Guy Srinivasan
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Measuring a card's 2p strength

Genie, for instance, has this capability. Here is at least one way to measure a quantity very, very close to the actual strength of a card. It converges slowly, which is problematic, but it does measure what we want. Can anyone think of a way that would converge faster? Or a way to measure what we want without modifying the game?

Step 1: Use ratings to predict win probabilities. IIRC difference in ELO, at small scales, is approximately proportional to log-odds of winning. You may need to discount for new players with unstable ratings, which is a very common fix implemented on most chess servers.

Step 2: Before the game begins, pick (say) 10 cards for each player, independently at random, which they will never draw during the game. Do not let the players know these cards. Any time the engine would let a player draw one of these cards, discard it instead and pick a new card from the deck (this prevents the other player from being much more likely to get it, which would be a pretty big confounder). Note that since the disallowed cards are chosen randomly and independently for both players, this does not affect their priors of winning much*.

Step 3: Set up a giant regression analysis of how much log-odds is added given that a specific card is guaranteed to not be drawn, given your rating, and given the ELO difference between you and your opponent.

So, explicitly, this comes extremely close to measuring the extra evidence-of-winning you get from knowing one player will not draw card X. I believe that ranking the cards according to this metric (for a given rating, for a given difference in rating between you and your opponent) will produce a ranking very close to the actual ranking of card strength (for a given rating, for a given rating difference).

This still doesn't measure well power of a card drawn in the early game, or in the endgame, or whatnot, but unless it's wildly bad to draw a certain card at one time or with some skill level or with some skill difference between you and your opponent, if it's very beneficial to draw that card given any subset of the givens, we'd be able to see it easily if we had enough data to feed into this metric. As I said, though, this will converge pretty slowly.

*except if it turns out that (for example) there are lots of cards which are broken at high levels of skill and only a few at low levels, and even then the probabilities of winning won't change much
 
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Dave J McWeasely
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I don't think you have to actively take card's out of a player's virtual deck, that information could just be computed at game end. How many games do you actually see Alien Tech Institute? 30%? For any given card there's more than enough games where its never seen.

You could also index cards to the turn that they're first seen. Alien Toy Shop on the last turn is ho-hum. On Turn 1, it is much stronger. Rebel Homeworld doesn't care so much.
 
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Guy Srinivasan
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Yes, measuring whether or not a card is seen would tell you something... but there's a huge confounder there, which is how many cards you look at. Trade strategies will see a lot more cards. I agree this wouldn't require a game change (just a code change), but I'm not sure how I would interpret the data I got from it.

Indexing cards to the turn they're first seen, again, correlates with some useful information, but I'm not sure how to factor out the effects of long versus short games. Any ideas?
 
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Dave J McWeasely
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GreedyAlgorithm wrote:
Yes, measuring whether or not a card is seen would tell you something... but there's a huge confounder there, which is how many cards you look at. Trade strategies will see a lot more cards. I agree this wouldn't require a game change (just a code change), but I'm not sure how I would interpret the data I got from it.

I'm probably just dense, but I don't see a problem with this. First of all, the set-aside can be simulated for 99.99% of games, since on any given game 30 cards are not going to be seen. On most games, the number unseen will be higher, so why not include the stats for them too? If it bothers you, ignore the stats for the cards beyond 30 that which are unseen.

If you want an index of time, I think turn number would be a good enough measure. But if you want something fuzzier, I think of the "opening" as the time up until the turn when a player makes their first trade (not counting Alpha Cent's or Ancient Race's initial trade). The midgame is the time from then until someone calls Consume-2x. I guess that's Prod-biased of me, and there should be a military equivalent, but I can't think of what that would be exactly.
 
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Serge Levert
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MrWeasely wrote:
The midgame is the time from then until someone calls Consume-2x. I guess that's Prod-biased of me, and there should be a military equivalent, but I can't think of what that would be exactly.


Perhaps 8 cards in tableau, like the new goal.
 
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I'm thinking outside the box here...

If us, as designers, have done our job correctly, then no individual card is siginificantly stronger than any other. So any measurement of the strength of individual cards is bound to have a high noise-to-signal ratio, and what would the point be, anyway?

