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Subject: Possible game balance fix for 4 player game: P3 to start with indigo instead of corn rss

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Chris Gibbs
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As part of my work in improving and adapting Tony Mitton's Puerto Rico Evolver in order to create some top-notch AI players for Tropic Euro (which are complete by the way, and should be online in around 1-2 weeks), I've also managed to speed it up by a factor of over 40 times, thus enabling me to quickly test out AI strategies over millions of games.

The 3 player game so far seems to be pretty much spot-on in terms of balance as the best players for each position win roughly the same amount of games. However, for 4 players it's a different story - there is a clear advantage for players in positions 3 and 4, with the best players for P3/P4 (P3 in particular) having nearly twice the chances of winning as P1.

Of course, this could just be down to my particular set of evolved strategies being more in favour of shipping and thus benefitting players that start with corn, especially as I've added some new advanced shipping evolution options to the evolver.

For those PR players amongst you with plenty of tournament experience: are P1 and P2 always considered to be a disadvantage, and is P3 generally considered to be the best starting position in tournament play?

I was curious to see how changing the starting plantation might affect things, so I ran some tests with the same set of evolved players except with P3 starting with indigo. This results in P1/P2/P3 becoming much more balanced (roughly equal chances of winning), but the only issue with this is now that P4 has an advantage over the other three positions (although not as large as the advantage of P3 over P1 in the standard setup).

Would anyone be interested in playing Tropic Euro with an option to swap the starting plantation for P3 from corn to indigo? It would be able to be set on a per-game basis.





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Werner Bär
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TropicEuro wrote:
For those PR players amongst you with plenty of tournament experience: are P1 and P2 always considered to be a disadvantage, and is P3 generally considered to be the best starting position in tournament play?
Let me quote a very experienced PR player:
jimc wrote:
In 4-player, seat 3 is strongest, followed by seat 1 and/or seat 4, with seat 2 being clearly the worst. The difference in strength between 2 and 3 is only about three points, though. By playing well or badly, players can readily overcome a three point disadvantage.

Seat 3 should win most, by a small amount. But if seat 4 wins much more often than seat 1, your AI's must act very one-sided.
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Chris Gibbs
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Werbaer wrote:
Seat 3 should win most, by a small amount. But if seat 4 wins much more often than seat 1, your AI's must act very one-sided.


True, although it's possible that the best seat 4 players have evolved some strategies that have not been used that much before (such as very aggressive shipping). Also, I've tested out the supposedly weaker P1 against the existing Tropic Euro AI and it beats it comfortably every time.

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Andy Leber
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Here's an interesting article I read a short while back, with some interesting comments at the bottom.

http://www.examiner.com/board-game-in-national/puerto-rico-b...


The stats apparently show that from every Puerto Rico tournament at the World Boardgaming Championships since they began in 2002, the winning percentage for seats is as follows:

1st seat: 19.9%
2nd seat: 19.5%
3rd seat: 30.8%
4th seat: 29.7%

They actually offer a bidding system now, where you can bid victory points at the beginning of the game to "buy" your starting position.

John Weber, who is mentioned in the article, actually comments at the bottom and says in the last 2-3 years the gaps between 2nd seat and 3rd seat have actually shrunk to where it's non-existant.


Edit: He also mentions other smaller things that can potentially affect the outcome. Such as an early corn allowing another player to greatly even themselves out with P3 or P4.

Another interesting one is that the average 4p game lasts 15 rounds, and who is the Governor in round 15? Player 3.

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Chris Gibbs
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Thanks - interesting article.

I've just done some further testing just playing the best evolver player so far for each position against each other (as opposed to putting them in a pool of 50-100 players for each position that are picked at random).

Without corn/indigo swap:

P1: 30.3%
P2: 12.5%
P3: 18.4%
P4: 38.7%

With corn/indigo swap for P3:

P1: 14.7%
P2: 24.3%
P3: 38.1%
P4: 22.7%

Each player sticks to a fairly tight strategy so I can't really conclude too much from these figures, especially as the win percentages are quite a bit different from when 50+ players from each position are played randomly against each other.

Perhaps the best source of stats will be to just to record outcomes from each game on the TE server involving human players. I'll also be using that data to replace underperforming AI players with new evolved players, so the AI should get better and better over time. Plus, each position for every size of game (2 to 5 players) will have a set of the "best" players so that should make it much more challenging, as the strategies for each player will be different (and with no indication to the user which particular evolved player is currently playing).

