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Subject: Twilight Imperium: Our leader board rss

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Christopher Halbower
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The Need for Statistical Reporting
We play Twilight Imperium. A lot. The idea of compiling our data and seeing what falls out has been bouncing around for a long time. So we started doing so on April 23, 2013. We have cataloged 17 games of Twilight Imperium since then, 15 regular games, two Fall of the Empire games. What follows will be a brief session report of our last game (yesterday) along with a "statistical" session report of all seventeen games. If users here on BGG enjoy following our statistical reportings (hereafter called, "leader board"), then I'll be inclined to keep you updated about our games and stats.

Rules Selection
First, the rules. We use the base game rules plus most of the add-ons from Shattered Empire (SE) and Shards of the Throne (SotT). We "roll for Speaker" ahead of time using a RNG. Also, everyone gets to pick the rule set for the game once. Once everyone has had their rule set played, then players are eligible to have their rule set played again.

The most common rules we play with are
SE Strategy Cards instead of base game (although the chosen player could choose either)
• Leaders about half the time
• Facilities about 20% of the time
• Flagships/Mercs/new races 100% of the time
• Simulated Early Turns about 10% of the time
• Pre-set maps about 25% of the time
• SE Objectives about 80% of the time

We also allow the chosen player to pick any house rules they want to implement.
• Memoryless space docks*
• Culled political deck
• Normalized Mech units (Mechs can be hit by PDS, bombardment...)
(*I'm not a huge fan of Memoryless space docks; we used it because people were new and it was a crutch; I'm hoping we move away from this in future games).


For race selection, we do something that is noncannon but is required to ensure statistical reliability: race selection/assignment.

Race selection/assignment
Players are offered their choice of two different races. The races must be races they have not yet played. Once a player has played all the other sixteen races, his seventeenth game will require him to play the only race he has not yet played. Once a player has played all races once, the process begins anew.

This ensures that everyone plays every race. You cannot just get into a comfort zone and shift into cruise control. You also will see every race played by every OTHER PLAYER. Both of these reasons are required to ensure the statistical reliability of our data. If Jon plays the Naalu Collective three out of four times, the data will be less representative of Twilight Imperium in general and less representative of Jon's skill set specifically.

Brief Session Report
Now that the rules for our leader board have been more or less established, what happened yesterday?

There were six of us: Chris, Dusty, Matt, Mongo, Zak and Kevin.

I was given the choice of the Barony of Letnev or the Emirates of Hacan. I chose the Baron. I'll save the Kitties for an 8 player game.

Dustin Shunta
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has yet to win a game since our leader board was implemented. He chose the Brotherhood of Yin.

Matthew Spencer
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chose the Universities of Jol Nar.

Mike Closz
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was the Arborec.

Kevin chose the bugs from Quinnara. Zak took on the role of the Ghosts of Creuss.

In the opening rounds, Dusty made his move to score his secret objective, "Threatening". He played a Sabotage against Matt when Matt played a seemingly benign "Ghost Ship". Dusty needed the space lanes close to his home system to be opened to that sector of space in order to score Threatening--well worth a precious Sabotage.

I thought I would be able to score my SO very easily (Usurper: 6 dudes on Mecatol Rex with a dock). I even had Hope's End in my hand. I thought I could put Hope's End next to Mecatol Rex and fart out troops every round and score my objective. But during galaxy creation, Matt placed an empty system where I wanted to put Hope's End. I lost focus after that and didn't strive for my secret objective like I should have. BIG MISTAKE! I won't make these mistakes next time.

Zak bought Hil Colish early. He move towards Mecatol Rex but didn't make his move when he could/should have.

Kevin fomented war in the galaxy when he took Bureaucracy. He purposely played warlike Objectives like "I won a space battle..." His reasoning: he's the Sardakk N'orr and war plays to his strength. Kevin didn't get Warsuns and chose to get extra carriers and dreadnaughts which would come back to haunt him.

Matt made some strong plays, keeping a nice pace with Dusty while flying under our proverbial radar. He was a strong threat to take the ancient throne.

Mike "Mongo" of the Arborec was too focused on Dusty's pesky diplomat on Primor (a system Dusty took to complete "Threatening") to achieve his secret objective. He wasn't in a strong position to push for his own win.

On the 2nd to last game round, Matt took Bureaucracy and played, "I now spend 10 resources or influence", in lieu of the first Stage II Objective.

