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Through the Ages: A New Story of Civilization» Forums » Strategy

Subject: Data-Driven strategy tips rss

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I-Sheng Yang
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I am trying to analyze past game data from boardgaming-online.
The current analysis is from the recent 10k+ games.
Results will be updated after more data are processed, and also with more aspects analyzed.

At this moment, I have not separated the "level" of games.
That means we are looking at the behavior of typical players, not only the good ones.
But well, since these people care enough to play online, we are probably not terrible.
In my next update, I will try to calculate ELO of each player, and repeat the analysis on games with good players only.
Tune in if you want to see the difference.

Please be aware that data can only tell us correlation, but not causation.
When I say a card or some action is "good", I mean "statistically, players who played that card or made that move is more likely to perform well."
However, we can never be certain that those moves are the reasons to victory.
When I say a card is bad, it may only mean that typical players do not use it correctly.
If anyone is interested to learn more, you can visit my blog http://spelguy.blogspot.ca/2018/02/a-data-driven-strategy-gu... for more information about the data and analysis.

Any game-related discussions are very welcomed here.
In particular, if you think a particular question is important for strategy and can be answered by data, please let me know.

For technical questions about data analysis, I think a discussion on my blog will be more appropriate.


Stage A and I.

1. Extra Civil Action (CA) is the top priority, followed by science.
Many people said this before, and they are correct. For this reason, Hammurabi appears to be the best Age A leader. Expectedly, Code of Laws (CoL) and Monarchy are both highly correlated with good performance throughout the game. In particular, at a similar cost, Code of Laws performs much better than Warfare. This shows that a early CA is better than an early Military Action (MA).

One curious fact is that Pyramids does not perform as well as Library of Alexandria.
Pyramids is taken 20% more often than other Age A wonders, and it is 10% more likely that people spend more CA on it. Nevertheless, the player with Library is slightly more likely to win. Thus, average players should probably not overspend CA on Pyramids.

This is not entirely surprising. Early science is also important. For example, Aristotle is the clear 2nd among leaders, while all other leaders are kind of equal. Library also gives you 10+ culture throughout the game, and the ability to hold one more card (which is part of the effect of an extra CA).


2. Go for Mine or Lab in Round 1.
It is pretty standard to build the 3rd Mine. If you have Urban Growth, you can also start with the 2nd Lab. These two choices perform equally well in the data I analyzed.
They are both better than all other choices, such as the 3rd Farm, or directly working on a Wonder. Of course, we already know the second point by simple counting. So this is like a sanity check of my data analysis.


3. Among all early Wonders, St. Peter's Basilica is the best.
In addition, Basilica seems to be the best Happiness solution throughout the game.
Some Wonders (like Colossus) actually has anti-correlation with good performance.
I start to think that when you cannot get Pyramids or Library, maybe it is a very good play to skip Age A Wonder such that you can grab Basilica cheap.
Of course, when you skip Age A Wonder, you should probably also refrain from playing early events. The extra resources from Age A events are more beneficial to players with a wonder to work on.


Stage I and II.

4. Use Leaders.
This is the single most important thing. You should never, never, be too busy doing "other things", and forgot to get a leader of the current age. Well, unless you are prevent a certain aggression. However, you should try to avoid that way ahead of time. A Leader does not use MA nor resources, so it should not interfere too much.

Among all things you can do, Leaders have the lowest cost. You grab them for a CA or 2, and play them free. For every other progress you want to make, you will need science/resource/population. The effect/cost ratio for Leaders is simply out of the chart. Also, leaders from the next Age are all statistically better than previous ones. You should always be happy enough to grab a general-purpose leader for 1 CA, or one highly synergizes with your setup for 2 CA.


5. Build Military with a small Science budget.
Opposite to the situation between CoL and Warfare, Strategy dominates over Justice System (JS). This might be obvious since JS is not really better than CoL. It's not worth spending extra science for those blue tokens.

It's not always possible to be the strongest civilization, but you should try your best to avoid being the obvious victim of aggressions. Strategy helps a lot on that, and so does Napoleon, who is statistically the best Age II leader.