The players I'm gonna be watching fairly closely are the ones for P1 for a 4 player game which start with builder/hospice, quite a few of the best evolved players so far use that strategy instead of settler/quarry so will be interesting to see how they get on against human opposition.

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Chris Gibbs
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Ignore those figures for a single AI player above, I just tried the much better approach of taking an average win percentage across all players in each position:

Without corn/indigo swap:

P1: 20.7%
P2: 18.9%
P3: 32.3%
P4: 28.7%

This is now very close indeed to the values reported by John Weber.

With corn/indigo swap for P3:

P1: 23.1%
P2: 21.3%
P3: 21.8%
P4: 33.6%

This seems to be too much of an advantage to P4, even though it balances out the other 3 players almost exactly. Perhaps a fixed handicap of say 1VP or 2VP could be applied to P4 to achieve an almost perfectly balanced 4P game?
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Andy Leber
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TropicEuro wrote:
Ignore those figures for a single AI player above, I just tried the much better approach of taking an average win percentage across all players in each position:

Without corn/indigo swap:

P1: 20.7%
P2: 18.9%
P3: 32.3%
P4: 28.7%

This is now very close indeed to the values reported by John Weber.

With corn/indigo swap for P3:

P1: 23.1%
P2: 21.3%
P3: 21.8%
P4: 33.6%

This seems to be too much of an advantage to P4, even though it balances out the other 3 players almost exactly. Perhaps a fixed handicap of say 1VP or 2VP could be applied to P4 to achieve an almost perfectly balanced 4P game?


Interesting. Even though the overall win percentages are higher, the article I quoted earlier did say the point spread at the end of the game was only 2-3 points... so you may be on to something there.
 
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TropicEuro wrote:
Ignore those figures for a single AI player above, I just tried the much better approach of taking an average win percentage across all players in each position:

Without corn/indigo swap:

P1: 20.7%
P2: 18.9%
P3: 32.3%
P4: 28.7%

This is now very close indeed to the values reported by John Weber.

With corn/indigo swap for P3:

P1: 23.1%
P2: 21.3%
P3: 21.8%
P4: 33.6%

This seems to be too much of an advantage to P4, even though it balances out the other 3 players almost exactly. Perhaps a fixed handicap of say 1VP or 2VP could be applied to P4 to achieve an almost perfectly balanced 4P game?


Try it! I'd try it with starting doubloons first, though. Do the analysis all over but instead of giving P4 3 doubloons to start, give him 1 or 2 depending on your results. You might even find a balance between tweaking his starting plantation and the doubloon adjustment.

I don't think I would do the VP handicap, though.
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Chris Gibbs
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markgravitygood wrote:

Try it! I'd try it with starting doubloons first, though. Do the analysis all over but instead of giving P4 3 doubloons to start, give him 1 or 2 depending on your results. You might even find a balance between tweaking his starting plantation and the doubloon adjustment.


Good idea - I've just done 4 more different configurations and tested them over a million games, here are the results:

1) Corn/Indigo swap for both P3 and P4 (so all players start with indigo):

P1: 27.0%
P2: 25.3%
P3: 25.3%
P4: 22.4%
(highest - lowest = 4.6%)


2) Corn/Indigo swap for P3, P4 starts with 2 coins:

P1: 25.4%
P2: 23.7%
P3: 24.4%
P4: 26.5%
(highest - lowest = 2.8%)


3) P3 and P4 start with 2 coins:

P1: 25.6%
P2: 23.6%
P3: 27.0%
P4: 23.6%
(highest - lowest = 3.4%)


4) Corn/Indigo swap for P3, P4 starts on -2 VP:

P1: 26.3%
P2: 24.3%
P3: 24.7%
P4: 24.7%
(highest - lowest = 2%)


So the 4th config is possibly the closest to perfect balance that can be achieved, but perhaps the 3rd config is the most elegant solution.

Also, it might be a bit disconcerting for the 4th player to start on -2 VP - the alternative is to take off 2VP at the end of the game, but this would be slightly more beneficial to P4 as the 2VP would be docked after any Customs House bonus.

