Go Articles Charles Matthews | ||

Setpiece Kos |

The way the ko in the Kato-Kobayashi game from the
previous article was unsettled for so long indicates
the slow and limited profitability of some kos. I think
*dan* players, anyway, will take the point. It
hung around because it wasn't urgent in the way other
parts of the board more truly were. Players coming up
to *dan* level do start to get a feel for the real
usefulness of *sente*, and often over-correct,
omitting plays they shouldn't. That is something to
get over, by balancing up notions of the positive
value of influential positions (typically completed
in *gote*) and the negative value of weak groups
arising from neglect of plays for security. What will be
said in this article is abstract, but it rather assumes
this sort of practical grasp as background.

The reason for thinking that the theory given here already isn't enough can be explained. So far the way to take into account the 'climate' in which a ko may be fought has in numerical terms comes down to intelligent application of miai counting. If there are alternate ways to play out a local position, which involve different tally changes, one does a price comparison on the basis of the value of a single play in the environment.

The trouble with this approach is that it is shopping, rather than playing a game. You may think that you are in charge of the game, and can direct play down a favourable avenue according to calculations. But your opponent may, just as likely, have alternate ways to play, and will vary strategy between them according to circumstances - in particular according to the current value of sente.

Therefore the phenomenon of break points demarcating
strategy A from strategy B in playing out a position
cannot in general be approached in a purely self-centred
way. I may calculate that strategy A is superior until
the temperature **t** in the game drops
below a certain value. But my opponent too is entitled
to switch strategies, and may do so quite rationally at
a different temperature.

This area of theory is what is meant by
*thermography* in Combinatorial Game Theory. I
am no great believer in mathematics in go for its own
sake, despite - perhaps because of - having a background
in mathematics myself. It doesn't have much to do with
professional players' normal approach, as far as I can
tell; and the kind of mathematics that denatures one's
intuition for the game can safely be taken as no good.
It is essentially impossible, though, to connect up
what I have written so far here to current ko research,
without tackling this area.

The step forward to take is to consider strategy over
the whole temperature range, up from **0**
to high-energy (anything over **20**
in normal go). Temperature **0** means
'penultimate plays': we can assume the game ends when
the local situation we are examining is played out. Many
games do end, for example, with a *minimal ko*
(Japanese *hanko*, the so-called 'half-point
ko'): and some are decided by its result.

What about the other end of the scale? General theory
as well as normal practice in counting the board
position indicate what to do: one counts territory on
the assumption that each side gets in its *sente*
endgame plays. No more and no less: no one can afford
to take *gote* if *sente* is large
enough. This is the exact opposite of the situation at
temperature **0**, where *sente* is
of no use to anyone.

It is in this setting that we wrestle with putting
plays in delimited regions of the board into context.
Strategy cannot be described in a way that is uniform
over temperature: there will be points of transition
from one way of playing to another. The new concept
to be introduced is first to plot a graph rather
than concentrate on calculation of the value of
**t** where a particular change takes
place.

The fundamental example is a *gote/sente* change.
Imagine an endgame position in which a choice of ways to
play offers itself: I can take *sente*, or accept
*gote* for a larger gain in points. Here is a
very common example:

We are looking only at the top edge here. Black can play
this way and take *sente*, at an early stage of
the endgame, because Black's cut at 6 is very big if
White fails to connect there.
On the other hand Black 1 here is a large *gote*
play, allowing the follow-up with Black 5 that ends in
*sente*, for a total swing of 14 points compared
to White playing the same way against Black.

Evidently Black's second way of playing gives up
*sente* for four extra points. We can express
that by saying that the *sente* play is worth
ten points. If we use miai counting, the first way
of playing is 10 points and the second way 7 points.
In a sense therefore one will expect that retaining
*sente* is preferred here, under normal
circumstances. But it seems worth pointing out that miai
counting is, on the face of it, a tool for comparing the
value of plays in different parts of the board. What are
we doing, exactly, when we apply it to the comparison of
different options in playing out the same position?

Let's draw what we'll call the profile of this position.

It will look like the graphic. The horizontal
line extending to the right should be at height
**10**, and the sloping line at the left
end should have gradient **-1**, crossing
the axis at **14**. What miai counting
tells us is to take into account the extra one black
stone in the first diagram compared to the second. In
this graphical method we draw the lines
**y = 14 - x** and
**y = 10**, which
therefore intersect at
**x = 4**. This
expresses what we could have said before: *sente*
can be given up for four extra points, only if the value
of *sente*, which is a name for temperature, is at
most 4.

