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[Karhumies]

So far, I have played exclusively 4-player Dominion games. Coming from that background, I have found much of the Dominion strategy on this site to be very heavily 2p-oriented, and not directly transferable/applicable to a 3-player or 4-player setting. Let me illustrate my point by simple arithmetics:

1) Provinces (average):

5-6p - 3 provinces/colonies per player
4p - 3 provinces/colonies per player
3p - 4 provinces/colonies per player
2p - 4 provinces/colonies per player

The switch from 2-3p to 4-6p affects game strategy significantly.
1.1 The game will last way less turns in 4-6p game. This tends to make big money strategies much better in 4-6p than in 2-3p games in a setting where the emptying of the province/colony pile is likely to be the end condition. Because engines have less turns of game time before the game ends.


2) Kingdom cards (average):

6p - 1.67 kingdom cards per player
5p - 2 kingdom cards per player
4p - 2.5 kingdom cards per player
3p - 3.33 kingdom cards per player
2p - 5 kingdom cards per player

Switch from any player number to any other player number will affect the game significantly.
2.1 The more players you have, the more people you have competing for the same engine parts.
2.1.1 Cards with combo with themselves (e.g. Fool's Gold) become weaker with more players. This can change the "must buys" into crap and vice versa in some kingdoms based on player amount.
2.1.2 Big money becomes a less risky / risk diversification option, as compared to competing for the limited engine parts with more players.

2.2 Fractional numbers imply advantage for the player going first, and disadvantage for player going last.

2.3 Attacks are more likely to hit (e.g Pirate Ship) rather than miss with more players.
2.3.1 The same attack is more likely to become "spammed" in a  game with more players, which can occasionally become problematic. Imagine 6 player game where everyone opens 5/2 with Torturer/Mountebank/Witch compared to 2 player game. Completely different game experience.

I would like the writers of future articles to acknowledge this more explicitly in their articles, e.g. "in a 2-player game, this strategy blahblah..." whenever and if they are assuming a 2p setting. Thank you.

The room is up to discussion now.

[jsh357]

Since most Dominion discussed here is played online, is swings toward 2p and always will.  It's easier to find one random opponent, there's less chance of a disconnect ruining the game, less chance of someone ragequitting and throwing off the balance, etc, etc...  There is also a much bigger swing factor and player order advantage in multiplayer, which makes some of the more high level players avoid it.

You are right, though.  More articles should acknowledge multiplayer games, even if not necessarily covering them.  It would be great if people who played with more players posted articles too, but they usually don't.

[brokoli]

Never play Dominion with 5+ player. This is awful.

[PenPen]

I really like how Dominion scales differently when it's 2P, 3P or more (I don't think I want to touch on 5P-6P...there's too much dead time for players who are waiting). Cards would interact and work differently under a different number of players. That's what make this game so fun.

I do mostly play 3 or 4P, and as jsh357 mentioned most of the strats here would be focusing on 2P since it's the prevalent game mode for Dominion. I'd love articles that expand on cards with 3 or 4P games, if I recall some of the articles posted on the main page do involve some discussions on multiplayer.

[DG]

I suspect that many authors of the articles are not deliberately omitting multi-player strategy, they are just not experienced enough to write very confidently on the subject. There are also some dangers from 'groupthink', where you can play with different groups of players and different strategies prevail. The group dictates the pace and style of the game. In terms of presentation it is much easier to show two player game logs and they are also much easier to understand by the reader.

[Karhumies]

Mini-article on Dominion game theory, 2p vs larger amounts of players, #1:

On the "mirror match aspect" advantage of going first

In chess, the player going first has an inherent advantage over the player going second due to the mirror match aspect. Example: the player going second (P2) mirros everything player going first (P1) does. However, P2 is prevented by game rules from mirroring the first check P1 makes, giving P1 an advantage already before the game starts.

In a 2p Dominion game, in a simplified Dominion kingdom without interactive cards (typically attacks), there is no innate mirror match advantage for player going first. Because there is 5/5 split of kingdom cards; 4/4 split of Provinces; and 6/6 split of alt. VP cards. If P1 and P2 always draw equal hands, do same actions and same buys, the game will always be a draw. The easiest way to simulate this is two very simple big money strategies, which should get roughly 50% win rate 1st vs 2nd after enough simulations.

