Monday, March 12, 2007

Game #4: Road Construction in Catan and the Value of Wheat

Another five-player game with Cinderella's family yesterday. 47 wood, 40 brick, 52 wheat, 75 sheep, 67 ore, and 21 discovery counters. (Once again we used jungles.)

Three times, players gave up cards when seven was rolled: 2 wood, a brick, 6 wheat, 2 sheep, and an ore. CindyBro and CindySis both used "Year of Plenty" cards to get a wood, 2 brick, and a wheat. CindyBro and Cinderella both had sheep ports, which came in handy in a very sheep-laden game, using them seven times for brick, thrice for wheat and ore, and twice for wood. CindySis used the ore port once, to get wood. Other trades with the bank were 4:1; ore for wheat twice, as well as ore for brick, ore for wood, and sheep for wood. No wood, no brick, 4 wheat, 34 sheep, and 18 ore were exchanged for 5 wood, 9 brick, 5 wheat, no sheep, and 3 ore. Including the "7" discards and the "YoP"s, that's 2 wood, 1 brick, 10 wheat, 36 sheep, and 19 ore surrendered, and 6 wood, 11 brick, 6 wheat, no sheep, and 3 ore obtained.

When a mixed group of counters and resources were exchanged for development cards, it was most common that the counters were used to replace wheat. In eight mixed purchases, seven counters were used for wheat, four for ore, and one for sheep. I still haven't figured out how to merge this data with the trading model I espouse, but it's a very good way to compare the relative values of wheat, sheep, and ore. Right now, I suspect that it is a mix of using the counters as substitutes for the high-value resources--usually wheat, and sometimes ore--and seeing an opportunity to obtain value while effectively discarding disposable sheep. (Four of the eight mixed purchases in this game were a sheep and two counters, and three were a counter, a sheep, and an ore. The remaining purchase was using the counter as a sheep, which is quite an outlier.)

The trading model indicates that in this game, brick was the most valuable resource, followed closely by wood. Three of the players spent the mid- to late game racing for the Longest Road bonus; CindyBro used up all 15 of his road pieces, and Cinderella, who won the race, used nearly all. CindySis would have used more than 11, but she no longer had room to catch back up with Cindy's road, so she stopped. CindyDad and I pursued a different strategies, though. I was built near both jungles, so I raised the Largest Army (eight soldiers), but had little success drawing other development cards--the only other type I drew was a single Longest Road. CindyDad tried to build in a balanced manner, but didn't build across the board like his children.

Cinderella proved victorious, with the Longest Road, a VP card, 2 cities, and 3 settlements. I had a chance to win, but drew too many soldiers, finishing with Largest Army, 2 cities, and 3 settlements. CindySis had 4 cities, CindyDad had 2 cities and 3 settlements, and CindyBro had 2 cities and 2 settlements. CindyBro was the first to get Longest Road, which put him ahead on points and drew the robber to the rich pasture that provided so many sheep for his port.

My problem: I wasn't able to build on a great wheat field early in the game. Instead, I chose to get a spot on the second jungle. This helped me buy development cards, but turning settlements into cities proved expensive, and slowed my economic growth. Cinderella, who, like her brother, used a "Sheep-O-Matic" strategy, was successful by having more than one pasture to rely on, and looking sufficiently non-threatening enough through the early game to keep from gaining too much attention. While CindyBro was leading, others were generally more willing to use her port to obtain needed resources, which prevented him from collecting the "user fee" that makes oft-used ports (especially the sheep port) valuable. This helped her catch up and build on more pastures, increasing her own flexibility. Lessons: in a five-player game, being able to build roads is important, as it allows a player to expand to other desirable locations, which are in abundance in such a game. Thus, brick and wood were the two most desirable resources. However, wheat is still important, and if pursuing a card-building or city-building strategy, it is the most important resource to have in abundance, as it is always valuable.

Sunday, March 04, 2007

Data and Observations from Games 1-3: Development Cards Are Nice.

