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Roll Through the Data

Introduction

It's the holiday break, and time for another busman's holiday for DangerDon. Following the Founding Fathers article, this year's target is Roll Through the Ages (The Bronze Age), designed by Matt Leacock and published by Gryphon Games This article assumes a good knowledge of the rules of RTTA.

This analysis is based on games played on Yucata.de, my favorite online board game site. All games played on Yucata.de are publicly available for replay and the format used is not overly difficult to parse for analyses like this one. I used all games played by the top 10 ranked RTTA players, plus my own (DangerDon) games, with resigned games thrown out. This adds up to 7,870 unique games of RTTA, 1,963 unique players, and 586,046 final dice played.



Players in a Game

RTTA can be played by 2, 3, or 4 players, but 2-player games are by far the most popular version played on Yucata, followed by 3 and 4--player games.

Players in Game

Yucata.de players prefer 2-player games overall, and for RTTA the overall length of games is no doubt a factor. There are two ways a RTTA game will end: a player builds five developments or all monuments are built by at least one player. For 2-player games, there are five monuments to build. 3-player games have six monuments and 4-player games have seven monuments. We should expect, then, that with more monuments to build, RTTA games with more players will have more rounds of play. 4-player games have slightly more rounds on average than 2-player games.

Avg Turns



Scoring

What makes a good score in RTTA? The following chart shows the average total scores by finish for 2, 3, and 4-player games. The "Winner" is the player(s) who won the game (including ties - there were 31 of them) and the "Runner-up" is everyone else.

Avg Score

Winners score around 25 points on average while runners-up score around 15 points. Note that the scores for everyone are lower when there are more players in the game. The extra points from additional monuments are offset by the additional disasters that will be sent your way by more opponents.

Scores in RTTA come from four sources: developments, monuments, bonus, and disasters penalties (negative scoring). Comparing the average scores in each of the categories provides some clues for better play.

Score Breakdown

Winners score more points in all categories, of course, but the higher average disaster penalty for Winners is unexpected. Winners don't shy away from disaster. Disasters penalize, but they're also the fastest way to gain goods.

In looking at the deltas between the Winners and the Runners-up in the various scoring categories, we can see that the biggest delta in in the monuments category. We can look at the scoring breakdown from a different angle.

The following two pie charts compare the average scoring breakdown for the sub-total score, which is the sum of developments, monuments, and bonus.

Breakdown Winners

Breakdown Runners-up

Winners emphasize monument building more than Runners-up do. Getting half of your points from developments seems to be a good guideline. Later in the article, we will see how the Winners' use of developments support this strategy.



Cities

Cities

In RTTA, workers are used for both building cities (4 to build plus 3 free cities) and monuments. More cities generate more resources in the form of dice, and monuments score points. A key question for RTTA players is how many cities is enough before the workers should be dedicated to monuments, keeping in mind that the first player to complete a monument scores twice as many points for that monument as everyone else. The following chart compares how Winners and Runners-up handle city building.

The most common number of cities built is 6, and Winners use 6 cities more than Runners-up. The optimal number of cities is 6.



Dice

DieDieDieDieDieDie

The most frequent decision made by RTTA players is deciding which dice to keep and which dice to re-roll. Each turn, players roll the number of six-sided dice equal to their city count. They then get up to two re-rolls, Yahtzee-style. Players have three ways to influence their final dice:

o Re-roll any or all of their dice, except for a Skull
o Stop re-rolling after 0, 1, or 2 re-rolls
o With Leadership, optionally re-roll any one die, including a Skull

The next two pie charts show the distribution of dice that players end up playing.

Dice Used Winners
Dice Used Runners-up

If the dice were completely random (which they would be if players never re-rolled the initial roll), then we would expect each result to be 1/6 or 16.67%. (The last two results, being player's choice, would add up to 1/6). Any result above 16.67% are results more often sought by the players (or in the case of the skulls, both sought and forced), and any result below 16.67% are results more often rejected, or re-rolled, by the players.