Dusty took Assembly so he would be able to take Bureaucracy next round. When the next round came to be: Matt informed us that the top Stage II Objective (the objective he saw but declined to play) was, in fact, Imperium Rex. Dusty had 6 points all of them hard points. Everyone else had fewer. If Dusty played Bureaucracy we would force the end of the game and win.

He chose Bureaucracy. Matt played, "Public Disgrace". Dusty cancelled with his agent (his second "Sabotage!")

Dusty could still win because he had more points then us. We could not defeat him by taking his home world. Our only recourse was to eliminate him. We planned his elimination.

I would take one system. Matt would take one. Kevin would take one. And Mongo would act last, attacking where we may fail.

1. I succeeded in taking one of Dusty's planets.
2. Matt succeeded in taking two of Dusty's planets!
3. Kevin played, "Flank Speed" to move his carriers and dreadnaughts into Dusty's home system. What's this? Dusty had ANOTHER SABOTAGE! Kevin didn't research Warsuns and still had sloooooooow ships. He couldn't attack Dusty.
4. Mongo was our last hope. He moved 10 ground forces to Dusty's home world. He dropped all of them onto Darien which was defended by 2 PDS's and 4 mech units. When the dust cleared, Dusty still controlled his home system.

Was the top card really "Imperium Rex" or was this a clever ruse by Matt to make us stomp Dusty?



It was really Imperium Rex. Dusty won.


Statistics from our Leader Board

Here is the most straight forward chart. Which races have won.
The Federation of Sol, the L1Z1X Mindnet and the Nekro Virus(!) have all gotten three wins.

Amazingly, the Yssaril Tribes, considered the most powerful race here on BGG, has only one win.



By average score, however, the Yssaril tribes are near the top of the chart. Only the Federation of Sol has a higher average. This suggests the Yssaril Tribe are indeed a strong race but that the players have a strong say in the outcome.



In this chart, we see the deviation (the average difference between the race's score and the winning score). This tells us how well the race fairs overall.

The Yin, Sol, Mindnet and Tribes are at the top. The Emirates of Hacan are at the bottom! However, there was one game we played that skewed this data. In that game, the Hacan player was skunked.

Let's remove that data point and look at the data



Now the bottom races are the Baron, the Hacan, the Clan and the Winnu. Perhaps there is some group think problems in our group? The Emirates of Hacan are a strong race yet they don't do so well in our games.

Going Forward
We are not quite to the point where anyone has played all 17 races. I have the most plays so far with 13. So in the coming months, we will have an entire "TI3 Cycle" completed where one or more of our group has played all the races once.

It will be interesting to see how the races play at that point. It will be interesting to see how often we score our SO's (a stat that we also track). Will we play these races better the second time? Will the average scores improve?

Please comment if you have any questions or comments. I am definitely looking for feedback. I think this data could be very useful to our community.

-Chris

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Christian Smith
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What is your group's perception in general about which races are strong? Perhaps it is over compensating for the Hacan. I'm also very surprised to see Saar so low.
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Necessary Evil
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I can provide data on the games played at the WBC each year. However I am not sure how statistically relevant they are since I allow for some choice in race (draw 3 pick one) and the "easy" races tend to get chosen more often.

I will say that the Xxcha do have one world championship to their name.

-M
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David Gagner
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I love statistics!

Do you keep track of player records as well?

In my group I keep track of my win stats, but not races played.
Nobody else keeps track of stats in my group.

I've always wished I could go back and get the stats for races played as well as I have almost 300 plays logged...

With that, I'm right near 50% win for my group.

I try to select each of the races, also.
It might be interesting to see which races are my weaker ones and so forth, but from the stand point of just my win/loss it seems like any race will do.

I would love to get to the World Boardgaming Championships eventually to play Twilight Imperium.
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Christopher Halbower
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CasMat wrote:
What is your group's perception in general about which races are strong? Perhaps it is over compensating for the Hacan. I'm also very surprised to see Saar so low.



Many of our group is on BGG so they can respond to their own perceptions of the Hacan.

Myself, I believe them to be a strong race. I'd have to dig into the stats a little more to see some trend. Perhaps they are strong in 6 player games but we've only played them in 8 player games. Something like that. I'd have to review the details to know for sure.

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Christopher Halbower
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DavidG55311 wrote:

Do you keep track of player records as well?


I do. I didn't post that information because I don't think most people here would care to know the win-loss records for Ben, Jon or our other TI3 regulars.

Perhaps I should upload the file so everyone can review the data.
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Roger Reisinger
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Very interesting, glad you posted your data. Your groups results are very different than my groups perception.

