War for Throne II: The Complete Opening Book (Remastered)

A video guide to this thread can be found here!

To get straight to the point: Wanting to investigate the fairness and balance of the variant custom position War for Throne II, I've put forth a list of three simple opening axioms to follow ensuring best play. Because the position is one of the few where a capture can be made on move one, it's essential that competitive players follow these axioms for the duration of the opening. As general rules, these are good principles to follow throughout the game, but middle- and end-games are varied in their complexity, and player judgement is more based on nuance in many situations. Following these rules leads to just eight possible openings, each with their own merit, which are listed, named, compared, and evaluated later in this post.

These axioms are simply a consequence of many basic capture/recapture and point maximisation principles of chess, along with opposite-cooperation theory (see an in-depth analysis by @GustavKlimtPaints), and have been developed during the course of over 4,000 game's experience in this particular variant.

Without further ado, the three axioms of opening theory in War for Throne II are (in order of precedence):

1. When you are under attack from more than one opponent, you must:
   1. capture one of them, or
   2. retreat to safety.
   3. Given a choice between 1a) and 1b), prioritize by (i) protection of your back rank first, (ii) protection from a check second, or (iii) maximising point value third
2. When your opposite is under attack by more than one opponent, without a turn in-between, you must attack one of your opponents.
3. Undefended pieces must always be:
   1. protected (your own), or
   2. captured (your opponents').
   3. Given a choice between 3a) and 3b), you may (i) choose equally with one exception: (ii) if the opponent to your left is also compelled to follow 3a) or 3b), choose the action against the opponent to your right.

Definitions:

• "Under attack" is used here to mean: (i) can have material captured, (ii) has had material captured without recapture, or (iii) can have an undefended piece threatened.
• The "opening" is concluded when any player is not faced with either axioms 1) or 2).

With these axioms, point values are rarely preserved, meaning that trades are usually unequal in point value. However, the potential consequences of preserving point value (i.e. only making equal trades) are far worse. Through careful analysis of the entire opening book (see below), and following from the EPG theory developed by @Hest1805, the point values of Kings and pawns during the opening should be thought of as roughly equal. In fact, the EPG of pawns and Kings are +5.25, and -5.25, respectively. These statistics are made equal when considering the value of a King to be exactly 12/11 that of a pawn (approximately 1.09 pawns).

Naïve play from you or your opposite can jeopardise the chance that either of you will place highly in the final standings. In other words, opposite-cooperation theory is essential for a chance at a prosperous outcome. In top-level play, the choice is clear: You can cooperate with your opposite, and have a chance at finishing first or second, or you can treat your opposite as just another opponent, and all but erase that as a possibility for either of you.

Having established the three axioms of opening play, the complete opening book contains eight possible openings. Yes, there are only eight. You'll notice I've named each, some with obvious derivations, some not so obvious. After all, what's the point of all this if I can't at least name something after myself? In these descriptions, a modified PGN4 notation is used with () after each player's move to indicate which axiom compels the decision.

The "Military" Opening: Blue relies on Green's support

1.r dxc4=K(1a)        .. 1.b 5xc4(3b)          .. 1.y dxc11=K(1a)       .. 1.g 4xk3=K(2)        

2.r jxk3(3b)          .. 2.b 10xc11(3b)        ...End of Line.

The "Autumn" Opening: Favours preserving material

1.r dxc4=K(1a)        .. 1.b 5xc4(3b)          .. 1.y dxc11=K(1a)       .. 1.g 11xk12=K(2)      

2.r d3(2)             .. 2.b 4xd3=K(1a/3a.ii)  .. 2.y Kd12(1b/3a.ii)    .. 2.g Kl11(1b/3a.ii)

3.r exd3(3b)          ...End of Line

The "Delayed Pinwheel" Opening: When Blue understands move order, it's his advantage

1.r dxc4=K(1a)        .. 1.b 5xc4(3b)          .. 1.y dxc11=K(1a)       .. 1.g 11xk12=K(2)      

2.r kxl4=K(2)         .. 2.b 10xc11(3b)        .. 2.y jxk12(3b)         .. 2.g 5xl4(3b)

...End of Line

The "Beginner's Left" Opening: "I like it. Simple. Easy to remember."

1.r dxc4=K(1a)        .. 1.b 5xc4(3b)          .. 1.y kxl11=K(1a)       .. 1.g 10xl11(3b)      

...End of Line

The "Pinwheel" Opening: And back around we go

1.r kxL4=K(1a)        .. 1.b 4xd3=K(2)         .. 1.y dxc11=K(1a)       .. 1.g 5xL4(3b)      

2.r exd3(3b)          .. 2.b 10xc11(3b)        ...End of Line

The "Grable" Opening: Is Green feeling the heat yet?