We see that the Skull result is the most common because it usually can't be re-rolled and it is a desirable result up to a certain point. The 3 Workers and 3 Food results are also sought after, with workers being preferred over food. The 1 Good, 7 Coins, and choice results are the least preferred.

It is notable that the distribution of dice results is about the same for Winners and Runners-up. All players seem to have the same sense as to the importance of various dice results. In the split between 3 Food and 3 Workers results, Winners prefer the Workers at a higher rate. This is consistent with the fact that Winners also end up with more Disaster deductions than Runners-up. Food's only purpose is to prevent famine (and to be sold for coins with Granaries).

How hard is it to get a certain number of skulls through re-rolls? The chart below shows the probably of having X skulls when rolling and re-rolling the dice from 6 cities (the optimal number of cities to have). It's assumed that the player always re-rolls all non-skull dice for both re-rolls.

Dice Skull Prob

We see, for example, that the probability of ending up with a Pestilence (3 skulls) after the 2 re-rolls is ~29%. This assumes that your strategy is maximizing the number of skulls. If you were purposefully targeting 3 skulls but not 4 or more, your strategy would be a bit different and result in a higher probability than 29%.

Note that Leadership does not have a dramatic effect in pursuing skulls. Leadership is more useful in removing skulls because you have a 5/6 chance of removing a skull with a single re-roll. For example, for the 16% of the time you end up with 4 skulls and you want Pestilence instead, you could re-roll a skull die with Leadership and add 16% x 5/6 to the 3 skull probability and end up with a 42% chance of getting Pestilence with 6 dice and leadership.

The dice results suggest part of a winning strategy, but they're not enough to put together a coherent plan. We'll have to keep digging.



Workers

Die

RTTA is in part a resource management game, and an understanding of the value of the various resources is useful. Key RTTA resources are workers (used to build stuff), food (used to avoid famine), and goods and coins (used to buy stuff).

Let's start with workers. The following chart shows the total workers used and the corresponding total points scored for every game in the data set.

Workers vs Score

Sharp-eyed readers will note that there is a cluster of data points where total workers equals 59. That's the maximum number of workers that can be used in a 2-player game. Building the four additional cities and the five monuments available in a 2-player takes 59 workers. Also notice that it is possible to finish a game having used zero workers. It's pretty rare, but in those games the players with zero workers won 9 games and lost 7.

The overall shape of the cloud of data suggests that more workers corresponds to more points. The formula for the best-fit line for these data points is Points = (Total Workers Used) * 0.37 + 7.59. The slope of this line, 0.37, suggests that each worker is worth 0.37 points. The coefficient of determination (R2) is 0.218. This is a measurement of how well the line fits the data. A value of 1.0 for R2 would be a perfect fit and a value of 0.0 would mean that the values for points were completely random in relationship with the values for workers. An R2 of 0.218 means that there is some correlation.

Does this mean that a worker can always be valued at 0.37 points? No, it doesn't. At any given point in the game, the value (or utility) of n workers depends on the game situation. For example, if you need three more workers to finish the great wall monument before your opponent does, three workers is worth 10 points to you right now, but will be worth less the next turn. Two workers, which leaves you short of finishing the great wall, doesn't give you any points this turn, and four workers doesn't get you any more points than three workers.

The set of points that maps the amount of one commodity (workers) to the amount of another commodity (points) that has equal utility (usefulness) to the player is called an indifference curve. That is, the player is indifferent to having one commodity or the other for every point on the curve. In games, economics, and real life, indifference curves are rarely linear and usually change over time (or game turns). For games with resource management, it's the challenge of understanding the set of changing indifference curves that makes it a game.

Since points are a proxy for victory perhaps we can derive some insight by looking at how workers used affect a player's finish.

Workers Used by Finish

This graph indicates that the more workers the better. Players that use approximately 40 or more workers win more than half of their games.



Food

Die

The following chart shows the total food used and the corresponding total points scored for every game in the data set.

Food vs Score

Note that it is not possible to use less than 3 food because each player starts with 3 food and that food is always spent on turn 1 to feed the 3 starting cities. No player in the set of games ever finished a game using less than 5 food.

