Just look how these people mock my genius.

But to continue, the two-dimensional evaluation system facilitates the application of alternative criteria to the selection of moves (good players do this already; it's only that the evaluation system calls attention to it).  Here are some possible criteria for White and the ranking of evaluations that follows from them (rankings are in order of decreasing desirability; also since all rankings start with WW, W and end with B, BB, I omit these).

1.  Maximize relative chances, maximize own chances among ties.  This gives: Wb, w, Zw, zw, =w, Z, z, =, ==, Zb, zb, =b, wB, b.

2.  Maximize relative chances, minimize opponent's chances (chance of defeat) among ties, maximize own chances among further ties (this last necessary to choose between Zb, zb and =b): w, Wb, =w, zw, Zw, ==, =, z, Z, Zb, zb, =b, b, wB.

3.  Maximize own chances, maximize relative chances among ties (almost like, White must win): Wb, Zw, Z, Zb, w, zw, z, zb, wB, =w, =, ==, =b, b.  But if White must win, then "==" should be shifted to the bottom.

4.  Minimize opponent's chances, maximize relative chances among ties (White must draw): ==, w, =w, =, =b, Wb, zw, z, zb, b, Zw, Z, Zb, wB.

Clearly these different criteria would give rise to very different move selections.  This already happens in practice, it's only the the marking system illustrates how. 

A single tree of variations marked with this method and "backsolved" according to some criterion would readily yield the optimal move selection under that criterion, but only if one assumed the criterion according to which each side is selecting its moves!  A standard assumption would be that the opponent is merely trying to thwart one's goal, but that isn't necessarily the case.  If one can assume that one's opponent must play for a win, while one's goal is to maximize one's relative chances, the opportunity may arise to steer the game (as White) into Wb territory since the opponent will prefer this even to =b.

My point is that to mark a given position in a tree of variations with fixed terminal evaluations, one needs to assume the criteria under which either side will select its move.  The mark assigned to any given position will depend on the two criteria assumed. 

This aspect of move selection is not revealed, but rather is concealed, by the standard, one-dimensional system of marking positions.  Under a one-dimensional system, for a given set of terminal marks, there is always a set of equally preferred, objectively optimal moves at the root position.

I will conclude by remarking that verbal annotations, while fully sufficient for conveying a message like Zb, do not suffice for ranking positions and backsolving trees of variations, or if they do suffice, there must be a very laborious process of translation and interpretation, which could have been avoided if the terminal positions had been marked according to the proposed method.

Here's a radical idea: how about some prose annotation, when a simple symbol doesn't suffice .

Maybe you could use an 8x8 grid labelled a-h horizontally and 1-8 on the vertical, to represent your terminology.

The a1 corner could be winning for White,
The h8 corner winning for Black.

tending towards the a8 corner could represent uncertain undynamic.
and tending towards h1 could represent uncertain undynamic.

You could even specify a range or block such as a1-a2 or a square d2-d4-f4-f2 to indicate how accurate the assessment is.

It would be a more visual representation than remembering symbols.

Eg; 1.d4 d5 could be assessed as:
1.d4 (c3-d4-c4) d5 (c4-d4)

Impressive. Or perhaps disturbing.

Can you sing all those to the tune of 'Tomorrow'?

So, notwithstanding the burning importance of the question, "Who will win Linares," I will continue to discourse upon this subject.

The two-dimensional scheme of positional evaluation proposed permits alternate rankings of positions according to different criteria, each of which leads to its own method "backsolving" trees of variations to obtain the preferred move in any given situation. 

I assume that the relative chances are the same in each of (==, =, z, Z), (=w, zw, Zw), and (Wb, w) and that these groups are in increasing order of relative advantage for White (and the analogous assumption when Black has the advantage).  So I obtain the preference ranking:

A: WW
B: W
C: Wb, w
D: =w, zw, Zw
E: ==, =, z, Z
F: =b, zb, Zb
G: wB, b
H: B
I: BB

where on each level there is indifference among alternatives in terms of relative chances.  White of course prefers the top, Black the bottom.  In my next post I'll exhibit some alternative criteria for ranking positions, and thus for "backsolving" trees of variations to obtain the preferred move.