Instead, it may be much more worthwhile to measure the strength of card *combos*, which seems to be the heart of game strategy anyway. Instead of measuring how often individual cards figure into victories, measure in some way how often pairs of cards appear in the same tableau, same victorious tableau, same losing tableau, etc. You'll have O(N^2) groups to keep track of instead of O(N), but I think you'll save a lot by getting usable signal. When you do have that data for the O(N^2) groups, you can probably then isolate cluster data and possibly even individual card data, if you want. Plus, you'll get information like "Card X works well when doing strategy Y" instead of a more generic "Card X works well", which is more useful for AI programming anyway.
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Rob Neuhaus
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onigame wrote:
I'm thinking outside the box here...

If us, as designers, have done our job correctly, then no individual card is significantly stronger than any other. So any measurement of the strength of individual cards is bound to have a high noise-to-signal ratio, and what would the point be, anyway?



The designers of Magic will disagree with you on the premise. From Making Magic,
Quote:
Diversity of Power Rewards the More Skilled Player

Mostly up to now I’ve explained why “bad cards” have to exist. I also want to point out that “bad cards” have some good effects on the game. I think the best reason to have a diversity of card power is it increases the skill in the game. A lesser player, for instance, is more likely to draft a sub-optimal card. A more novice player will put questionable spells into his decks. Both occurrences increase the chances that the better player will win.


I definitely agree the magic designer about card power knowledge leading to skill. My willingness to throw away apparently good combinations (Genetics Lab + Galactic Genome Project say) for just damn good cards (Terraforming Guild) is one the reasons I am a good race player. However, this is mostly tangential, and I applaud you for taking the higher ground and not intentionally making weak cards.

However, I simply disagree that no card in Race is significantly stronger than the other. The redesign of Gambling World is a flat out admission that it was underpowered, no? There are many other examples of bad cards, just perhaps none quite as bad as GW. The twin contenders of suck, Expanding Colony and Terraformed World. Yes, yes, I know you can come up with situations in which either are the exact right play. The problem is those situations do not arise frequently enough to make them worthwhile cards in general.

Quote:
Instead, it may be much more worthwhile to measure the strength of card *combos*, which seems to be the heart of game strategy anyway. Instead of measuring how often individual cards figure into victories, measure in some way how often pairs of cards appear in the same tableau, same victorious tableau, same losing tableau, etc. You'll have O(N^2) groups to keep track of instead of O(N), but I think you'll save a lot by getting usable signal. When you do have that data for the O(N^2) groups, you can probably then isolate cluster data and possibly even individual card data, if you want. Plus, you'll get information like "Card X works well when doing strategy Y" instead of a more generic "Card X works well", which is more useful for AI programming anyway.


I completely agree that card combos are useful. I disagree that finding powerful combos is much easier and less prone to noise than finding powerful individual cards. The reason is two-fold. First, n choose 2 >> n, for n ~= 100. There will be fewer games played with pairs of cards than on individual cards, which makes the pair learning problem inherently harder. Simply put, there is less signal. Furthermore, given a good ranking on pairs of cards, I can create a good ranking on individual cards by simply aggregating the pair scores for each individual. The better you get at ranking pairs, the better I get at ranking individual cards.

EDIT: Link the the Goa post Tom mentions below. http://www.boardgamegeek.com/article/1202589#1202589
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Tom Lehmann
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I would also disagree with the statement that my goal as a designer was to make "all cards balanced". I would certainly agree with the statements that I wanted "no broken cards" and that all cards should be "useful", though some might only be useful in fairly specialized circumstances (i.e. Pilgrimage World).

However, there is some space between "all cards balanced" and "no broken cards" and I believe this is a very important design space (what I have referred to as "texture" in other similar discussions, see my long post in a Goa thread as an example).

From the very start, I wanted a few cards that were "a strategy in a box" and a bit stronger than others. In the base game, Consumer Markets and Galactic Federation would be the two prime examples. There was a lot of discussion about Galactic Federation during the early development of RFTG as to whether its -2 discount was "broken" or within the range of "ok"; I came down on the side of "not broken", on the "high side of ok", and good for the game as it was "a strategy in a box" for the development path.