There is even less of a correlation between food and points than in the case of workers. An R2 of 0.11 means that there is some correlation, but not a strong one.

Astute observers may ask, "if food is only good for avoiding famine, where one food prevents one point of deduction, why isn't the slope of the best fit line closer to 1.0?" There are two main reasons for this: the indifference curve and opportunity cost. Food is only useful if you can use it to feed your cities. Food at the end of the game is worth nothing. (Winners average 0.49 units of food at the end of the game, Runners-up 0.83). Opportunity cost means that the food you're using to prevent famine (and saving a point) could have been something else that would have gained you points.

Let's take a look at food used and victories.

Food used by Finish

Using more food does not lead to more victories. This is consistent with the previous conclusions that Winners do not shy away from disasters and that Winners prefer the worker die rolls to the food die rolls.



Goods

Die

In RTTA, goods, along with coins, are the purchasing power used to acquire developments. When you collect a given number of goods in a turn, each good is added to a subsequent row, starting with row one (wood). The fifth good is added to row five, the sixth good is added to the first row again, and so on.

This chart is not derived from the data set; it's just math.

Purchasing Power

This lines in this chart show the cumulative power of goods in RTTA. The purchasing power of a number of goods on a given row is BaseValue * Count * (Count + 1) / 2. This is the sum of the numbers from 1 to Count, multiplied by the base value. The lines on the chart shows how each good collected in a turn adds more and more to your purchasing power until the goods wrap around back to row one for six goods. Starting the turn with goods in your inventory means that new goods will add even more purchasing power.

The bars on the chart show the "push your lock" aspect of RTTA. For calculating disasters in the chart, it is assumed that the player collects goods using the fewest dice possible. That means 1 good is achieved with a single "1 good" die, 2 goods are achieved with a single "1 skull and 2 goods" die, 3 goods are achieved with one of each, and so on. Goods up to 3 have no disaster impact. A set of 4 goods comes with a 2 point deduction from drought. 6 goods comes with a pestilence that costs everyone else 3 points. 8 goods comes with an invasion, costing you 4 points.

There are developments that modify each of these disasters. In later sections, we'll see how often these disasters occur and are prevented in practice.



Developments

Developments are the "tech tree" of RTTA and offer the most opportunity for strategic play. Players can acquire up to 5 of the 13 developments. Each of the developments has a cost in coins and is worth a certain number of points. Additionally, each development enables a capability or effect that is beneficial to the player. The following table shows some stats for the 13 developments:

Development Base Points Per Coin Ability Ability Benefit Per Build
Leadership 0.20 Re-rolls
Irrigation 0.20 Points saved 1.87
Agriculture 0.20 Amount of extra food 5.10
Quarrying 0.20 Amount of extra stone 2.62
Medicine 0.20 Points saved 2.48
Coinage 0.20 Additional Purchasing Power 15.23
Caravans 0.20 Goods saved 1.18
Religion 0.30 Revolts redirected 0.04
Granaries 0.20 Coins made from food sales 0.72
Masonry 0.20 Extra workers 5.95
Engineering 0.15 Purchased workers 10.94
Architecture 0.16 Bonus points 2.54
Empire 0.13 Bonus points 6.22

The first column shows each development's cost in coins, name, ability, and points scored. Column two is the development's points divided by cost. Column's three and four show, on average, how much each development's ability comes into play each time you build it. Column four is the number and column three is the unit of measure. Leadership has no entry in column four because re-rolls are not recorded in the game histories (or I couldn't find it).

The Base Points Per Coin reveals that 0.2 points per coin is the magic number with four exceptions. The first exception is Religion, which has the biggest bang for the buck: 6 points for 20 coins. That is offset by the fact that Religion's ability, redirecting revolts, are very rare. On average, you will use Religion's ability once every 24 builds.

