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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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kn1tt3r

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #25 on: January 18, 2012, 02:21:41 am »
0

Treasure map/wareouse is a typical opening that I aim for only as p2. The increased variance helps.
I don't get this. TM/Warehouse is such a hugh combo, you need very good reasons not to go for it.

I MIGHT agree on simple TMs on BMesque boards against a potentially better player, but with Warehouse support, 1st/2nd player shouldn't really play a big role in your decision making.
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Mean Mr Mustard

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #26 on: January 18, 2012, 07:30:42 am »
0

I think that it is safe to say that even the best players, playing at top form, lose somewhere in the neighborhood of twenty-five percent of their games because of bad shuffle luck or by losing on turns, which is complicated by Isotropic's matching system that forces steady winners to play out of second player more often.  Games that are lost out of error in tactics or strategy do not effect that twenty-five percent, those losses subtract from the rest of the pie.

As to which is worse, variance or second player disadvantage?  Seems like a wash to me.  Both play a role regardless of skill differential.  In a close to evenly skilled match if the shuffles are equal the first player will usually win, and any subtle differences in builds reflect the skill differential.  A highly skilled player may win against a weaker opponent despite both bad shuffles and position, but if they do, again it is reflected in the seventy-five percent or so of games that it is possible to win.

This is the rub of Dominion.
« Last Edit: January 18, 2012, 07:34:51 am by Mean Mr Mustard »
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #27 on: January 18, 2012, 07:46:23 am »
+1

In the long run the variance won't matter, but there will always be the second player disadvantage.
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Mean Mr Mustard

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #28 on: January 18, 2012, 07:47:52 am »
0

What do you mean?  Clarify?
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Empathy

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #29 on: January 18, 2012, 08:14:53 am »
0

Treasure map/wareouse is a typical opening that I aim for only as p2. The increased variance helps.
I don't get this. TM/Warehouse is such a hugh combo, you need very good reasons not to go for it.

I MIGHT agree on simple TMs on BMesque boards against a potentially better player, but with Warehouse support, 1st/2nd player shouldn't really play a big role in your decision making.

TM/warehouse is a good combo, but doesn't dominate any board its present on. I'd definitely open most ambassador openings rather than TM/warehouse on p1, but not on p2, though it depends on support for amb.

example: http://councilroom.com/game?game_id=game-20120116-065836-18fd44e7.html

That peddler deck my opponent played was *sweet*. Warehouse made sure that devel actually met a useful card to devel (rather than crappy copper).  I'm sure he could have made it explode T16 if it wasn't for me getting an early lead through TM luck, forcing him to find alternative green.

I still think my argument for 5/2 vs p1 advantage holds: if you are going to react to an opponent's 5/2 opening (changing your strategy because he got an unexpected HP/witch/mint), then so should you adapt to playing p2. The fact that any advantage gets "averaged out" after a high number of games doesn't change the fact that you can improve your play that way. And as Geronimoo mentioned, the p2 advantage will actually remain post-averaging if you play p2 more often than p1 (which is the case for a lot of us).
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #30 on: January 18, 2012, 09:05:35 am »
0

Treasure map/wareouse is a typical opening that I aim for only as p2. The increased variance helps.
I don't get this. TM/Warehouse is such a hugh combo, you need very good reasons not to go for it.

I MIGHT agree on simple TMs on BMesque boards against a potentially better player, but with Warehouse support, 1st/2nd player shouldn't really play a big role in your decision making.

TM/warehouse is a good combo, but doesn't dominate any board its present on. I'd definitely open most ambassador openings rather than TM/warehouse on p1, but not on p2, though it depends on support for amb.

Sure it does. Just because it doesn't dominate ALL boards it's on doesn't mean it's very very strong on many of them.
And yes, ambassador probably provides a reason to not go for it.

kn1tt3r

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #31 on: January 18, 2012, 09:12:07 am »
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I'm sure he could have made it explode T16 if it wasn't for me getting an early lead through TM luck, forcing him to find alternative green.
Turn 6 for a successful TM/Warehouse combo isn't extraordinary lucky.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #32 on: January 18, 2012, 09:35:31 am »
0

I think that it is safe to say that even the best players, playing at top form, lose somewhere in the neighborhood of twenty-five percent of their games because of bad shuffle luck or by losing on turns, which is complicated by Isotropic's matching system that forces steady winners to play out of second player more often.  Games that are lost out of error in tactics or strategy do not effect that twenty-five percent, those losses subtract from the rest of the pie.

As to which is worse, variance or second player disadvantage?  Seems like a wash to me.  Both play a role regardless of skill differential.  In a close to evenly skilled match if the shuffles are equal the first player will usually win, and any subtle differences in builds reflect the skill differential.  A highly skilled player may win against a weaker opponent despite both bad shuffles and position, but if they do, again it is reflected in the seventy-five percent or so of games that it is possible to win.

This is the rub of Dominion.

But one of the points that has come out in our discussion so far is that first-player advantage (ignoring attacks for the moment) is essentially this: P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards. I've mentioned before that attack cards can change this drastically, as can +buy to a lesser extent.
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WanderingWinder

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #33 on: January 18, 2012, 09:44:56 am »
0

I think that it is safe to say that even the best players, playing at top form, lose somewhere in the neighborhood of twenty-five percent of their games because of bad shuffle luck or by losing on turns, which is complicated by Isotropic's matching system that forces steady winners to play out of second player more often.  Games that are lost out of error in tactics or strategy do not effect that twenty-five percent, those losses subtract from the rest of the pie.

As to which is worse, variance or second player disadvantage?  Seems like a wash to me.  Both play a role regardless of skill differential.  In a close to evenly skilled match if the shuffles are equal the first player will usually win, and any subtle differences in builds reflect the skill differential.  A highly skilled player may win against a weaker opponent despite both bad shuffles and position, but if they do, again it is reflected in the seventy-five percent or so of games that it is possible to win.

This is the rub of Dominion.

But one of the points that has come out in our discussion so far is that first-player advantage (ignoring attacks for the moment) is essentially this: P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards. I've mentioned before that attack cards can change this drastically, as can +buy to a lesser extent.
Okay, but that means that there IS 1st player advantage, even without attacks, buys, etc........
Furthermore, if you're saying that it's easier for P1 to get lucky... isn't that less lucky somehow?

Mean Mr Mustard

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #34 on: January 18, 2012, 10:13:20 am »
0

I think I am in over my head here, and should bow out.  Explain this, and I'll shove off: if a poor player has to get extremely lucky to beat an elite player, despite turn advantage, how exactly is variance getting cancelled out?
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rinkworks

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #35 on: January 18, 2012, 11:06:29 am »
0

P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards.

This is just not mathematically sound.  One player benefitting more from lucky shuffling and the other being more vulnerable to unlucky shuffling aren't effects that cancel each other out but in fact compound each other.

Let's assume that each player has an equal chance, each turn, of missing an opportunity to buy a key card.  We'll count the number of missed opportunities for each player over the course of, let's say, 8 turns.  (It doesn't matter how many, really.)  Let's say each player has a 20% chance of a turn being a missed opportunity.  (The exact percentage affects the magnitude of the resulting advantage but not whether or not there is one.)  Let's let M1 = the number of P1's missed opportunities.  Let M2 = the number of P2's missed opportunities.

If we simulate this, then the values of M1 and M2 we come up with will determine how the key card splits.  If M1 < M2, then P1 wins the split.  If M1 is equal to or one more than M2, then the split ties.  For P2 to win the split, M2 must be at most two less than M1.