This implies that in a 2-player game, the winner is determined by
1. Interactive cards (e.g. P1 plays Militia -> P2 is most likely unable to mirror everything on his turn)
2. Differentiation strategy rather than mirroring (P2 chooses a more/less effective strategy than P1)
3. Luck, especially in post-shuffle draws. Luck can make a player e.g. "miss Province purchase", giving advantage to the other player.

However, in a 3-player game the Kingdom card split in a mirror is 4/3/3 (Provinces 4/4/4; alt. VPs 4/4/4). Giving inherent advantage to player going first in the mirror aspect of the game: when he purchases the 4th copy of a kingdom card (last in stack), player #2 and #3 are unable to mirror this move. Of course, the value of this advantage depends on the kingdom cards available, whether any players "miss a buy" (intentionally or unintentionally) -> 4/3/3 can become 3/4/3, etc.

4-player game: kingdom cards 3/3/2/2, first two players have advantage. last two players have disadvantage. Colonies 3/3/3 (no adv.); alt. VPs 4/4/4 (no adv.).

5-player game: kingdom 2/2/2/2/2 (no advantage); colonies 3/3/3/3/3 (no advantage); alt. VPs 3/2/2/2/2 (very marginal advantage to first, IMO negligible)

6-player game: kingdom 2/2/2/2/1/1 (advantage to P1-4, disadvantage to P5-6; also 1-2 kingdom cards average per deck is often perceived as "less fun" by engine builders); colonies 3/3/3/3/3/3 (no advantage); alt. VPs 2/2/2/2/2/2 (no advantage)

I will later write mini-articles about
- "Signal-giving advantage of going first" and compare this to MtG draft signals. Discussing why this affects mirror vs diversification strategies across the table, and why this advantage becomes smaller as the number of players go up.
- "Information advantage of going last" and compare this to Texas hold 'em poker. Discussing why going last in a (6-player) game can be a reward instead of a punishment. And why this advantage becomes larger as the number of players go up.

These are some really fundamental aspects of Dominion game which could have - and in my opinion should have - been discussed already when the base set came out before Intrigue. And much more integral part of the game than single card analysis/strategy. Because single card strategy can only be good/bad within the confines of these underlying aspects of the game. If you ignore these assumptions, you can not discuss single card strategy effectively. Therefore, if you mean a 2p game, you have to say it is a 2p game because for any other amount of players, the real game situation is different because the assumptions are different already when the game begins.

[loppo]

In a 2p Dominion game, in a simplified Dominion kingdom without interactive cards (typically attacks), there is no innate mirror match advantage for player going first. Because there is 5/5 split of kingdom cards; 4/4 split of Provinces; and 6/6 split of alt. VP cards. If P1 and P2 always draw equal hands, do same actions and same buys, the game will always be a draw. The easiest way to simulate this is two very simple big money strategies, which should get roughly 50% win rate 1st vs 2nd after enough simulations.

This implies that in a 2-player game, the winner is determined by
1. Interactive cards (e.g. P1 plays Militia -> P2 is most likely unable to mirror everything on his turn)
2. Differentiation strategy rather than mirroring (P2 chooses a more/less effective strategy than P1)
3. Luck, especially in post-shuffle draws. Luck can make a player e.g. "miss Province purchase", giving advantage to the other player.

+buys and gainers also add a big part to P1 advantage, because they allow splits of 5/3 (victory cards) and 6/4 (kingdom cards). If P1 buys/gains the last two cards P2 has nothing to mirror, and this factor is huge in my Opinion.

[Dsell]

I should have one of my RL friends write an article on how to play 4P Dominion. I can beat her in 90% of the 2-player games we play (and so can a couple of my other friends) but she is just so good at 4P, winning at least as much as I do and often more. We each have hot and cold streaks but her playstyle is just built for 4P and it works. And I'm the only one of my RL friends who reads much f.ds!