Game #1
In Thursday night’s game of The Settlers of Catan, played with Cinderella, FD&C Yellow Dye #4, and Eto Camoe, I recorded the resources produced by the board, as well as the trades made with the bank, discarded resources from 7 being rolled, cards taken from the bank on “Year of Plenty” cards, and the cards used in concert with Discovery counters obtained from the jungle.* The resources produced were 48 wood, 14 brick, 53 wheat, 47 sheep, 31 ore, and 5 discovery counters. Only once did a player (me) have to discard resources on a seven; I threw away 3 wood and a sheep. The “Year of Plenty” was used to produce two ore, which were used on that turn to construct a city—a typical use of a YoP. The discovery counters were used alone once, then two were used with a wheat once.

Without a port, trading with the bank requires trading four resources of one type for one of another. This steep cost means that many players are unable or unwilling to make such a trade. However, three times, wood was traded for brick, and once each was wood traded for sheep, wheat for brick, and wheat for sheep. The four “generic” ports allow 3:1 trading, and it was used once to trade brick for ore. (Notably, this came immediately after Cinderella used a “Monopoly” card to take everyone’s brick. Otherwise, giving up three bricks, even to get a valuable ore, would not have made sense.) Resource-specific ports played a very large role in the game; unsurprising in a game with so little brick and ore. Yellow Dye #4 started play with the “Sheep-O-Matic” strategy; he had settlements next to two great pastures: 6 and 8. Obviously, having the sheep port helped YD4 greatly. Trading two sheep for another resource happened often: four times each for brick and ore, twice for wheat, and once for wood. Cinderella soon built the wheat port, which she used thrice for Brick, and once each for wood, ore, and sheep. I built the ore port, but never used it. YD4 also built the wood port, and used it to get ore three times. In total, 22 wood, 3 brick, 20 wheat, 22 sheep, and no ore were traded to the bank for 2 wood, 11 brick, 2 wheat, 3 sheep, and 9 ore.

When combined with the previous instances, this sums to 25 wood, 3 brick, 21 wheat, 23 sheep, and 0 ore were sent to the bank and 2 wood, 11 brick, 2 grain, 3 sheep, and 11 ore were collected from the bank through the decisions of the players. Knowing how players chose to give and take cards from the bank shows the difference between optimal and actual resource production. Obviously, in a game like this, the players believed that wood was produced far too much when compared with its perfect complement, brick. (Not a surprise.) This lack of road-building material drove up the production of cities and development cards through a substitution effect. Since players couldn’t build as many roads, they built other things. By the end of the game, an unusually high 14 development cards were bought, Eto Camoe and YD4 had built all four of their allowed cities (and one more settlement each), and I had built three cities. Cinderella, who had taken brick with a monopoly card, had no ore production, so she used her wheat port primarily to buttress road and settlement building, and finished with only one city, but used all five of her settlements.** Since her strategy ran contrary to the city-building of the rest of us, it was not difficult for her to expand over much of one side of the board and gain the Longest Road.

In this game, ore was the most valuable resource, used for the cities and development cards that often prove useful. Only slightly less valuable was brick, the perfect compliment to the plentiful wood. Trading to get brick is also easier to immediately use than ore, in many cases; if a player would like to build a city, it takes three ore; to build a road, only one brick and one of the plentiful wood are needed. Since wheat and sheep were also plentiful, the same circumstances apply to settlement-building. This made bricks valuable in the early game, when building only one road was often enough to get to a new settlement site. Later, though, when the best sites were taken and Cinderella had Monopolized brick, the other players focused on cities.

The other resources were far less valuable, but wheat was still more valuable than sheep and wood, likely because of its many possible uses; everything but roads requires wheat. Sheep were actually slightly more valuable than wood in this game, but it took wood being produced at about three and a half times the rate of brick for this to happen. Although wood is usually more common than brick (since there are three hills and four forests in a 4-player game), it is usually less than even the 33% difference that would be randomly predicted, because players will recognize the relative scarcity of brick while placing their initial settlements.