The next exception is Engineering, which comes in at 0.15 points per coin. The game designer must have felt that the ability to exchange stone for workers was worth something extra. Engineering builders purchased an average of 10.9 workers per game, and we'll see later how this affects victories. Architecture and Empire have lower base values for points per coin, but the bonus points make up for it. For builders of of Architecture, this averages 2.54 points (for 2.54 monuments) per build. Empire builders average 6.22 bonus points for cities. Since players start with 3 cities, any Empire build is worth at least 11 points.

The average ability benefit per build gives us additional info on how to measure the value of a development's ability. Ability benefits are in several different units, so making comparisons is difficult. Let's start with the easy ones. We've already covered Architecture and Empire, which add bonus points. The ability for Irrigation is also expressed in points, where each Irrigation builder saves an average of 1.87 points, slightly less than one drought averted. Medicine also saves points.

The other units of measure for development abilities are food, stone, coins, goods, workers, revolts, and the ability to exchange commodities. It is tempting to convert these units into a common unit, say points, to make the comparison between the developments easier, but we're going to resist this temptation. For such a units conversion to be meaningful, we would have to assume that the exchange rate between any two commodities were constant. We know that this is not the case. In fact, we saw in the earlier sections that while more points is associated with more victories and that more food is associated with more points, we also saw that more food is associated with fewer victories.

How, then, do we measure the strategic value of the various developments? As in the case with workers and food, we take a step back and compare how Winners and Runners-up build developments. The following table is for 2-player games.

Development Winners Pct Runners-up Pct Winners Pref
Empire 52.46% 32.99% 59.03%
Engineering 32.03% 22.35% 43.35%
Architecture 24.80% 18.30% 35.53%
Religion 65.22% 49.04% 33.00%
Quarrying 92.24% 75.47% 22.21%
Granaries 7.57% 6.42% 17.94%
Leadership 31.71% 28.51% 11.21%
Masonry 43.07% 39.70% 8.47%
Coinage 13.23% 12.81% 3.26%
Agriculture 16.88% 17.64% -4.29%
Medicine 32.37% 35.54% -8.91%
Irrigation 27.94% 42.13% -33.68%
Caravans 6.47% 10.05% -35.61%

The 2nd column, "Winners Pct" is the percentage of times the Winner built the development in question. The 3rd column, "Runners-up Pct" is the percentage of times a runner-up built the development. The "Winners Pref" column is ([Winners Pref] - [Runners-up Pct]) / [Runners-up Pct]. This is the column that illustrates how Winners and Runners-up differ in their development play. Positive numbers indicate that Winners play the development more often and negative numbers indicate that Winners play the development less often than Runners-up. The table is sorted by the "Winners Pref" column from high to low.

Empire is at the top of the list, and given that it is on average the highest scoring single play in RTTA, it's position should not be a surprise. Since Empire is so expensive to build, pursuing Empire is not really a strategic choice beyond "score a lot of points." The same could be said for Architecture at #3. The real strategic insights come from looking at the development preferences of the less expensive items where players have more choices.

Building and using Engineering is a key winning strategy due to its ability. Winners build Engineering 43% more often than Runners-up, and Winners acquire 13.1 workers per Engineering build compared to 8.1 for Runners-up. Religion is also a key development for winning due to its points. We saw in an earlier table that Religion has the best bang for the buck of the developments. Quarrying is the most popular development, being built by 92% of Winners and 75% of Runners-up.

Although Winners build Granaries more often than Runners-up, Granaries is the least popular development. We saw in the earlier table that Caravans builders average 1.2 goods saved per build. This is a weak ability and Winners build Caravans 35.6% less than Runners-up. Building Caravans is not recommended. Irrigation is also at the bottom of the list. Remember that Irrigation's only function is to save the player from a two point drought deduction, and Winners don't shy away from disaster. Building Irrigation has an opportunity cost. In 2-player games, build Leadership instead of Irrigation.

The remaining developments, Masonry, Coinage, Agriculture, and Medicine, are preferred almost equally by Winners and Runners-up.

