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Author Topic: Reasons for the 1st-player vs 2nd-player advantage  (Read 10263 times)

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WanderingWinder

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Reasons for the 1st-player vs 2nd-player advantage
« on: January 16, 2012, 03:51:47 pm »
+4

That there is generally a first-turn advantage in dominion is pretty well understood. That it is magnified for some kinds of cards in comparison to others is also fairly well known. But the precise reasons for why the advantage exists and why some cards exhibit them more than others are not. It is my goal to answer these questions here.

I believe that there are essentially two reasons why first-player advantage exists.
The first is the better understood of the two - in a race for depleting piles, the first player is, at some points, going to have had more turns, and thus more chances to get a majority of a particular card (or, in the case of curses, to dish the majority of that card out). This is the case with any card that has the chance of running out before the game ends. Notably, in most every game, provinces. Which is what gives the first-turn advantage for big money (well, ok, duchies sometimes too). But this is more pronounced in games where 'winning the split' on another card. So we see bigger 1st-turn advantage on cards like minion, peddler, hunting party...
Related to this, strategies which go for a three-pile ending tend to have a bigger-than-you'd-have-otherwise first-turn advantage. Moreover, mega-turn strategies have a significant first-player advantage. Apart from the large possibility these have for three pile endings, the first player just gets the chance to 'go off' first. And if the second player 'goes off' before the first player, they must have had fewer preparatory turns, which means that second player will 'go off' smaller.

The second major reason for first-turn advantage is in reshuffle timing. This comes up in player interactions - most normally attacks. Essentially, the deal here is that when player one attacks player two, he attacks 'the same turn' - i.e., if player one attacks on turn 7, he hits player two's turn 7 hand/deck. If player two attacks on turn 7, he hits player one's turn 8 hand/deck. Moreover, if both players would reshuffle between those turns (as is often the case for turn 7/8), that makes a significant difference. If it's a cursing attack, Player two's curse doesn't hit player 1's deck until the 3rd reshuffle, whilst player one's curse hits player 2's deck on the second. Advantage first player.
If, on the other hand, it's a handsize attack, player two is at an advantage, as he's hitting that 3rd-reshuffle hand is more painful than hitting a 2nd-reshuffle hand.

Now, let's look at some concrete, albeit hypothetical examples to try to demonstrate my point a little more clearly.
You both open 2/5 on an IGG board. You both open copper/IGG. Player 1 reshuffles after his second turn, with a deck of 1 IGG, 8 copper, 3 estates. Player 2's deck after the first reshuffle is 1 IGG, 8 copper, 3 estates, and 1 curse, because player 1 dished out that curse before player two was able to reshuffle. Massive first-player advantage.
You both do exactly the same things, opening witch/hamlet. You both play a hamlet on turn 3 and witch on turn 4. Each witch triggers a reshuffle with its draw. But player 1's witch hits player 2 with a curse before that reshuffle, whereas player 2's hits only after.
Hopefully you get the idea.

Now here's some notable cards or types of cards and where the first-player advantage lies with them (and if it's tricky, why). Keep in mind though, that there's an overarching principle that the early turns are more important than the later ones, which adds a little extra wrinkle for first player in general. Also, when I say second player gets an advantage, that's only relative to the 'average case', i.e. what the first-turn advantage would be without these cards. That's almost certainly not going to be enough to overcome the inherent 1st-player advantage in the game. But also keep in mind that 2nd player has his advantages too - the tiebreak rule and being able to adjust his strategy to the opponent. Even so however... I've yet to see a board where 2nd player can really hold his own.

Cutpurse - pretty good 1st-player advantage, because the earlier reshuffles you're more likely to hit a copper on (and it's more likely to matter, too)
Bureaucrat - decent 1st-player advantage, as the earlier reshuffles have more estates in them, and though that gets reversed later on, earlier turns are more important.
Handsize-reducers - second-player advantage, as they attack more powerful hands
Curse-givers - massive 1st-person advantage
Sea Hag - Still 1st-person advantage, for the first major reason (i.e. being able to win the curse race is HUGE), but much less so than other cursers, as it's more likely that second-player forces 1st to discard something good. Plus putting the curse on top of the deck somewhat mitigates the extra-reshuffle-with-a-curse-in-your-deck thing. Not much, but a little.
Jester - well it's board dependent, but I think in general it's second-player advantage. I think the real power of Jester is in grabbing good cards from your opponent (and it also skips them!), and 2nd player will have slightly better chances to grab them, as well as better things to grab.
Bishop, vault - 2nd-player advantage, as when they use it, the opponent's hands are stronger, mitigating the drawback. Embassy goes similarly.
Trash-for-benefit - 1st-player advantage. One major thing these can do is shorten the game, which helps the person in the lead - generally 1st player.

ackack

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #1 on: January 16, 2012, 04:07:47 pm »
+1

If, on the other hand, it's a handsize attack, player two is at an advantage, as he's hitting that 3rd-reshuffle hand is more painful than hitting a 2nd-reshuffle hand.