I stand by that design decision.
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The redesign of Gambling World occurred not because it was "underpowered", but because it was both "unpopular" *and* viewed as "not thematic" (there was no "gamble" involved). We didn't make Gambling World more complex in RFTG or TGS, as we didn't want that extra complexity in the game or to slow down the game too much for a single card. With experienced players in RvI, those issues are not a problem.
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Guy Srinivasan
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Looking at 2 cards at once is a very good point if the goal is to make a decent AI. You really need that dependence information. And you're absolutely right, as soon as you've got that most basic level of dependence, you can cluster and then check (I'm guessing) more like O(n) than O(2^n) likely combos.


One of the big differences between Magic and Race is that Magic has to play well constructed, sealed, and booster draft. Race needs to play well out of the box and draft. Clearly I don't know all the implications of these differences, but at least one, it seems, is that throwing in genuinely bad cards in Magic is a better idea than in Race.

If there were a variant of Magic which forced arbitrary cards into your possible draws, genuinely bad cards would not be a good design decision.
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Tom Lehmann wrote:
However, there is some space between "all cards balanced" and "no broken cards" and I believe this is a very important design space (what I have referred to as "texture" in other similar discussions, see my long post in a Goa thread as an example).

To quote my friend Jonathan: A game should have lumpy bumpy texture.

Tom Lehmann wrote:
The redesign of Gambling World occurred not because it was "underpowered", but because it was both "unpopular" *and* viewed as "not thematic" (there was no "gamble" involved).

Gambling World doesn't break theme so much as it breaks the rules. This is not a game like Agricola where each card is a new rule. In RftG, each card has a mix of powers, but those powers are shared with other cards. The 6 devs have unique score rules, but those only affect final score. Gambling World is its own special rule that you need to follow during the game.

The only other card as unique as GW is Improved Logistics. However IL is an important part of the game; it helps a military strategy to outrun a produce/consume strategy. GW adds little to the game; too little for the amount of rule space it occupies.
 
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Barticus88 wrote:
Tom Lehmann wrote:
However, there is some space between "all cards balanced" and "no broken cards" and I believe this is a very important design space (what I have referred to as "texture" in other similar discussions, see my long post in a Goa thread as an example).

To quote my friend Jonathan: A game should have lumpy bumpy texture.


The cards with extremely narrow envelopes of acceptable use are a problem for me. There are already plenty of opportunities for skill in the game. Replicant Robots is definitely a power card with a generous envelope, but players that reflexively deploy them without regard to its interaction to the rest of their tableau will lose games because of them. That's a skill-based difference, and doesn't involve any deliberately weak cards.

Each card should have its moment in the sun. That moment should not be pathologically hard to construct; it should occur with a measurable frequency. I'd say Terraformed World and Research Labs are examples of cards that are overpriced for what you get. It is very rarely an acceptable play, and almost never a brilliant one.

That's bad.

Skill should be coming from knowing when to use a card. No card should be quite as bad as Terraforming Robots was in the base game.
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MrWeasely wrote:
The cards with extremely narrow envelopes of acceptable use are a problem for me.


I actually like the fact these cards exist. 70% of cards seen are used as discards anyway, so i think there's a lot of room for variety and narrow cards. I believe they make the game better.

Terraformed World is great for a trade strategy. Research Labs is certainly a weird one, feels overcosted because we are paying for 3 abilites, 2 of which we usually have no use for. I still find myself playing the card on occasion, more often than Terraformed World.
 
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MrWeasely wrote:
I'd say Terraformed World and Research Labs are examples of cards that are overpriced for what you get.


I've never understood the general dislike of Terraformed World. Sure, it doesn't do anything cool, but it's one of the highest scoring non-military worlds in the game (only a couple of alien worlds are higher). This makes it great for (a) end of game, (b) if you have lots of available cards (as mentioned above, in a trade strategy), (c) if you have replicant robots, (d) if you have military and use Imperium Cloaking. In any of those situations, it's 5 vps make it very good.
 
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Gshc wrote:
MrWeasely wrote:
I'd say Terraformed World and Research Labs are examples of cards that are overpriced for what you get.


I've never understood the general dislike of Terraformed World. Sure, it doesn't do anything cool, but it's one of the highest scoring non-military worlds in the game (only a couple of alien worlds are higher). This makes it great for (a) end of game, (b) if you have lots of available cards (as mentioned above, in a trade strategy), (c) if you have replicant robots, (d) if you have military and use Imperium Cloaking. In any of those situations, it's 5 vps make it very good.