[Rabid]

In a 2p Dominion game, in a simplified Dominion kingdom without interactive cards (typically attacks), there is no innate mirror match advantage for player going first. Because there is 5/5 split of kingdom cards; 4/4 split of Provinces; and 6/6 split of alt. VP cards. If P1 and P2 always draw equal hands, do same actions and same buys, the game will always be a draw. The easiest way to simulate this is two very simple big money strategies, which should get roughly 50% win rate 1st vs 2nd after enough simulations.

This implies that in a 2-player game, the winner is determined by
1. Interactive cards (e.g. P1 plays Militia -> P2 is most likely unable to mirror everything on his turn)
2. Differentiation strategy rather than mirroring (P2 chooses a more/less effective strategy than P1)
3. Luck, especially in post-shuffle draws. Luck can make a player e.g. "miss Province purchase", giving advantage to the other player.

Glad to see more 3p+ discussion.
But is disagree with the above quote.
Extensive simulations of Big Money show a significant first player advantage in 2p games.
I think mostly due to "luck" favouring P1 as they get more turns on which to be lucky.

[Octo]

I think mostly due to "luck" favouring P1 as they get more turns on which to be lucky.

Just to expand this with a bit more detail (if I understood you correctly that is) - if player 1 gets lucky then they have created a real advantage. If player 2 gets lucky then usually they have just gained a tempo of sorts - while P1 was originally going first, now it's P2 leading the buys. This is exemplified directly in loppo's comment about +buyers: At 3 kingdom cards each, P1 buys their 4th, then P2 lucks out and buys both their fourth and fifth in one turn. P1 buys their 5th next turn and it's all square at 5 each - P2 didn't really gain a whole lot there, they just switched the lead buy around. Compare with having 3 each and P1 lucking out this time and getting 4th and 5th in one turn. P2 can only buy 4th, then P1 finishes up with 6 to P2's 4. P2 needs 2 bursts of luck to get a real advantage, while P1 only needs 1.

I see the appeal of 2P and also how it leads to a more clean analysis of the strategy, but I find a lot of the advice on here is often wasted on me as I rarely play 2P (though it took me a while to figure out why the strats just didn't seem to work as strongly as people suggested). Attacks are just a whole different ball game with more players, as well 3-pile endings. Some of the change comes from it being harder to distract the opponent from their strategy - with 3 opponents at least one will likely just stick to their guns and not get dragged into some alt-VP-who-wants-to-make-the-final-push standoff type affair that is more common in 2P.

[Duchess Ninja]

These are just some thoughts on 3-4 player Dominion (I echo the Never Ever Play 5+ Player Dominion! sentiment - it just doesn't work). I don't have any specific card advice, so this is just about what to expect (based on my experience, of course) on the general differences of 2 vs 3+ player and the tendencies of players. There will be no math. In fact, I don't think the math matters as much in 3+ player, because you're much more likely to have players who don't play at nearly the same skill level.

My experience with Dominion is mostly live (i.e., not on iso) and mostly four-player. I too read most of the posts and articles on this site, and I've wondered why more people don't post more about 3+ player. I think that with two-player, it's generally on iso where rankings are involved, so people take it more seriously. In my multiplayer games (I realize two-player is "multi", but I'll use the term here for 3+ player games), it's almost always more casual.

In the usual groups in which I play multi-player, I estimate that I win 50-60% of the time. This is not due to my being a Dominion Genius, but rather that I know the "real" value of most cards (from reading this site and playing), I generally can stick to my opening strategy, and my opponents generally are a bit more casual about their play. Multi-player also brings out the "let's have fun!" aspect of the game more than the "must-win" aspect, so playing a winning strategy is generally easier if you can anticiapate that yes, the Pirate Ship or Swindler pile will be the first to be depleted.

The main difference that I've noticed from actual play is that you are very unlikely to have a unique strategy for any given game, despite players likely starting with wildly different buys. The people that I play with are smart, and if they see me or someone else do something that hadn't occurred to them before (Vault for a Grand Market, or Bishop the same Fortress a couple of times in a turn, for example), they will immediately copy it. In a two player game, there is generally less time to do this, and if your opponent already has a couple of pieces of his engine, he's less likely to mirror you because he's essentially committed. Copying your strategy usually does not apply to alt-vp three-piling strategies in multi-player, though.