I managed to eke out a victory in the game—10 points to the 9 of every other player, on 3 cities, Largest Army (five soldiers [one unplayed]), and two victory point cards. Eto Camoe and YD4 had 4 cities and a settlement, and Cinderella had a city, five settlements, and Longest Road. Close game, fun game. I always enjoy playing with people that know what they’re doing. This game also highlighted the importance of buying development cards; the eight I purchased were vital. Cinderella’s “Monopoly” kept Eto Camoe from winning on his next turn at that point of the game. Moral of the story: development cards are awesome. When you’ve got four points, start considering the purchase of development cards. When you’ve got five points, start buying if you haven’t yet. When you’ve got six, development cards need to be your top priority.***

*We played a variant that involved a “jungle” tile, rather than a desert. The jungle doesn’t produce resource cards; when the number on its chit is rolled, it produces a counter that can be used as a wild card for wheat, sheep, or ore when purchasing a development card. They may not be traded or used to purchase any other item.

**Each player has only four cities, five settlements, and 15 roads. Using all of the roads is rare, but it is relatively common for a player to have all of their city or settlement pieces in play. When a player has built all of their cities, they may not upgrade another settlement; when they have built five settlements, they must upgrade one to a city in order to once again be able to place a settlement.

***Unless you’re playing the road-building strategy. Then you’ve got to start thinking about cities once you’ve got three settlements, and making it a priority when you’ve got four settlements, because once you’ve got five, you’re a bit stuck. The way to win here is five settlements, Longest Road, a city, and a victory point. (Or a second city instead of one of those settlements.) The problem with this strategy is that the points are obvious to every other player, which attracts attention. If you play this way, you need to be an expert politically, able to convince other players that the development cards face-down in front of card-building players like The Franchise are victory points.

Game #2
Friday, Cinderella and I played with her younger siblings. This game had far more statistical balance: 31 Wood, 35 Brick, 50 Wheat, 36 Sheep, and 35 Ore. Wheat was something of an outlier, but due to its value, its frequency was welcome. In a game like this, with relatively equal production amounts, the trades made with the bank were more due to short-term trends than the overall production during the game, in serious contrast to Thursday’s game. This game was standard Settlers, without any variant rules. Most of the resources that players elected to give or take from the bank were in port trades, but I once gave 4 wood to the bank on a 7,* and gained 2 wheat from a “Year of Plenty” card.

CindyBro chose to start the game with the wheat port, so the port trades in the early game were mostly conducted by him, trading to get brick and wood once each, and ore four times. Cinderella built the brick port, and traded for brick thrice, wheat twice, and ore once. CindySis built the sheep port, but since she had low sheep production, and built the port when I was shifting my focus to development cards, so demand for sheep had increased. As such, she used the sheep port once each to get wheat and brick. Strangely, she used the ore port more frequently, twice for wheat, and also for brick and wood. Cinderella once used a generic port to trade sheep for wheat. 4:1 trades, which were not present on Thursday, were relatively common, as players traded sheep for wood, wood for wheat, and sheep for wheat.

In sum 4 wood (8 including discards), 12 brick, 12 wheat, 15 sheep, and 8 ore were exchanged for 6 wood, 3 brick, 8 wheat (10 including “Year of Plenty”), no sheep, and 5 ore. Clearly, sheep were the least valuable resource; no player needed to trade for sheep at any point. Despite the outsized wheat production in the game, it still seems to have been slightly more valuable than ore. Unsurprisingly, in a game that had roughly equal production of wood and brick, the two were roughly equal in value; with the slightly rarer wood having a slightly higher value over the course of the game.

CindyBro and CindySis played balanced strategies; she had 3 cities and 2 settlements, he had 2 of each. Cinderella played a road-building strategy, with a city, 4 settlements, and Longest Road. If I had not won on my turn, she would have had enough to build a city on her turn and win. At the start of my final turn, I had three cities, a settlement, and Largest Army. I had bought six development cards (four soldiers, [one unplayed] a “Year of Plenty,” and an unplayed “Road Building”), and each of the other players had one soldier.