I agree. I try to play terraformed world when I draw it just because it's a quick 5vps...

 
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Gshc wrote:
I've never understood the general dislike of Terraformed World. Sure, it doesn't do anything cool, but it's one of the highest scoring non-military worlds in the game (only a couple of alien worlds are higher). This makes it great for (a) end of game, (b) if you have lots of available cards (as mentioned above, in a trade strategy), (c) if you have replicant robots, (d) if you have military and use Imperium Cloaking. In any of those situations, it's 5 vps make it very good.


Galactic Trendsetters is just so much more interesting than Terraformed World. When you throw down Trendsetters, people sit up in their chairs. Trendsetters equalizes Terraformed World on VP after the very first 2x. Trendsetters has the odd pairing with Galactic Renaissance. Terraformed World fails to do anything interesting with any 6-dev.

But also compare it to the high-cost alien worlds. Those come with a trade good, and a very potent one at that. Those cards have all kinds of interactions with other cards and goals. Terraformed world is just an inert rockball. It only scores with SETI and New Economy, and that's not saying much.

I'm not saying I hate Terraformed World. It just bugs me that its not trying.

PS: On point (d), I think you mean that "Terraformed world is arguably the biggest prize for cards like Colony Ship". I seldomly see that transition, since TFW is an endgame card, and by the endgame I usually have plenty of cashflow and not enough time to mess with those dinky little cards. Lost Speicies Ark World seems to be still the prefered target, though Trendsetters, Merchant World, and Studios all are popular.
 
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MrWeasely wrote:
Galactic Trendsetters is just so much more interesting than Terraformed World. When you throw down Trendsetters, people sit up in their chairs. Trendsetters equalizes Terraformed World on VP after the very first 2x. Trendsetters has the odd pairing with Galactic Renaissance. Terraformed World fails to do anything interesting with any 6-dev.


Well said. The only situation that Terraformed World is better that the Trendsetters are when settling on the last turn, and using its consume power at 1x -> TW gets 6 VPs, GT gets 5 VPs. Considering how useful the Trendsetters can be earlier in a quick consume/produce strategy, however, GT >>>> TW in every other situation.

Quote:
But also compare it to the high-cost alien worlds. Those come with a trade good, and a very potent one at that. Those cards have all kinds of interactions with other cards and goals. Terraformed world is just an inert rockball. It only scores with SETI and New Economy, and that's not saying much.


Also remember that there are 3 cards that give -2 discounts to Alien worlds, which would make those worlds cost 4 as opposed to 6 if used, plus a solid trade or produce good.

Quote:
I'm not saying I hate Terraformed World. It just bugs me that its not trying.


I just find it to be a boring card, that I rarely find any use for. The only card I can think of as a 4+-cost non-military world that I wouldn't use Doomed World for.
 
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MrWeasely wrote:
GreedyAlgorithm wrote:
Yes, measuring whether or not a card is seen would tell you something... but there's a huge confounder there, which is how many cards you look at. Trade strategies will see a lot more cards. I agree this wouldn't require a game change (just a code change), but I'm not sure how I would interpret the data I got from it.

I'm probably just dense, but I don't see a problem with this. First of all, the set-aside can be simulated for 99.99% of games, since on any given game 30 cards are not going to be seen. On most games, the number unseen will be higher, so why not include the stats for them too? If it bothers you, ignore the stats for the cards beyond 30 that which are unseen.

I could be wrong! Here's a toy game to find out. There are 10 cards labeled 1-10 and two players. Each player gets two turns. In their first turn they draw a card and look at it. Second turn they simultaneously and independently decide whether to repeat the first action or gain 1.5 points. Then anyone with cards adding up to 7+ gains points equal to their highest card. Whoever has the most points wins.

I think this shares all the relevant features with RftG that we need, and it's simple enough to analyze. Assuming good poker faces, correct strategy is to take the 1.5 if you drew a 10, 9, 8, or 7 on the first turn. Now to code it up and see win rates given that one player didn't see a 1, or given that one player's draw chances are changed to never receive a 1...

P(W|~1) = 55.19%
P(W|do(~1)) = 53.45%

(over 100 million runs)

This particular difference is because when you see worse cards early on, you look for more cards later, so bad hands are disproportionately likely to contain a 1 and not contribute to ~1 tallies. Whereas in the do(~1) case, where we guarantee not seeing a 1, having a bad hand without a 1 early doesn't make it more likely you have a bad hand with a 1 later which will be thrown out of the accounting.

