Path of Civilization is a medium-crunch point-salad Euro-style Civilization Game ¹. My group has played it a few times and one of the the things that makes it interesting is the core mechanic is you have 5 cards in your hand representing technologies. Each technology card has a left hand side which represents how the technology produces tokens that interacts with the game board (warriors, philosophers, population, military power etc.). There are 8 types and this is the primary way you score VP. It also has a right hand side that generates research points into technology which allows you to buy more advanced technology cards. There are 5 flavors of technology and they each have their own research type. This is the core of the tech tree.

Now any system with a known tech tree is going to make a gamer ask “What’s the optimal path to buy the techs in”. After the second play our group sat and discussed for a few minutes and tried to think of what the optimal tech tree would be. It’s not obvious and we spent about 30 minutes discussing it but failed ². Part of the problem is that the shared game board has leaders, wonders and philosophy levels that you can buy,in addition to end of turn Events/Wars and these all change game to game. All of these primarily are for VP, which could influence your choices, and many of them produce tech research which could speed up your purchases. But ignoring these factore I wanted to know what was the optimal tech tree?

• Game Background
  • Tech Tree
• Implementation Details
  • First few turns
    • Turn 1
    • Turn 2
  • Limiting the search space
  • Scoring
• Results & Insights
  • There are a larger number of plays within the same tech level of each other.
  • Our original belief about there being a standard opening was wrong
  • Focusing on yellow and brown early seem to help
  • Rainbow wins
  • So what’s the best sequence of moves

Game Background

Here’s a full review but the core turn sequence is

1. Place your cards
   • Choose two cards to produce the left hand side
   • Choose two other cards to produce the right hand side
   • Permanently discard the remaining fifth card
2. Left hand card produces tokens
3. Use the tokens, including saved ones from previous turn, to interact with the world including buying wonders, leaders, philosophy levels
4. Right hand cards produce technology research
5. Use the research points, including saved ones from previous turn, to buy a card. This also gives you a bonus of a a combination of tokens/tech research
6. Score for end of turn event/war. VP is scored

Experience gamers know with this type of game the key element is building your engine in the early game so that you’re producing increasingly ludicrous amounts of tokens late game to win the larger VP Wars/Events that happen later in the game. This means early to mid game optimizing for steps 4 and 5 so that you can buy the best technology cards. To the games credit, and unlike a lot of Civilization games, it makes exclusively focusing on tech a bad strategy as the other mechanics are also important. But it’s still the primary focus early to mid game.

Tech Tree

What makes the tech tree in this game so interesting is that the 5 tech colors (brown, red, blue, yellow, green) don’t make tech research points for their own color. Instead when played for their right hand research they produce research for the adjacent colors. By adjacent I mean the colors are lined up left to right and produce for their neighbors, wrapping around. So brown technologies produce research for red and green, red produces research for brown and blue etc.

brown cards makes green & red research

red cards makes brown & blue research

blue cards makes red & yellow research

yellow cards makes blue & green research

green cards makes yellow & brown research

What’s even more interesting is that it’s sometimes asymetrical.

• The level 1 blue tech costs 1 and makes 1 red and 1 yellow tech research
• The level 2 blue tech costs 4 and makes 2 red and 1 yellow tech research. You get 1 green research when buying it
• The level 3 blue tech costs 7 and makes 2 red and 2 yellow tech research. You get 1 green research when buying it
• The level 4 blue tech costs 10 and makes 2 red and 3 yellow tech research.You get 1 green research and 1 brown research when buying it

All the colors have this pattern where the research they produce less of at level 2 is the one which they produce more of at level 4. As an additional complication some of the techs each give you a one time bonus when you buy them. These are generally tokens. But buying yellow and to a lesser degree blue and green tech cards also produce tech research points in colors that they are not adjacent to.

All of this makes the tech tree more of a tech web

In addition to the tech tree you have the other cards that interact with the game board mainly as a way of scoring VP. This includes buying leaders and wonders, winning Wars, philosophy level and Events. But some of these cards also produce tech research.

Implementation Details

If you just ignore the wonders/leaders/philosophy bonus/Events/Wars though and just consider the generic tech tree cases it seems feasible to compute what the optimal play is. The 5 technologies that you start with, level 0 techs, each only produce 1 research of a single tech. This further limits the possibilities in the early game.For programmers you can find the source code here.

First few turns

Turn 1

You have just the 5 starting cards you need to choose 2 of them to play on the left hand side of the board, on the first turn this will produce nothing. Of the remaining 3 you need to choose 2 to play on the right hand side, which will produce 2 different colored research. The last card you discard. Every turn there are only 30 card placements

(5 choose 2 for the left)x(3 choose 2 for the right)x(1 choose 1 for the discard)

With the level 0 techs only producing 1 research each it means in the first turn each of these 30 possibilities can only possibly buy 2 level 1. Meaning that the first turn only has 60 possiblities³.

