0s! Really, I don't want to look at either, which is why I don't think orange rooms are very good in late game.

      If you need to get lucky to win you'd rather have some 10s to hit. But I'd rather be in a winning position before we get to that point so I don't need to hit a 10 to win.

  2. Have you considered an analysis in terms of something like the Black-Scholes-Merton formula for option pricing?