I don't think this is in general true. I think the mechanism is very similar to Cutpurse, which we agree is a 1st player favoring card. Militia is a good 1st player card because 1st player gets two chances at disrupting pre turn 5 buys, whereas 2nd player only gets one. The cumulative advantage of having stronger cards earlier is pretty significant, and I suspect it is more significant than the effect that you mention here. (Part of that last bit is that the effect becomes reversed in the late game, when later shuffles will be greener than earlier shuffles. There's no compensation for the getting denied early economy.)
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Geronimoo

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #2 on: January 16, 2012, 04:33:21 pm »
+1

There's one case I'm certain 2nd player has an advantage: Ill-Gotten Gains rush with Smugglers.
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #3 on: January 16, 2012, 04:59:55 pm »
0

Well, smugglers itself probably deserves a spot for 2nd-player advantage.

As for the militia thing... don't know what to tell you. Militia definitely exhibits 2nd-player advantage relative to the average card (which is all I'll claim; smugglers/IGG may be a case where there's true 2nd-player advantage, but they're extremely rare), unless you can get an engine built so as to play it every turn. If you look at it this way: yeah, there's twice as much chance to hit you before that first reshuffle, but then there's more chances for 2nd player to hit 1st before the next reshuffle... and getting hit by a militia on the first reshuffle usually doesn't hurt so much - yes, we all remember those golds that got dashed, but on average, you're discarding what? A copper and an estate?

LastFootnote

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #4 on: January 16, 2012, 05:14:04 pm »
+3

As for the militia thing... don't know what to tell you. Militia definitely exhibits 2nd-player advantage relative to the average card (which is all I'll claim; smugglers/IGG may be a case where there's true 2nd-player advantage, but they're extremely rare), unless you can get an engine built so as to play it every turn. If you look at it this way: yeah, there's twice as much chance to hit you before that first reshuffle, but then there's more chances for 2nd player to hit 1st before the next reshuffle... and getting hit by a militia on the first reshuffle usually doesn't hurt so much - yes, we all remember those golds that got dashed, but on average, you're discarding what? A copper and an estate?

Dude, that Copper you discard almost certainly brings your coin total for turn 3 or 4 down from $6 to $5 or from $5 to $4. That has ripple effects that last the rest of the game. I'll wager that the fact that player 2 is more likely to disrupt a slightly better hand does not make up for that. As you say, those early buys are the most important.
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ackack

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #5 on: January 16, 2012, 05:14:29 pm »
0

As for the militia thing... don't know what to tell you. Militia definitely exhibits 2nd-player advantage relative to the average card (which is all I'll claim; smugglers/IGG may be a case where there's true 2nd-player advantage, but they're extremely rare), unless you can get an engine built so as to play it every turn.

I'm inferring from the way that you've written this that your certainty is coming from simulations. While that would be a surprising result to me, I'd have some questions before I'd sign off on it altogether. Most pointedly, I'm assuming that the simulation is pure Big Money + Militia and ignores the other elements of the kingdom. (Even if the simulator is creating random kingdoms each time, my understanding of how buy rules are specified is that the rest of the kingdom would be functionally invisible unless you've written a strategy to account for it.) But that's a big deal for the kind of argument I'm making - if there are good $5s that we're not taking into account (and there often are) then a Big Money simulation will understate the first player advantage here.
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #6 on: January 16, 2012, 05:33:21 pm »
0

As for the militia thing... don't know what to tell you. Militia definitely exhibits 2nd-player advantage relative to the average card (which is all I'll claim; smugglers/IGG may be a case where there's true 2nd-player advantage, but they're extremely rare), unless you can get an engine built so as to play it every turn.

