Ty Wilson
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Before delving into the heart of the thread, I wanted to provide a brief introduction and little context for it. To cut to the chase, skip to the 3rd paragraph. I am a new member of BGG, and first time poster, but a longtime boardgamer and card player (more of the "bridge variety" than the "poker variety"). I've lurked on BGG for the past 5 years or so to gather information on new games to recommend to my regular gaming group of the past 20 years.

As my gaming group has gotten older, we've come to appreciate substantial games that can be finished in a reasonable amount of time. We just don't have the energy anymore for marathon games of Divine Right, Axis & Allies, and Republic of Rome. Many of us have also learned to appreciate the euro-style of gameplay, in spite of cutting our teeth on some of the AT games of Avalon Hill. But there again, it's tough to set aside a really long evening for another game of 1870. So, I have great admiration for RftG and its expansions. It does a nice job of removing much of the "fiddliness" of a full-blown board game without skimping on the strategic heft.

I am by no means an expert player. However, after roughly 100 games (95% against Keldon's AI, in progression through all the expansions with most in BoW, and the remaining 5% IRL in base) I feel I have played and beaten the AI enough to have some understanding of the range of strategies in BoW that can be employed and when to use them. I am also an inveterate "data and stats" lover, so I particularly enjoyed rrenaud's thread on his analysis of the Keldon server data. I believe, like others, that there is something to be learned about effective strategies by examining these data. By profession, I spend a lot of time analyzing ecological data. When presented with something like the Keldon dataset, I tend to look at it like an ecosystem. So, the analysis that follows comes at the problem somewhat differently, and I hope is complementary to Rob's. One of the tools that ecologists use to understand the relationships amongst species (e.g. tree species) and their environment (e.g. climate and landscape position) is ordination. In a nutshell, it's a tool for ordering species along environmental gradients in the hopes of understanding some of the important factors that influence where certain species occur.

I've applied this tool to the Keldon dataset, treating the cards played in a tableau as the "species" data. There are also two items about the "environment" that likely influenced which cards were played that are known a priori and readily available from the data: the player of the cards and goals achieved in the game (for most GS, RvI, & BoW games, at least). It appears that there was a bug in the game log script used to compile the data because many of the games that had no goals listed actually did have some achieved when I checked them out individually on the Keldon server. This analysis only uses games with at least one goal achieved and assumes that this bug is essentially random in nature and does not systematically impact only certain types of games. There are actually two other "environmental variables" to consider, number of players and expansion version, but I have dealt with those by limiting the analysis to 2-player advanced BoW games. Because of concerns that have been expressed about the influence of the AI on the dataset, I have restricted it to human vs. human (HvH) games. And because of concerns about the BoW learning curve, I have further restricted the analysis to tableaus of players with at least 50 2PA BoW HvH games to their credit, which includes almost all of the @30 players rated better than [AI] Data for 2PA BoW on Rob's stats page.

In summary, the analysis is based on 21,871 tableaus from 219 "experienced" players of BoW 2PA HvH games with at least one goal achieved. There were, in truth, roughly twice that many tableaus that could have been used, but because of the technical specs of my home computer and the size of the dataset, that's all that could be processed. Even with just half, that's still a rich dataset, and I am confident that a similar analysis of the other half would provide comparable results. BTW, these 219 experienced players account for almost 75% of the tableaus meeting the BoW 2PA HvH goals criteria. To put this into better perspective, there are almost 2,700 different names of players associated with games meeting the BoW 2PA HvH goals criteria. In other words, the majority of these players have only played a few such games on the Keldon server, many only one or two.

Here's Figure 1.



There's a lot to this, but I'll try to explain the keys points. The "Loadings" plot has to do with the "environmental variables". In order to make it somewhat legible, I had to use abbreviated names for the labels. For example, V17 is the goal "Most Military", V19 is the goal "Most Prod worlds", and V3 is the goal "First 3 aliens". You'll notice that the arrows point in different directions and have different lengths. This is related to the magnitude and direction of their influence on the cards that get played. Those that point in the same direction have similar influence, opposites have opposing influences, and perpendicular arrows are uncorrelated. The "Species" plot shows where cards occur along the environmental gradients defined by the loadings. Again, these are abbreviations. For example, V189 is "Uplift Mercenary Force". You'll notice that it's over in the vicinity of where that "Most Military" arrow was pointing. Finally, the "Scores and predictions" plot indicates where each tableau occurs along the environmental gradients. Tableau 29799 corresponds to Player 1's tableau in http://www.keldon.net/rftg/showgame.cgi?gid=99750. I would characterize that as a fairly military tableau and would conclude that a1cibiades employed a military strategy for that particular game. You'll notice a roughly equilateral triangle shape to these data. The vertices of this triangle correspond to the three primary strategies employed in these games: military (right), production/consumption (bottom), and development/prestige (top). All other strategies employed appear to be a mixture of these three elemental strategies.

Some players might have a proclivity for a certain style of play, or tend to play certain strategies more than others. That's where Figure 2 comes in.