The result of averaging this calculation over many simulations should be apparent, but just for fun I simulated this 10,000 times and got the following results:

* P1 wins the split: 3789 times
* Split is even: 4494 times
* P2 wins the split: 1717 times

In other words, P1 is more than twice as likely to win the split as P2 is.  Not that I recommend reading too much into these specific numbers!  Obviously I've made a lot of assumptions on how many turns it will take to exhaust the supply pile of the key card, and the likelihood of missing an opportunity to purchase one.  Moreover, the chance of missing any one opportunity is correlated with the chances of missing any of the others, because having a bad hand means possibly having a good hand next.  Still, you can finagle these numbers any way you want to:  ultimately, the odds MUST favor P1, simply because P1 can win the race after making more mistakes than P2 can make in order for him to.
« Last Edit: January 18, 2012, 12:02:45 pm by rinkworks »
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jomini

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #36 on: January 18, 2012, 11:12:06 am »
+1

Ttlyod:

Think of it this way for all cards in dominion except provinces there is an even number of cards. Say there is a dominant strat involving a dominant card, and each player has a 95% chance of acquiring that card (one copy) each turn. Statistical odds of P2 hitting that 5 times in a row are ~ 77.4%. Now what about P1? Well they also will hit 77.4% of the time on their first 5 runs, but they will also hit on their 6th shot 95% of the time. So of that 22.6% of games, 95% of those will also result in parity for P1 in other words P1 has  ~98.9% chance of reaching parity or better in the dominant strategy (I'm ignoring multiple misses as they are rather unlikely). P2 starts the game with a 21.5% disadvantage. Note - I'm using 95% odds which is remarkedly low variance (i.e. shuffle luck is close to non-existant) AND I'm treating each hand as independent (in reality if a player misses on one hand they generally are somewhat more likely to hit on the next one, which makes a P1 miss even less damaging relative to a P2 miss). Note that this scenario has no attacks, no +buys (or other card gain), no player interaction (aside from competition for scarce resources) and no preferential shuffling.

Now what about when there isn't a competition for the split on cards? Well say there is a card (like forge) where you want exactly one and where acquiring said card is the dominant strategy. Okay let's say we reach a parity point in the game where each player has a 75% chance of getting forge each turn (say with apothecaries & villages). Now P1 will hit it 75% of the time and 25% of the time P2 will not hit it. In 18.8% of games P1 has a solid lead. Now suppose P1 misses and P2 hits it, P2 doesn't get a solid lead unless P1 misses again. This will only occur in 4.7% of games. This leaves P1 with an advantage solely due to turn order in 14.1% of games.

There are just too many types of dominion games where it is a race to some critical threshold (e.g. 5) or some critical limited resource (e.g. curses, minions, cities, etc.), in either case there is a real P1 advantage. It gets tempered by shuffling (as we don't shuffle after every turn players have 2 or 3 turns to tie a race scenario), but shuffling dynamics also let P1 win out in a lot of interaction elements.

The long and the short of it is that the only scenarios where I want to be second player are those where I can adjust my play to interact with my opponent (e.g. going tunnel if he goes militia & going militia if he doesn't go tunnel). You can get rid of the +buys, the bonus card gains, all the attacks, and all the shuffle timing (e.g. by having a board where chancellor is dominant) and I will STILL take P1 even facing a tie-breaker penalty.
« Last Edit: January 18, 2012, 02:22:55 pm by jomini »
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #37 on: January 18, 2012, 05:06:51 pm »
0

P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards.

This is just not mathematically sound.  One player benefitting more from lucky shuffling and the other being more vulnerable to unlucky shuffling aren't effects that cancel each other out but in fact compound each other.

Let's assume that each player has an equal chance, each turn, of missing an opportunity to buy a key card.  We'll count the number of missed opportunities for each player over the course of, let's say, 8 turns.  (It doesn't matter how many, really.)  Let's say each player has a 20% chance of a turn being a missed opportunity.  (The exact percentage affects the magnitude of the resulting advantage but not whether or not there is one.)  Let's let M1 = the number of P1's missed opportunities.  Let M2 = the number of P2's missed opportunities.

If we simulate this, then the values of M1 and M2 we come up with will determine how the key card splits.  If M1 < M2, then P1 wins the split.  If M1 is equal to or one more than M2, then the split ties.  For P2 to win the split, M2 must be at most two less than M1.

The result of averaging this calculation over many simulations should be apparent, but just for fun I simulated this 10,000 times and got the following results:

* P1 wins the split: 3789 times
* Split is even: 4494 times
* P2 wins the split: 1717 times

In other words, P1 is more than twice as likely to win the split as P2 is.  Not that I recommend reading too much into these specific numbers!  Obviously I've made a lot of assumptions on how many turns it will take to exhaust the supply pile of the key card, and the likelihood of missing an opportunity to purchase one.  Moreover, the chance of missing any one opportunity is correlated with the chances of missing any of the others, because having a bad hand means possibly having a good hand next.  Still, you can finagle these numbers any way you want to:  ultimately, the odds MUST favor P1, simply because P1 can win the race after making more mistakes than P2 can make in order for him to.

That is a very impressive refutation of something I didn't say. My first comments in this discussion were to the effect that going first (aside from attack cards) isn't an advantage because the cards will split evenly. I was wrong, as has been explained, because my argument assumed similar shuffling between P1 and P2. Once you allow for different degrees of shuffle luck, you can see that P1 has a greater upside from shuffle luck while P2 has a greater downside. I never said these effects cancelled out. Claiming that they compound each other, however, is nonsensical -- the two effects are really just the two sides of the same coin. My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck. If P1 and P2 have similar shuffling outcomes, P1 will have no advantage in terms of splitting key cards. Given, however, that P1 and P2 are unlikely to have similar shuffling outcomes, on average P1 will be at an advantage, since P1 is more likely than P2 to benefit from better shuffle luck.
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Axxle

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #38 on: January 18, 2012, 07:10:19 pm »
0

P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards.

This is just not mathematically sound.  One player benefitting more from lucky shuffling and the other being more vulnerable to unlucky shuffling aren't effects that cancel each other out but in fact compound each other.

Let's assume that each player has an equal chance, each turn, of missing an opportunity to buy a key card.  We'll count the number of missed opportunities for each player over the course of, let's say, 8 turns.  (It doesn't matter how many, really.)  Let's say each player has a 20% chance of a turn being a missed opportunity.  (The exact percentage affects the magnitude of the resulting advantage but not whether or not there is one.)  Let's let M1 = the number of P1's missed opportunities.  Let M2 = the number of P2's missed opportunities.

If we simulate this, then the values of M1 and M2 we come up with will determine how the key card splits.  If M1 < M2, then P1 wins the split.  If M1 is equal to or one more than M2, then the split ties.  For P2 to win the split, M2 must be at most two less than M1.

The result of averaging this calculation over many simulations should be apparent, but just for fun I simulated this 10,000 times and got the following results:

* P1 wins the split: 3789 times
* Split is even: 4494 times
* P2 wins the split: 1717 times

In other words, P1 is more than twice as likely to win the split as P2 is.  Not that I recommend reading too much into these specific numbers!  Obviously I've made a lot of assumptions on how many turns it will take to exhaust the supply pile of the key card, and the likelihood of missing an opportunity to purchase one.  Moreover, the chance of missing any one opportunity is correlated with the chances of missing any of the others, because having a bad hand means possibly having a good hand next.  Still, you can finagle these numbers any way you want to:  ultimately, the odds MUST favor P1, simply because P1 can win the race after making more mistakes than P2 can make in order for him to.

That is a very impressive refutation of something I didn't say. My first comments in this discussion were to the effect that going first (aside from attack cards) isn't an advantage because the cards will split evenly. I was wrong, as has been explained, because my argument assumed similar shuffling between P1 and P2. Once you allow for different degrees of shuffle luck, you can see that P1 has a greater upside from shuffle luck while P2 has a greater downside. I never said these effects cancelled out. Claiming that they compound each other, however, is nonsensical -- the two effects are really just the two sides of the same coin. My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck. If P1 and P2 have similar shuffling outcomes, P1 will have no advantage in terms of splitting key cards. Given, however, that P1 and P2 are unlikely to have similar shuffling outcomes, on average P1 will be at an advantage, since P1 is more likely than P2 to benefit from better shuffle luck.

edit: sorry, didn't read all of tlloyd's responses before this.

P1: Buys one key card
P2: Buys one key card
P1: Buys two key cards
P2: Buys two key cards
P1: Buys three key cards
P2: Only one key card left, buys it.

