In strategic games like Go and Chess, if you commit too much to one strategy, your opponent can play the counter strategy, like in Rock Paper Scissors . Losing options is one way of being too committed. Optimal flexibility means keeping open as many options as possible, never committing before it is necessary. In the most extreme case, high-level strategies might have to be chosen randomly.

Flexibility is an important concept in strategic games like Go and Chess. If you commit too much to one strategy, your opponent can play the counter strategy, like in Rock Paper Scissors , where always choosing rock is defeated by always choosing paper. But even just losing options is an example of being too committed, because you can no longer play the lost option. Examples for losing/keeping options in Go & Chess:

• Wasting Kō threats
• Playing Sente moves without good reason, e.g. Atari whenever possible or peeping
• Aji Keshi
• Invading usually excludes the option of reducing, and vice versa
• Light shapes are flexible, making your opponent heavy takes away options
• Ability and willingness to sacrifice
• Multi-purpose moves
• Miai
• Probing moves
• Komoku , takamoku and mokuhazushi commit towards one direction, hoshi does not
• From Chess: the threat is stronger than the execution

On average, having more options increases the probability that among those extra options is one that is better than what you already have. That is particularly important for strong players. Playing without commitment is a hallmark of the strongest AIs.

And so a perfect Go player might have to choose high-level strategies randomly, like in Rock Paper Scissors.

If you think long enough about strategy in Go, you will discover the importance of flexibility.

When you have just started learning Go, most high-level concepts and strategies are probably not yet known to you and you will try to stick with the concepts you already know, master them and hope to win that way. For example, in a handicap game, you might just defend and try not to lose a group of stones. But then your opponent plays a lot of moves that appear to be Sente and he just makes more territory. This is just one example of how it is easy for a strong player to exploit commitment to one strategy.

Lose by always choosing the same strategy

It shouldn't be surprising that in Go there is no single strategic concept that will guarantee a win. Otherwise the game would be far too simple. Everybody would just use this strategy and be at least equal to an opponent using the same strategy. Tic Tac Toe is like that. In complex games, there are multiple strategic concepts that compete with each other, and you can't rely on just one. In Rock Paper Scissors , if you choose one strategy, e.g. just playing rock all the time, your opponent will play the perfect counter strategy, namely always playing paper. If you alternate between rock and paper, your opponent will answer with paper when you choose rock and with scissors when you choose paper. No simple strategy can win in this game in the long run. In fact, optimal is to choose between rock, paper and scissors randomly. Other games can also 3 strategies A, B, C such that A always defeats B, B always defeats C, C always defeats A. Or maybe A wins over B just 55% of the time, like in intransitive dice , but that's still an advantage.

Go has a lot of strategic concepts and applying them correctly requires not committing too much to any single one of them. If you commit too much to one direction, it is like always playing rock in Rock Paper Scissors, your opponent just has to play the equivalent of paper. After enough time studying your gameplay, your opponent will choose the appropriate counter strategy. But even in just 1 game, if you repeat the same pattern often enough, your opponent might find a way to exploit that right away.

Examples

Wasting Kō threats is a good example of limiting your flexibility. You'll miss them in the next Kō fight, you also can't use them any more to gain thinking time and you choose one course of action that might remove other ways to play in that area. If you lose too many threats, you might even have to avoid a Kō fight later. In general, playing Sente moves without good reason usually removes options. This often manifests as giving Atari whenever possible. Playing peeps is another common example. A one-point jump is a popular target for peeps, but after the peep exchange, you can't cut any more. A trumpet connection is often peeped as well, but peeping on one side usually loses the option to peep on the other side. The worst kind of Sente exchange is Aji Keshi. But even in Gote, Aji Keshi is bad because it removes options.

Invading usually excludes the option of reducing, and vice versa. You have to choose, but don't choose before you know the right choice. Playing light shapes is an example of flexibility and making your opponent's group heavy is an example of taking away options, namely sacrificing. In general, the ability and willingness to sacrifice stones is an example of playing flexibly. Katago is still better at sacrificing than the best humans. Judicious sacrificing also explains why strong AIs prefer a 3-3 invasion under a hoshi stone early - they are better at handling the opponent's influence. The same for strong Chess AIs: in the match between AlphaZero versus Stockfish , AlphaZero happily sacrificed multiple pawns for long-term strategic gains and won several games by paralyzing Stockfish's position, i.e. taking away Stockfish's options.

Multi-purpose moves, like the ear-reddening move , are moves that give you multiple ways of continuing and no way for your opponent to take away all of them at once. Miai moves contain the flexibility in their definition. Your group being enclosed is an example of being forced to live locally, losing the option of running away. Probing moves are multi-purpose moves and they also tell you which of two options is the right one. Answering the probing move commits you to one course of action and that is valuable information to your opponent, which is why sometimes strong players don't answer probing moves.

In the opening, hoshi is popular because it is symmetric and thus gives you the option to extend to either side, whereas komoku , takamoku and mokuhazushi are committed towards one direction. More generally, many jōseki choices are incomprehensible if you can't see that they leave open as many options as possible. Many non-jōseki sequences are suboptimal simply because they have committed too much towards one choice. Chess has the idiom that the threat is stronger than the execution. Having the threat means having the option to execute, but not the obligation. After you execute the threat, the option to not do so has disappeared.