Our Strongest races:

Yssaril
Hacan
L1Z1X
Jol Nar
Naalu
Mentak
Barony
Embers
Cruess

Weakest:

Sol
Yin
Xxcha
Winnu
Norr
Saar
Nekro
Arborek

When we play we ban Yssaril and deal 2 races per player, we always play with 6.
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Jim Lederer
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I won as the Hacan last time I played.

Player skill is a HUGE factor. For example, one of my good friends who plays has never won a game, and generally is never in contention. He just doesn't focus on completing an objective every round, nor on taking e.g. the Imperium II card (when we play with that and Age of Empires, which we typically do when new players join the game) so he can claim multiple objectives in a single round (as well as potentially getting a point for holding Mecatol Rex). He also fails to complete his secret objective.

Jim
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Christopher Halbower
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I just reviewed the data a bit more in depth. So far, the Emirates of Hacan have been played five times. So far, three of the five times were by newbis (
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David Gagner
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It may not be interesting until there is more data, but I would be interested to see who has the greater win % and whether they are using "top tier" races or if it is more the player.

The regulars in my group have all played well over 100 games each, so none of us are strangers. We do have new people who come and go as the years roll on though. I would say anyone who has played over 100 games will do better with any race than nearly all newer players would be better races.

I do feel that this game is extremely well done. Nearly everything you do has a counter reaction to prevent/stop/react to it. Sometimes its just, "if you attack him, I attack you" type threats. That's why this game is great.

halbower wrote:
DavidG55311 wrote:

Do you keep track of player records as well?


I do. I didn't post that information because I don't think most people here would care to know the win-loss records for Ben, Jon or our other TI3 regulars.

Perhaps I should upload the file so everyone can review the data.
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bob mackenzie
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yssiral tribe, is extremly powerful if played correctly... we had a noob play him yesterday and because of his skill, didnt know how to utilise the race to its potential. However i quite like what you have done with race selection.

I would actually put the races wins down to players skill and game day luck then strength of them alone. also have you guys thought about placing 'high tier' races in one game and then 'lower tier' races in another game? to make it more of an evenly balanced match?

looking good soo far though
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DavidG55311 wrote:
I would love to get to the World Boardgaming Championships eventually to play Twilight Imperium.

YES YES!!!!

Oh, and I find these statistics fascinating, but haven't played the game enough to add much to the discussion.

Chris, have you ever played with a variant race?
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Shawn Garbett
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I am a professional statistician. Send me raw data. I will run the numbers.
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Christopher Halbower
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akinfantryman wrote:
DavidG55311 wrote:
I would love to get to the World Boardgaming Championships eventually to play Twilight Imperium.

YES YES!!!!

Oh, and I find these statistics fascinating, but haven't played the game enough to add much to the discussion.

Chris, have you ever played with a variant race?


I have not.
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Christopher Halbower
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CyberGarp wrote:
I am a professional statistician. Send me raw data. I will run the numbers.


I'll upload the file to the file area.
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Dustin Shunta
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CasMat wrote:
What is your group's perception in general about which races are strong? Perhaps it is over compensating for the Hacan. I'm also very surprised to see Saar so low.


My personal opinion is that the races with stall mechanics are generally better than the other races. The ability to stall is powerful and lets you set up situations where it is difficult to retaliate.

I also believe that races that rely on their race specific technology are in a bit of a pickle. If a game only lasts 5 or 6 turns and you spend resources obtaining that technology it can make it difficult to obtain colored technologies for objectives.

Although I believe some of the races are inherently more powerful than others, I do not believe that the power disparity is very large. I always feel like I have a shot when I start the game.

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CasMat wrote:
What is your group's perception in general about which races are strong? Perhaps it is over compensating for the Hacan. I'm also very surprised to see Saar so low.

+1 to everything jimbov said.

Galaxy setup is more important in most games than race choice, but stalling is very useful for scoring objectives. Of course, my favorite race is the Naalu, 'cause I'm a seize the initiative kind of guy. Except when I seize something else, like warfare II. Then I'm more of a seize the warfare II kind of guy.

I haven't bought a single race tech in I don't know how many games. I generally go tech-light in general unless the objectives force me to do otherwise. Ships are just more important than tech in most instances.

I think that all of the disparities in TI3 (galaxy setup, Secret Objective, races, etc.) can be overcome by good negotiation. In fact, in a recent game, I came within a gnat's ass of winning with the Saar after a neighbor, on a stardate which will live in infamy, blew up both of my space docks on turn 3. I talked the rest of the table into letting me rebuild on Mecatol, played people against each other, and in the end lost only due to a misplay on timing that I'm still kicking myself over.