1.r kxL4=K(1a)        .. 1.b 4xd3=K(2)         .. 1.y kxl11=K(1a)       .. 1.g 10xl11(1a.ii)    

2.r Kk3(1b)           .. 2.b Kc4(1b)           ...End of Line

The "Greedy Grable" Opening: "Take what ya' can. Give nothing back."

1.r kxL4=K(1a)        .. 1.b 4xd3=K(2)         .. 1.y kxl11=K(1a)       .. 1.g 10xl11(1a.ii)    

2.r exd3(1a)          .. 2.b 11xd12=K(2)       .. 2.y exd12(3b)         .. 2.g 5xl4(3b)

...End of Line

The "Beginner's Right" Opening: For fans of "Beginner's Left," this one's "Much more better."

1.r kxl4=K(1a)        .. 1.b 11xd12=K(2)       .. 1.y exd12(3b)         .. 1.g 5xl4(3b)      

...End of Line

Evaluations:

Speaking strictly point-wise, not all openings are created equal. In the table below, the openings are listed by row, player by column, and each cell shows the points gained per piece lost. Between all eight openings, and all four players, the average of this value is about 2.0, so the resulting points-per-piece number is compared to 2.0. i.e., if a player lost one pawn and one king (two pieces), and gained five points for doing so, his value would be +0.5, because (5 pts.) / (2 pcs.) = 2.5, and 2.5 - 2 = 0.5. However, if the same player only gained three points for the same investment of two pieces, his value would be -0.5, because (3 pts.) / (2 pcs.) = 1.5, and 1.5 - 2 = -0.5.

This analysis makes a shocking revelation! In contrast to what most observers of this position have posited, Blue and Green actually have strong relative advantages in the opening versus Red and Yellow, with perhaps the strongest player in most outcomes being Green. This trend continues when the opposite's strength is also considered. To get a better picture of how each player fairs in the opening versus the adjacent players, an "advantage" metric can also be used. For the table below, each player's points-per-piece value (from the table above), is weighted with their opposite's (at a ratio of 2:1), then compared with the average from their adjacent's. Stated mathematically,

Relative Advantage / Disadvantage = 

(2/3)*(Your Pts./Pc. - 2) + (1/3)*(Opp. Pts./Pc. - 2) - (1/2)* (Adj. Pts./Pc. - 2) - (1/2)* (Adj. Pts./Pc. - 2)

So as mentioned, Blue and Green are doing strong individually, but considering their combined strength in most openings, and Red/Yellow's combined weakness, the difference almost amounts to a full point advantage! That isn't to say, however, that Red and Yellow are always disadvantaged. Red is in control of move one, and Blue for move two, et cetra. Given the opportunity, these players will, in succession, determine which opening is played based off of what works best for them.

For example: Acting in his immediate best interest, Red plays 1.r kxl4=K, knowing that the average evaluation for Red is higher for all lines starting with a capture on Green. Now having the choice, Blue plays 1.b 4xd3=K, knowing the average evaluation for Blue is higher when capturing Red. Following the same logic, Yellow plays 1.y kxl11, and after Green recaptures, Red goes for the "Greedy Grable."

Mapping these decisions, with calculated evaluations, a decision tree is given below, showing all opening lines and the average evaluation for resulting lines when decided upon by the player who's on move (i.e., On move 1.r, Red can choose to capture Blue or Green. Averaging the possible lines resulting from a capture of Blue gives the evaluation for Red of -1.0. The average evaluation for Red after capturing Green is -0.6):

Before concluding, there must be a brief discussion of what to do with deviation from the axioms. After all, not all games are played out by experienced players with a keen understanding of the principles of solid play. There are countless openings not listed here in which one or more players deviates from the axioms. In these situations, it remains best practice to continue following them yourself. Uncooperative opposites can especially lead to your position being all but lost out of the opening, but such is the case with four-player chess. Do not fool yourself into thinking that best play spells a win (or even second place) in every game, although it frequently will. Good things usually happen. Bad things sometimes happen. After all, this is just the opening. Even with a disastrous first three moves, there will still be plenty of time to make up for a disadvantageous position.

In conclusion, play the axioms for this variant, especially in the opening. Think of yourself first, your opposite second, know the possible choices your opponents will face, and anticipate their decisions. In the opening realise that a piece is a piece, and if you can get more than one point for giving up a king, you're doing well. This mindset should be shifted immediately when the opening is concluded, and no more moves are forced. The middle- and end-game have other guidances, although it's doubtful that one can concisely state in simple language any set of governing axioms for post-opening play. For subsequent moves, let experience be your guide.

An earlier version of this post contained a few erroneous calculations. This new version has fixed those minor mistakes, and introduces a more robust evaluation analysis method.