Another very important difference is that attacks are never ignored completely and this will likely make the game longer than the average two-player game. In two-player, ignoring Jester or Young Witch may be prudent. In multi-player, there will always (always!) be at least one player who buys the attack(s). It could be a good strategy (since more opponents are affected) or it could be that he feels he has no chance and just wants to screw people. Always plan for the attack(s) in the kingdom. Consider the attacks yourself before anyone else does it if there isn't a good defense.

Three-piling, at least in my experience, is less common, but I would love to know if this is true in general. With four-player especially, there will likely be one or two more casual dominion players that just don't consider that option very often - they want big money and big actions and all the damned Colonies. Unless there is a very obvious three-pile strategy (like Silk Road/Island/Workshop) or heavy cursing and no trashing, the Province/Colony path to victory is very likely. You generally cannot plan on your opponents helping you empty piles. And with twelve victory cards per pile, alt-vp strategies are less viable unless someone is mirroring you. This could happen, but I see people shy away from these cards in multi-player precisely because it helps the person who started that strategy by speeding up the three-pile.

As stated earlier, the "luck" swings are wider. With more people playing, one is likely to get luckier and one is likely to get decidedly unluckier than the others. This is probably another very good reason the "serious" players prefer two-player. This often means that someone may have to switch strategies to something much riskier; Treasure Map for example. (I know this happens in two-player, but it is much more likely to happen to someone in multi-player). There isn't much you can do, but desperate players can do silly things and mess with the flow of your game. Of course, it may be that you are the one who has to take risks, but it's usually pretty obvious what you need to do in that case.

[timchen]

The biggest difference to me is the "cursing dilemma" which happens with 3 players or more, evident with 4 players.

[blueblimp]

In a 2p Dominion game, in a simplified Dominion kingdom without interactive cards (typically attacks), there is no innate mirror match advantage for player going first. Because there is 5/5 split of kingdom cards; 4/4 split of Provinces; and 6/6 split of alt. VP cards. If P1 and P2 always draw equal hands, do same actions and same buys, the game will always be a draw. The easiest way to simulate this is two very simple big money strategies, which should get roughly 50% win rate 1st vs 2nd after enough simulations.

This doesn't work (unless you assume exactly equal draws, which is too unrealistic), because P2 might fail to mirror if P1 ends the game on his turn. This matters even in games when both players take equal numbers of turns, because P2 must consider the possibility of the game ending on P1's turn.

For example, the PPR. It's P2's turn, down 2 points, and there are two provinces remaining. If P2 knew the game wouldn't end on P1's turn, it would be correct to buy a province, because it gives the most points. But since P1 might buy the final province to win in that case, the actual correct move is to buy a duchy.

To get in that situation, there must have been some non-mirroring in previous turns, but in BM games, that will universally be true, so IMO it's not worth discussing the alternative.

[Karhumies]

Got a lot more games under my belt tonight. I got to say, I love the addition of Hinterlands in 4p games.

Some notable things:
Hinterlands + Dominion 4p, set-up "Highway robbery" ended up by 3-piling Throne Room, Oasis and Embassy (with only 4 cards left in the Copper pile). Throne Room + Noble Brigand made everyone somewhat equally poor, dishing out 5-6 coppers from the pile in one turn. Also, people did not buy enough Libraries, choosing to answer the Margrave attacks with their own Margrave instead.

5p game (last game of the night with remaining players from all 3 tables combined; random kingdom) with Stables + Scheme + Bishop engines in a table without +buys (except Talisman). Half of the Provinces ended up in the Trash pile, since no one had enough copies of Stables to filter through their Province buys. Long chains ending up with 7 gold + Baron for 8th, trashing a Province for 5VP. First time I see that happening. Largely due to people trashing too many coppers and/or thinking they could get one more copy of the stables (which they didn't).

Important mental notes
1. 4p, 5p, 6p dishing out curses and/or coppers is brutal.
2. 5p can work amazingly well, provided that the kingdom does not contain +buys to rub the starter advantage into the other players' faces.

[ftl]

Do you mean Bishop and not Baron?

[Octo]

@blueblimp: your comparison doesn't really work.

