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[liopoil]

The challenge is simple. Design a kingdom where the second player has a higher chance of winning than the first player with optimal play. You may not pick starting hands, nor can you choose the order of the knights pile, the ruins pile, or the black market deck. You can choose the bane, the contents of the ruins pile, and the contents of the black market deck.

I don't actually have a kingdom in mind, but I have some ideas and I think it could lead to some interesting discussion. Unfortunately it is hard to verify solutions, so try to make the benefit for the second player as large as possible.

[jomini]

There are several options I can see to start from:
1. A rock/paper/scissors type board. The classic example is a dreadfully boring Militia/Library/Council Room BM game with Baker at the bottom of the Black Market Deck (with all all the other horrible cards above it, basically each player can open 5/3 but Bm is a horrid play and Baker unlikely to be drawn nicely). So if you both open 5/3, P1 can take any of the three cards listed ... and P2 can counter nicely. E.g. P1 buys Militia then P2 buys Lib. If P1 buys Cr, then P2 buys Militia. If P1 buys Lib, then P2 buys Cr. In theory, you don't need the wonky stuff with the Baker coin, but the point is easier to see without having concerns with the price point. Another case would be something like Young Witch/Bane/Silver. With a decent enough bane you can easily have opening Yw loses to bane which loses to silver which loses to Yw.
2. Reactions. Say there are decent attacks out (like Oracle, Militia, and Noble Brigand) and you have decent options in reactions like Beggar and Watchtower. Knowing which attack the other guy is committed to for the first few hands is an advantage if there are near equal strategies with the reactions (E.g. Watchtower is a lot better as dead draw against Militia, while Beggar is not as hot against Noble Brigand).
3. Embargo. Mostly this is for potion cards. If P1 commits to a potion buy for something he needs to spam on the potion side - like Apothecary, Alchemist, Scrying Pool, or Vineyards, you might be in a better position if you can use embargo to tank that and then follow a marginally less bad strategy.
4. Enabled strategies. Like with reactions, some setups key off player interaction. For instance, Embassy & Mint make for a first mover problem. Mint with a free silver is pretty good, Mint without a Silver is pretty terrible. If you P1 with 5, you have to play the odds on P2 getting Mint as well; if you are P2 you can safely open Embassy without worrying about easy Minting. Marauder/Death Cart can also be a bit of a thing here.
5. Higher player counts with limiting cards. P1 tips his strategy, P2 can then avoid competing for cards and take the 2nd best strategy. This forces P3 to fight P1 (possibly moving both's odds below P2) or to take the 3rd best strategy (high odds of not winning). If P1 opens Remake/Squire to go for a Remake/Lib/Sqr/Spy/Jester run then P2 can opt instead for a weaker option (like B-crat/Silk Road) hoping that fighting for Sqr will limit P1 and P3. Or, even more simply something like opening Yw (horrid bane) for P1 and P2 going for Beggar/Feoda - P3 either goes Yw (nerfing P1 and powering up P2's Beggars) or mostly concedes the game to P1.

[Holger]

This old quote is relevant here:

Has there ever been a kingdom empirically shown to have second-player advantage? I think this may be one of the things that don't depend on the board.

A board where BM-Noble brigand is best has a slight 2nd player advantage, by my research, though I could be missing something of course. I assume it's possible to engineer a kingdom to have this be the best strategy, though obviously it would be difficult.

But he never expanded on his research, AFAIK; it's very hard to prove that a strategy is optimal for some board, even with a single kingdom card. Also, the optimal strategy can easily depend on whether you're starting player.

jomini: The problem with your options 1-4 is that P1 can generally just do nothing on his 1st turn (or do something "generically good" like buying a Silver), putting P2 into the bad "have to act first" position.
Therefore, I don't expect there to be any board where P2 has an advantage over P1 in 2-player games when ignoring P2's tiebreak rule advantage, and possibly not even when the tiebreak rule is accounted for.

The best chance may be with positive interaction cards, which certainly reduce first-player advantage.

[liopoil]

I don't need a rigorous proof. A good simulation with both players playing fairly close to optimally will do.

[GeoLib]

jomini: The problem with your options 1-4 is that P1 can generally just do nothing on his 1st turn (or do something "generically good" like buying a Silver), putting P2 into the bad "have to act first" position.