Even if both players get the exact same shuffle luck, at the very least P1 gets an advantage in splitting any key card if there is a way to obtain more than one in a turn.
« Last Edit: January 19, 2012, 05:50:52 pm by Axxle »
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #39 on: January 18, 2012, 07:45:13 pm »
0

(e.g. going tunnel if he goes militia & going militia if he doesn't go tunnel).
I saw this and just had to point out http://councilroom.com/game?game_id=game-20120105-165629-8d87aaab.html.  He went tunnel because i went militia, and he actually continued with tunnel.  I just didn't play militia, so he bought a blank card on turn 2 and turn 3.  Admittedly, I also had a semi-blank card and he had a 4 vp advantage, but I wanted to bring it up because it was a one in a thousand kind of game involving tactical counterplay. 
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Anon79

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #40 on: January 18, 2012, 09:20:56 pm »
0

Would have been interesting if your opponent had gotten Lookout earlier, say by opening Lookout/Tunnel in response to Militia. Now Tunnel isn't a blank card.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #41 on: January 18, 2012, 09:50:46 pm »
0

My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck.
But that's like saying that King's Court isn't that good because you need good actions to multiply with it. King's Court IS of course good, because usually those actions are out there. Just as P1's advantage is... well I'm not sure who you're saying is overstating it, because as far as I can tell, all anybody's claimed is that it exists and it isn't negligible... but P1's advantage exists and is, well, not negligible, because there quite often is differential shuffle luck.
And there's usually something else to compound it, too, whether attacks, or gains, or three piles, or plus buy, or a mega-turn...
So... is your point just that it's dependent on differential shuffle luck? What I'm describing as type I in the article certainly is, sans extra gains or buys, which I don't think anyone's disputing. What I'm describing as type II is not, as I demonstrate in the article.
If that's not your point... I'm not sure what your point actually is.

Empathy

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #42 on: January 18, 2012, 10:07:06 pm »
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I'm sure he could have made it explode T16 if it wasn't for me getting an early lead through TM luck, forcing him to find alternative green.
Turn 6 for a successful TM/Warehouse combo isn't extraordinary lucky.

Agreed, but I felt somewhat more on the lucky end of my warehouse draw outcomes/mandarin plays.

My point was more that, if given the choice between his or my deck, I would prefer his. I think the odds are pretty close, probably almost 50-50, but biased towards the peddler deck. I also think I would have lost the peddler split if I had try to compete with him playing as p2, and that my odds playing his deck as p2 were easily 40-60.

The argument that this p1 advantage averages out as you play more games is true, but that also holds for 5/2 split advantages. There is no reason for you to not play as many 5/2 openings on advantageous boards than your opponent, given enough games. Actually, it is *more* the case than p1/p2 outcomes, given that these are biased by your skill level.

That doesn't mean you shouldn't micro-manage each single game according to the outcome "I have 3/4 split and my opponent has 5/2 split". Similarly, the outcome "I play second" should be weighted in your decision process.

The only way to nullify this argument is if the optimal p1 and p2 strategy always coincide. I am a firm believer that they don't, at least on some select boards (maybe 20% of them?). These boards are typically ones that favor p1.

My favorite example is chapel when there is no militia. Chapel without attacks makes for *very* deterministic games, especially if there is a solid engine involved. chapel/grand market, chapel/KC and chapel/tournament are so heavily biased towards p1 that I struggle to find ways to counter those openings.

If my opponent goes chapel/tournament, I will definitely not follow suitl. The only way I can see myself saving that game is by getting a lucky province => baron/coppersmith.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #43 on: January 19, 2012, 03:12:43 am »
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Not quite the same as the current topic, but I wouldn't be surprised if, in a game between a good player and a mediocre player, the mediocre player had a 2nd-player advantage. In the following sense: he can simply copy the optimal strategy and win by luck, and that may be the most likely way for him to win.
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jomini

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #44 on: January 19, 2012, 10:59:04 am »
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P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards.

This is just not mathematically sound.  One player benefitting more from lucky shuffling and the other being more vulnerable to unlucky shuffling aren't effects that cancel each other out but in fact compound each other.

Let's assume that each player has an equal chance, each turn, of missing an opportunity to buy a key card.  We'll count the number of missed opportunities for each player over the course of, let's say, 8 turns.  (It doesn't matter how many, really.)  Let's say each player has a 20% chance of a turn being a missed opportunity.  (The exact percentage affects the magnitude of the resulting advantage but not whether or not there is one.)  Let's let M1 = the number of P1's missed opportunities.  Let M2 = the number of P2's missed opportunities.

If we simulate this, then the values of M1 and M2 we come up with will determine how the key card splits.  If M1 < M2, then P1 wins the split.  If M1 is equal to or one more than M2, then the split ties.  For P2 to win the split, M2 must be at most two less than M1.

The result of averaging this calculation over many simulations should be apparent, but just for fun I simulated this 10,000 times and got the following results:

* P1 wins the split: 3789 times
* Split is even: 4494 times
* P2 wins the split: 1717 times

In other words, P1 is more than twice as likely to win the split as P2 is.  Not that I recommend reading too much into these specific numbers!  Obviously I've made a lot of assumptions on how many turns it will take to exhaust the supply pile of the key card, and the likelihood of missing an opportunity to purchase one.  Moreover, the chance of missing any one opportunity is correlated with the chances of missing any of the others, because having a bad hand means possibly having a good hand next.  Still, you can finagle these numbers any way you want to:  ultimately, the odds MUST favor P1, simply because P1 can win the race after making more mistakes than P2 can make in order for him to.

That is a very impressive refutation of something I didn't say. My first comments in this discussion were to the effect that going first (aside from attack cards) isn't an advantage because the cards will split evenly. I was wrong, as has been explained, because my argument assumed similar shuffling between P1 and P2. Once you allow for different degrees of shuffle luck, you can see that P1 has a greater upside from shuffle luck while P2 has a greater downside. I never said these effects cancelled out. Claiming that they compound each other, however, is nonsensical -- the two effects are really just the two sides of the same coin. My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck. If P1 and P2 have similar shuffling outcomes, P1 will have no advantage in terms of splitting key cards. Given, however, that P1 and P2 are unlikely to have similar shuffling outcomes, on average P1 will be at an advantage, since P1 is more likely than P2 to benefit from better shuffle luck.

You are still wrong. As I showed, P1, solely by dint of turn position also has an advantage in a race - first to get a card that dominates. Even when you aren't fighting to split cards (say in in forge/KC x2 /Monument x3  game - nothing will pile), you still are fighting to be the first one to a forge & the first one to start scoring megapoints regularly. This happens whenever you have a key card (like goons, golem, possession, forge, witch, followers, mint, KC, etc.) that greatly benefits playing it first but has high cost.

Further, you are mistaking what "shuffle luck" actually entails here. When we do these probabilistic analyses we are giving the players the EXACT SAME distribution of luck. The distribution of hands in my analysis is IDENTICAL for both players. P1 is just a likely as P2 to luck out and get the good cards on any given hand, the ONLY difference is he gets to play an extra hand a statistically significant percent of the time.

This as actually an unrealistic assumption. Once a player breaks parity and pulls ahead, they tend to have better odds of getting whatever it is they want (key kingdom card, province, etc.). So take a minion race. Both open chancellor/silver (everything else is crap or too pricey). If P1 enters T3 (newly shuffled from chancellor) with 2 minions and P2 enters T3 with 1 minion, then P1 has better odds of acquiring a minion on T3 (if he has to minion for 4 he has better odds of hitting 5 coin the next 4 card hand or trying his luck again). When the card you buy makes buying the next card you want more likely, we witness a (normally slight) compounding of odds.

This is not to say that the P1 advantage is overwhelming, but that it will account for a good percentage of the odds of winning a game.
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DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #45 on: January 19, 2012, 11:14:25 am »
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@jomini:
The point is you are talking about something else the he is:

He said: Given both player at the current game get "equally lucky" (in the sense of: missing their target equally often), then, given no +buy (which I think was not mentioned in this posts but was explicitly mentioned in all posts before, so we can savely assume that this assumption is still valid), player 1 has no advantage in this split. But p1 profits more or earlier for a lucky outcome than p2 does.
Which is exactly what's happening.