So far this is only a way of presenting the same data as a picture. It does make it easier to explain some points:

- The difference of 4 points here is the important number: we have calculated points values relative to White's hane-connection here, which matters greatly if we are comparing this endgame position to another, but in comparing two local sequences just moves the graph up and down.
- The dashed lines represent strategies not played. We are taking the upper piecewise-linear graph made from two straight-line graphs. If there were more lines, for example of gradient -2 for a tally change of two stones (the case of capturing and then filling a ko), the generalisation as a picture is obvious; but not something so easily described in words or symbols.

The general relation to tally reckoning and miai
counting is clear enough: comparing a gain of
**a** points with **m**
extra stones, with **b** points by using
**n** extra stones, we must look somehow at
**(a - b)/(m - n)**,
the profit per additional stone assuming **m** and
**n** are different. We can do this by
equating **a - mt** with
**b - nt**, and solving.
The graphical method does
this, while fixing a degree of freedom within the model.
The gradient of the line for a *gote* play is set
to be -1: that is, we regard a *gote* play as
costing **one** extra stone. That is the
reasonable convention to take here, because then the
lines for *sente* plays come out horizontal. But
it does again indicate that this is a shift from the
usage for miai counting for comparisons in different
areas: we treating the figure of 7 points miai for the
gote play in the example as a quite different style of
accounting for what is going on, highly relevant to
**why** play in that position, but not a
number that relates directly to **how** to
play. It tells us about the vertical positioning of two
lines that move up and down by the same amount according
to a reference diagram like the second one above, but
played out for White rather than Black.

This may seem a long and perhaps too apologetic discussion of a simple profile graph. The fact is that profiles may in interesting cases become rather complex, and then the interpretation becomes much tougher unless the basic type of example is something felt to be quite natural.

The decision-making process based on profiles takes
the temperature **t** as a variable about
which the players are fully informed (of course this
is already an assumption about powerful reading and
counting); and therefore available for use in a given
position. The way profiles are combined isn't mysterious
- it's the ordinary minimax idea but with the variable
**t** carried along. That is, considering
the upper region of the profile graph as defining it,
these regions intersect when there are different options
to combine. Because 'up' in a zero-sum game is assigned
to the player given positive scores (conventionally
Black), we have to understand that White is allowed to
read 'upper' with the opposite sense.

The temperature itself controlled by the course of the
game: in the conventional way of playing the opening
and endgame it is fairly reasonable to think of it
as tied closely to the occupation of 'big points',
but in the middlegame phase that is much harder to
accept. One **expects** temperature to
decrease, with some excursions caused by threat plays
which are effectively forcing, or fights that break out
and have to be resolved before the game can proceed
further in other areas. That actually doesn't say very
much - the trend of temperature is to decrease, but in
a running fight it may be increasing for a while as
players add stones to unsettled groups they can't afford
to sacrifice, and then when the groups become safe the
temperature will drop back. The proper explanation of
triple ko and other similar repetitive positions is that
a long-cycle repetition occurs (which it can as soon as
there are three kos on the board) which prevents the
temperature dropping, a much rarer phenomenon than the
mere possibility of simple looping.

This article has been intended as a gentle introduction
to thermography (for my own benefit as much as anyone
else's). The definitive presentation is in terms of a
*thermograph*, which is made by taking Black's
and White's profiles of a position, re-orienting them
so that the ultimate horizontal lines become vertical
and gluing them along that line (then called the
mast). The effect is, for example, to take a line of
gradient **-2** for Black's play, as we
would draw for a profile, and to make it have gradient
**1/2** - slow uphill work. This brings
the language used in this article to the point of
compatibility with that of the games theory classic
*Winning Ways* by Berlekamp, Conway and Guy.

I (unlike a number of enthusiasts for theory in go) don't recommend anyone to start off reading the literature on Combinatorial Game Theory (CGT), as the theory is now called, in the search of insight into go. I was an academic colleague of John Conway's for a number of years, and a go player, and never then made the leap of connecting go with CGT, though others in my club and department such as the late Frank Adams did. I come to this quite late, in order to catch up with the research being carried on by Elwyn Berlekamp and his school. Combinatorics isn't a soft option, whatever anyone may tell you otherwise. The piece of history attributing to watching go being played some of the inspiration behind Conway's input into CGT seems well-attested. It would be nice to think that CGT could return the compliment to go; but what combinatorics models most effectively is often related to its precursor across a serious transition (for example, formal languages and natural languages).

The collaborators on the
Sensei's Library site,
notably Bill Spight, have been wrestling with the
expository problem posed by the gulf between go and CGT
ways of thinking. This is still work in progress, but
see the CGT path
there if you'd like to approach this area and are wisely
reluctant to read hundred of pages of theory first.

Posted 4 November 2002. Copyright © 2002 Charles Matthews | ||

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