Despite many players here favor Constitutional Monarchy (CM) due to its extra MA, Republic is statistically better. Here is my theory. The costs on Stage II governments are designed to encourage revolutions to Republic, but peaceful transitions to CM. This is somewhat historically accurate. However, since the science cost for revolution to Republic is so small, it opens up many other options. For example, it helps you to get Strategy, which is more than enough to negate the MA difference. Besides, Napoleon or Maximillien can sustain your MA for a while. Republic allows you to reaching the maximum CA from a government so early is quite amazing.


7. Be opportunistic on Farm/Mine/Lab/Units.
Many people debate over the importance of Stage I and II techs here. It turns out that none of them makes statistically significant difference. For Farm/Mine/Lab, that means you should not overspend CA to get something specific. Just take whatever is cheap, and make up what you lack with yellow cards. For units, just go for whatever works with available tactics.


End Game.

8. Military is Key.
If I have to ask one question to determine the winner, it will be "who uses the most MA in the last 4 rounds".
It could be waging wars for profit, or producing more units to defend your culture lead. Either way, if you are in the military game, your chance to win is much higher. It seems like there is rarely a large enough culture lead that cannot be wiped out by 2+ wars.

Also, Air Force is the single Military Tech that actually makes a difference statistically. It works all the time, and it works for everyone. It's worthwhile even just to deny your opponents.


9. Aim for Wonders.
First of all, you need a lot of CA to take and complete them. Ideally, you also want a big resource production, something like 4 Iron or 3 Coals. Although a timely collection of Reserve/Engineering Genius cards can help with that. It is often hard to squeeze out more population at this point. These are the ways to get points without population. If you are up for it, Engineering or a lower building tech may help you build 2 of those big wonders. Quite often enough to negate early culture leads.


10. Don't count on Gandhi.
Leaders are still good with 1 exception. There might be cases that he's only way to protect your early culture lead, but that is not very often sufficient to secure your victory.


11. Democracy.
Statistically, this is the Age III government you want to have. It is not a must though. Air Force and Civil Service probably deserves your science more.
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I-Sheng Yang
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Major Update 02/12/2018

1. I now have data for 30k+ games.
That number will stop growing though.
I had to keep my desktop running for 10+ nights to scrape those data, and it's no longer worth the effort keep doing that just to increase data by an order 1 factor.

Let's hope there will be more convenient way to get the data in the near future.

This, however, makes me comfortable enough to split the data according to player counts or player skills.

2. I calculate the TrueSkill for each player.
This allows us to look at more interesting statistics.

I repeated the previous analysis on games which average player skill is higher than global average. Namely, games with good players. The result does not change by too much. Most notably:
(A) Caesar becomes the worst Age A leader.
(B) Age II leaders becomes slightly less important than Age I leaders.
These 2 changes are already mild, and nothing else stands out.

On the other hand, I can repeat the previous analysis to predict "player skill" instead of the "game outcome". If a card is strongly correlated with player skill, but not strongly correlated with the game outcome, then it suggests a common mistake among good players.

I perform the analysis on all Age A and Age I cards, for 3ers and 4ers. Here are 4 such mistakes I noticed. (An new post on my blog has the same content with pictures. http://spelguy.blogspot.ca/2018/02/a-data-driven-strategy-gu... )

Pyramids vs Library of Alexandria.
Stronger players slightly prefer Pyramids over Library, more than weaker players do. However, Library performs significantly better for winning the game. I cannot see any alternative interpretation here and must conclude that this is a mistake made by strong players. Note that strong players must have performed better in many other places to make up the difference. However, there is very little doubt that at a 1-to-1 comparison, Library is better. Start to use it more!

I don't think that in the long run, Pyramids' ability is weaker. I looked at the statistics of how early do people build Age A Wonders. Pyramids and Library tend to be built as early as possible, which means delaying the 2nd Philosophy. 1 CA is better than 1 Science later in the game. However, this early in the game, Science might be slightly better, or at least equal. 1 turn early into any Age I technology can have a compounding effect that snowballs your economy. Combining with other benefits from Library makes it somewhat more powerful than Pyramids built at the same time.

Thus, maybe we should stop rushing to complete Pyramids. I know, it is a tempting package deal to get that CA early, delay population growth and get that 1 extra food. But maybe it's not worth delaying the extra Science production.