Not much of a difference in this toy game, sure, but I just don't know how much or in what direction the difference would be in the real game.
 
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rrenaud wrote:


However, I simply disagree that no card in Race is significantly stronger than the other. The redesign of Gambling World is a flat out admission that it was underpowered, no? There are many other examples of bad cards, just perhaps none quite as bad as GW. The twin contenders of suck, Expanding Colony and Terraformed World. Yes, yes, I know you can come up with situations in which either are the exact right play.
To me, the appeal of (the old) Gambling World was mainly getting a consume for VP power. That and it only costs 1 and was worth 1 VP. Getting cards on the other power is merely an added bonus.

Plus, if you're trying to get Innovation Leader, consume powers can often times be the one power that's really hit or miss.

Similar to San Juan's Gold Mine. Getting a card during the Prospector role is a bonus. Seeing what cards get flipped up is the main thing to me. Getting cards is always nice, but there's a reason it only costs 1.

.

Expanding Colony ain't great either, but this one's also nice for if you just want a Consume power. Producing on a blue windfall is good for those several cheap blue windfall worlds. Free Trade Assoc. also gives +2 pts for this world, and this also helps Innovation Leader to get that elusive Consume power.


Barticus88 wrote:
The only other card as unique as GW is Improved Logistics. However IL is an important part of the game; it helps a military strategy to outrun a produce/consume strategy. GW adds little to the game; too little for the amount of rule space it occupies.
Hidden Fortress and Pan Galactic Research aren't "groundbreaking" in uniqueness, but they do sidestep the hard rules of the game ending at 12 cards in tableau and a hand limit of 10.


...

I too prefer Gal. Trendsetters overall to Terraformed World, but they both feed off each other's weaknesses. If you have Col. Ship or Doomed World, it probably wouldn't be worth it to save that card and wait for GTrend. if you've got TW right there in your hand. Furthermore, TW works good if built lategame, but it can still be useful early game. People of course prefer GT early game, but if this is built lategame, you're hardpressed to score much out of it anyways.
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I noticed that measuring a homeworld's strength as a homeworld doesn't have any of the problems that measuring other cards do, since we can see which homeworld you started with by your ending tableau. So I did a an analysis of the Genie data by generating 2p ratings and finding a log-prob-maximizing, single amount of evidence per homeworld you should add to the evidence given by rating difference between you and your opponent upon seeing you're playing that homeworld. I'm certain this overfits the data somewhat. But here are the interesting parts of the results anyway.

For reference: if you have a 10% or 50% chance to win, adding 0.5 bits of evidence gives you a 13.5% or 57.5%.

2p games without the expansion:
{EE, ELC, AC} > {NS} > {OE}. Additionally, when looking only at games between 1500+ players, Old Earth is a good 0.35 bits worse than when looking at all games.

With expansion, without goals:
{AC, ELC} > {DW, SC, EE, OE} > {NS, AR, DAF}. Or, with only 1500+,
{ELC} > {AC} > {DAF, SC} > {DW, NS} > {AR} > {EE} > {OE}
In this setup DAF gets 0.52 bits better between 1500+, and EE 0.41 bits worse.

With expansion, with goals (I think this has the most overfitting):
{AC, NS} > {ELC} > {SC} > {AR} > {DW} > {OE, DAF}. Or, with only 1500+,
{NS} >> {AR} > {ELC} > {SC} > {OE} > {AC, EE, DAF} >> {DW}

ETA: (this looks wrong, hold on a sec) This one says that if all you know is that two players, both rated 1500+ on Genie, are about to play with goals and the first has New Sparta and the second Doomed World, you should expect NS to win 69.5% of the time. Seems extreme, and there aren't too many games with this setup, which is why I expect overfitting. I'm not allowed to subtract things like I did there. P(~Win|Homeworld2)/P(~Win|~Homeworld2) is decidedly not the reciprocal of P(Win|Homeworld2)/P(Win|~Homeworld2).
 