A bonus of the 3rd config is that I could just change the rule set permanently without having to add a separate option, as it requires only seemingly minor adjustment to the game (i.e. no plantation changes).
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Chris Gibbs
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It also turns out that the 3 player game is similarly unbalanced, I was only looking at the best players before, but now using the better average method of all players:

P1: 29.7%
P2: 28.7%
P3: 41.6%

Now with P3 starting with 1 coin:

P1: 33.5%
P2: 32.7%
P3: 33.6%


Looks like we have a winner! I'll do some further tests on the 5p game and if the same applies then it would be a very simple and elegant solution to achieve much better balance across all game sizes:

1) All players that start with corn have 1 coin less
2) Factory / University price swap
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TropicEuro wrote:
markgravitygood wrote:

Try it! I'd try it with starting doubloons first, though. Do the analysis all over but instead of giving P4 3 doubloons to start, give him 1 or 2 depending on your results. You might even find a balance between tweaking his starting plantation and the doubloon adjustment.


Good idea - I've just done 4 more different configurations and tested them over a million games, here are the results:

1) Corn/Indigo swap for both P3 and P4 (so all players start with indigo):

P1: 27.0%
P2: 25.3%
P3: 25.3%
P4: 22.4%
(highest - lowest = 4.6%)


2) Corn/Indigo swap for P3, P4 starts with 2 coins:

P1: 25.4%
P2: 23.7%
P3: 24.4%
P4: 26.5%
(highest - lowest = 2.8%)


3) P3 and P4 start with 2 coins:

P1: 25.6%
P2: 23.6%
P3: 27.0%
P4: 23.6%
(highest - lowest = 3.4%)


4) Corn/Indigo swap for P3, P4 starts on -2 VP:

P1: 26.3%
P2: 24.3%
P3: 24.7%
P4: 24.7%
(highest - lowest = 2%)


So the 4th config is possibly the closest to perfect balance that can be achieved, but perhaps the 3rd config is the most elegant solution.

Also, it might be a bit disconcerting for the 4th player to start on -2 VP - the alternative is to take off 2VP at the end of the game, but this would be slightly more beneficial to P4 as the 2VP would be docked after any Customs House bonus.

A bonus of the 3rd config is that I could just change the rule set permanently without having to add a separate option, as it requires only seemingly minor adjustment to the game (i.e. no plantation changes).


For me, I'd go with #2 as I am not for VP handicaps - getting to the end of the game knowing you are in the hole 2VP is a little much, and the difference is only .8% - nominal.
 
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Chris Gibbs
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Surely #3 is the best overall, as it's almost as good as #2 and is much more elegant and can be applied identically to 3p and possibly 2p/5p games as well?

Also note that the figures are accurate only for my particular set of "best" players in each position, and might differ by 1-2% from the stats for another set and for all human players as a whole (i.e. #3 could be better than #2 for a larger data set)
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Chris Gibbs
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Some of the best 2p and 5p AI players are still in development, but my preliminary tests for the 2 player game show it to be unbalanced:

P1: 43.5%
P2: 56.5%

Now with P2 starting with 2 coins instead of 3 achieves almost perfect balance (within the accuracy of my preliminary test):

P1: 48.1%
P2: 51.9%


And for the 5 player game:

P1: 18.4%
P2: 17.4%
P3: 20.9%
P4: 22.5%
P5: 20.9%
(highest - lowest = 5.1%)

Now with P4 and P5 starting with 1 less coin:

P1: 19.4%
P2: 19.2%
P3: 21.9%
P4: 20.9%
P5: 18.7%
(highest - lowest = 3.2%)


So, overall it looks as though a very elegant fix indeed can be achieved across all game sizes by removing 1 coin from any player starting with corn.

Intuitively this kinda makes sense anyway, as it costs the players with indigo 1 more than the corn players to start production (and/or a production delay of 1-2 turns vs corn).
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TropicEuro wrote:



3) P3 and P4 start with 2 coins:

P1: 25.6%
P2: 23.6%
P3: 27.0%
P4: 23.6%
(highest - lowest = 3.4%)



What about the 3rd option, but P3 starts with 1 coin, P4 with 2 coins. How do those numbers pan out?
 
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Chris Gibbs
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KeyserSoze wrote:
What about the 3rd option, but P3 starts with 1 coin, P4 with 2 coins. How do those numbers pan out?


Good suggestion, here are the results for P3 (1 coin), P4 (2 coins):

P1: 28.2%
P2: 25.9%
P3: 20.2%
P4: 25.7%
(highest - lowest = 8%)

So this puts P3 at too much of a disadvantage. It also might confuse some players as P3 might wonder why they should start with 1 less coin than P4 who seemingly is in a similar starting situation.