Age I Wonders.
Stronger players do not build Age I Wonders more often than weaker players. However, except for the Great Wall, all other Age I Wonders seem to win a lot. The interpretation here requires a bit more steps. Recall that if we use Age III cards to predict the game result, Wonders performs exceptionally well. That does not mean those cards win the game. They simply "indicate" that the player has enough CA, science and resources to complete a big Wonder. Leveraging those into other developments probably wins the game, too.

We did not have to consider this "post selection" effect for Age A Wonders, because everybody starts at roughly the same condition. For Age I, we should ask whether there could be significant differences in infrastructure, which determines a player's ability to build a Wonder.

That does not seem to be the case to me. Even if someone upgrades Iron early, it will take 3 rounds before an upgraded mine to generate a net resource gain. During the prime time to build Age I Wonders, a few timely Yellow Cards probably give you more resource advantage, and that is not much. Thus, it seems like these Wonders are really contributing to victory, instead of just "indicating" someone's exiting advantage. Furthermore, if that were the case, Great Wall should have been an equally good "indicator", but it is not.

Thus, I again conclude that Taj Mahal, University, and Basilica, are under-valued by good players. Investing your early resources and CA in them seems to be a good deal, compare to other things you might have done. This is probably because all other ways to use resources require population and science. You are likely in shortage of either or both during Age I-II transition.

Code of Law vs Warfare.
Good players value them almost equally. However, CoL appears to perform significantly better. I again do not see an alternative explanation here. They have the same number of cards, and the science costs differ by only 1. Thus again, good players seem to over-value Warfare. Probably a bit paranoid and trying too hard to prevent early aggression. I won't say that's wrong. If you are indeed the better player in a game, then the most likely way for you to lose is probably a devastating early aggression. It is a good circumstantial strategy to be slightly paranoid about that and go with the mediocre Warfare. As long as a stronger player do not suffer from an early aggression, she can count on later moves to cover the lost ground.

Knights vs Swordsmen.
Good players value them almost equally, but Swordsmen performs better than Knights. This, even to me, is a surprise. Knights open up more Age I tactics, but Swordsmen are easier to discover and build. At Age II, such difference is almost gone. I would have expected them to perform equally well. My only explanation is that on average, people do spend 15% more CA to take Knights from the card row, and that is bad enough to undermine Knights' record.


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I-Sheng Yang
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Space reserved for future updates.
 
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Shawn Fox
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This closely matches my own opinions developed over 100+ plays of through the ages, especially recently with the app.

Have you looked at multiple card correlations yet? For example barbarossa may not be that great by himself, but what if you have barbarossa + great wall? I'd think there would be some strong correlations between wonders and leaders or other specific technologies. An even more obvious one is the very powerful combination of michelangelo + saint peters.

[edit] Also you really need to do separate analysis for 2 player vs. 3/4 player as I'd expect to see substantial differences due to the higher availability of yellow cards in lower player games as well as the zero sum aspect of 2 player compared to 3/4 player.
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This is an interesting approach.

I think a statistical analysis of TtA needs to include some calculation of Expected Value (EV), possibly in terms of end-game culture production? I'm not exactly sure of the model here, but win rate seems a little more opaque unless it's with duels.

TtA is not a deterministic game like Go or Chess or Terra Mystica; it has degrees of variance akin to poker. True, an Age III culture lead an be wiped out in two wars, but in a duel the odds of drawing and being able to play two wars before game end impact whether this is will result in a more frequent win-rate vs. alternative paths.

I'm also not sure Elo ratings are particularly useful in this game. Elo is a results-oriented system designed for deterministic games. You can make every correct decision in a game of TtA for the specific game state and still lose to an inferior player because the level of variance is high. That may lead to Elo ratings that don't really reflect the actual skill of the players.

That's not to say that a superior player won't beat an inferior one more often, but the idea that at higher levels of play, a person with a higher Elo-based TtA rating will beat one with a lower rating every time is just not true.

In other words: I don't believe (strictly-speaking) results-based rankings are appropriate for a game like TtA.

Regardless, I find the analysis interesting and I will be following this discussion closely.

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I-Sheng Yang
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sfox wrote:
This closely matches my own opinions developed over 100+ plays of through the ages, especially recently with the app.