I don't think this is necessarily true. This is the "strategy stealing" argument and it only works if any move cannot decrease the favorability of P1's position. By buying silver P1 loses an opportunity and does not necessarily have at least as good of a position. If P1 has 5/2 and opens silver to avoid making a choice, then I think P2 now has an advantage on many boards.

This old quote is relevant here:

Has there ever been a kingdom empirically shown to have second-player advantage? I think this may be one of the things that don't depend on the board.

A board where BM-Noble brigand is best has a slight 2nd player advantage, by my research, though I could be missing something of course. I assume it's possible to engineer a kingdom to have this be the best strategy, though obviously it would be difficult.

But he never expanded on his research, AFAIK; it's very hard to prove that a strategy is optimal for some board, even with a single kingdom card. Also, the optimal strategy can easily depend on whether you're starting player.

In the kingdom of

Code: [Select]

Noble Brigand, Feast, Transmute, Scout, Coppersmith, Pirate Ship, Adventurer, Treasure Map, Outpost, Harvest
I think both players play BM+NB and if that gives P2 an advantage (which I believe because of the advantage of potential T1 silver stealing), then we're probably good. We could at least test the proposition that BM+NB gives P2 an advantage with the simulators.

[Holger]

jomini: The problem with your options 1-4 is that P1 can generally just do nothing on his 1st turn (or do something "generically good" like buying a Silver), putting P2 into the bad "have to act first" position.

I don't think this is necessarily true. This is the "strategy stealing" argument and it only works if any move cannot decrease the favorability of P1's position. By buying silver P1 loses an opportunity and does not necessarily have at least as good of a position. If P1 has 5/2 and opens silver to avoid making a choice, then I think P2 now has an advantage on many boards.

Probably, if the important choice is about a $5 card. But in this case P2 is almost as likely to have to commit to a choice before P1 anyway: If P2 has a 5/2 start and P1 doesn't, or if P1 starts 2/5 and P2 starts 5/2, P2 has a disadvantage e.g. in the Embassy/Mint situation.
(The probability of both players starting 5/2 or both starting 2/5 is about 1.4%, probably far too little to overcome the "generic" P1 advantage.)

In the kingdom of

Code: [Select]

Noble Brigand, Feast, Transmute, Scout, Coppersmith, Pirate Ship, Adventurer, Treasure Map, Outpost, Harvest
I think both players play BM+NB and if that gives P2 an advantage (which I believe because of the advantage of potential T1 silver stealing), then we're probably good. We could at least test the proposition that BM+NB gives P2 an advantage with the simulators.

Might Pirate Ship counter NB? Probably not, and there's certainly some board where BM+NB is optimal. So what's the optimal BM+NB strategy, and is it implemented in some simulator?

[jomini]

Probably, if the important choice is about a $5 card. But in this case P2 is almost as likely to have to commit to a choice before P1 anyway: If P2 has a 5/2 start and P1 doesn't, or if P1 starts 2/5 and P2 starts 5/2, P2 has a disadvantage e.g. in the Embassy/Mint situation.
(The probability of both players starting 5/2 or both starting 2/5 is about 1.4%, probably far too little to overcome the "generic" P1 advantage.)

Doing nothing, however, is not an equivalent option. For instance, say P1 passes cold on T1. P2 passed cold on T1. P1 passes cold on T2. P2 buys X. P2 will most likely (10/11) see X next shuffle. P1, if they buy something > X on T3 won't see it until T5. Because of the shuffle, waiting can also be hazardous.

This gets worse when we consider things like a Baker buried in a truly awful Bm deck. Now P1 has 50% odds of hitting a "$5" on T1; passing may allow P2 to buy the "rock" (e.g. Embassy) without having a chance to buy the "paper" (e.g. Mint). In the other 50% of games, half of those will feature P2 having their "$5" in T2 as well. This means we can engineer a situation where P1 has a huge information deficit 75% of the time. In like manner, stuff like Potion/Embargo can play out. 75% of the time P1 has to make a Potion/no Potion call before P2; we should be able to set up something that makes this poisonous (e.g. Pot > X > Emb > Pot).

[Holger]

Good points, but I'm not sure if those situations are enough to overcome P1's half-turn advantage.

With the Baker token, opening Mint becomes viable even without the Embassy's Silver - whether you open 5/2 or 4/3, you can always buy the Silver yourself on turn 1/2.