You're trying to make a point about mirror matches, but the example you use has P2 having gained a tempo on P1 (i.e. he has the opportunity to take the province lead) yet they're still down 2VPs some how, so it's not even nearly a mirror match, which totally confuses your point. To illustrate this, the situation you describe could equally apply to the players the other way round, and in fact it will be P1 who has to deal with the PPR normally in a hypothetical mirror match, not P2.

I don't really think Karhumies (stop me if I'm wrong) was thinking that mirror actually happened enough to have strategy ready, but was talking in terms of statistical probabilities/advatnages in the same way that the simulators do - an 80 - 20 win ratio is still no guarantee of a win, but it's still worth discussing right? As it turns out, P1 does have an advantage in a 2P game, but the (flawed) assumption does bring some interesting points to light.

[Karhumies]

Mini-article on Dominion game theory, 2p vs larger amounts of players, #2:

- "Signal-giving advantage of going first" and compare this to MtG draft signals. Discussing why this affects mirror vs diversification strategies across the table, and why this advantage becomes smaller as the number of players go up.

Assuming an environment with veteran Dominion players only. No newbies. If that is not your gaming environment, then tough luck, this mini-article will be as useful to you as the 2p strategy articles are to me.

Player going first can, especially in kingdoms with a lot of valid 3/4 split options, give a signal to the other players about his/her opening strategy. This works ONLY when 1-3 ALL HOLD TRUE:
1) the person going early knows how to play the most obvious good strategies in the kingdom
2) the players going later know some basics about Dominion strategies
3) the players going later know how to read the signal the early players are giving

By giving out a signal with early buys, the early player basically informs the late players that "I choose to exercise my first player advantage" / mirror advantage for this strategy. If you copy this strategy, you will be disadvantaged compared to a situation where you choose an alternative strategy (provided there is one). Because of reasons discussed above in this thread.

This advantage is the most significant in 2p, since most kingdoms have more than 1 valid/effective strategy, but less than 5-6 valid/effective strategies. Meaning that in especially in 5-6 players' games late players have to copy the strategy of someone who has played earlier than them, thus typically hindering both of their strategy less effective while making the other players' less contested strategies more effective. The early players in 5-6p games typically perceive this as "not fun", because it involves an element uncontrollable to them compared to a 2-3p (sometimes 4p) environment, where these signals do matter and make a difference.

Additionally, in 4p, 5p and 6p, players are less likely to focus on the opposing turns, meaning less attention to signals. This causes frustration to players used to giving out signals which turn out to be meaningful in 2p and 3p.

Basically,
I. Going first = freedom to pick your strategy first. This is an advantage if you pick a superior strategy. Can turn into disadvantage if you pick suboptimal strategy or your strategy becomes contested by late players in 3p, 4p, 5p, 6p.
II. Contesting a strategy chosen by an earlier player = mirroring = bad in the long run* for you. In 3p, 4p, 5p, 6p, also bad in the long run for the early player who became contested. In 2p, gives an advantage to P1.
III. Going late = less freedom to pick uncontested strategy. Disadvantage when the best strategies have been taken. Becomes huge advantage if you can identify a superior strategy which is still uncontested and you can see and capitalize on that.

IV. Newbie players are likely to copy strategy by player they see as "experienced". In 2p, this gives an advantage to the other player (mirroring); in 3+ players, this gives an advantage to the other, uncontested players. Experienced players going early can perceive this follower-mirroring disadvantage in multiplayer as "not fun".
 
V. Experienced players can attempt to "rob" a strategy in multiplayer by intentionally contesting it. If the early player gives up the strategy in fear of ruining their game with contest, this can turn around the mirror to the late player's favor.
VI. Experienced players may also give an intentional false signal of contesting the strategy. Not in hopes of robbing said strategy, but in hopes of making the early player dilute that strategy by protective measures. Very close to robbing in practise; only the underlying motivation is different.

*Short run = 1 deck revolve or less (early game: 1-2 turns mirroring max.)
Long run = longer than 1 deck revolve

Do you mean Bishop and not Baron?

Yes, I do.