Have you looked at multiple card correlations yet? For example barbarossa may not be that great by himself, but what if you have barbarossa + great wall? I'd think there would be some strong correlations between wonders and leaders or other specific technologies. An even more obvious one is the very powerful combination of michelangelo + saint peters.

[edit] Also you really need to do separate analysis for 2 player vs. 3/4 player as I'd expect to see substantial differences due to the higher availability of yellow cards in lower player games as well as the zero sum aspect of 2 player compared to 3/4 player.
Indeed, I was very interested in combos.
The current results are negative.
Per player per game, every card is picked less than 20% of the time.
That means a 2-card combo likely appears less than 4% of the time.
I will need more data to make any combo effect statistically significant.

As for separating multiplayer games, indeed, I will soon have enough data to make that statistically significant. I also look forward to seeing any difference.
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Brent Celmins
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Jingking wrote:


4. Use Leaders.
This is the single most important thing. You should never, never, be too busy doing "other things", and forgot to get a leader of the current age. Well, unless you are prevent a certain aggression. However, you should try to avoid that way ahead of time. A Leader does not use MA nor resources, so it should not interfere too much.

Among all things you can do, Leaders have the lowest cost. You grab them for a CA or 2, and play them free. For every other progress you want to make, you will need science/resource/population. The effect/cost ratio for Leaders is simply out of the chart. Also, leaders from the next Age are all statistically better than previous ones. You should always be happy enough to grab a general-purpose leader for 1 CA, or one highly synergizes with your setup for 2 CA.[/b]
Curious as to this particular analysis in more depth for specific leaders. People have a tendency to shirk the Age II culture leaders (specifically Shakespeare and Bach), so I'm interested in learning more about their impact vs. the Age I leaders that people may ride out until the start of Age III. I read the blog, and it appears that you may be driving toward the conclusion that Age II Leaders are superior to Age I leaders overall, even if it means replacing Leonardo for Shakespeare early?

Also: Columbus, Homer, and Alexander are special cases since they all make taking another leader more expensive. Columbus, especially, is basically Opportunity Cost: The Card.


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Harvey Cohen
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Thank you for doing this analysis. I look forward to reading more of your analysis.
 
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Some other interesting questions:

What is the effect of taking and playing each leader on the first turn of a new age?

What is the average number of turns players keep each leader?

More generally, how is a player's score impacted when they have specific leaders in play for X number of turns?

What I'm trying to get to is which leaders (if any) are worth keeping in play all the way through the next age instead of replacing them.
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I-Sheng Yang
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Here is the rough answer to those leader questions.
I have not done detail analysis of "how people use their leaders".
Since I have not scraped a lot of data yet, questions which are too specific will not have statistically significant answers.

I would suspect that "how long should I keep a leader" is a very tactical question.
For example, many people probably will be very happy to keep Aristotle to almost the end of Age I, take Columbus, and use it on Vast Colony right before Age II.
Those kind of situations do not happen often enough to show statistical significance.
All I can see at this moment is that if you did not play an Age I leader, you chance to win is lower (relatively to someone who did). The same thing applies for an Age II leader. And the difference is much greater than any other type of cards.

There are also difference between leaders, roughly at 2-3 error bars between the best and the worst.
Pure culture generators are indeed weaker than their peers.
Michelangelo being the worst in Age I, and Bach in Age II.
People seem to get the right impression, since they are also picked less often.
All I can say is that, they should probably be picked even less often. I am looking forward to see whether "good players" indeed behaves that way.
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Last year, I tracked the data for 100 games I played solitaire, using four players. I haven't really had time to do a write-up of the results yet. Obviously tracking data from over 10,000 games is much better. My analysis did have the advantage that the skill level of the "players" was equal.

However, my findings tracked yours reasonably well.

On Age A, I had Library of Alexandria as the best Age A wonders, and much better than the Pyramids. Actually my data indicated that players were probably better off not building an Age A wonder if they couldn't build the Library, the other three had negative correlations with winning. Though the 5th CA is obviously important, my data indicated that it was better to get it through Code of Laws than through either the Pyramids or monarchy. Homer was the best Age A leader, followed by Ham, Aristotle did not perform that impressively.