I think that Hacan have the potential to be amazeballz, as lots of Trade Goods should be helpful for achieving most objectives, but we haven't seen them played really well since we started collecting data. I bet they'd have a stronger showing if we could include games from 2012 and earlier.
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Christopher Halbower
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CyberGarp wrote:
I am a professional statistician. Send me raw data. I will run the numbers.


I tried to upload the data to the file section but my request was denied.
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Christopher Halbower
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I've thought some more about the data and what it means. I've come to a few conclusions.

1. Our race selection process versus the rules-as-written has a huge effect on the game. Because everyone will have to cycle through all 17 races before being allowed to play the same race a second time, the data will have more meaning that simply dealing out a race at random. This will help prevent group-think also.

2. I think more groups do not use the rules-as-written for race selection. Is this true? Do groups deal out a couple of races and allow players a little bit of choice? I know it's not RAW, but I really believe this is one house rule that MOST players use: players have a little bit of choice in what race they play.

If this is true, then players are more likely to choose Yssaril Tribes or the Emirates of Hacan over the Xxcha Kingdom or the Brotherhood of Yin. This will aggravate the current point-of-view that the Yssaril Tribes et. al. are overpowered since they are not just straight-up powerful but also more chosen than weaker races.

Am I off base in my assumption about how the race selection process of other groups?
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Shawn Garbett
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I went through and created a statistical model. First of all, the term "statistically significant" means that there is enough data to discern a difference. This is commonly misinterpreted as denoting size, as significant in common parlance means that. In statistics, significant just means the signal is stronger than the noise. I will use the term "discernible" to mean statistically significant henceforth to hopefully eliminate confusion. My apologies to Fisher.

The problem statistically with this data set, is that while 95 results seems like a lot, there's a lot of possible levels of each factor (person / race). This means that overall, it's hard to say if it's just random luck or an actual effect without probably around 200 results.

I used R, and henceforth will take a fairly liberal alpha value (it is a game after all), of 0.1. I first constructed the model.


setwd("~/Projects/game-stats")

> d = read.csv("ti3-leader.csv")

> m = lm(Raw.Score ~ Race + Player + Strategy.Card, data=d)

> summary(aov(m))
Df Sum Sq Mean Sq F value Pr(>F)
Race 16 76.49 4.781 1.538 0.1159
Player 16 121.85 7.616 2.449 0.0063 **
Strategy.Card 1 4.59 4.592 1.477 0.2290
Residuals 61 189.66 3.109
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
14 observations deleted due to missingness


This says that player skill makes a discernible impact on final score (0.0063 < 0.1). However, race does not have a discernible impact. The Strategy Card choice does not have a discernible impact. But, due to external knowledge of game rules, we know it does.


> summary(m)

Call:
lm(formula = Raw.Score ~ Race + Player + Strategy.Card, data = d)