You say that obviously both player have the same distribution of luck, which is of course save to assume.  But that's not what he is talking about. He is not talking about what's happening when the random variable $LUCK has the same distribution for both players, but what is happening when the actual realization of $LUCK is the same (or different) for both players. To understand in which of this cases p1 has an advantage over p2, and in which not.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #46 on: January 19, 2012, 11:38:57 am »
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It's not that interesting to consider only games in which p1 and p2 have the same exact "luck"  outcome, because that will almost never happen.  This is setting aside the matter of measuring luck given divergent strategies.

It's like saying.  First, let's restrict our statement to the less than 1% of games in which the sequence of multiple random outcomes is the same for both players, then X.  Sure, I'll agree with you on that.  But it says very little because it rarely applies.

OTOH, considering comparable levels of luck is interesting.  The 6-4 split when p1 gets a little luckier than p2 getting matched to a 5-5 split when p2 is a little luckier than p1 is indeed insightful.
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DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #47 on: January 19, 2012, 11:53:35 am »
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I anyway think that is getting a bit offtopic:
We started with the thesis that for racing for a pile a firstplayer advantage does not exists, because ... I don't know anymore .... We explained to whomever that this is wrong, because if both player does not have the same shuffleluck, than there really is an advantage. Then whoever realized what was the mistake, namely that his analysis only holds in the case with equal shuffleluck, and wrote a post where he for my understanding explained exactly that. Then there was a post that critics exactly that you may not only consider the cases with equal shuffleluck, but whatever.

I think that is clear to everybody since at least yesterday, leading to my last post, which obviously did exactly the opposite of what it should do. m(
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #48 on: January 19, 2012, 11:58:05 am »
+1

How about the following toy model:

Take two Gaussian r.v. for the outcome of "luck". All you affect is therefore the mean and variance of your "luck"/deck.

Now every turn, you draw a random sample from your Gaussian, and add it up. Whoever gets first to some threshold number (say 40ish points) wins.

It makes sense that, to minimize the average number of turns to attain the threshold, you simply maximize the mean of your deck. To *win* however, you need to do something more subtle: maximize the probability of getting there before your opponent.

Now it's pretty clear that the first player has an edge in games where the variance of the deck is small: at the limit where sigma ->0, he always wins! This advantage decreases drastically as sigma becomes large.

Now, I am sure someone can come up with a pair of 2 means and variances such that p1 goes for a high mean, low variance deck, and p2 increases his odds by going with a lower mean, higher variance.

Actually, just take mean = 39, variance = 0, mean = 0 , variance = 100000000000000.

p1 wins deterministically on T2, but p2 has 1/2 of winning on t1. Best odds p2 can hope to get in this toy model.

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jomini

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #49 on: January 19, 2012, 12:12:08 pm »
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Dtsu:

As I noted very early on, parity rules only apply in a deterministic setting. E.g. where each player gets equivalent hands at all times. By this I mean if P1 and P2 will both always hit or miss at the same time then parity is maintained and there can be no other way to split cards than 5:5. However if P1 and P2 merely hit or miss at the same rate, then P1 has an advantage.

The first condition is generally not applicable, players will have differing hands from time to time. The second condition is a good first order approximation. P1 is not getting any luckier than P2, he has no deviation from expected hand performance, the "luck" component is indeed identical. P1 simply has an extra turn.

Now in a lot of games, P1 has additional advantages that WW was pointing out (like being able to witch on T3 before P2 shuffles) and in some games he has fewer (e.g. smugglers with something like IGG). Nonetheless, P1 retains an inherent advantage in winning the split on limited cards and in winning the race to a key card thanks to his extra turn (or half turn if you want an expectation value). In some very simple games (like big money/smithy) this is mitigated by P2 winning ties ... but in a lot of games (those involving card gainers, +buys, interactive cards, those involving dominant high value cards, etc.) P1's advantage is real and explains a good bit of win distribution in matches.
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DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #50 on: January 19, 2012, 12:15:19 pm »
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I never doubted that. I only doubted that somebody wanted to say something else.
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dondon151

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #51 on: January 19, 2012, 12:17:43 pm »
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Nonetheless, P1 retains an inherent advantage in winning the split on limited cards and in winning the race to a key card thanks to his extra turn (or half turn if you want an expectation value).

This reminds me of that puzzle posted a little while back where you had to find ways for player 2 to win an FG split. Needless to say, the chances of that are slim to none if both are going for the optimal strategy. Part of this is because FG is so easy to get, but you can extrapolate this to any key card that one would like to have a lot of.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #52 on: January 19, 2012, 02:37:05 pm »
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My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck.
But that's like saying that King's Court isn't that good because you need good actions to multiply with it. King's Court IS of course good, because usually those actions are out there. Just as P1's advantage is... well I'm not sure who you're saying is overstating it, because as far as I can tell, all anybody's claimed is that it exists and it isn't negligible... but P1's advantage exists and is, well, not negligible, because there quite often is differential shuffle luck.
And there's usually something else to compound it, too, whether attacks, or gains, or three piles, or plus buy, or a mega-turn...
So... is your point just that it's dependent on differential shuffle luck? What I'm describing as type I in the article certainly is, sans extra gains or buys, which I don't think anyone's disputing. What I'm describing as type II is not, as I demonstrate in the article.
If that's not your point... I'm not sure what your point actually is.

That is exactly my point. And the "sans extra gains or buys" was an important part of my point which has been overlooked by some in this discussion. Ditto for the distinction between types I and II. I have already admitted to understating the advantage, but I am still seeing in here comments to the effect that the first person to get a particular card has an advantage -- to which I reply, only if that card can interfere with the other player's deck (generally an attack, but also province in a tournament game) or allows for multiple gains/buys.
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #53 on: January 19, 2012, 02:45:57 pm »
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P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards.

This is just not mathematically sound.  One player benefitting more from lucky shuffling and the other being more vulnerable to unlucky shuffling aren't effects that cancel each other out but in fact compound each other.

Let's assume that each player has an equal chance, each turn, of missing an opportunity to buy a key card.  We'll count the number of missed opportunities for each player over the course of, let's say, 8 turns.  (It doesn't matter how many, really.)  Let's say each player has a 20% chance of a turn being a missed opportunity.  (The exact percentage affects the magnitude of the resulting advantage but not whether or not there is one.)  Let's let M1 = the number of P1's missed opportunities.  Let M2 = the number of P2's missed opportunities.

If we simulate this, then the values of M1 and M2 we come up with will determine how the key card splits.  If M1 < M2, then P1 wins the split.  If M1 is equal to or one more than M2, then the split ties.  For P2 to win the split, M2 must be at most two less than M1.

The result of averaging this calculation over many simulations should be apparent, but just for fun I simulated this 10,000 times and got the following results:

* P1 wins the split: 3789 times
* Split is even: 4494 times
* P2 wins the split: 1717 times

In other words, P1 is more than twice as likely to win the split as P2 is.  Not that I recommend reading too much into these specific numbers!  Obviously I've made a lot of assumptions on how many turns it will take to exhaust the supply pile of the key card, and the likelihood of missing an opportunity to purchase one.  Moreover, the chance of missing any one opportunity is correlated with the chances of missing any of the others, because having a bad hand means possibly having a good hand next.  Still, you can finagle these numbers any way you want to:  ultimately, the odds MUST favor P1, simply because P1 can win the race after making more mistakes than P2 can make in order for him to.