With Age I there is a bigger difference, as my data had the Basilica as the worst wonder, with the University scoring most high among Age I wonders. As with Constitutional Mnoarchy/ Republic, my data indicated that CM was slightly better, but taking Republic wouldn't kill you by any means.

With the late game, yes my data indicated that Democracy had a very high correlation with winning. However, only staying on Despotism until the end or Fundamentalism were almost sure losers. And yes, being able to fight and win, or at least not lose too badly, a war on culture was critical. Gandhi did OK among the leaders, but I think this is mainly due to my having a tendency to pick up Gandhi when I have Napoleon (the best Age II leader), to keep another player from getting Gandhi, and they plopping Gandhi in for the last turn after winning through culture wars.
 
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esabatine wrote:
Last year, I tracked the data for 100 games I played solitaire, using four players. I haven't really had time to do a write-up of the results yet. Obviously tracking data from over 10,000 games is much better. My analysis did have the advantage that the skill level of the "players" was equal.
I wouldn't accept those data for any instructive purposes. In a solitaire context, each "player" can literally anticipate what the other is going to do with perfect certainty.

And even if we assume proxying level top players making the optimal move in each and every game state in those 100 solitaire games (which would require a near-superhuman level of mastery of the game, in and of itself), the fact that you knew what each player held in their military hands and which events & impacts were seeded makes the data particularly problematic.
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SWxNW wrote:
I'm also not sure Elo ratings are particularly useful in this game. Elo is a results-oriented system designed for deterministic games. You can make every correct decision in a game of TtA for the specific game state and still lose to an inferior player because the level of variance is high. That may lead to Elo ratings that don't really reflect the actual skill of the players.
Thanks for the good advice.
After a bit of reading, I decided to use Microsoft TrueSkill instead.
It is at least a probabilistic model that takes the uncertainty into account.
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While there are definitely problems with this approach, I don't think being able to see all the political hands is one of them. Commentators here seem to overweight the effects of the political events.

In any case, in any particular game, a political event will come out a certain percentage of the time. For example, Civil Unrest going by memory seems to come out 3% of the time. When deciding how urgently you want to fix your unhappiness problems, there is no need to know whether Civil Unrest is in another players' hand. You just expect it to come out 3% of the time and decide whether you are willing to take that risk. You can also build up lots of culture and a weak military and hope that the strong military player draws a War on Culture. That is a bad risk to run, but it actually happens that way in about 10% of the games according to my data.

I didn't have the analytical tools to download and go through the data of the Boardgame geek games, however, so I did what I could. I'm pleasantly surprise actually that my findings wound up matching up so well. I even checked my data for the Constitutional Monarchy/ Republic thing and found that my data indicated a slight but not overwhelming advantage to Constitutional Monarchy (CM used twice as much as Republic and twice as much first place finishes) and the discrepancy is probably more experienced players knowing how to "pull off" using Republic.
 
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esabatine wrote:
W Commentators here seem to overweight the effects of the political events.
Respectfully disagree.

For example, knowing if Ravages of Time or Iconoclasm has been seeded (or more specifically, knowing that it definitely hasn't been seeded) has a massive influence on a player's decision-making. Only the players that draw those cards can know for sure, and behaviors for the players in the balance of the game will reflect that decision.

In a 4p game, there is a lot of culture earned from Impacts. All players knowing all impacts in the deck will definitely cause players to make different decisions than with the information hidden.
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It might be helpful to separate games into 2p and 3/4p, as the value of some cards (Colossus, Gandhi...) changes quite significantly, and the game is quite different when pacts are a thing.

I am surprised that Bread and Circuses is listed as one of the worst cards in the game, and that Alchemy is rated so low (in addition to the ConMon surprise).
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I would be very interested to see the analysis further broken down to different player counts, as I found that the value of different cards varies greatly among different player counts.
 
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esabatine wrote:
While there are definitely problems with this approach, I don't think being able to see all the political hands is one of them.
Groupthink might be a more important issue with a group of a single person.

Already, on this board we can see the effect of groupthink. Many people claim that Pyramids is far superior to other age A wonders. The statistical study shows that it's not so good, I suspect some players overestimate its strength, and take it with a poor timing which ends up hurting their game.