Residuals:
Min 1Q Median 3Q Max
-3.6740 -0.9460 0.1885 0.9356 3.1037

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.93445 1.62892 3.643 0.000558 ***
RaceBarony of Letnev 0.22472 1.32738 0.169 0.866124
RaceBrotherhood of Yin 1.47245 1.37630 1.070 0.288898
RaceClan of Saar 0.02738 1.17722 0.023 0.981523
RaceEmbers of Muatt 0.87704 1.16367 0.754 0.453939
RaceEmirates of Hacan 1.20590 1.37422 0.878 0.383653
RaceFederation of Sol 2.99544 1.26062 2.376 0.020649 *
RaceGhosts of Creuss 0.06455 1.28825 0.050 0.960198
RaceL1Z1X Mindnet 1.65650 1.21085 1.368 0.176315
RaceMentak Coalition 1.01226 1.24049 0.816 0.417669
RaceNaalu Collective 1.10917 1.10972 1.000 0.321497
RaceNekro Virus 0.46738 1.12659 0.415 0.679699
RaceSardakk N'orr 0.61092 1.23285 0.496 0.622001
RaceUniversities of Jol Nar 1.25406 1.15369 1.087 0.281316
RaceWinnu -0.03689 1.29677 -0.028 0.977399
RaceXxcha Kingdom 1.43358 1.14805 1.249 0.216542
RaceYssaril Tribes 2.23165 1.20277 1.855 0.068369 .
PlayerBen Burkholder 0.26833 1.47386 0.182 0.856140
PlayerBubba Bartels -7.14035 2.45038 -2.914 0.004983 **
PlayerCasey Bousho -6.18851 2.21531 -2.794 0.006957 **
PlayerChris Halbower -0.29101 1.44065 -0.202 0.840592
PlayerDavid Mitchell -2.05920 1.85222 -1.112 0.270610
PlayerDusty Shunta 0.01266 1.44848 0.009 0.993055
PlayerJeremy Scott Pyne -0.77372 2.02999 -0.381 0.704421
PlayerJoe Veldhuis -2.18174 1.65230 -1.320 0.191627
PlayerJon Horne -0.71922 1.59908 -0.450 0.654471
PlayerJoshua Hannebohn -2.97531 2.52980 -1.176 0.244123
PlayerKevin Raliegh -1.87814 1.48507 -1.265 0.210796
PlayerMatt Bigham -1.97971 1.65957 -1.193 0.237524
PlayerMatt Spencer -0.50557 1.46473 -0.345 0.731162
PlayerMongo Closz -0.72779 1.44011 -0.505 0.615121
PlayerRocky Thompson 0.14978 1.54391 0.097 0.923036
PlayerZack Anderson -2.98989 1.71389 -1.745 0.086107 .
Strategy.CardImperial II 0.81613 0.67158 1.215 0.228953
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.763 on 61 degrees of freedom
(14 observations deleted due to missingness)
Multiple R-squared: 0.5169, Adjusted R-squared: 0.2555
F-statistic: 1.978 on 33 and 61 DF, p-value: 0.01057


Interpretation:

The (intercept) value is the reference performance of Adam Mitchell playing the Arborec. All other estimates are additive to this reference value. The p-value of the (intercept) is meaningless in the context of the questions being asked, it just says that there is strong evidence that players score above zero. Without this result for the p-value of the intercept, I would question the entire model as essentially worthless.

Note the estimate of the Imperial II card factor is an additional 0.81 points on average. This is in line with what is known from the game mechanics, and serves as an internal validation of the estimates. It's as stated before not statisically discernible, but it's quite close to exactly what was expected.

Bubba Bartels and Casey Bousho are way behind the pack and perform poorly in scoring (-7, -6!). Zack Anderson is tailing the pack a bit (-3 points). The rest of the players are in a pack together and perform at about the same skill level. More data might differentiate them further, but at present, it's not enough to tell.

As far as races go, the Federation of Sol is super strong compared with all others. Just having this race, on average gave players in this group a 3 point advantage. The second race that made a discernible difference was Yssaril Tribes, which allowed a 2.2 point advantage. The other races were all within statistical noise of each other, and differences were not discernible. However, the earlier aov results, said overall we can't trust these numbers -- but with more data maybe.

The adjusted R-squared is 25%. I.e., the factors of race and player explain 25% of final outcomes. 75% is due to another factor (luck?). It's a common debate about a game, how much is skill and how much is luck. In this analysis, 25% is skill and 75% is luck would be a possible view from the data.

Further, the model was checked to make sure statistical assumptions were met, and they were without any issues. I.e., you can trust these results, the methods were applied appropriately.
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Christopher Halbower
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Thanks for doing this Shawn. As we accrue more data, I'll update the file.
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Shawn Garbett
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I'm going to modify a previous statement about skill and luck. I think the adjusted R^2 value is not appropriate in this circumstance. The adjustment is trying to account for the number of variables, and reflects the quality of the model, which we already knew was suspect. The unmodified value is how much of the variance is explained by the model, this value was 52%. I.e., Half the game is luck, and half the game is skill is indicated by this analysis.
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Christopher Halbower
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CyberGarp wrote:
I'm going to modify a previous statement about skill and luck. I think the adjusted R^2 value is not appropriate in this circumstance. The adjustment is trying to account for the number of variables, and reflects the quality of the model, which we already knew was suspect. The unmodified value is how much of the variance is explained by the model, this value was 52%. I.e., Half the game is luck, and half the game is skill is indicated by this analysis.


Shawn:

Is there any indication of our skills improving with time? The data does include the dates we played. As new players gain experience, does the luck/chance factor go down?
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Shawn Garbett
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Odd result for that question. I created a game number as an ordered factor for each player over time.