Turn 2

Each of those 60 would similarly have either 60, 90 or 120 possibilities. This is because there are 30 different card placements, but now you have

1. the research you didn’t use from the turn 1
2. if you place that level 1 tech on the right hand side you get 2 different tech research points
3. the level 0 tech you played as the other right hand hard.

This means you could have the ability to buy 4 different level 1 techs at turn 2. While this increase the number of possibilities, two factors decrease it.

Limiting the search space

First you have duplicate scenarios due to the level 0 techs not producing anything when played on the left hand side. Assuming all the other card placements are the same it doesn’t matter if your turn sequence is

Turn 1
   	 Discard:green0
   	 Left:blue0 collecting nothing
Turn 2
   	 Discard:blue0

or

Turn 1
   	 Discard:blue0
   	 Left:green0 collecting nothing
Turn 2
   	 Discard:green0

Either way they’re effectively the same.

Secondly, some of the moves are just bad. In the early game, when you still have starting level 0 techs buying a tech just to discard it is a bad idea. This could change in later turns, as cards could be bought just for bonus or VP, but for the first 3 turns you should not see the sequence

Turn 1
   	 Discard:yelllow0
   	 Left:brown0 collecting
   	 Left:green0 collecting
   	 Right:blue0 collecting BLUE:1
   	 Right:red0 collecting RED:1
   	 Bought:blue1 collecting PHILOSOPHY:1
   	 At end of turn have RED:1 PHILOSOPHY:1
    Turn 2
   	 Discard:blue1

Adding deduping and limiting obviously stupid moves made me think that I could limits made me initially think I could generate all the scenarios. While the code runs fine for the first few turns the number of scenarios at the start of turn 4 is 58,820 and each of those has 30 placements and ~11 different cards you can buy. People who’ve worked with Big Data know this isn’t that large but it was running fairly slow on my 8 year old computer. I didn’t want to rewrite the code to be more performant or to run on the cloud. So I had to limit the results that I evaluated for the next turn.

Scoring

I had already written a scoring function for final (turn 7) scoring. Each card in hand scored 3 points for the tech it could produce which meant.

• level 0 tech card in hand 3 points
• level 1 tech card in hand 6 points
• level 2 tech card in hand 9 points
• level 3 tech card in hand 12 points
• level 4 tech card in hand 15 points

Each remaining type of tech research scored based on if the value was close enough to buy a higher level tech on the next turn. So if you were 1 research away from buying a level 2 tech it scored you 1 point. 1 research away from level 3, 2 points etc⁴.

By limiting the possible scenarios each turn to to the top 1,000 tech scores ⁵ and then sampling 9,000 additional from the remaining scenarios I was able to run the code in a reasonable amount of time. There’s probably further optimizations that should be done to both the turn scoring and the final scoring. I’m happy to accept pull requests :)

Results & Insights

After 7 turns of simulation some interesting results played out. You can see the results for the top 1000 here and the full source code here

There are a larger number of plays within the same tech level of each other.

There are more than 500 sets of sequences that basically score within 3 points of each other using the criteria above. What this means is that it’s a well balanced game with no obvious ‘right’ move. This is a credit to the game design and means that you need to look at the Events, Wars and what the other players are doing to decide what to do.

Our original belief about there being a standard opening was wrong

To actually figure out the optimal play you would need to add in the Events and Wars at minimum. Even then it would still depend on the Philosophy Bonuses, Leaders and Wonders. The reason is that all of these give you some tech research. And this could change how early you could buy tech. In any case where you’re only a single point of research away there could be a card that gives you that research.

For example it’s easy to get 3 brown research by the end of turn 2. But you need 4 to buy the level 2 brown tech card. If there were a leader or event that gave you one more brown research in the first two turns it would change your entire game.

Focusing on yellow and brown early seem to help

Getting yellow level 1 tech (or 2 of them) and brown in early turns helps. This makes sense because they both give you bonus research points, which makes it easier to build up. An insight a friend had was that Wars only give you easy VP if no one else focuses on this. Since this simulation doesn’t take War VPs into account when ranking technological progress, it makes sense that red is underrepresented. There is a subset of the moves that takes the red level 1 tech in the first turn, which then generates brown.