Conclusions: once again, it was clear to me that whenever possible, one should start buying development cards when at about 5 VPs. This game strongly suggests wheat is the most valuable resource; even when it was considerably more common than other resources, it was still in high demand by most players, as evidenced by it being the most common objective of elective bank transactions. Wood, ore, and brick were of similar values, with only very slight differences in value, but seem to be valued in that order, likely due to wood’s slight scarcity. Relatively new players also seem to have a greater predilection for road-building, which likely increased the value of wood and brick to Brother and Sister, who were each playing in their third game.

*On my final turn, I had a wood, 2 brick, 2 wheat, 2 sheep, and 3 ore when I rolled a 7. Since I could win by either building a settlement or a city, and could do either one, I am omitting my choice from the data. Either way, a brick and a sheep would have been discarded, but the other three cards would depend on what I would use to win. Either way, I would definitely keep a wheat (probably both).

Game #3
We had another game Saturday afternoon. Same cast as the last one, except with Cinderella’s Dad as well. Only his second game, but the guy has a mind like a steel trap, so I was not surprised that he became the strongest opponent. I ended up having to throw him under the bus to ensure my win, encouraging CindyBro to get Longest Road. (Sorry about that one, CindyDad.) Playing with 5-6 players requires a larger island, so there are two more of each productive land type, and another desert. Or jungle, in this case, as I love that variant, since it means that more development cards are in play, which makes the game more complex. Always a good thing, I think. (It also strengthens my “card builder” strategy, so I’m always in support of it…) There are also more ports on a 5-6 player map; in addition to the five resource-specific and four generic ports of the standard game, there’s a second sheep port and a fifth generic port. Large games also allow players a “special build phase” between players as well, when they can build if they have the exact resources needed, but they may not trade with other players or the bank. Largely, this rule is to help players from having to discard too frequently when sevens are rolled, but it also introduces some odd side effects.*

Unlike the last game, this match saw a return of uneven production, with 51 wood, 36 brick, 56 wheat, 71 sheep, 23 ore, and 19 discovery counters. Thus, it is no surprise that ore was the most sought-after resource in the game, as players needed it badly to produce development cards and cities. With a larger map and more players, trading with the bank was also rarer, as people were more likely to get what they needed from one another.

Trades with the bank were mainly to obtain ore. Cindy had sheep ports, so she traded in sheep often; six times for ore, thrice each for brick and wood, and twice for wheat. CindyDad’s wheat port was used once, for ore. He also made 4:1 trades with sheep and wood to get ore. In sum, 4 wood, no brick, 2 wheat, 32 sheep, and no ore were traded for 3 wood, 3 brick, 2 wheat, no sheep, and 9 ore. On sevens, 6 wood, a wheat, and 3 sheep were discarded, and CindyBro once used a “Year of Plenty” to get wheat and wood. In total, 10 wood, no brick, 3 wheat, 35 sheep, and no ore were voluntarily given to the bank, and players elected to collect 4 wood, 3 brick, 3 wheat, no sheep, and 9 ore.

CindyBro and I were the ones built by the jungle, so we frequently used discovery counters to buy more development cards. When development cards were purchased with a mix of counters and cards, four times it was a wheat and 2 counters, and once each it was a wheat, a sheep, and a counter; a sheep, an ore, and a counter; and a sheep and 2 counters. In mixed sets, counters were used for 2 wheat, 4 sheep, and 6 ore. I’m still not sure how to account for this data in relation to the resources given/chosen model, but it’s definitely important, especially in further highlighting the value of ore during this game. The mild surprise here is that sheep were not the least-substituted of the resources, but this was largely due to my unusually high wheat production during the endgame, when I was buying a development card approximately once per turn, (I had a city on one jungle and two settlements on the other, so I would get two counters on such rolls, and could spend them very rapidly) and CindyBro’s predilection for only getting development cards once he had three counters.

When ore is produced at such a low rate by the board, it is unsurprising players used most of their bank exchanges to augment their ore supplies. Having jungles in such circumstances is highly useful for producing development cards, which saves the valuable ore for use in city-building. Having jungles makes a “card building” strategy very effective, even when wheat or ore are relatively rare.