I'm inferring from the way that you've written this that your certainty is coming from simulations. While that would be a surprising result to me, I'd have some questions before I'd sign off on it altogether. Most pointedly, I'm assuming that the simulation is pure Big Money + Militia and ignores the other elements of the kingdom. (Even if the simulator is creating random kingdoms each time, my understanding of how buy rules are specified is that the rest of the kingdom would be functionally invisible unless you've written a strategy to account for it.) But that's a big deal for the kind of argument I'm making - if there are good $5s that we're not taking into account (and there often are) then a Big Money simulation will understate the first player advantage here.
Good 5s will definitely push an advantage more towards 1st player here. Possibly even over the top. Yes, it's kingdom-dependent (surprise surprise).

ehunt

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #7 on: January 16, 2012, 05:36:37 pm »
0

i essentially always veto militia as 2nd player when there's not something more pressing to veto because of the 50% differential in the probability that militia ruins one of the ubercritical turn 3/4 hands. are there statistics suggesting that militia is in fact less pro-first player than the average card? that really surprises me.
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #8 on: January 16, 2012, 05:40:11 pm »
0

As for the militia thing... don't know what to tell you. Militia definitely exhibits 2nd-player advantage relative to the average card (which is all I'll claim; smugglers/IGG may be a case where there's true 2nd-player advantage, but they're extremely rare), unless you can get an engine built so as to play it every turn. If you look at it this way: yeah, there's twice as much chance to hit you before that first reshuffle, but then there's more chances for 2nd player to hit 1st before the next reshuffle... and getting hit by a militia on the first reshuffle usually doesn't hurt so much - yes, we all remember those golds that got dashed, but on average, you're discarding what? A copper and an estate?

Dude, that Copper you discard almost certainly brings your coin total for turn 3 or 4 down from $6 to $5 or from $5 to $4. That has ripple effects that last the rest of the game. I'll wager that the fact that player 2 is more likely to disrupt a slightly better hand does not make up for that. As you say, those early buys are the most important.
http://dominionstrategy.com/2011/03/09/basic-opening-probabilities/
5 to 4 decently often, 6 to 5 very rarely, 6 to 4 pretty rare too.

WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #9 on: January 16, 2012, 05:41:45 pm »
0

i essentially always veto militia as 2nd player when there's not something more pressing to veto because of the 50% differential in the probability that militia ruins one of the ubercritical turn 3/4 hands. are there statistics suggesting that militia is in fact less pro-first player than the average card? that really surprises me.
Yeah, and actually that's more from game data than it is from simulations. Thing is, in a militia game, $4 isn't really a bad amount to have (generally) in comparison to $5. $6 is what you really want, and you're much, much more likely to get a $6 hand ruined on the 2nd reshuffle than on the first.

DG

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #10 on: January 16, 2012, 05:46:00 pm »
0

Tournaments exhibit most of the reasons for first turn advantage
 - the first player may be able to claim unique prizes first
 - the first player is more likely to shuffle provinces into the draw deck and reveal them from hand first
 - the first player can attack earlier with followers and get a decisive advantage
 
Tournaments usually play well with some trashing of the starting cards. This amplifies all the above advantage.
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ackack

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #11 on: January 16, 2012, 05:48:18 pm »
0

Yeah, and actually that's more from game data than it is from simulations.

What game data? This was going to be one of my other comments: while I think appeals to the CR data are also a little overused, this is a spot where if there were a way to peel this particular statistic off I think it would be a pretty good measure. I'd think you'd want a pretty large sample without too much bias towards particular players. Is there a way I'm unaware of to get this in CR, or are you compiling stuff yourself from the tarballs?
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #12 on: January 16, 2012, 06:01:30 pm »
0

Yeah, and actually that's more from game data than it is from simulations.

What game data? This was going to be one of my other comments: while I think appeals to the CR data are also a little overused, this is a spot where if there were a way to peel this particular statistic off I think it would be a pretty good measure. I'd think you'd want a pretty large sample without too much bias towards particular players. Is there a way I'm unaware of to get this in CR, or are you compiling stuff yourself from the tarballs?