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(calculated values used in previous post: evidence is given in hundredths of a bit, so -24 is -0.24 bits)

===2p games, without the expansion===

Epsilon Eridani: 9
Earth's Lost Colony: 8
Alpha Centauri: 3
New Sparta: -6
Old Earth: -34


===2p games, without the expansion, ratings > 1500===

Epsilon Eridani: 13
Earth's Lost Colony: 10
Alpha Centauri: 8
New Sparta: 7
Old Earth: -70


===2p games, with the expansion, without goals===

Alpha Centauri: 38
Earth's Lost Colony: 32
Doomed World: 7
Separatist Colony: -2
Epsilon Eridani: -6
Old Earth: -6
New Sparta: -15
Ancient Race: -19
Damaged Alien Factory: -24


===2p games, with the expansion, without goals, ratings > 1500===

Earth's Lost Colony: 54
Alpha Centauri: 36
Damaged Alien Factory: 28
Separatist Colony: 17
Doomed World: 2
New Sparta: -2
Ancient Race: -17
Epsilon Eridani: -47
Old Earth: -68


===2p games, with the expansion, with goals===

Alpha Centauri: 65
New Sparta: 65
Earth's Lost Colony: 48
Separatist Colony: 20
Ancient Race: 3
Epsilon Eridani: -21
Doomed World: -38
Old Earth: -54
Damaged Alien Factory: -64


===2p games, with the expansion, with goals, ratings > 1500===

New Sparta: 106
Ancient Race: 56
Earth's Lost Colony: 39
Separatist Colony: 12
Old Earth: -5
Alpha Centauri: -17
Epsilon Eridani: -26
Damaged Alien Factory: -31
Doomed World: -122
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I wish i had a statistics degree so i could understand any of that. :) In particular, how "bits" can be grokked as a layman.

I did at least learn that NS gains a hell of a lot from goals.
 
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These bits are (or would like to be! ) equal to

log2(P(Player has Homeworld X | Player wins)/P(Player has Homeworld X | Player loses))

So if we're talking coin flips, and a coin weighted to land heads 3/4 of the time is like a homeworld, and you get dealt the coin 1/9 of the time (the rest of the time you get a fair coin) and win if you flip heads, then this quantity is equal to log2(P(unfair|win)/P(unfair|lose)). This can be found by

P(unfair|win)/P(unfair|lose) = P(win|unfair)/P(lose|unfair) * P(win)/P(lose) = (3/4)/(1/4) * ((3/4)(1/9)+(1/2)(8/9))/((1/4)(1/9)+(1/2)(8/9)) = 3.35ish, whose log2 is 1.74.

Which means translating from bits back into probability is going to be a little weird with the homeworlds... I'll do the math later. They should still rank the homeworlds about right.
 
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Rob Neuhaus
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BTW, I updated the data and added support for determing whether a game was advanced or not, so you might want to re-run your analysis.

http://rftgstats.googlecode.com/svn/trunk/genie_analysis/con...
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Rob Neuhaus
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Larry (from genie) did a similiar analysis. He posted it at bgholic here, http://www.bgholic.com/story/race-galaxy-homeworld-statistic... but gave me permission to repost here verbatim.

Quote:



Using Rob's R4TG statistics, courtesy of Genie, I ran Glicko-2 on the home worlds. There has been enough games (>300 each) that the deviation is mostly in the same range, so I will leave them out.

These are all two-players, advanced.


No goals, The Gathering Storm:
Homeworld Rating Wins Games
Earth's Lost Colony 1534.13 193.5 363
Old Earth 1528.75 189.5 359
Epsilon Eridani 1515.26 191.5 372
Alpha Centauri 1505.67 184 364
Ancient Race 1493.43 194 393
New Sparta 1492.20 195.5 397
Separatist Colony 1485.38 171.5 353
Doomed world 1481.47 161 334
Damaged Alien Factory 1461.56 161.5 349


Goals, The Gathering Storm:
Homeworld Rating Wins Games
Alpha Centauri 1600.79 205 343
New Sparta 1574.36 194.5 340
Ancient Race 1520.38 171 329
Epsilon Eridani 1506.00 174 344
Earth's Lost Colony 1501.54 168.5 336
Separatist Colony 1478.58 150 313
Old Earth 1452.40 152.5 336
Damaged Alien Factory 1433.81 149.5 343
Doomed world 1427.22 140 326


For each of these ratings, the deviation is less than 30. This means that there's a 95% confidence that each of these ratings is within 60 of their true rating.
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