Rainbow wins

Looking at the turn by turn breakdown it’s rare for the top sequence of turns to not have either all the colors or only be missing 1 color for a turn. This is more prominent in mid game (turns 3-6) while early on when insufficient tech is being produced they still are only missing a color for 2 turns. The majority of turn 7 hands had all 5 colors with a smattering of double yellow, double browns and double blues. I think double reds

So what’s the best sequence of moves

With all that being said the best move (not taking any of the Events/Wars/other bonuses into account) with a score of 55

Turn 1
   	 Discard:green0
   	 Left:blue0 collecting
   	 Left:brown0 collecting
   	 Right:red0 collecting RED:1
   	 Right:yellow0 collecting YELLOW:1
   	 Bought:red1 collecting MILITARY_POWER:1
   	 At end of turn have YELLOW:1 MILITARY_POWER:1
    Turn 2
   	 Discard:yellow0
   	 Left:blue0 collecting
   	 Left:red0 collecting
   	 Right:brown0 collecting BROWN:1
   	 Right:red1 collecting BROWN:1 BLUE:1
   	 Bought:yellow1 collecting BROWN:1
   	 At end of turn have BROWN:3 BLUE:1 MILITARY_POWER:1
    Turn 3
   	 Discard:brown0
   	 Left:blue0 collecting
   	 Left:red0 collecting
   	 Right:red1 collecting BROWN:1 BLUE:1
   	 Right:yellow1 collecting BLUE:1 GREEN:1
   	 Bought:brown2 collecting BUILDER:2 MILITARY_POWER:1
   	 Scoring for event with 0 scribes
   	 At end of turn have GREEN:1 BLUE:3 BUILDER:2 MILITARY_POWER:2
    Turn 4
   	 Discard:red0
   	 Left:blue0 collecting
   	 Left:red1 collecting WARRIOR:1
   	 Right:brown2 collecting GREEN:2 RED:1
   	 Right:yellow1 collecting BLUE:1 GREEN:1
   	 Bought:blue2 collecting PHILOSOPHY:2 GREEN:1
   	 Scoring for war with 4 warriors and MILITARY_POWER
   	 At end of turn have GREEN:5 RED:1 PHILOSOPHY:2 BUILDER:2 
        MILITARY_POWER:2
    Turn 5
   	 Discard:blue0
   	 Left:red1 collecting WARRIOR:1
   	 Left:yellow1 collecting SCRIBE:1
   	 Right:blue2 collecting RED:2 YELLOW:1
   	 Right:brown2 collecting GREEN:2 RED:1
   	 Increasing collectable limit to 6
   	 Bought:green3 collecting BLUE:1 POP:3 BUILDER:1
   	 Scoring for event with 1 scribes
   	 At end of turn have YELLOW:1 BLUE:1 RED:4 WARRIOR:1 
        AMBASSADOR:1 PHILOSOPHY:2 BUILDER:3 MILITARY_POWER:2 POP:3
    Turn 6
   	 Discard:yellow1
   	 Left:brown2 collecting BUILDER:1 AMBASSADOR:1
   	 Left:red1 collecting WARRIOR:1
   	 Right:blue2 collecting RED:2 YELLOW:1
   	 Right:green3 collecting YELLOW:2 BROWN:2
   	 Increasing collectable limit to 7
   	 Bought:yellow2 collecting BROWN:1 RED:1 POP:1
   	 Scoring for war with 6 warriors and MILITARY_POWER
   	 At end of turn have BROWN:3 BLUE:1 RED:7 
        AMBASSADOR:2 PHILOSOPHY:2 BUILDER:4 MILITARY_POWER:2 POP:4
    Turn 7
   	 Discard:red1
   	 Left:green3 collecting AMBASSADOR:2 SCRIBE:1
   	 Left:yellow2 collecting PHILOSOPHY:1 SCRIBE:1
   	 Right:blue2 collecting RED:2 YELLOW:1
   	 Right:brown2 collecting GREEN:2 RED:1
   	 Bought:red4 collecting MILITARY_POWER:4 PHILOSOPHY:2 WARRIOR:1
   	 Scoring for event with 2 scribes
   	 At end of turn have BROWN:3 GREEN:2 YELLOW:1 BLUE:1 WARRIOR:1 
        AMBASSADOR:4 PHILOSOPHY:5 BUILDER:4 MILITARY_POWER:6 POP:4
Hand: ['blue2', 'brown2', 'green3', 'red4', 'yellow2']
Tech:
 BROWN:3 GREEN:2 YELLOW:1 BLUE:1
Cubes:
 WARRIOR:1 AMBASSADOR:4 PHILOSOPHY:5 BUILDER:4 MILITARY_POWER:6 POP:4

⁶

So at the end of turn 7 you have the level 2 tech card for blue, brown and yellow, the level 3 for green and the level 4 for red. In addition to that you have tech research and cubes shown. Again while this would be the optimal play without the Events/War/Leaders/etc it almost certainly wouldn’t be with them. By turn 3 you easily could have gathered an additional Blue research and had the choice to buy the blue level 2 tech card early etc.

Thanks to Conrad, Josh, Rich and Lucas for discussions that led to this