This is actually from a study originally done quite some time ago. I certainly have never touched a tarball. theory referenced it in his annotated video.... I'll see if I can't find it (if somebody doesn't bail me out first? I really don't know where I'm looking)

WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #13 on: January 16, 2012, 06:14:04 pm »
0

Well, I'm apparently better at searching than I remember. Original militia data stems from here: http://forum.dominionstrategy.com/index.php?topic=20.msg238#msg238
There's more good resources on this here: http://forum.dominionstrategy.com/index.php?topic=946.0
If you're interested, Donald X's comments on first turn are here: http://forum.dominionstrategy.com/index.php?topic=91.0

To clarify and sum up, it seems to me that militia has somewhat of a first-turn advantage, roughly that of plain old money, less than that of the average card, and certainly much much less than what people seem to think.

DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #14 on: January 17, 2012, 02:27:32 am »
0

This is actually from a study originally done quite some time ago. I certainly have never touched a tarball. theory referenced it in his annotated video.... I'll see if I can't find it (if somebody doesn't bail me out first? I really don't know where I'm looking)


Too add some sims, it seems that also there BM-Militia has about the same first-player-advantage as BMU: 51-42.
Militia-Lab (buy 2 Militias and Lab at $5), has 53-39, standard-Lab is 51-39.
Militia+2Mountebank starting from 4/3 has 56-40, the standard Mountebank-bot also.

Not sure what to take from this, often the Militia seems not to have an additonal first-player advantage, but it does also not favour p2 compared to BM.  Even IF there are important $5s. The Labs ... first I would guess that is because the Militia gives you a better chance to win the split, but there are rarely more than 4 Labs gone (total!), so I don't think the split really matters. Maybe it's because the game is faster?
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ackack

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #15 on: January 17, 2012, 09:36:12 am »
0

Well, I'm apparently better at searching than I remember. Original militia data stems from here: http://forum.dominionstrategy.com/index.php?topic=20.msg238#msg238
There's more good resources on this here: http://forum.dominionstrategy.com/index.php?topic=946.0

The collection of a bunch of games is interesting. Thanks for the link.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #16 on: January 17, 2012, 12:11:08 pm »
0

I think your claim about the first source of 1st-player advantage (better to go first in a race for scarce resources) is not as uncategorical as you imply. In a two-player game, there is an equal number of kingdom cards (4 or 5) for each player. So P1 is at no advantage, because while P1 buys the first card, P2 buys the last.

This changes with +buys, since if you buy (for example, Fool's Gold) two at a time, they will split 6/4 instead of 5/5. At the extreme, as you mention, P1 wins in a landslide if both players draw KC x2 Bridge x3 on the same turn. In the absence of +buys, P1 is more likely to win a split only if his deck performs better, in which case this advantage is really just a consequence of the second type (getting to attack earlier turns).

Tournament has a massive P1 advantage even without +buys, partly because there is only one of each prize. (The other advantage for P1 is what makes the card really painful. Just yesterday I had Tournament plus $7 for a province, but P1 bought a Province, shuffled, and drew it into his next hand. :()

So in a game without +buys or attacks, I think we could safely conclude that there is no first-player advantage. In fact P2 would have the advantage from more info at each turn, and even more advantage given cards like Smugglers.
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DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #17 on: January 17, 2012, 12:14:35 pm »
+5

I think your claim about the first source of 1st-player advantage (better to go first in a race for scarce resources) is not as uncategorical as you imply. In a two-player game, there is an equal number of kingdom cards (4 or 5) for each player. So P1 is at no advantage, because while P1 buys the first card, P2 buys the last.

But usually it's not that you are guaranteed to buy an important card each turn. When P1 misses one opportunity more than P2, the cards still split 5/5, while if P2 misses an opportunity, they split 6/4.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #18 on: January 17, 2012, 01:03:02 pm »
0

I think your claim about the first source of 1st-player advantage (better to go first in a race for scarce resources) is not as uncategorical as you imply. In a two-player game, there is an equal number of kingdom cards (4 or 5) for each player. So P1 is at no advantage, because while P1 buys the first card, P2 buys the last.

But usually it's not that you are guaranteed to buy an important card each turn. When P1 misses one opportunity more than P2, the cards still split 5/5, while if P2 misses an opportunity, they split 6/4.

You may be right about this, but I think it emphasizes the degree to which first-player advantage is dwarfed by shuffle luck.
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jomini

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #19 on: January 17, 2012, 01:48:01 pm »
0

But shuffle luck is part of first player advantage, by dint of having an extra turn, P1 gets .5 extra turns with a fractional number more shuffles. Purchasing parity is only valid when there is 100% chance of matching parity; and that just isn't the case too often (for instance even with well built draw engine you can find yourself with no initial +action or even just all your green). The biggest advantage for P1 is that they get an extra turn 50% of the time.