On the opening of my last turn, I had 2 cities, 2 settlements, Largest Army (6 soldiers), and a Victory Point card. CindyDad had a city and five settlements; Cindy had 2 cities, 3 settlements, and a VP card; CindyBro had a city, 3 settlements, and Longest Road (9 segments); and CindySis had 2 cities and 2 settlements.

When I placed settlements for this game, I had no access to brick or wood, (I bordered two fields, a pasture, two mountains, and a jungle) but strangely, it was not especially difficult to expand to two additional sites. The ease with which I did so was largely because of my access to wheat and ore—wheat due to its value to all strategies, and ore because of how rare it was in this game. As a result, it was quite easy to get 1:1 trades with other players to collect building materials, and combined with early use of soldiers, which allowed more access to more resources, I was able to keep the robber away from my own productive sites and even, on occasion, steal the resources I needed to expand. I had actually intended to upgrade my first expansion site to a city before worrying about the second, but when I managed to get the wood and brick I needed without having to overpay, I did, which had the added effect of denying other players a solid expansion site.

*Players cannot use a development card purchased during that turn, but they can buy a card during the build phase immediately before that turn, then use the card before rolling. (Of course, using a non-soldier card does not offer a benefit.)


Starting points and destinations (More Catan)

Collecting data—in only three games—has had a strange side effect on me; it has become more obvious than ever how difficult it is to get the ten point needed to win without getting extra points with Longest Road, Largest Army, and/or VP cards.* Points “on the board” in the form of cities (2 points) and settlements (1 point) are useful because they produce more resources, but they are often more expensive than the cost of a few road segments or a couple development cards. Buying many development cards or building roads to far-off locations early delays one’s own economy, and means productive capacity will be outstripped by rivals, but trying to win with 4 cities and 2 settlements or 3 cities and 4 settlements is essentially impossible; even getting that many sites is difficult, especially in a four-player game.** Usually, those that try such a strategy lose with about seven or eight points, as others are able to get cheaper but unproductive points more rapidly in the endgame. Thus, knowing when to switch from building up one’s productive capacity to buying the non-productive points is vital in The Settlers of Catan. This has an interesting parallel to how individuals and nations use their own money and economies—how big a focus should be placed on short-term consumption, and how much should be invested toward the future?

This naturally leads to a discussion of the basic placement and play strategies available at the outset of the game. The ultimate beginner’s strategy is to place settlements on the sites that have the most resources available—the tiles surrounding them have a lot of pips. This often means that a player does not get all of the resources they need very often, so making such a strategy work relies on adept trading or poor decisions by other players. (Actually, the ultimate beginner’s strategy is to place settlements completely at random, but that lacks any hallmarks of “strategy.”)

Soon enough, though, players recognize that they need resources that can work in concert with one another, which leads to three plans. The first is to place to get some of each resource: a balanced approach. This allows players to be more self-reliant, which is good for players that are not as adept at trading; they can avoid doing so very often. The drawback to this strategy is that it can be difficult to build up an army or gain longest road this way, and chasing both usually means accomplishing neither, but it does provide a great deal of flexibility, whereas the other strategies are less so. If it becomes evident that no players are pursuing one of the two +2 VP bonuses, a balanced player can take advantage of such a circumstance, and if they move quickly, such a player can sometimes get both.

The first specialized strategy people become interested in is usually “road building,” emphasizing brick and wood. Building long roads and numerous settlements can be a great way to get ahead, especially early in a game. To get to ten points, this strategy requires either one city, five settlements, Longest Road, and a victory point; or two cities, four settlements, and Longest Road. This strategy can often be difficult to get past nine points on, but a player with this strategy does not get “boxed in.” This strategy is often less damaged by the robber, which is good, since it generates fewer soldiers to chase the robber away.

The second specialized strategy is “city building,” emphasizing ore and wheat, and upgrading the initial settlements to cities as soon as possible. This strategy generally requires building to one or two more settlement sites and upgrading those to cities, as well as buying enough development cards to get Largest Army. Winning with this strategy usually entails three or four cities, Largest Army, and possibly 2 Victory Point cards. Having the robber placed on a productive site can be very damaging to someone pursuing this strategy, so building up an army to chase the robber away is important.