This study is interesting because on one hand, it can reveal the effect of current groupthink (as with the Pyramid example), on the other hand, it might strengthen it on other topics. For example, the infamous "culture leaders" Shakespear and Bach, which are considered the worst in many strategy guides. it leads most players to avoid them. Probably mostly beginners pick them and the data ends up confirming the current groupthink.

But TTA is a game of tactical opportunity. Card value cannot be measured absolutely, it always depends on the context. What players consider a good card is one that is often good. That doesn't make it always the right choice. And on the other hand, a card considered bad might be more situational (or specialized). Sometimes the "bad" card is the best choice.

I would suggest everybody to keep that in mind when analysing statistical data. Playing only the "good" cards might make you a good player, but knowing when and how to play the more situational cards is probably what makes the difference between a good and a great player.

Disclaimer: I'm not a great player.
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A better approach to the "bad" leaders may limited use instead of outright avoidance. I agree that the "bad" leaders are pretty much the "pure" culture generators, because of the problem of losing all that culture anyway through wars.

Outright avoidance may lead to switching to higher age leaders quickly enough and/ or taking leaders for too many CAs. But the only leaders you are stuck with for lots of turns are the Age I ones. Most the others (Cook being the biggest exception) can be dumped quickly after they do a few specific things for you, or in the case of Age III brought in for the last turn to get a few cheap culture points and/ or help with the Age III wonders.

For example, Bach basically gives a discount in discovering theater techs. so lately if I'm using him he is often there for less than a turn, to discover Opera or (better) Movies, and maybe upgrade that temple to a theater. He is bad news if you let him hang around.

Likewise, Aristotle, who is considered to be a good leader, has a serious problem in inducing players to clog up their hand, so can hurt you unless you are disciplined enough to grab the techs you really think you need and them dump him for an Age I leader as quickly as possible.
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esabatine wrote:

For example, Bach basically gives a discount in discovering theater techs. so lately if I'm using him he is often there for less than a turn, to discover Opera or (better) Movies, and maybe upgrade that temple to a theater. He is bad news if you let him hang around.
Bach does generate marginal culture and there is nothing about him as your leader that prevents you from keeping yourself protected militarily (unless you are moving your arenas over to theaters, which seems like a bad idea anyway, unless you have a good strength edge and the Basilica). Considering how powerful Chaplin is in the right game state, I don't agree that keeping Bach for an extra turn or two is automatically bad for your game.

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Likewise, Aristotle, who is considered to be a good leader, has a serious problem in inducing players to clog up their hand, so can hurt you unless you are disciplined enough to grab the techs you really think you need and them dump him for an Age I leader as quickly as possible.
I have no idea why you would couch this as a serious problem with Aristotle.

The Aristotle-hand-clogging issue is a common one for new players only. It's one of the first light bulb moments that new players will experience upon subsequent plays: "Oh, using a leader's ability every time I can isn't necessarily a great idea."

If, game after game, a player continually drafts techs they don't intend to use and clog up their hand, it's not a serious problem with Aristotle-- it's a serious problem with that individual player not learning from their mistakes.

I wouldn't expect a beginner to understand almost anything about the overarching workings of the systems of game this complex, much less the minutiae of using arguably the best Age A leader.
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Riku Koskinen
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Interesting summary. It seems the data supports the general consensus pretty well, e.g. Hammurabi/Aristotle being the top age A leaders as well as Napoleon, Strategy, and Air Forces being very strong.

Jingking wrote:
Pyramids is taken 20% more often than other Age A wonders, and it is 10% more likely that people spend more CA on it. Nevertheless, the player with Library is slightly more likely to win. Thus, average players should probably not overspend CA on Pyramids.

This is not entirely surprising. Early science is also important. For example, Aristotle is the clear 2nd among leaders, while all other leaders are kind of equal. Library also gives you 10+ culture throughout the game, and the ability to hold one more card (which is part of the effect of an extra CA).
I'd mention LoA's pseudo-warfare ability as well, which is arguably more useful than the extra civil card hand space. Keeping at least 3 military cards in hand gives good flexibility, but it is hard to measure how strong this actually is.

