Benjamin Geiger: I guess that go players might agree that 80% of thickness or territory comes from 20% of the moves...
Elroch: Beware the Pareto principle, which is one of the most prominent examples of abuse of statistics (which would make Pareto turn in his grave). The authoritative sounding name encourages people to use it as a way of fabricating statistics when none are available. The general statistical fact is that when a quantity of something is shared among a population, there is some minimum number X between 0 and 50 such that (100-X)% of the quantity is shared among X% of the population. There are real examples where X has any value from almost 0 to exactly 50.
Seriously, if the size of go plays decreases linearly, we get this kind of picture:
|\ | \ | \ | \ Value | \ | |\ | 80 | \ | | \ | |20 \ ------------- 0 .55 P P Plays
In that case, about 55% of the plays produce 80% of the territory.
My impression is that the curve is flatter on both ends, with large plays predominating before the endgame, followed by a rather quick drop to smaller plays. Then you have maybe 65% of the plays producing 80% of the territory.
Evpsych: The territory? Do you mean unclaimed? Undefended? Uninvadable?
At move 1, how much territory is claimed? X=komi? The whole board? Some inscrutable CGT temperature/pressure/quantum entanglement measure? :)
If there is a capture, was that a refuted claim? Is it a land rush? A battle for grabbing the other's territory?
I'm trying to figure out what you mean by territory, as simple as that sounds! Thanks for any clarification...
Elroch: Perhaps the best theoretical definition of the size of a move is based simply on the result of a game with perfect play afterwards. This gives a precise number to each point in a game, and to each alternative move. The size of a single move can be obtained by simply comparing the result of a perfectly played game after it, to the result of a perfectly played game after a pass. Of course, in practice the precise numbers cannot be worked out, except in the endgame, but the concept is handy.
Bill: Traditional go evaluation, as well as modern CGT evaluation, takes a different approach that considers independent parts of the board separately. Since play may shift between these independent parts, you cannot just take the value of a local position or play by considering only the result of perfect local play. See miai value and CGT.
Elroch: When the endgame has been reached, local analysis and global analysis give the same value for a move, as the game has become disconnected into a sum of small relatively simple games. Before this stage, the only real value of a move is the global value as the less obvious consequences of a move, involving more remote interactions, or deep lines, may be as important as the obvious consequences of a move (the game is not disconnected yet). Of course the global value of a move may be essentially impossible to calculate. I speak from a position of substantially better knowledge of CGT than of go, which sometimes allows me to understand the theoretical truth a lot more easily than I can apply it. However, Bill probably has a greater knowledge than I do about CGT, as well as incomparably more experienced at go. :-) It is also worth pointing out that even the late endgame is much more complicated than is generally realised, as Berlekamp and Wolfe have shown. This is because when games are summed (local endgame positions combined), the values of the local games may not be numbers, complicating the analysis. A striking demonstration of this was Berlekamp's finding endgame examples where the strongest go pros fail to find the correct move, but CGT does.
Bill: All I was trying to indicate was that traditional go evaluation (as well as CGT evaluation) is different from the evaluation you propose. If I understand you correctly, your evaluation is global. Here is a small example to illustrate the difference, as I understand it.
Suppose that the rest of the board is settled with a net score of 0, and that there are two gote left on the board, one of which gains 3 points for the player who takes it and the other of which gains 1 point. If Black plays first he will gain 3 points and then White will gain 1 point, for a net score of +2 (for Black). If White plays after a Black pass she will gain 3 points and then Black will gain 1 point, for a net score of -2 (for Black). By your evaluation the move is worth 4 points. By traditional evaluation the top play is worth 3 points (miai value).
Elroch: You are right of course. I was talking about the value of the move, or of various possible moves, and this is not related in a simple way to the local miai value of the play, as it depends on the whole gamut of things that could happen afterwards. As a consequence the miai value is much more practical to calculate, even though the effect on the final score is of principal interest to the player. One area where a more holistic view seems to be useful is that of tedomari, getting the last play before a significant drop in temperature.
Charles Matthews Suppose you have a game in which in the middle game the exchange of a few plays Black/White/Black/.../White does not leave the current scores about where they were. Then the game will look like a mismatch, assuming the swing in the score is always one way. That's how games go when the players are half-a-dozen stones apart in level. So the straight line graph is a reasonable evened-out model for well-played games at higher handicaps, where White catches up gradually. For even games, you might instead expect a flat portion in a graph based on positional judgement, from about move 50 to 150. Various exchanges occur in the amounts of territory sketched out in the opening, with framework -> influence -> territory being a natural process.
I think what Bill may be hypothesising is about consolidation of territory: the split (for one player) in positional judgement between solid territory and influence ends up as 100% territory at the end of the game, and one could graph the derivative of the territory component.
Bill: To answer Evpsych's question, Benjamin started off talking about "thickness or territory". I took him to mean value in a general sense, not specifying which form it takes. I just said "territory", since thickness is eventualy translated into territory. I labeled one dimension of my graph, "value". By "territory" I meant the net value of a whole board position.
At the end of the game each player may have around 60 points, for instance. However, each player has gained more than a dozen times that. Most such gains have been countered by plays that nearly erased them, so the actual territory at the end of the game is a mere fraction of the gains made during play.
Perhaps Benjamin was thinking about the plays that finally consolidate territory or thickness. Indeed, there are very few of those. Sometimes they are huge plays, but for the most part they are relatively unimportant. Oba do not make territory yet, but they add much value to your position. They, and other large plays, are what this saying should be about. To use the business metaphor, we give credit to the salesman who makes the sale, not the cashier who takes the money.