[Karhumies]

I don't really think Karhumies (stop me if I'm wrong) was thinking that mirror actually happened enough to have strategy ready, but was talking in terms of statistical probabilities/advatnages in the same way that the simulators do - an 80 - 20 win ratio is still no guarantee of a win, but it's still worth discussing right? As it turns out, P1 does have an advantage in a 2P game, but the (flawed) assumption does bring some interesting points to light.

I was talking from a theoretical standpoint. There are kingdoms in Dominion in which exact mirror = tie. The mirror concept is artificial and the probability of a true mirror match is very, very unlikely in a Dominion setting due to the probabilities involved in repeated shuffles and draws. But it illustrates fundamental theory. My point was, and still is, that 2P turn-by-turn exact mirroring without any random variable, such as different draws or actions for P1 and P2, in Dominion can only give P2 maximum 50% win rate, and even that is in a really, really simplified kingdom which we are very unlikely to see in real life unles we artificially construct one. Which we could, if it illustrates fundamental theory enough for us to learn something valuable. I hope you see my point better now.

Essentially, I recognized that among the factors "breaking" the mirror 50-50 probability are
1. interaction
2. differentiation
(3. luck = random variable, the arch-nmesis of real-life turn-by-turn mirroring)

...but I did not realize that 4. +buy / +gain also breaks the 50-50 probability, because that can easily lead to 6-4 piles instead of 5-5 when buying either #8 and #9 / #9 and #10 simultaneously (earlier +buy does not break the mirror because, well, it can be mirrored by P2). I argue that empty piles are not THAT common, and that this does not negate most of my theoretical points from applying. After all, most of the game time piles have 4+ cards each (I am intentionally excluding triple buy/triple gain here).

In other words, +buy means that mirror strategy win percent for P2 in 2p Dominion, in practice, even in vanilla kingdoms without interaction, is much less than 50%. This should not be a surprise result for anyone...but had you previously really thought about WHY it is so? Well, by now you should have.

Also, interaction strategies and differentiation strategies should be employed by P2 in order to "break away" from the 50-50 mirror and turn the game into their advantage = to win despite having to go 2nd. ; +buy is better used by P1 in order to get into "runaway leader" position. Given that P2 is mirroring P1, it should not occur that P2 is double buying something P1 has not previously double bought already. I mean, otherwise it's not true mirroring.

[RisingJaguar]

@blueblimp: your comparison doesn't really work.

You're trying to make a point about mirror matches, but the example you use has P2 having gained a tempo on P1 (i.e. he has the opportunity to take the province lead) yet they're still down 2VPs some how, so it's not even nearly a mirror match, which totally confuses your point. To illustrate this, the situation you describe could equally apply to the players the other way round, and in fact it will be P1 who has to deal with the PPR normally in a hypothetical mirror match, not P2.

I don't really think Karhumies (stop me if I'm wrong) was thinking that mirror actually happened enough to have strategy ready, but was talking in terms of statistical probabilities/advatnages in the same way that the simulators do - an 80 - 20 win ratio is still no guarantee of a win, but it's still worth discussing right? As it turns out, P1 does have an advantage in a 2P game, but the (flawed) assumption does bring some interesting points to light.

I think the point blueblimp was trying to make was if both are going into their next shuffle, tied and with exactly similar deck contents.  The two following situation could happens. 

Player 1 Coins: $5, $8
Player 2 Coins: $8, $5

Player 1 would buy a duchy. If player 2 buys a province, 2 risks having Player 1 buying a province first, despite have less coins left in their deck and "half a turn" (or whatever you want to call it) behind.  Player 1 gets advantage

If the reverse occurred...  so:

Player 1 Coins: $8, $5
Player 2 Coins: $5, $8

This game would end up in a tie, even though it was just the complete reverse situation.  Player 1 isn't put into the same bind that player 2 was with Coin disbursement.  No such advantage

This is a situation that player 1 has net advantage in two opposite situations.  Thus, it's not really a luck thing (as opposed to say a 5/2 start vs. 4/3 start).

EDIT: I remember this being somewhere in the forums, I just don't remember where...

[Karhumies]

Player 1 Coins: $5, $8
Player 2 Coins: $8, $5

This is a practical game occurrence which someone involved may perceive or call as "mirroring". Has very little/nothing to do with earlier post about theoretical mirroring.