Also note that since I'm using one particular set of around 150 "best" evolved players in each position, it might not be completely representative of all types of players using every strategy possible, so the values are probably accurate to +/- 1% or so - i.e. it's possible that the 1 coin fix is in general even more effective than my stats suggest, and it's almost certain that it's more balanced than the standard setup.

I'm still a bit stunned that the 1 coin fix seems to work almost equally as well for all game sizes
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TropicEuro wrote:


Good suggestion, here are the results for P3 (1 coin), P4 (2 coins):

P1: 28.2%
P2: 25.9%
P3: 20.2%
P4: 25.7%
(highest - lowest = 8%)

So this puts P3 at too much of a disadvantage. It also might confuse some players as P3 might wonder why they should start with 1 less coin than P4 who seemingly is in a similar starting situation.

Also note that since I'm using one particular set of around 150 "best" evolved players in each position, it might not be completely representative of all types of players using every strategy possible, so the values are probably accurate to +/- 1% or so - i.e. it's possible that the 1 coin fix is in general even more effective than my stats suggest, and it's almost certain that it's more balanced than the standard setup.

I'm still a bit stunned that the 1 coin fix seems to work almost equally as well for all game sizes


Thanks. It really made a bigger swing against the P3 than I'd expected.
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Chris Gibbs
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For creating the final set of "best" players to go into TE, I tested a wider range of different players/strategies over a much larger number of games (25 million) and the results are in:

2 Players
P1: 48.9%
P2: 51.1%

3 Players
P1: 32.5%
P2: 32.8%
P3: 34.7%

4 Players
P1: 26.1%
P2: 23.5%
P3: 27.2%
P4: 23.2%

5 Players
P1: 19.6%
P2: 19.4%
P3: 19.6%
P4: 20.7%
P5: 20.7%

This proves beyond a shadow of a doubt that the 1 coin fix is more balanced, and creates almost perfect balance for the 2p, 3p and 5p games.

Also note that in most cases the players that start with corn still have an advantage over the indigo players, although the advantage is minimal.
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So will there be an official rule change in the next print of PR? ;)
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Chris Gibbs
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Bernaar wrote:

So will there be an official rule change in the next print of PR?


I've contacted the board game publishers
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Juha K
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Thanks guys!
This is extremely useful info.
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TropicEuro wrote:
Bernaar wrote:

So will there be an official rule change in the next print of PR?


I've contacted the board game publishers


Was there ever a response?
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TropicEuro wrote:
Bernaar wrote:

So will there be an official rule change in the next print of PR?


I've contacted the board game publishers


Would be awesome if they accepted the change. AND made the Factory-University cost swap official. thumbsup

Nice work, Chris. While I'm not in favor of making every little change for change's sake (too often these are insufficiently playtested and/or are just trying to make up for strategic weaknesses from the players crying for them), even near-perfect games like PR still have a few obvious flaws to iron out. Seat position would seem to be one of those, and it looks like you researched this one thoroughly.

I could see my group adopting the '-1 doubloon for players starting with corn' change, which would be only the second change we've accepted to the game, along with the Factory-Uni cost swap. Kudos. cool


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Holmes108 wrote:

John Weber, who is mentioned in the article, actually comments at the bottom and says in the last 2-3 years the gaps between 2nd seat and 3rd seat have actually shrunk to where it's non-existant.


You may've misread that. What Weber actually says is that the gap between Player 3 and Player 4 (both starting corn players) is what's shrunk:

Andy, this is John Weber, the guy mentioned in Skip's original article. I have a couple of theories about why the stats say what they say, keeping in mind that over the last 2-3 years, the gap between the 2d best seat (#4 or 2d corn) and the supposed best seat (#3 or first corn) has shrunk to where it's non-existent.

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SystemShock wrote:
Holmes108 wrote:

John Weber, who is mentioned in the article, actually comments at the bottom and says in the last 2-3 years the gaps between 2nd seat and 3rd seat have actually shrunk to where it's non-existant.


You may've misread that. What Weber actually says is that the gap between Player 3 and Player 4 (both starting corn players) is what's shrunk:

Andy, this is John Weber, the guy mentioned in Skip's original article. I have a couple of theories about why the stats say what they say, keeping in mind that over the last 2-3 years, the gap between the 2d best seat (#4 or 2d corn) and the supposed best seat (#3 or first corn) has shrunk to where it's non-existent.



Nice catch, thank you! It's been so long since I wrote that, I'm not sure anymore if I misread the original response I got in those comments, or if I just mistyped when summarizing it here... lol. I think I did misread it the first time though.

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