A couple of other cards have second player advantage:
Noble Brigand - if you buy this a P1 on T1 you risk giving your opponent a turn 2 with a 3 coin or higher card. For P2, you have a modest chance of discarding P1's T1 buy.

Tribute - Tribute works better the more diverse your opponent's deck becomes; the real payout coming from 6 coin cards like nobles & harem. You have slightly better odds of getting better tribute hits with later hands as they almost always have more diversity.

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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #20 on: January 17, 2012, 02:42:25 pm »
0

But shuffle luck is part of first player advantage, by dint of having an extra turn, P1 gets .5 extra turns with a fractional number more shuffles. Purchasing parity is only valid when there is 100% chance of matching parity; and that just isn't the case too often (for instance even with well built draw engine you can find yourself with no initial +action or even just all your green). The biggest advantage for P1 is that they get an extra turn 50% of the time.

This is circular logic. The extra turn for P1 (the final turn) is not the source of any advantage -- it is the result of that advantage (among other much more significant things like skill and shuffle luck). You can't just point to the extra turn and say that is an advantage -- you have to explain how P1 got that extra turn and show that it is not the result of better play or shuffle luck. Without attacks, or multiple-buy turns leading to an uneven split of key cards, the inherent advantage to P1 is negligble.
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rinkworks

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #21 on: January 17, 2012, 04:14:22 pm »
0

Without attacks, or multiple-buy turns leading to an uneven split of key cards, the inherent advantage to P1 is negligble.

That's an inexplicable claim.  I was going to respond to your earlier post about this, but DStu said it flawlessly:

But usually it's not that you are guaranteed to buy an important card each turn. When P1 misses one opportunity more than P2, the cards still split 5/5, while if P2 misses an opportunity, they split 6/4.

In all probability, both players will miss opportunities.  The number of opportunities they miss will vary based on the variance of shuffle luck.  But that variance is not so great as to render insignificant the fact that the first player can miss a whole opportunity for free and still split evenly.  How many turns before all the Minions (for example) are bought up?   I'd guess by turn 9-11 or so?  Player 1 being able to miss an extra opportunity allows him some 25-50% more missed opportunities than Player 2 and still split Minions at least evenly, assuming 4/3 openings (so as not to count turns 1-2 as opportunities to buy Minions).  Huge.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #22 on: January 17, 2012, 04:41:12 pm »
0

Perhaps I have understated the P1 advantage, but not nearly to the degree it is usually overstated. My point is that unlike, say, a game of chess, a single game of dominion is not a great indicator of the relative skill of the opponents, given how much shuffle luck contributes to the outcome. And the solution - playing a series of games - happens to effectively address whatever inherent advantage comes with going first, so why stress? I believe it was Theory who once said he'd be happy to go second if he could control how the first reshuffle turned out.
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rinkworks

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #23 on: January 17, 2012, 07:46:13 pm »
0

Ah, I see.  Well there I'm with you.  While the first player advantage is statistically observable and significant, it takes a lot of games averaged together to see it.  I'm sure shuffle luck dwarfs the advantage for any single game, and play skill usually dwarfs both.

But I was approaching this discussion of turn advantage from a point of having already accounted for skill and shuffle luck and analyzing the turn advantage effects that remain.  Which, as I said, I think are still significant enough to think about and explore.
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Empathy

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #24 on: January 17, 2012, 10:53:04 pm »
0

Perhaps I have understated the P1 advantage, but not nearly to the degree it is usually overstated. My point is that unlike, say, a game of chess, a single game of dominion is not a great indicator of the relative skill of the opponents, given how much shuffle luck contributes to the outcome. And the solution - playing a series of games - happens to effectively address whatever inherent advantage comes with going first, so why stress? I believe it was Theory who once said he'd be happy to go second if he could control how the first reshuffle turned out.

I think the fact that the first player advantage both affects veto decisions and sometimes even opening decisions is non-negligible.

I believe that a lot of my recent level/skill gain has come mostly from recognizing when I should change my opening depending on my position, and what to veto in what position.

Treasure map/wareouse is a typical opening that I aim for only as p2. The increased variance helps.

Saying that p1 advantage is negligible because you can average it out (but not react to it in one single game) would be like saying that you don't react to a 5/2 opening.
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