That is a very impressive refutation of something I didn't say. My first comments in this discussion were to the effect that going first (aside from attack cards) isn't an advantage because the cards will split evenly. I was wrong, as has been explained, because my argument assumed similar shuffling between P1 and P2. Once you allow for different degrees of shuffle luck, you can see that P1 has a greater upside from shuffle luck while P2 has a greater downside. I never said these effects cancelled out. Claiming that they compound each other, however, is nonsensical -- the two effects are really just the two sides of the same coin. My point all along has been that P1's advantage is often overstated, since it really is dependent on differential shuffle luck. If P1 and P2 have similar shuffling outcomes, P1 will have no advantage in terms of splitting key cards. Given, however, that P1 and P2 are unlikely to have similar shuffling outcomes, on average P1 will be at an advantage, since P1 is more likely than P2 to benefit from better shuffle luck.

You are still wrong. As I showed, P1, solely by dint of turn position also has an advantage in a race - first to get a card that dominates. Even when you aren't fighting to split cards (say in in forge/KC x2 /Monument x3  game - nothing will pile), you still are fighting to be the first one to a forge & the first one to start scoring megapoints regularly. This happens whenever you have a key card (like goons, golem, possession, forge, witch, followers, mint, KC, etc.) that greatly benefits playing it first but has high cost.

Further, you are mistaking what "shuffle luck" actually entails here. When we do these probabilistic analyses we are giving the players the EXACT SAME distribution of luck. The distribution of hands in my analysis is IDENTICAL for both players. P1 is just a likely as P2 to luck out and get the good cards on any given hand, the ONLY difference is he gets to play an extra hand a statistically significant percent of the time.

This as actually an unrealistic assumption. Once a player breaks parity and pulls ahead, they tend to have better odds of getting whatever it is they want (key kingdom card, province, etc.). So take a minion race. Both open chancellor/silver (everything else is crap or too pricey). If P1 enters T3 (newly shuffled from chancellor) with 2 minions and P2 enters T3 with 1 minion, then P1 has better odds of acquiring a minion on T3 (if he has to minion for 4 he has better odds of hitting 5 coin the next 4 card hand or trying his luck again). When the card you buy makes buying the next card you want more likely, we witness a (normally slight) compounding of odds.

This is not to say that the P1 advantage is overwhelming, but that it will account for a good percentage of the odds of winning a game.

You need to read the earlier discussion more carefully. The majority of the cards you give as examples are either attacks, in which case you are conflating two distinct types of P1 advantage, or allow for multiple gains/buys, which I have said from the beginning can grant P1 an advantage even with identical shuffling. The other cards you mention do not provide the advantage to P1 that you claim. Why would P1 buying a mint on his third turn provide P1 an advantage if P2 also buys a mint on his third turn?
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #54 on: January 19, 2012, 03:24:27 pm »
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A question for anyone (if anyone is still reading this  :P):

How does the distinction between luck distribution and outcomes apply given the shuffling mechanic in dominion? Take turns 3 and 4, for example. It clearly doesn't undermine my point if we allow P1's turn 3 to match P2's turn 4, since the cards gained on both turns generally get shuffled into the deck at the same time. P1 gets no advantage from buying a gold on T3 if P2 gets a gold on T4. But it would of course matter if P1 got his $6 hand before the second reshuffle while P2 didn't get $6 until T5. That is the type of shuffle-luck I'm excluding. Theory's point was that such disparate luck outweighs who goes first. My point is that if this type of disparate luck is necessary to give P1 a significant type-I advantage, we might as well chalk it up to luck as to positional advantage, since P2 has just as much chance to be the lucky one and would be equally advantaged by it.
« Last Edit: January 20, 2012, 10:22:15 am by tlloyd »
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jomini

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #55 on: January 19, 2012, 04:51:21 pm »
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Quote
You need to read the earlier discussion more carefully. The majority of the cards you give as examples are either attacks, in which case you are conflating two distinct types of P1 advantage, or allow for multiple gains/buys, which I have said from the beginning can grant P1 an advantage even with identical shuffling. The other cards you mention do not provide the advantage to P1 that you claim. Why would P1 buying a mint on his third turn provide P1 an advantage if P2 also buys a mint on his third turn?

I have given no examples of using attacks in any mathematically worked example. Further NONE of my analysis has looked at the impact of playing the attack, merely who gets there first or who wins the split. For instance with Forge I have not even looked at its gain; I literally leave it as a breakpoint and analyze who gets there first (literally if all the forge did was forge curses & coppers into coppers my analysis STILL holds) . This "critique" is spurious and, frankly, I am disappointed that you could not argue without making it.

As far as mint goes. Let's say that both players open chancellor/mining village with a goal of minting to big money (which for whatever reason is dominant here). To simplify analysis the chancellor will always hit for both players (no luck advantage there). Now suppose P1 hits mint on T2 & chancellors. Then he has trashed 3 (4) coppers & has a mint. This will happen X% of the time. Now P2 may or may not hit mint & chancellor as well. He, likewise has X% odds. (X^2)% of the time we will have a degenerate match. However, X(1-X)% of the time, P1 will be playing with a short deck and have a strong advantage. P2 will be in the reverse situation only (1-X)^2*X% of the time. Hence the odds of P2 having an extra turn with a thin deck are much lower than those for P1.

The scenarios are these:
Most often: P1 & P2 buy mints on T3 or P1 misses & P2 misses (no advantage)
Next most often: P1 buys a mint on T3 & P2 misses on T3 (advantage P1)
Next most often: P1 misses a mint on T3, buys it on T4 & P2 gets a mint on T3 (no advantage)
Least often: P1 misses a mint on T3 & T4 and P2 buys a mint on T3 (advantage P2)

Now you correctly note that for many decks purchases on T3 or T4 are equivalent, this is true, but for the purposes of this discussion that can be largely abstracted by saying what are the odds that P1 hits something on T3 or T4 and what are the odds that P2 hits something on T3 or T4. We could run the entire analysis again without the lucky chancellor condition, but the net effect is this degeneracy diminishes, but does not eliminate the first player advantage from having a probable extra turn.


This is a matter of statistics. We could, in theory, derive correlation coefficients or do an ANOVA regression to determine how much the variance in player achievement is driven by turn position. "Chalking it up to luck" is bad mathematics and incorrect. Some of the variation in player achievement comes from skill, some comes from random card distribution, and some comes from position.

I believe that skill > card distribution > position; but these are three separate quantities which can be observed in the long run. Further I believe that if skill and card distribution are held equal, then position would be determining in all but the simplest engines where the final determinant - the tie breaker rule - would dominate.

« Last Edit: January 19, 2012, 04:55:33 pm by jomini »
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A_S00

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #56 on: January 19, 2012, 05:32:33 pm »
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Further I believe that if skill and card distribution are held equal, then position would be determining in all but the simplest engines where the final determinant - the tie breaker rule - would dominate.
Strictly speaking, if two players play the same (which I assume is what you mean by "skill held equal") and card distribution is held equal, I don't believe the tie-breaker rule will ever come into play.  Either the setup will be such that one player can pull off something like a double Province buy to end the game (in which case position is the determinant), or the players will perfectly mirror one another and the game will be a draw.  The tiebreaker rule only comes into play when the second player gets ahead - through inequalities in skill or luck, which you are saying are held equal here - and then the first player leverages positional advantage to tie the score.