> m
> summary(m)

Call:
lm(formula = Raw.Score ~ game + Race + Player + Strategy.Card,
data = ordered.d)

Residuals:
Min 1Q Median 3Q Max
-3.7569 -0.9466 0.1788 0.9198 3.0461

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.917272 1.641870 3.604 0.000638 ***
game 0.021294 0.065927 0.323 0.747822
RaceBarony of Letnev 0.164726 1.350075 0.122 0.903297
RaceBrotherhood of Yin 1.511335 1.391734 1.086 0.281851
RaceClan of Saar 0.092288 1.202867 0.077 0.939099
RaceEmbers of Muatt 0.861267 1.173327 0.734 0.465784
RaceEmirates of Hacan 1.253446 1.392231 0.900 0.371552
RaceFederation of Sol 3.006639 1.270456 2.367 0.021198 *
RaceGhosts of Creuss 0.081519 1.298873 0.063 0.950165
RaceL1Z1X Mindnet 1.670468 1.220604 1.369 0.176240
RaceMentak Coalition 1.031191 1.251076 0.824 0.413066
RaceNaalu Collective 1.145205 1.123506 1.019 0.312147
RaceNekro Virus 0.494954 1.138162 0.435 0.665216
RaceSardakk N'orr 0.639967 1.245253 0.514 0.609192
RaceUniversities of Jol Nar 1.240306 1.163037 1.066 0.290498
RaceWinnu -0.003937 1.310376 -0.003 0.997613
RaceXxcha Kingdom 1.480006 1.165469 1.270 0.209030
RaceYssaril Tribes 2.240828 1.212032 1.849 0.069413 .
PlayerBen Burkholder 0.138729 1.538064 0.090 0.928431
PlayerBubba Bartels -7.192012 2.473751 -2.907 0.005102 **
PlayerCasey Bousho -6.178872 2.231953 -2.768 0.007483 **
PlayerChris Halbower -0.448806 1.531365 -0.293 0.770476
PlayerDavid Mitchell -2.072686 1.866440 -1.111 0.271213
PlayerDusty Shunta -0.129591 1.524246 -0.085 0.932529
PlayerJeremy Scott Pyne -0.822351 2.050593 -0.401 0.689822
PlayerJoe Veldhuis -2.242395 1.675129 -1.339 0.185738
PlayerJon Horne -0.800651 1.630563 -0.491 0.625198
PlayerJoshua Hannebohn -2.925654 2.553209 -1.146 0.256397
PlayerKevin Raliegh -2.005410 1.547111 -1.296 0.199860
PlayerMatt Bigham -2.059874 1.690207 -1.219 0.227724
PlayerMatt Spencer -0.628689 1.524047 -0.413 0.681434
PlayerMongo Closz -0.868247 1.514574 -0.573 0.568610
PlayerRocky Thompson 0.085080 1.568217 0.054 0.956914
PlayerZack Anderson -2.995698 1.726703 -1.735 0.087889 .
Strategy.CardImperial II 0.822362 0.676836 1.215 0.229123
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.776 on 60 degrees of freedom
(14 observations deleted due to missingness)
Multiple R-squared: 0.5177, Adjusted R-squared: 0.2445
F-statistic: 1.895 on 34 and 60 DF, p-value: 0.01513

> summary(aov(m))
Df Sum Sq Mean Sq F value Pr(>F)
game 1 15.96 15.961 5.058 0.0282 *
Race 16 71.94 4.496 1.425 0.1613
Player 16 110.70 6.919 2.193 0.0149 *
Strategy.Card 1 4.66 4.658 1.476 0.2291
Residuals 60 189.33 3.156
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
14 observations deleted due to missingness
> plot(m)


The analysis of variance says that game number has an influence, i.e. each player can learn. The linear regression says the effect is very small, and hard to estimate, i.e. 0.02 points per game increase on average. Very little change in previous results.
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David Gagner
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My group will usually deal out all the races into equal stacks. Sometimes we makes stacks equal to # of players +1. Then we choose our race from there.

My group also has 15 or so custom races we added so we have a ton of options. Most of the custom races were found here or elsewhere, but we've personally created around 4 or 5 so far.

We don't have many other house rules, except that the number of facilities in play is always equal to double the # of players. This has helped eliminate the problem of second turn, the first 3 players buy up all the refineries. The other one we will do is run 3 strategy cards when playing with 3 players or 2 strategy cards when playing with 5 players. We use the Shattered Empires strategy cards and the extras are chosen at random from the base set #2-7. Races such as Jol-Nar that can use Tech better, must choose only one strategy card to use their ability on when both are chosen for a round. Rounds where double warfare are chosen can be quite interesting.
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