Thus, it's not really a luck thing

Yes, I should have called that "random variation" thing. My fault for not being precise enough while talking generic theory stuff.

I was talking about theoretical level mirroring. Concepts such as "What is the maximum gain from full mirroring everything in 2p Dominion (answer: max. outcome for 2P is 50-50 in theory, much less in practice)".

Theoretical level stuff is basically about finding the minimum and maximum values of given strategies. These are limits which can not be exceeded even in theory. They are theoretical, and unlikely to occur in real life (very small probability). It's about minmaxing, in a way.

Repeated simulation stuff is about finding real life averages of given strategies in order to negate the variation inherent in Dominion. Average strategy is often completely impossible to happen in real life: 3.2 VP in one turn is not possible under Dominion rules in practice under whatever rules because the cards have full numbers. If average always happened (which never does outside from calculus), there would be no random variation.

Both approaches have their uses. But I have not seen that much theoretical stuff around here. Except, well, about minmaxing first 2 buys because that is very much practice-related.

But even then, you could go into some hardcore probability trees of e.g. first two deck revolves by each player in a 4 player game, given that what they have been doing on each of their turns. In real life situations, it is often not feasible to calculate this while playing. But having at least a rough idea of how probability and "given that"-assumptions work (in 3+p) can significantly improve one's game. Additionally, based on some reports that I have read, there are people who have done some hardcore early game probabilities in fixed kingdom games they have setup to lure people in the game. In this kind of a situation, P2 can use that probability advantage to turn the game around into almost 90% win despite going 2nd since they know all "correct" responses to P1 first two deck revulsions.

[RisingJaguar]

Player 1 Coins: $5, $8
Player 2 Coins: $8, $5

This is a practical game occurrence which someone involved may perceive or call as "mirroring". Has very little/nothing to do with earlier post about theoretical mirroring.

I was talking about theoretical level mirroring. Concepts such as "What is the maximum gain from full mirroring everything in 2p Dominion (answer: max. outcome for 2P is 50-50 in theory, much less in practice)".

Theoretical level stuff is basically about finding the minimum and maximum values of given strategies. These are limits which can not be exceeded even in theory. They are theoretical, and unlikely to occur in real life (very small probability). It's about minmaxing, in a way.

Repeated simulation stuff is about finding real life averages of given strategies in order to negate the variation inherent in Dominion. Average strategy is often completely impossible to happen in real life: 3.2 VP in one turn is not possible under Dominion rules in practice under whatever rules because the cards have full numbers. If average always happened (which never does outside from calculus), there would be no random variation.

Both approaches have their uses. But I have not seen that much theoretical stuff around here. Except, well, about minmaxing first 2 buys because that is very much practice-related.

There's a good chance I do not understand this properly, but I saw you make a comparison between theoretical mirroring and simulating a bazillion times to show that P2 and P1 in basic big money is 50% win rates both ways. 

My understanding is that this is not the case.  In certain situations as above, that has nothing to do with either of the three... "conditions" you previously mentioned, player 1 has inherent advantages given to them.  This was what Blueblimp was alluding to. 

Now my situation probably doesn't affect a huge amount of games, but this sort of situation where if Player 1 "misses out" early in the deck shuffle, then Player 1 doesn't miss out said resource split.  However I THINK this is outside the realm you're talking about.  I just think the basic idea that Player 1 can end games with an extra turn is an actual advantage in this theoretical mirroring we are talking about. 

EDIT:

Thus, it's not really a luck thing

Yes, I should have called that "random variation" thing. My fault for not being precise enough while talking generic theory stuff.

My point is that I take is you think random variation will balance out.  Sometimes P2 gets advantage, hooray P2 wins.  Vice versa, all balances out.  However, there are times, mainly in PPR, where one advantage, then the opposite nets out to advantage in P1.  Thus, expected win rate of P1 > P2 in straight up BM. 

[DStu]

Player 1 Coins: $5, $8
Player 2 Coins: $8, $5

This is a practical game occurrence which someone involved may perceive or call as "mirroring". Has very little/nothing to do with earlier post about theoretical mirroring.