I guess there could be situations where two equally skilled, equally lucky players don't play the same (say, because the optimal strategy differs depending on position...like buying a turn-2 IGG is very good for player 1 but only pretty good for player 2), but I don't think those cases can be summarized as "only the simplest engines."
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DStu

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #57 on: January 20, 2012, 02:04:22 am »
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I begin to feel I'm trolling here, but let me summerize the thread again:
tlloyed stated here
Quote
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.
[...]
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.
(where the context was "racing to piles"). Which is of course wrong. Which was discussed in the next posts, ending with him stating here
Quote
But one of the points that has come out in our discussion so far is that first-player advantage (ignoring attacks for the moment) is essentially this: P1 is in a better position to benefit from relatively lucky shuffling, while P2 is more vulnerable to relatively unlucky shuffling. If P1 and P2 have similar shuffle luck, then P1 has no advantage when it comes to splitting key cards. I've mentioned before that attack cards can change this drastically, as can +buy to a lesser extent.
Which, in the parts that where wrong about the first statement, is exactly what was wrong before. And that's where you start trying to correct him on either totally different topics, or (your last post) with an example which illustrates exactly this principle...
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tlloyd

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #58 on: February 18, 2012, 11:28:57 pm »
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Sorry to revive this conversation, but we've had a some recent evidence that people (even very high-ranked players) perceive a first player advantage where (at least from my understanding) there is no such advantage. I've been wrong more than once in this discussion alone, so if I'm still wrong someone please help me! Below is the quote from Geronimoo and my response on the blog:

“WW’s turn three clearly indicates his strategy for this game: Masquerade Big Money (with Ventures). This strategy is very strong and easy to play. And WW is sure not going to make any mistakes with it, greening very soon and not giving me any opportunities to catch up. I could follow his lead here with a Silver, but most likely I’ll lose due to 1st player advantage. ”

Okay, so I started a big fight about this on the forum, but I think this idea is simply false. How could Masq + BM put player 1 at an advantage? Put aside Masq for a moment. First-player advantage (if there are no attacks and no possibility for gaining multiple cards on one turn) consists in P1′s chance of getting a majority of a key card (for example, a race for Fool’s Gold or Minions). P1 is expected to evenly split the key cards even if P1 has slightly worse luck, whereas if P1 has slightly better luck P1 might win the split 6/4. P2 can only win the split for a key card with significantly better shuffle luck. But notice that P1 is able to get an even split despite worse luck only by having “an extra turn”–i.e., P1 has a crappy turn and misses his fifth Minion on turn X, P2 gets his fifth Minion on turn X, and P1 still get his fifth (even split) on turn X + 1. But if both players play straight-up money strategies, the only pile that is likely to run out is the Provinces. And if P1 gets an even split of the Provinces by buying the last one on his turn (the “extra” turn), P1 loses! P2 is just as likely as P1 to get 4 Provinces by turn X, so there is no first-player advantage.

Now consider Masquerade: an almost-attack. Attacks can create a first-player advantage, but does Masquerade? Consider that when P1 plays Masquerade, he trades a card from his turn X hand for a card from P2′s turn X hand. In terms of probability, it’s an even trade (although I’ve many times been forced to give a copper for an estate). But when P2 plays Masq, he trades a card from his turn X hand for a card from P1′s turn X+1 hand. Assuming that the quality of the cards in your hands generally increases over the relevant part of the game, the trade is to P2′s advantage!

So, yes there is such a thing as first-player advantage, but there was no such advantage in this game.
« Last Edit: February 18, 2012, 11:31:24 pm by tlloyd »
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dondon151

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #59 on: February 18, 2012, 11:40:52 pm »
0

I'm sure that Geronimoo was referring to the fact that P1 always has an innate advantage due to being able to end the game with an extra turn compared to P2, not because the cards in the kingdom exaggerated the advantage in any specific way.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #60 on: February 18, 2012, 11:56:37 pm »
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The third paragraph is false. In a discussion a long time ago on BGG, I think we have concluded that if the only scoring card in the game is Province, then there would actually be a slight second player advantage (if there is no other card contest involved, in the current context.) When there are other lower VPs, it depends on how easy they are to get. Usually there is a first player advantage though. The point being that the first player can pretty much always buy the highest VP he can afford; the second player however, is more often forced to buy the lower one due to the PPR rule. Or just to view it in a general way: barring the tiny portion of the games such that P2 wins by turns, there are certainly some games ended with P1 played one turn more. The extra turn is certainly an advantage for P1. It is just in some peculiar corner cases (such as the one when the Province is the only VP) that this extra turn does not translate.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #61 on: February 19, 2012, 12:00:20 am »
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Consider the given game of Masq-BM vs a mega-turn engine. Masq-bm needs to get 7 provinces to have more than half the available VP and thus win. Suppose it takes X turns to do this (don't want to say a number because that would sidetrack the conversation.) (Also, let's say that that's X turns to have 'more than half the VP' in whatever combination - 6 provinces and 2 duchies, 5 provinces and 3 duchies and 2 estates, whatever.)

If the engine player goes first, then they have to have their mega-turn go off on turn X. Because then on turn X, they've had X turns, the masq player has had X-1 turns and hasn't gotten that 7th province.

If the engine player goes second, then they have to have their mega-turn go off on turn X-1, because if their megaturn goes off on turn X, then the masq player already has more than half the VP.

There's your first player advantage. If the engine guy goes first, he has X turns to set up his mega-turn to win; if he goes second, he has X-1 turns. The same gameplay result - a mega-turn explosion buying all remaining VP on turn X - loses as p2 and wins as p1.

This is all averages, of course. You could get lucky or unlucky as either player. But on average, given such strategy choices, you have to be 1 turn faster as p2 than as p1 to win.
« Last Edit: February 19, 2012, 12:05:54 am by ftl »
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #62 on: February 19, 2012, 12:16:04 am »
+1

(vaguely remembered statistic) ~15X more games end with player 1 having taken an extra turn than end in a draw.

Even if (wild guess) ~33% of the time player 1 is a moron and ends the game in his own defeat, there is a 10-fold difference....Saying nothing of the fact that the statistics clearly show that substantially greater than 50% of games end in player 1's favor.

I spent an awful lot of time looking through CR at cards that people presupposed gave additional player 1 or player 2 advantage, but there was no significant deviation.  The (pretty accurately remembered statistic) ~10% bonus for going first is "just" the bonus you get due to the fact that you can end the game at a point where you've taken 1 more turn than your opponent. 

I still think that player 2 should get a "free turn" after p1 ends the game, even if he doesn't get to buy phantom provinces a-la ascension.
« Last Edit: February 19, 2012, 12:18:40 am by rod- »
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #63 on: February 19, 2012, 12:46:55 am »
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Sorry to revive this conversation, but we've had a some recent evidence that people (even very high-ranked players) perceive a first player advantage where (at least from my understanding) there is no such advantage. I've been wrong more than once in this discussion alone, so if I'm still wrong someone please help me! Below is the quote from Geronimoo and my response on the blog:

“WW’s turn three clearly indicates his strategy for this game: Masquerade Big Money (with Ventures). This strategy is very strong and easy to play. And WW is sure not going to make any mistakes with it, greening very soon and not giving me any opportunities to catch up. I could follow his lead here with a Silver, but most likely I’ll lose due to 1st player advantage. ”

Okay, so I started a big fight about this on the forum, but I think this idea is simply false. How could Masq + BM put player 1 at an advantage? Put aside Masq for a moment. First-player advantage (if there are no attacks and no possibility for gaining multiple cards on one turn) consists in P1′s chance of getting a majority of a key card (for example, a race for Fool’s Gold or Minions). P1 is expected to evenly split the key cards even if P1 has slightly worse luck, whereas if P1 has slightly better luck P1 might win the split 6/4. P2 can only win the split for a key card with significantly better shuffle luck. But notice that P1 is able to get an even split despite worse luck only by having “an extra turn”–i.e., P1 has a crappy turn and misses his fifth Minion on turn X, P2 gets his fifth Minion on turn X, and P1 still get his fifth (even split) on turn X + 1. But if both players play straight-up money strategies, the only pile that is likely to run out is the Provinces. And if P1 gets an even split of the Provinces by buying the last one on his turn (the “extra” turn), P1 loses! P2 is just as likely as P1 to get 4 Provinces by turn X, so there is no first-player advantage.

Now consider Masquerade: an almost-attack. Attacks can create a first-player advantage, but does Masquerade? Consider that when P1 plays Masquerade, he trades a card from his turn X hand for a card from P2′s turn X hand. In terms of probability, it’s an even trade (although I’ve many times been forced to give a copper for an estate). But when P2 plays Masq, he trades a card from his turn X hand for a card from P1′s turn X+1 hand. Assuming that the quality of the cards in your hands generally increases over the relevant part of the game, the trade is to P2′s advantage!

