Dominion Strategy Forum

Please login or register.

Login with username, password and session length
Pages: 1 2 [3] 4  All

Author Topic: Reasons for the 1st-player vs 2nd-player advantage  (Read 49785 times)

0 Members and 1 Guest are viewing this topic.

DStu

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2627
  • Respect: +1490
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #50 on: January 19, 2012, 12:15:19 pm »
0

I never doubted that. I only doubted that somebody wanted to say something else.
Logged

dondon151

  • 2012 US Champion
  • *
  • Offline Offline
  • Posts: 2522
  • Respect: +1856
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #51 on: January 19, 2012, 12:17:43 pm »
0

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.
Logged

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #52 on: January 19, 2012, 02:37:05 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.

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.
Logged

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #53 on: January 19, 2012, 02:45:57 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.

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?
Logged

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #54 on: January 19, 2012, 03:24:27 pm »
0

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 »
Logged

jomini

  • Saboteur
  • *****
  • Offline Offline
  • Posts: 1060
  • Respect: +766
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #55 on: January 19, 2012, 04:51:21 pm »
0

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 »
Logged

A_S00

  • Spy
  • ****
  • Offline Offline
  • Posts: 84
  • Respect: +41
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #56 on: January 19, 2012, 05:32:33 pm »
0

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."
Logged

DStu

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2627
  • Respect: +1490
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #57 on: January 20, 2012, 02:04:22 am »
0

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...
Logged

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #58 on: February 18, 2012, 11:28:57 pm »
0

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 »
Logged

dondon151

  • 2012 US Champion
  • *
  • Offline Offline
  • Posts: 2522
  • Respect: +1856
    • View Profile
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.
Logged

timchen

  • Minion
  • *****
  • Offline Offline
  • Posts: 704
  • Shuffle iT Username: allfail
  • Respect: +234
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #60 on: February 18, 2012, 11:56:37 pm »
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.
Logged

ftl

  • Mountebank
  • *****
  • Offline Offline
  • Posts: 2056
  • Shuffle iT Username: ftl
  • Respect: +1345
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #61 on: February 19, 2012, 12:00:20 am »
0

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 »
Logged

rod-

  • Conspirator
  • ****
  • Offline Offline
  • Posts: 213
  • Respect: +49
    • View Profile
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- »
Logged

WanderingWinder

  • Adventurer
  • ******
  • Offline Offline
  • Posts: 5275
  • ...doesn't really matter to me
  • Respect: +4381
    • View Profile
    • WanderingWinder YouTube Page
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #63 on: February 19, 2012, 12:46:55 am »
0

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.

WanderingWinder

  • Adventurer
  • ******
  • Offline Offline
  • Posts: 5275
  • ...doesn't really matter to me
  • Respect: +4381
    • View Profile
    • WanderingWinder YouTube Page
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #64 on: February 19, 2012, 12:50:03 am »
0

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.

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #65 on: February 19, 2012, 01:10:38 am »
0

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 »
Logged

tlloyd

  • Tactician
  • *****
  • Offline Offline
  • Posts: 404
  • Respect: +84
    • View Profile
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?   ;))
Logged

timchen

  • Minion
  • *****
  • Offline Offline
  • Posts: 704
  • Shuffle iT Username: allfail
  • Respect: +234
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #67 on: February 19, 2012, 02:37:44 am »
0

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.
Logged

dondon151

  • 2012 US Champion
  • *
  • Offline Offline
  • Posts: 2522
  • Respect: +1856
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #68 on: February 19, 2012, 03:42:55 am »
0

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 »
Logged

rod-

  • Conspirator
  • ****
  • Offline Offline
  • Posts: 213
  • Respect: +49
    • View Profile
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.
Logged

DG

  • Governor
  • *****
  • Offline Offline
  • Posts: 4074
  • Respect: +2624
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #70 on: February 19, 2012, 10:49:45 am »
0

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.
Logged

Robz888

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2644
  • Shuffle iT Username: Robz888
  • Respect: +3388
    • View Profile
Re: Reasons for the 1st-player vs 2nd-player advantage
« Reply #71 on: February 20, 2012, 03:21:13 am »
0

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.
Logged
I have been forced to accept that lackluster play is a town tell for you.

MasterAir

  • Alchemist
  • ***
  • Offline Offline
  • Posts: 39
  • Respect: +8
    • View Profile
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.
Logged

Captain_Frisk

  • Saboteur
  • *****
  • Offline Offline
  • Posts: 1257
  • Respect: +1263
    • View Profile
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.



Logged
I support funsockets.... taking as much time as they need to get it right.

Ozle

  • Cartographer
  • *****
  • Offline Offline
  • Posts: 3625
  • Sorry, this text is personal.
  • Respect: +3360
    • View Profile
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
Logged
Try the Ozle Google Map Challenge!
http://forum.dominionstrategy.com/index.php?topic=7466.0

Sullying players Enjoyment of Innovation since 2013 Apparently!
Pages: 1 2 [3] 4  All
 

Page created in 0.06 seconds with 21 queries.