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2. Go for Mine or Lab in Round 1.
It is pretty standard to build the 3rd Mine. If you have Urban Growth, you can also start with the 2nd Lab. These two choices perform equally well in the data I analyzed.
I guess you mean round 2, since technically round 1 is the one when you only pick cards. But how about round 3? Mine/lab round 2 and the other on round 3 has been the norm for as long as I remember. Sometimes you can finish a wonder on r3 instead if you got resources from events, especially if you got Engineering Genius, but you still want the 3rd mine/2nd lab asap. Although Development of Warfare/Religion can postpone this even further.

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3. Among all early Wonders, St. Peter's Basilica is the best.
In addition, Basilica seems to be the best Happiness solution throughout the game.
Some Wonders (like Colossus) actually has anti-correlation with good performance.
I start to think that when you cannot get Pyramids or Library, maybe it is a very good play to skip Age A Wonder such that you can grab Basilica cheap.
Of course, when you skip Age A Wonder, you should probably also refrain from playing early events. The extra resources from Age A events are more beneficial to players with a wonder to work on.
Basilica is very strong indeed. If you can secure it from Ravages of Time by having another age A/I wonder, there are no happiness issues for you at all in the game. They hardly nerfed it at all with this New Story edition (hello Transcontinental Railroad), and now it sometimes is even better than the old version thanks to giving 1 happiness by itself. To not make it stand out so easily, they easily could have made it 3 steps to finish and maybe have Taj Mahal be at 2 steps of 4+4.

It's an interesting observation about the two weaker age A wonders. I wonder (pun intended) how many Colossus pickers tried too hard by picking Cartography as well, postponing the discovery of their other, more important techs. I have always considered Colossus's (New Story) main advantage to be the 2 strength, giving good edge for all the strength events. It's good for Hammurabi as well to not worry about early defense. Anyway, the mini-Cartography effect is just a small bonus, not the reason to pick the wonder.

I would never skip an age A wonder just because of the events potentially spamming stuff. This could be manageable with 2 players, but you are very likely to pick either of the good wonders anyway, and even if not, Colossus is pretty decent because any extra strength is so much better in duels. You either are the weakest or strongest when evaluating events; there is no middle ground.

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Stage I and II.

Also, leaders from the next Age are all statistically better than previous ones. You should always be happy enough to grab a general-purpose leader for 1 CA, or one highly synergizes with your setup for 2 CA.
I don't quite agree that this could be a general rule of thumb. There usually are only a few age II leaders you are interested in, and you don't even necessarily want to switch to him right away.

-Bach literally does nothing with many setups.
-Shakespeare just gives 1 happy face unless you go for libraries and theaters, which you can't do very often.
-Cook with zero colonies is a bad idea even if you seeded a few.
-Robespierre's revolution is not always needed, at least age II: You can spend the 6 extra science for Const.Mon. and not skip drawing military cards, or you have Monarchy and there is no reason to change to age II govt. Although his MA is a good enough reason to pick and play him if you are still at 2 MAs.
-Newton could give 0, but usually 1 extra science at first, and a few CAs. Useful, but I can imagine many scenarios where the age I leader is better for a couple of more rounds.
-Napoleon is the only leader that I could see switching to as soon as I get the chance. And he certainly is worth 3 CAs.

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5. Build Military with a small Science budget.
Opposite to the situation between CoL and Warfare, Strategy dominates over Justice System (JS). This might be obvious since JS is not really better than CoL. It's not worth spending extra science for those blue tokens.
Strategy is certainly much stronger than JS, and they cost almost equally much. But that doesn't mean JS is bad. It just comes at a time when you should already have your CA count fixed. I would happily pay one extra for Code of Laws if it came with 3 blue tokens. Sometimes when I have Monarchy and good science production, I go for Justice System. Or if I'm for some reason still at 4 CAs, then of course it's good. And it could explain why JS doesn't perform well in your data; it is often picked by those who are a bit behind in progress due to having had only 4 CAs until that point.