So, yes there is such a thing as first-player advantage, but there was no such advantage in this game.
No. There WAS such a thing in that game. There's basically always a card you're racing for - provinces. In the simplest form of the game, with only money, 1st player needs to get lucky once more than their opponent to win. 2nd player needs to do it twice more than 1st player. So there's your big money first player advantage. With how things stack out, it's about a 10% advantage, meaning 1st player wins actually about 20% more often (very roughly 50-40-10).
Masquerade's actually a really complicated card, because there's a lot of game theory with the passing aspect. But generally, 1st player gets better control of things, because 2nd player has to react. And while overall, the passing of cards from 1p to 2p is higher quality cards passed than go the other way, in practice, you almost always have a spare copper or so to toss over, so it's not a big deal. And masquerade speeds the game up, which is really important, because the shorter the game, the less likely 2p is going to get lucky twice more often than 1p.

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #64 on: February 19, 2012, 12:50:03 am »
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I spent an awful lot of time looking through CR at cards that people presupposed gave additional player 1 or player 2 advantage, but there was no significant deviation.  The (pretty accurately remembered statistic) ~10% bonus for going first is "just" the bonus you get due to the fact that you can end the game at a point where you've taken 1 more turn than your opponent. 
I've spent roughly no time doing the same, and I can pretty confidently conclude that you're flatly wrong, simply because you have to look board by board. In the aggregate, people take different strategies, and other factors on the board are largely going to make a lot of the stuff wash out. Probably more important, the whole thing gets hidden in the difference in skill between the players (let's not forget that p2 is in general quite a bit stronger than p1 here). Doing studies on things with many fewer confounding variables, we can see that there's very significant difference between 1st-player advantage in different kinds of decks.

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #65 on: February 19, 2012, 01:10:38 am »
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Sorry to revive this conversation, but we've had a some recent evidence that people (even very high-ranked players) perceive a first player advantage where (at least from my understanding) there is no such advantage. I've been wrong more than once in this discussion alone, so if I'm still wrong someone please help me! Below is the quote from Geronimoo and my response on the blog:

“WW’s turn three clearly indicates his strategy for this game: Masquerade Big Money (with Ventures). This strategy is very strong and easy to play. And WW is sure not going to make any mistakes with it, greening very soon and not giving me any opportunities to catch up. I could follow his lead here with a Silver, but most likely I’ll lose due to 1st player advantage. ”

Okay, so I started a big fight about this on the forum, but I think this idea is simply false. How could Masq + BM put player 1 at an advantage? Put aside Masq for a moment. First-player advantage (if there are no attacks and no possibility for gaining multiple cards on one turn) consists in P1′s chance of getting a majority of a key card (for example, a race for Fool’s Gold or Minions). P1 is expected to evenly split the key cards even if P1 has slightly worse luck, whereas if P1 has slightly better luck P1 might win the split 6/4. P2 can only win the split for a key card with significantly better shuffle luck. But notice that P1 is able to get an even split despite worse luck only by having “an extra turn”–i.e., P1 has a crappy turn and misses his fifth Minion on turn X, P2 gets his fifth Minion on turn X, and P1 still get his fifth (even split) on turn X + 1. But if both players play straight-up money strategies, the only pile that is likely to run out is the Provinces. And if P1 gets an even split of the Provinces by buying the last one on his turn (the “extra” turn), P1 loses! P2 is just as likely as P1 to get 4 Provinces by turn X, so there is no first-player advantage.

Now consider Masquerade: an almost-attack. Attacks can create a first-player advantage, but does Masquerade? Consider that when P1 plays Masquerade, he trades a card from his turn X hand for a card from P2′s turn X hand. In terms of probability, it’s an even trade (although I’ve many times been forced to give a copper for an estate). But when P2 plays Masq, he trades a card from his turn X hand for a card from P1′s turn X+1 hand. Assuming that the quality of the cards in your hands generally increases over the relevant part of the game, the trade is to P2′s advantage!

So, yes there is such a thing as first-player advantage, but there was no such advantage in this game.
No. There WAS such a thing in that game. There's basically always a card you're racing for - provinces. In the simplest form of the game, with only money, 1st player needs to get lucky once more than their opponent to win. 2nd player needs to do it twice more than 1st player. So there's your big money first player advantage. With how things stack out, it's about a 10% advantage, meaning 1st player wins actually about 20% more often (very roughly 50-40-10).

But P1 can't end the game in a tie, so if P1 has three Provinces and P2 has four, P1 can't buy the Province even if he can afford it. That means P2 only has to get lucky once more to win. It gets more complicated when you consider Duchies, but it doesn't change the basic principle.

EDIT: It is more complicated with Duchies, and it does change the basic principle. See Timchen's comment and my response.
« Last Edit: February 19, 2012, 02:27:36 am by tlloyd »
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #66 on: February 19, 2012, 02:25:05 am »
0

The third paragraph is false. In a discussion a long time ago on BGG, I think we have concluded that if the only scoring card in the game is Province, then there would actually be a slight second player advantage (if there is no other card contest involved, in the current context.) When there are other lower VPs, it depends on how easy they are to get. Usually there is a first player advantage though. The point being that the first player can pretty much always buy the highest VP he can afford; the second player however, is more often forced to buy the lower one due to the PPR rule. Or just to view it in a general way: barring the tiny portion of the games such that P2 wins by turns, there are certainly some games ended with P1 played one turn more. The extra turn is certainly an advantage for P1. It is just in some peculiar corner cases (such as the one when the Province is the only VP) that this extra turn does not translate.

Okay, I think I may see what you're saying here. I don't have any sophisticated way of imposing "equal shuffle luck," but here's my simplistic way of viewing it: Obviously if a player wins because his deck consistently hits $8 while the others' hits only $5, that's not (immediately) due to first player advantage. But it's not realistic to constrain the two decks to have identical turns throughout. So P1 and P2 should get an equal number of $8 hands and an equal number of $5 hands, although they may not come in the same order. (This of course also ignores the possibility that decks can produce $7/$6 or $9/$4 instead of the nice $8/$5, but I don't think that is relevant to first player advantage, other than to suggest how inconsequential it can be compared with the sheer randomness of a game based on shuffling).

Okay, given this setup, we assume that P1 and P2 have each purchased 3 Provinces, and it is now P1's turn. If P1's next two hands are $5/$8, while P2's hands are $8/$5, then P2 is in a bind, despite the fact that the players have had comparable shuffle luck. The crux of the problem is that P2 has to spend his $8 hand on a Duchy to avoid losing. P1 is just as likely as P2 to have an $8 hand on the following turn, so P1 is at an advantage.

My question is, is there any situation in which P1 is similarly forced to use an $8 hand on a Duchy, and how much more/less probable is that scenario than the one just described? The only one I can think of is if the players had the following distributions of coin across three turns (with 6 Provinces gone already) -- P1: $1/$8/$8; P2: $8/$8/$1. Are there any more plausible scenarios?

So thanks TimChen - I now see why the presence of lower VP cards like Duchy and even Estate can (I would suggest rarely) give P1 the win despite P2's equal-performance deck.

However, I would bet that in Geronimoo's game, Masquerade at the very least compensates for this P1 advantage. Any of you simulator folk want to prove me wrong on that one? (Geronimoo? want to defend your honor?   ;))
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timchen

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #67 on: February 19, 2012, 02:37:44 am »
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Yes you got it!

Let me add that your case for P1 to use the PPR rule is actually an improvement over P1's chances to the game with only Provinces, where he would have lost already; whereas in the first case the PPR rule for P2 is only partially saving the game where he is originally winning (by turns), but now have a real possibility to lose.

In this sense the relative probability of the two cases is not important. They both move toward a first player advantage compared to the case with Province only.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #68 on: February 19, 2012, 03:42:55 am »
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So thanks TimChen - I now see why the presence of lower VP cards like Duchy and even Estate can (I would suggest rarely) give P1 the win despite P2's equal-performance deck.

However, I would bet that in Geronimoo's game, Masquerade at the very least compensates for this P1 advantage. Any of you simulator folk want to prove me wrong on that one? (Geronimoo? want to defend your honor?   ;))

Well, I don't think that this will actually answer your question, but will more likely corroborate the first paragraph that I quoted above (although it may answer your question, I don't know).

