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Tedomari
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    Keywords: EndGame, Strategy, Go term

Tedomari means the last play. It is used not only for the last play of the game, but for the last play at different stages, and for the last play before the size of plays (temperature) makes a significant drop. As a general rule you want to try to get tedomari. There is a saying about the last gainful play of the game, Tedomari is worth double.

Getting tedomari is related to go infinitesimals and how to play them.

--BillSpight

See Tedomari Discussion.

Tedomari problems

Here is a simple tedomari problem I (Bill Spight) composed.

[Diagram]
Tedomari problem 1

How large are the plays, a, b, and c?
What is the optimal order of play when Black plays first, and what is the result?
What is the optimal order of play when White plays first, and what is the result?

(The optimal order of play obtains the best result for each player.)

Tedomari Problem 1 Solution



A very nice example of tedomari that you can work out yourself are the problems [ext] 365 and [ext] 368 at goproblems.com.

--dnerra

Bill: They are from Mathematical Go.

Robert Pauli: Here's a cute little tedomari example (DGoZ 5/03):

[Diagram]
White's turn. No komi. No captives.

Locally best play isn't globally best play, as everyone will find out sooner or later.



I like these miai values (even if I'm still struggling with them), however, would they help White in any way ??

Bill: Gee, maybe I am missing something, but the best local play seems like the best global play to me. (OIC. It depends on what you mean by best play. I think you mean optimal play. I had orthodox play in mind. The normally best local play differs from the best global play at or below temperature 1. They are the same above that.)

Robert Pauli:
Hmm, I'm ahead of the solution in DGoZ 6/03, but...

[Diagram]
White wins by one[1]

[Diagram]
Only jigo

Locally best for the lower part, but missing the win.



Are you using some kind of mathematical never-ever-draw rules, Bill, or what am now I missing? ;-)

So, how do miai values direct White to choose the "inferior" move?

Bill: Here is what I mean:

[Diagram]
Normal local play

After W1, White has W3 - B6 with sente, and Black has B8 - W9 with sente. This is 2 points better than the result above, starting with W8. With rare exceptions, W should play W1 unless the remaining plays are very small.

(W3 - W5 is sente because of possible later threats against Black's group. But even if it is a 2 2/3 point gote instead of a 3 point sente, W1 is still the normal local play.)

W1 was my first thought, and then I wondered why this was a problem. ;-)


[1]

[Diagram]
Better play for Black

Bill: A fine point: B2 is better technique than playing the sente at B8 first, because it gives White the chance to make a mistake.


[Diagram]
White's mistake

If White makes a mistake and plays W5, Black gets tedomari, for jigo.



Robert Pauli:
Thanks for your reply, Bill, even if it crushes my beliefs. Somehow my impression was that the binary tree used to compute miai values was made up of locally, say, minimax-best plays, but that doesn't seem to be the case. What you're saying sounds like miai values themself ranking the moves to be used in the tree. Boy, I'm really confused. Not that I'm new to recursion, but where's the ground case here? Temperature zero? At least it can be checked with minimax, and since nothing can be gained the miai value of each move should be zero in that case. Then I can slowly work my way back...let me instanly hurry to my chamber and contemplate... :-)

Bill: This may help. It's from my talk at the Computers and Games 2002 conference. The slides are [ext] on my home page.

Suppose that we have the following game tree, with / indicating a play by Left (Black) and \ indicating a play by Right (White):

               G
              / \
             /   \
            H    -7
           / \
          /   \
        -2    -4

Suppose that Left (Black) plays first. We get the following backed-up minimax values.

              -4
              / \
             /   \
           -4    -7
           / \
          /   \
        -2    -4

For calculating the miai value of G, we add the parameter, t.

               G
              / \
             /   \
            H   -7+t
           / \
          /   \
       -2-2t  -4

Then we get these backed up thermographic values:

      max(min(-4,-3-t),-5)
              / \
             /   \
   min(-4,-3-t)  -7+t
           / \
          /   \
       -2-2t  -4

The top value represents the left wall of the thermograph of G. Solving for t in

     min(-4,-3-t) = -7+t

we get

     t = 2

So the miai value of G is 2.

That's kind of sketchy, but I hope you get the idea.

(BTW, we are going to have to move some of this discussion. :-))

Robert Pauli:
Wow. Be sure - I don't get a single word. Dummy me thought that it simply would go by propagating averages and then to look at the distance between root and one of its daughters:

            = -5      -
              / \     | = 2
             /   \    |
         = -3    -7   -
           / \
          /   \
        -2    -4

Just an incident that it turns out to be the same ?

Do I need to swallow thermography to get miai values ?

(Move it whereever you want, Bill, I'm lost anyway. :-)

Bill: Sorry, Robert. I was way too sketchy.

You do not need to know thermography to figure out miai values. People, including myself, have been doing that for many, many years. :-)

However, the thermograph (TG) gives more information than just the count and miai value. One thing it can indicate is which plays (each line in the TG is associated with at least one line of play) to choose under which circumstances. That is one question in the DGZ problem. To answer this kind of question you have to look at the walls of the TG.

You brought up the question of minimax. The walls of the TG can be expressed in minimax terms, and that is what I showed.

Going back to the problem. The line of play to use when White plays first to figure out the count and miai value for the bottom right corner is W1 on the 2-3 point. That was obvious to me. The line of play to use when play is restricted to the corner is W1 on the 3-4 point, solidifying the wall. (That is also the one to use at temperature zero, OC.)

The right wall of the TG for the corner tells us that, as a rule, the time to switch between those plays is at temperature 1. We could work that out without the TG by considering that playing on the 2-3 allows Black to make a 3 point reverse sente, while playing on the 3-4 is only 2 points worse, on average, than playing on the 2-3. We can gain one point by playing on the 3-4, at the cost of one move.

Now, since the remaining play on the board has a miai value of one point, thermography does not give us a clue which play is right. However, as it happens, the play that is correct under most circumstances is still correct for the problem.

The thing is, when you are looking for the best play in an area of the board, you do not as a rule look for the best play at temperature 0. That is not realistic in a real game. Usually you want the play that defines the miai value. That's why my eye immediately went to the 2-3 point.

As for figuring miai values by taking averages, you realize that that does not work in the following example.

           = -3.5     -
              / \     | = 3.5
             /   \    |
         =  0    -7   -
           / \
          /   \
        +4    -4


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This is a copy of the living page "Tedomari" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.