Why do asymmetric ko fights exist?

Question

When I play teaching games with much stronger go players, situations often come up where one side must win a ko fight in order to survive.

In these situations, my opponent will explain to me that winning this ko is worth more or is more important to the side which must win to survive.

However, this never made sense to me. At the end of the game, points are compared, so it seems that a loss by one side should be equivalent to a gain by the other. With this understanding, it seems that the side trying to kill should care just as much about the ko as the side trying to survive.

Why is it not the case that both sides should always care equally about the outcome of a ko fight?

Answer 1

I think the easiest way to think of this is in terms of options, and the freedom to take whatever options benefit me.

In your example, you have no options: You need to win this ko, or you will lose.

However, in my case, killing you might not be my only option. I could kill you and swing things thirty points in my favor, or if I felt like it I could grab that empty corner over there worth twenty points and strengthen my position on the right, or I could go back and reinforce my moyo on the bottom to help me with the late game fighting that hasn't started yet, or I can pick a fight with those weak stones in the centre that you forgot to connect five turns ago. I have options. You don't.

As for why the ko is asymmetric in the first place, I would be weighing not only the relative value of winning the ko, but also the actual chances I have of winning it. In most "asymmetric ko" situations, you're probably the one with the first opportunity to win the ko, and I probably won't have any threats on the board big enough for me to significantly threaten you with: Nothing I do is likely to stop you from winning it, because you need to win this ko, or you will lose. If it was just a question of "I can definitely kill you and get a hundred points, or I can definitely take two hundred points over here instead while you win the ko" that's not really an asymmetric ko.

Even if there are big enough ko threats, or somehow I had the first opportunity to win the ko, depending on how close the score is and how strong our respective positions are there might still be no real need for me to be greedy; I can forego the risk and cost of the ko (maybe those high-value ko threats will be more useful later when there's a more important fight going on) and take a lesser but safer advantage elsewhere and still maintain a comfortable lead. Why should I use up valuable aji and give you the opportunity to carve out a chunk of my territory when I don't need to?

Let's take, for the sake of example, the following situation with a 70-point life-or-death ko on the board:

• I play a 70-point ko threat.
• You respond.
• I take the ko.
• You play a 60-point ko threat.
• I win the ko.
• You follow through on your threat.

Now, this whole exchange, I'm ahead by ten points, and I've used up a 70-point ko threat to get there. This…isn't terrible, but not really an incredible outcome; if I had just played a ten-point ko threat in the first place and let you ignore it, I'd be in the same place score-wise, but with less wasted aji. Or I could've played any number of other ko threats that may have given me less points but otherwise strengthened my position. Options.

You, on the other hand, don't really have the opportunity to play a ten-point ko threat since, well, I'll just take the ko: That's now a 60-point advantage for me. You're the one who needs the big threats here, not me. Whether you live or die, I'm still coming out ahead while you still have to go back and secure your tenuous life situation. I'm totally okay with this.

In other words, you're painting yourself into a corner where you give your opponent every chance to dictate which battlefield he wants to fight on. This is generally a bad idea.

Answer 2

Short answer

How can a ko fight be “asymmetric”? By offering one player a strong incentive to start it.

When a ko is asymmetric

You are right that once the ko is started, the result is equally valuable to both sides. Where the asymmetry comes in is when one player, considering whether to start the ko, compares three local outcomes:

• They start the ko and win. (local score W)
• They avoid the ko and make less profit. (local score A)
• They start the ko and lose. (local score L)

This is asymmetric if A is not midway between W and L, that is to say the extra profit from winning the ko is significantly different from that lost by losing it, i.e. W - A ≠ A - L.

Advantageous asymmetry

If the profit is significantly greater than the loss – W - A ≫ A - L – this player will probably want to start the ko, as they do well out of it unless they either win by ignoring a threat even bigger than W - A or lose and are only compensated by following up on a ko threat even smaller than A - L.

Your opponents’ comment that “winning this ko is worth more” or “is more important to the side which must win to survive” is misleading; the difference is that starting the ko puts at risk more of the score that side would have if the ko were not played.

An example

Here is an (idealised) example from the endgame, where Black has pushed into White’s territory and White has blocked; should Black start a ko with a or connect at b?

$$Bm1 An endgame ko
$$ _XXXOOOOO_
$$ _..XO...._
$$ _c.XOa..._
$$ _.Xb12.X._
$$ ----------

This depends on how much damage the side that wins the ko can do to the other’s territory. Perhaps Black can do a lot of damage to White by winning, but only needs to defend at (e.g. c) if they lose; if so, this is a good asymmetric ko for Black to start.

Disadvantageous asymmetry

A ko can also be asymmetric in the opposite direction: the player starting it has a lot to lose and little to gain. One should, of course, avoid starting such kos. This could arise in the above position if Black were somewhat weaker and White stronger.

Kos that are not asymmetric

A ko is not asymmetric if avoiding the ko is not an option; in this case the only outcomes that matter are winning or losing the ko. For example, this is the case if making a ko is the only way to save a group.

In the situation you describe – having to win a ko to survive – the defender has no options and the ko is no longer asymmetric, but the attacker may have had a different way to attack without a ko, yielding a small profit; in that case it was asymmetric at that point.

A different sort of asymmetry

Finally, though you are right to say the result is equally valuable to both sides, one player may start a ko on the grounds that though their chances are faint, it is the only way they can see to win the game! This also applies, of course, to various other risky plays, such as deep invasions.

Answer 3

A do or die ko is often the biggest thing on the board. The person who will die basically has to ignore everything else and win the ko.

The person who won't die can "settle" for just a little less. Say the ko is worth 30 points. The one person has to fill the 30 point ko. The other person can make a 25 point ko threat and have it ignored, and lose five points on the transaction. As long as the loss is limited (5 points, not 10-15), the second person will usually win the game.