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Despite many players here favor Constitutional Monarchy (CM) due to its extra MA, Republic is statistically better. Here is my theory. The costs on Stage II governments are designed to encourage revolutions to Republic, but peaceful transitions to CM. This is somewhat historically accurate. However, since the science cost for revolution to Republic is so small, it opens up many other options. For example, it helps you to get Strategy, which is more than enough to negate the MA difference. Besides, Napoleon or Maximillien can sustain your MA for a while. Republic allows you to reaching the maximum CA from a government so early is quite amazing.
This is interesting. Science costs aside, 6 CAs and 4 MAs is clearly better than 7 CAs and 2 MAs. And if the average player has 5 CAs at that point, 7 CAs and 4 MAs versus 8 CAs and 2 MAs is better with a bigger margin than in the first case, for the CM player.

Republic can cost 3 science, whereas CM usually is paid in full like you said, so that 9 science points spared could mean something, even if that comes at cost of 2 corrupted resources.

Or could it just be so that due to the popularity of CM, people are less eager to pick Republic? And Republic is maybe picked more often by experienced players than less-experienced ones, whereas the pick rate of CM is high for all player skill levels.
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One of the questions I have around the Basilica is: given how low on the ratings the happiness urban techs are in the analysis, how do winning players adjust when they don't get the Basilica?

Clearly, people win without the Basilica.

Homer is a potential response, but the decision around leader selection in Age A can't be that informed due to the uncertainty of when the wonders will come out.

There are only four other happy faces attached to wonders, one of which (Hanging Gardens) is even worse than the Homer "prescience" problem since it's Ravages of Time fodder.

It may be too difficult to control for this in a statistical model. The tactical nature of the game might give false correlations (specifically, if I have the Basilica, then it stands to reason I'm less likely to draft Arenas and Temples. Does this actually mean I'm more likely to win as a result?).

Still, brings up interesting questions regarding Temples and Arenas since they are the go-to happiness solution for a lot of players.

 
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SWxNW wrote:
One of the questions I have around the Basilica is: given how low on the ratings the happiness urban techs are in the analysis, how do winning players adjust when they don't get the Basilica?

Clearly, people win without the Basilica.

Homer is a potential response, but the decision around leader selection in Age A can't be that informed due to the uncertainty of when the wonders will come out.

There are only four other happy faces attached to wonders, one of which (Hanging Gardens) is even worse than the Homer "prescience" problem since it's Ravages of Time fodder.

It may be too difficult to control for this in a statistical model. The tactical nature of the game might give false correlations (specifically, if I have the Basilica, then it stands to reason I'm less likely to draft Arenas and Temples. Does this actually mean I'm more likely to win as a result?).

Still, brings up interesting questions regarding Temples and Arenas since they are the go-to happiness solution for a lot of players.
I am indeed also struggling with this.
Before seeing the data, I had always been happy picking Arena at 1CA when I don't have other solutions to my happiness problem.
Now I am a bit lost.

In terms of techs, the 2nd best happiness solution is Organized Religion, which many people knew without looking at data.

In practice, in a game with typical players, I think the best happiness solution (even better than Basilica) is Vast/Inhabited Territories.
Many players seed these while they should not, and do not bid high enough for them.
(I don't have data analysis to support this though.)
People who gets those may not even realize that they just solved the happiness problem (at least partially).

This could lead to what you call the "false correlation", or simply in my term "correlation does not imply causation".
The players who still have a happiness problem to solve are already weak players, thus whatever they pick tend to appear negatively correlated with victory.

Edit:

Another viable solution is simply "be unhappy", as someone already recommended in a strategy guide.
There are only a few events which rewards happiness or punish unhappiness.
Thus, if the solution is marginal (only gives you 1 more worker elsewhere and not immediately), maybe it's not worth the science cost.
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Jingking wrote:
Another viable solution is simply "be unhappy", as someone already recommended in a strategy guide.
There are only a few events which rewards happiness or punish unhappiness.
Thus, if the solution is marginal (only gives you 1 more worker elsewhere and not immediately), maybe it's not worth the science cost.
But this isn't really a long-term solution, is it? I've always interpreted this to mean building a Religion without the Age A event isn't that much better than having a discontented worker early in the game, but at some point you need to deploy workers somewhere.

You can't just have three or four discontented workers sitting around in your worker pool doing nothing all game long.
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How about player order? I've always been under the impression that going first is best (and going second better than third).

What's the distribution for 2/3/4-player games?
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