This is actually the "Loadings" plot again, this time leaving the goals off of the figure so you can focus on the influence of the players on cards played instead. Each point corresponds to one of the 219 most experienced players. The same interpretations apply here as to the tableaus: military (right), production/consumption (bottom), and development/prestige (top). Most fall somewhere in the middle, which isn't too surprising for experienced players. The red points that are labeled are the @30 highest rated players. You'll notice that most of them are also in the middle, but tending towards the top. The ones that I find most interesting are V148 and V8, which correspond to raistlin and ascolta. These are both highly rated players who have found success playing predominantly development/prestige strategies. Their graphs on Rob's stats page bears this out as well (http://rftgstats.com/bow/player_raistlin.html and http://rftgstats.com/bow/player_ascolta.html).

What I take away from all of this is that the data support my general understanding of the game and available strategy space. I don't think these results suggest that prestige is overpowered, but that BoW has rebalanced the game towards development strategies and away from, for example, production/consumption in the base game. So, nothing earth-shattering here, but it is interesting to see some experienced players who have enjoyed success by adopting strategies towards the edges of the overall strategy space. Maybe they're on to something that we should be paying attention to.
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Rob Neuhaus
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Cool!

Are these plots from R? Is it possible to substitute the goal images/card icons for the labels and make the plots a lot bigger?

How long does the analysis take to run? What is the bottleneck? If it's just memory and you can do this stuff over ssh, I'll happily give you access to a machine with 16 GB of memory.
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Ty Wilson
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Yes, the analysis was done in R and these plots were produced using the "ade4" package. I haven't tried to use images within plots before, but if you have a sample script that I could use, I'd be willing to give it a try. What's the best resource for the card images? I'll also take a look at the docs to see how straightforward the process is. I can certainly make the plots much larger.

Thanks for your offer of access to a better system. The technical issue is both RAM and CPU related. The matrix that needs to be manipulated for the half-sample dataset is roughly 22,000 x 240. With the overhead of Win XP (yes, it's an older machine) and the intermediate R objects that get created in the process, that's pushing the limits of my 4GB RAM. And with a slower processor, it takes about an hour or so to run, if I recall correctly. However, I really don't think the results would suggest anything different than what we're seeing in the one that I posted. I'll run it for the other half just to confirm that. If they differ significantly, I'll take you up on your offer and run on it the entire set of tableaus.
 
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Ty Wilson
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Sorry, I responded too hastily on the go. I see now that I misread your suggestion about the plots. You were referring to card icons and goal images, not card images. That's certainly doable. I'll try to model them after your rftgstats plots, but it will take some time. R is immensely flexible, but can be tedious when formatting graphics.
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Rob Neuhaus
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One thing that might be easy is to dump point data from R in json, and make the graphs with the same javascript.

It could also be useful to do some bootstrapping to get some feel for the variance in the per player data.
 
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Ty Wilson
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Both are good suggestions. I'll leave the plots to you and will send you the coordinates as JSON. I've thought of a slightly different way to formulate the problem and analysis that will let me use all of the tableaus and compute a variance in position for players. Once that's done, I'll GeekMail you the files.
 
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Rob Neuhaus
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Sounds good, email is mildly preferable, rrenaud@gmail.com
 
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Ty Wilson
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Sorry, this took longer than I expected. I did the analysis over again removing players as predictors. This allowed me to use all tableaus from all players and to compute a (co)variance in their tableaus. The "card positions" and "goal vectors" files have been e-mailed to your gmail address. Also, I now have all player positions (not just the top 200 or so) along the two gradients depicted in the earlier figures, as well as 95% confidence ellipses. I'll work on a new figure over the weekend showing where Keldon 2PA BoW HvH players are located within this strategy space, based on the average of all tableaus that they've played.
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Ty Wilson
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Here is a link to an SVG file

https://dl.dropbox.com/u/25279235/bow_2pa_hvh.svg

that shows where the top 718 ranked 2PA BoW players meeting the criteria discussed in this thread are located in "strategy space". Assuming you're using a newer browser, you should be able to view it within the browser itself. There is some javascript embedded that allows you to pan and zoom around the image. To pan, left click and drag. To zoom, use your mouse wheel. Because it's SVG and not an image, you can use the "find in page" functionality of your browser to locate a particular name. It's kind of interesting to see who you're neighbors are in this space, folks with a similar style of play. This analysis is a little different than the first, but the same directions more or less apply: military to the right, produce/consume to the left, development to the top, and alien(!) to the bottom. I think this might be because of the inclusion of all tableaus, including those of inexperienced players. Perhaps they overvalue alien cards because of their high trade value, or are easily swayed by the alien goal when it's in the game.

[Edit: I've updated the SVG file so that as you mouse over a name, it is highlighted, and the corresponding 95% confidence ellipse of the player's position is displayed.]

[Edit: I've updated the SVG file so that it graphically indicates the general area of the four main strategy sectors. Also, you can now toggle between displaying player names and player ratings.]
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