Sometimes, the result of the above strategies is an overabundance of another resource. If the port for that resource is in a good location, a player can sometimes place on that port and another good spot to generate the chosen resource. Since sheep are often a plentiful, but less useful commodity, this strategy is often called “Sheep-O-Matic.” Using it for other resources is generally less successful, but I’ve seen wood utilized as well, when most players were pursuing city-building strategies and brick was in short supply. Such a strategy can allow the player to exact a “tax” or “user fee” on their turn, using their port by proxy for another player, at the cost of another resource card.

The final, and riskiest, placement strategy is to seek out the rarest resource—usually brick or ore, as they have only three tiles—and building on the best spots available for that resource. When other players need that resource, it will be possible to get advantageous trade offers. Unfortunately, it may also attract the robber’s attention, as an opponent may try to steal the valued resource.

*Victory Point development cards are worth 1 point each, Longest Road and Largest Army are worth two points each, but come at minimum costs of 5 road segments or 3 soldiers. To take one of these titles away from another player requires building a longer road or a larger army; in the case of a tie, the initial player to build to that size retains the points. This makes Largest Army especially secure; the player with it knows exactly how large each players’ army is, and can buy more development cards when they notice a rival doing the same, and only one development card can be used by a player on any specific turn. Longest Road is harder to keep, since players can build as much road as they can afford on any turn.

**There are only four city and five settlement pieces available to each player, so a five city win is not possible.


More Catan: Mechanics of the Robber and Why Everyone Should Love Development Cards.

The robber, as well as the purchase and use of development cards, are subjects I have not yet covered in detail in my Catan crash course. When a seven is rolled at the outset of a player’s turn, all players with more than seven resource cards must discard half (rounded down). The player that rolled then must move the robber to another location. (Interestingly, this is the only time that a game mechanic is tied to the player rolling the dice.) The robber prevents the tile he is on from producing resources, even when its number is rolled, so it is a good way to keep the player that is in the lead from getting the resource one believe they would find most helpful. Additionally, when a player moves the robber, they can choose to steal a resource card at random from any player with a settlement or city bordering the new location of the robber.

Fortunately, players can do something about the pesky robber always being around them: they can build an army. Doing so requires “development cards,” purchased at a cost of a wheat, a sheep, and an ore. Unlike the resource cards, these cards can’t be traded, and only one can be used each turn—and not on the turn it was purchased. Of the twenty-five development cards, five are Victory Point cards. These are an exception to the limitations on development cards; they can be played on the same turn as other development cards, and can be played immediately. These cards should only be played on the turn that a player wins; they are very useful because holding these cards is less obvious than having another settlement or city on the board in plain view. Six cards are special “Discovery” cards of three varieties: “Monopoly,” “Road Building,” and “Year of Plenty.” “Monopoly” is commonly believed to be the strongest card in the game; it allows a player to name a resource and confiscate all of that type from the other players. “Road Building” allows a player to put down two additional roads anywhere on the board, and “Year of Plenty” allows a player to take two resources of any kind(s) from the bank, which is often useful for big purchases, when a player needs to get two resources that are particularly scarce in that game. For those counting along at home, this leaves 14 development cards to fall into the “soldier” category. These brave men can be called upon at any point in one’s turn—even before rolling the dice—to chase the robber off to a distant place. Each soldier can only be used once, but keep his card after playing it, because having the Largest Army (minimum 3 soldiers) is worth a bonus of 2 Victory Points. Playing a soldier card triggers all of the mechanics of the robber—one must choose a new location for the robber, allowing one to steal a card from a neighbor of that tile and blocking future production.*

*In the 5-6 player expansion, there are nine more cards—six soldiers and one each of each “discovery” type. There is also a blank card that can be used to represent the place of a lost/damaged card, or stand in as another Victory Point. However, I’ve concluded that doing so makes the strategies I like slightly more powerful, so I’m going to avoid such play in the future and recommend against it to others.