I'm not sure if that analysis is very helpfull. Because considering the non-random plain BigMoney, it also gives you a method to always guarantee a tie as Player2. Namely mirroring. With same draws, and without actions, that gives you a tie.

Also, you somehow make a non-random argument to in the end get a probablilisitc result, namely a 0.5 winchance.  Which does not seem very trustworthy...

[Karhumies]

Mini-article on Dominion game theory, 2p vs larger amounts of players, #3:

- "Information advantage of going last" and compare this to Texas hold 'em poker. Discussing why going last in a (6-player) game can be a reward instead of a punishment. And why this advantage becomes larger as the number of players go up.

In large player number games, going late (or even last) can be better than going first. Quite often. The late players can see which strategies are uncontested, and choose one of them. Or choose to contest the strategy of a specific player. In many environments, the above can be better than going first = choosing any of the multiple valid strategies, and blindly hoping that no one contests it. Note here, that in 2p P1 gains a significant advantage if P2 chooses to mirror P1. But in 3+p, if P1 is copied by P2, the advantage from both of their strategies being diluted down is gained by P3 and the others. And so forth.

Taking into consideration real-life variables.
1. First player is time-constrained in real life, and therefore likely to choose a really good but obvious strategy.
1.1 Based on my empirical experience, obvious strategies are more often contested than missed by everyone in the table. Especially when there are less really good strategies than there are players, it tends to be so that the really obvious strategies are the ones becoming contested.
 
2. It is easier to copy than to figure out something new or different.
2.1 P1 is the most likely player in the game to be copied. This often leads to a mirror disadvantage in MP for P1. Something which never happens in 2p, where mirror = P1 mirror advantage. This is amongst the biggest differences between 2p and MP Dominion, IMO. If P1 does not realize this essential difference and keeps on being mirrored, this can turn out bad for him.

3. Last player has a lot of time to think about strategy, or "choosing his poison" while the other players act.
3.1 Last player has the most knowledge about other players' strategies before making a choice.
3.2 Last player is the most likely to think about alternative strategy because he has the most thinking time before his turn (especially the crucial early turns).
3.3 Last player is the least likely to be contested in the game because he is the last player to act in the game, and everyone else has chosen their strategy first already.
3.4  Despite having less (obvious) really good options available when last player's turn arrives, he knows a lot about how each strategy alternative is likely to play out because he knows who he is contesting with. ; In contrast, P1 usually picks something (obvious) and blindly hopes that no one copies (someone usually does).

4. If the last player intentionally decides to contest P1 strategy (which has a good chance of being the best strategy, if the players are experienced enough), P1 has very limited time to respond to this because his turn is next. This can throw him/her off from his game because he most likely does not want a mirror game in multiplayer. While last player again gets a full round of reaction time.

[blueblimp]

Thanks RisingJaguar for salvaging a usable point from my post.

I think the theoretical mirror is too unrealistic for whole-game scenarios, but it's useful for modeling two players rushing a single pile, because those fairly often end up in buy-1-per-turn scenarios. I think it's useful to look at bad luck resistance, here. For example, when rushing Duchies (for a Duke strategy), player 1 gets a "free miss"; not making $5 for one turn can still allow an even Duchy split. Player 2 doesn't get this comfort zone.

[Karhumies]

My point is that I take is you think random variation will balance out.  Sometimes P2 gets advantage, hooray P2 wins.  Vice versa, all balances out. However, there are times, mainly in PPR, where one advantage, then the opposite nets out to advantage in P1.  Thus, expected win rate of P1 > P2 in straight up BM.

Yes, that is an incorrect assumption I made previously, since I have 0 games experience from playing 2p Dominion. I stand corrected: P1 has a significant advantage in 2p mirrors. Which means that instead of 50-50 average for P2, it kind of becomes 50-50 max. for P2. And in real life, much lower than that. I should probably use more hedging language.

I'm not sure if that analysis is very helpfull.

It probably isn't. But it's a starting point.

Also, you somehow make a non-random argument to in the end get a probablilisitc result, namely a 0.5 winchance.  Which does not seem very trustworthy...

It wasn't. My included assumptions were wrong (missing the effect of +buy et cetera from the proposed theoretical model), as shown above.