4 trials of a single Masquerade mirror match in Geronimoo's simulator yielded the following results:
Player 1 (54.1%) - Player 2 (40.6%)
Player 1 (53.8%) - Player 2 (40.7%)
Player 1 (54.2%) - Player 2 (40.9%)
Player 1 (54.4%) - Player 2 (40.6%)

The same uneven split happens for other BM strategies, like DoubleJack:
Player 1 (54.9%) - Player 2 (38.4%)

And Courtyard:
Player 1 (52.0%) - Player 2 (38.4%)

Then just for comparison, I edited the Masquerade bot to buy neither Duchies nor Estates:
Player 1 (45.0%) - Player 2 (42.7%)

So the uneven split is much smaller, and it seems to suggest that the presence of lower VP cards significantly augments the first player advantage. Notice that player 2 isn't really winning less often; rather, it's player 1 that's winning more of the ties that occurred in the Province-only games.
« Last Edit: February 19, 2012, 03:44:56 am by dondon151 »
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rod-

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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #69 on: February 19, 2012, 10:39:44 am »
+1

I spent an awful lot of time looking through CR at cards that people presupposed gave additional player 1 or player 2 advantage, but there was no significant deviation.  The (pretty accurately remembered statistic) ~10% bonus for going first is "just" the bonus you get due to the fact that you can end the game at a point where you've taken 1 more turn than your opponent. 
I've spent roughly no time doing the same, and I can pretty confidently conclude that you're flatly wrong, simply because you have to look board by board. In the aggregate, people take different strategies, and other factors on the board are largely going to make a lot of the stuff wash out. Probably more important, the whole thing gets hidden in the difference in skill between the players (let's not forget that p2 is in general quite a bit stronger than p1 here). Doing studies on things with many fewer confounding variables, we can see that there's very significant difference between 1st-player advantage in different kinds of decks.
Show me those studies?  The simulator results that have been posted on this page show the exact same distribution of mirror match results:  54/40 +/- 2.  This is completely on par with the councilroom data, although the proportion of ties is considerably lower on isotropic. (presumably, real people have a better implementation for endgame and an aversion to ties) 

I'd be happy to parse through the tarball again if you would help me ask the right question.  Should i be looking for a very specific board, or cutting all players with level less than X, or what?  Keep in mind that finding >5000 games is still going to be a requirement.  I've gotten much better with R over the last month, so given the right questions, I'd be a lot more likely to be able to find something of interest.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #70 on: February 19, 2012, 10:49:45 am »
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Masquerade games are not perfect mirrors since you can retain the last estate as a tie-breaker and not trash it.  I was going to post up something about the simulator being unable to manage this but it actually can. On its own though this doesn't increase win rate. I suspect there's another sim challenge there to perfect the management of the last estate.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #71 on: February 20, 2012, 03:21:13 am »
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Don't mean to sidetrack the argument going on, but I was trying to come up with other cards that benefit second player, since most people seem to agree they are rarer.

Embargo. As second player, you have a good chance of knowing your opponent's strategy, and you may be able to disrupt his purchase of a key card. The most obvious situation is when first player opens Potion, seeking Alchemist, or Scrying Pool, or Familiar.

City. Though racking up key cards benefits first player--because first player has a better shot at the 6-4 split--City mitigates this sort of, because second player gets the first upgraded City turn if they split unevenly. And second player can always choose not to buy the last one, and deny first player the first upgraded City turn. So I would say that it is better for second player, at least relative to other cards where you and your opponent race to get all of them.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #72 on: February 20, 2012, 05:29:36 am »
+1

I still think that player 2 should get a "free turn" after p1 ends the game, even if he doesn't get to buy phantom provinces a-la ascension.

I like this idea, on most boards it probably doesn't swing the pendulum too far the other way.  In fact there is probably still a slight first player advantage.  It does detract from the simplicity of dominion somewhat, which in many ways is its biggest selling point.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #73 on: February 20, 2012, 10:47:10 am »
+1

I still think that player 2 should get a "free turn" after p1 ends the game, even if he doesn't get to buy phantom provinces a-la ascension.

I like this idea, on most boards it probably doesn't swing the pendulum too far the other way.  In fact there is probably still a slight first player advantage.  It does detract from the simplicity of dominion somewhat, which in many ways is its biggest selling point.

For the longest time, I agreed with this... EQUAL TURNS!  Then I started to think about the consequences, particularly in boards that end on piles.  I think in general, it would result in games going longer, as we'd need theory to write us a new rule on how not to end the game as P1 with less than a duchy lead.

It's an interesting variant, and while I suspect it will bring the gap closer together, I think it may end up leaving people with a bad taste in their mouths.  Every time P1 ends the game, they are going to have to sit there and sweat... did I end it too soon?  And when they were wrong, instead of just having lost, P1 is going to feel like they shouldn't have started the end game.  P2 is going to feel like everything comes down to luck of the draw on their final turn.  If they make it, its going to feel cheap, if they don't, its going to feel lucky.



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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #74 on: February 20, 2012, 11:05:52 am »
0

Yeah, I dont think it would of worked too well either.

For example: Provinces, if a P1 is going to win by taking the last province, it is usually because he has the lead, and likely has a 5-3 Province lead. So the extra turn is unlikely to have any effect for the next player as he would need more dutchys to compensate (obviously not always the case)

Plus it requiresso much more brain hurting thought when ending three piles, and the game ending 3 pile when you know you are in front is the 'end the game quick way' when you dont think you would win if opponent can get the provinces

I think on Isotropic these things I mentioned above are not really relevant as the level of play is higher, and there is a lot of min/maxxing going on with simulators. But for the thousands and thousands that don't play on Isotropic then these things will matter as they wont know things like the optimum amount of dukes to dutchys, or how many Silvers to buy before greening, etc..

Essentially i think it would please the hard corers (which in my view is anybody who regularily visits forums about Dom) and would impact the game for the casuals
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #75 on: February 20, 2012, 11:35:54 am »
0

Embargo. As second player, you have a good chance of knowing your opponent's strategy, and you may be able to disrupt his purchase of a key card. The most obvious situation is when first player opens Potion, seeking Alchemist, or Scrying Pool, or Familiar.
~1/3 of the time, your 1p opponent will get their 1st alchemist/scrying pool/familiar before you get your embargo.  Is it worth making your starting silver a one-shot to hand out a curse 2/3 of the time?  Often(alchemist, yes, others...probably not)...but the 1st player has a higher chance of making such a strong embargo play, although it's possible they will have to make their embargo buy "in the dark" as to their opponent's strategic plans.   
Quote
Essentially i think it would please the hard corers (which in my view is anybody who regularily visits forums about Dom) and would impact the game for the casuals
As far as casual players are concerned, every casual game of dominion i've played has resulted in the game ending with almost NO idea who is in the lead, with the player who ends the game just doing it without any real concern for the score except to say "well i got a province this turn".    Not much difference there.  And as for "feeling dirty" for winning a close game as 2nd player, I ALWAYS feel dirty winning a close game as 1st player.  When i squeak out a win (on provinces) by less than 7 points, I know that it probably wasn't deserved.  I've grown used to it.
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Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #76 on: February 20, 2012, 09:59:07 pm »
+1

A big effect it would have is making ending the game on piles not make sense, ever. Because if you're in a remotely close game, if you spend a buy to end the game on piles, the other person will have presumably an equal turn, and not spend any buys on anything but green.

Makes mega-turn games really weird - you can't pull the trigger until you can buy up more than half the points, not until you can end the game on piles as now.

Actually, would give p2 a BIG advantage in mega-turn games! Because p2 can end the game on piles while buying just the minimum amount of green, whereas P1 would never have that option.

Would be interesting, but I don't think it would necessarily make the game more fair. I suspect that p1 vs p2 advantage would then vary based on the cards in the set.

Dominion is a short enough game that an easy way to to balance out first-player advantage is to 'play to X games, loser of the previous game goes first, have to win by two.'
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