3-player and 4-player games change the equation for some developments. Let's take a look at development preference for 4-player games:

Development Winners Pct Runners-up Pct Winners Pref
Empire 45.75% 24.90% 83.72%
Engineering 20.46% 14.42% 41.87%
Quarrying 81.85% 65.52% 24.92%
Religion 56.18% 48.51% 15.80%
Medicine 72.20% 63.20% 14.25%
Architecture 15.44% 13.65% 13.16%
Masonry 35.91% 33.44% 7.37%
Agriculture 16.80% 17.85% -5.92%
Granaries 5.60% 6.40% -12.57%
Coinage 7.53% 9.83% -23.42%
Leadership 24.32% 31.95% -23.88%
Caravans 5.98% 8.15% -26.57%
Irrigation 27.41% 43.98% -37.68%

Empire is still at the top of the list and Irrigation is still at the bottom. Medicine is a recommended development for 4-player games. It's near the top of the table instead of 3rd from the bottom. In a 4-player game, there are three opponents instead of one to send you their Pestilence.



Monuments

Great Wall

Monuments don't offer nearly as many strategic decisions as developments do. With the exception of the Great Wall, they only score points and you can build as many as you want.

Monument Workers Points for First Points for Second Points Per Worker (First) Num Players
Great Pyramid 15 12 6 0.80 3,4
Great Wall 13 10 5 0.77 2,3,4
Hanging Gardens 11 8 4 0.73 2,3,4
Oblesisk 9 6 3 0.67 2,3,4
Temple 7 4 2 0.57 4
Stone Circle 5 2 1 0.40 2,3,4
Step Pyramid 3 1 0 0.33 2,3,4

The more expensive (in worker cost) monuments also reward the most number of points per worker. The downside of expensive monuments? The more workers required, the longer the monument takes to build, and the greater chance that another player will build it first and cut your reward in half. Winners score 39% of their points from monuments compared to 34% for Runners-up.

Monuments Built

In two-player games, the Great Wall is the biggest monument and the most popular. The other monuments are less and less popular until the Step Pyramid, which looks like an anomaly for Winners. The reason for the popularity of the Step Pyramid is that the game ends when each monument has been built at least once by any of the players in the game. Since Step Pyramid is usually the last monument built, the winning player often has the opportunity to end the game by building Step Pyramid.

Monuments Built

The same chart for 4-player games is a bit more interesting. We see that the Great Wall is the most popular monument regardless of how many players in the game. Both Stone Circle and Step Pyramid are more popular for Winners because of their ability to end the game.



Disasters

Die

Disasters are associated with both point deductions and additional goods. Disasters can affect the player, affect the opponent(s), be prevented, or be redirected. The average number of times each of these occur during a game for a player is shown in the next two charts.

Disaster Events

As expected, disasters that take more skull dice are rarer. Revolts and invasions are very rare. Winners are more aggressive (and/or) more lucky in getting Pestilence, which only affects opponents. Part of pursuing Pestilence is knowing what dice to re-roll and when to stop rolling. The other part is using Leadership, which Winners build more often in 2-player games. Winners prevent Pestilence less often. That comes from Winners favoring other developments over Medicine in 2-player games.

The "damn the disasters, full steam ahead" theme for Winners is continued for Droughts. Winners also prevent Droughts less often. This come from Winners being less likely to build Irrigation.

Disaster Events

The pattern is repeated for 4-player games with one exception. Winners prevent Pestilence more often than Runners-up. In 4-player games, Medicine is a recommended development.



Outliers

Stat Min Observed Max Observed Theoretical Max
Monument Score 0 40 43
Development Score 0 32 34
Bonus Score 0 12 14
Disasters Deduction 0 52
Total Score (Winner) 3 54 91
Total Score (Runner-up) -22 39 91
Total Score (All) -22 54 91

This table shows some interesting minimum and maximums in the 7,870 games. Players have come pretty close to achieving the max score in monuments, developments, and bonuses. The highest score overall is 54 out of a theoretical max of 91.

Respectfully submitted,

DangerDon (Don Laabs)

P.S. If you'd like to comment on the analysis, you can do so on this BoardGameGeek thread.