Losing options costs points

On average, having more options increases the probability that among those extra options is one that is better than what you already have. Conversely, losing options increases the probability that your continuation is suboptimal, because the optimal path was among the lost options. That statement is so general that it applies no matter how many options we are talking about. If you are very strong, gaining even just 1 additional point can be quite impactful and thus losing options is even more problematic for strong players. In fast games, the increased complexity that comes as the cost of more options has to be balanced against the benefit of more options, but in slow games I would expect the balance to go towards more options and more complexity.

This idea became more apparent to me after watching games of Katago versus Katago. Both players seem to display an almost ridiculous aversion to commitment, always playing tenuki and constantly changing their minds about saving a group or sacrificing it. Strong humans try that too, but Katago carries it to extremes. And given how strong Katago is, this is clearly not just a bad habit. It must be a necessary component of really strong play. My statement about losing options potentially forcing you to take suboptimal paths probably explains it.

The Go board is big and a Go game is long, meaning even tiny advantages can accumulate over time and options can have influence over long distances.

Committing to one strategy is a loss of options

Remembering the comparison with Rock Paper Scissors, even a small inclination towards one particular strategy may cost you several points over the course of a game if your opponent is as strong as Katago. And so a perfect Go player might have to choose high-level strategies randomly, like in Rock Paper Scissors.

As an entirely incidental observation: Buddhism thinks attachment is bad, this article opines that attachment to one strategy in Go is bad and both Buddhism and Go seem to be popular in the same geographical area.

If you think long enough about what strategies help you win in Go (and Chess and other sufficiently difficult strategic board games) , you will eventually discover the importance of flexibility. Flexibility here simply means having many options, i.e. not being forced to commit to any single one option.

When you have just started learning Go, most high-level concepts and strategies (e.g. overconcentration , leaning attacks , probing moves , inducing moves ) are probably not yet known to you and you will try to stick with the concepts you already know, master them and hope to win that way. For example, strong players may tell you that with enough handicap , you just have to defend everywhere and you will win as long as you don't lose a group of stones. But then your opponent abuses your commitment to defense by playing a lot of moves that appear to be Sente and your territories shrink while his territories grow. You may not have lost a group, but you still lost the game. This is just one example of how it is easy for a strong player to exploit commitment to one strategy.

Lose by always choosing the same strategy

It shouldn't be surprising that in Go there is no single strategic concept that will guarantee a win. Otherwise the game would be far too simple. Everybody would just use this strategy and be at least equal to an opponent using the same strategy. Games like Tic Tac Toe are like that - simple, but also boring. In a game of at least some complexity, you must assume that multiple strategic concepts exist that compete with each other, so that you can't rely on just one. A relatively simple example is Rock Paper Scissors . If you choose one strategy, e.g. just playing rock all the time, your opponent will be able to find the perfect counter strategy, namely always playing paper. Even if you try to be a little more creative by alternating between rock and paper, your opponent will eventually figure it out and always answer with paper when you choose rock and with scissors when you choose paper. No simple strategy can win in this game in the long run. If you give your opponent enough time to observe, your opponent will find the perfect counter strategy. In fact, the optimal strategy for Rock Paper Scissors is to choose between the three options randomly. In other games, it might not be as obvious, but they might also have their equivalents of 3 strategies A, B, C such that A always defeats B, B always defeats C, C always defeats A. If A, B & C do not always win, they still might win more than 50% of the time, which is enough to gain an advantage. An example would be a game using intransitive dice .

Go has a lot of strategic concepts and even if you know all of them, applying them correctly requires not committing too much to any single one of them. If you commit too much to one direction, it is like always playing rock in Rock Paper Scissors, your opponent just has to play the equivalent of paper. Of course it requires some time. If I play against an opponent for the first time, I might not know his strengths and weaknesses yet, but after a dozen games or so, I will be able to figure that out and correspondingly choose counter strategies. But even in just 1 game, if you repeat the same pattern often enough, you might betray your inflexibility and allow your opponent to start exploiting that before the game is over.

Examples

This all sounds pretty abstract, so I want to give a few examples for losing/keeping options. Wasting Kō threats is a good example of limiting your flexibility. You won't just miss them in the next Kō fight, you also can't use them any more to gain thinking time and you choose one course of action that might remove other ways to play in that area. If you lose too many threats, you might even have to avoid a Kō fight later on and choose a suboptimal line as a consequence. In general, playing Sente moves without good reason, just because you can, usually removes options. For double-digit-kyu players, this often manifests as giving Atari whenever possible. Playing peeps is another common example. A one-point jump is a popular target for peeps, but once the peep has been answered, cutting is not longer an option. A trumpet connection (i.e. two tiger mouths back to back) is often peeped as well, but you usually have to choose one side to peep, and after the peep is answered, the other one is usually no longer Sente. The worst kind of Sente exchange is Aji Keshi. Aji keshi does not have to be a Sente exchange of course, in can be Gote, too. But once again, Aji Keshi removes options.

If you choose to play an invasion , that usually excludes the option of reducing the Moyō instead, and vice versa. Eventually you have to make the choice, but if you choose too early, you may figure out that the other choice would have been better later on. Playing light shapes is an example of flexibility and making your opponent's group heavy is an example of taking away options of your opponent, namely the option of sacrificing parts of the group. In general, the ability and willingness to sacrifice stones is an example of playing flexibly. Humans probably still have to learn a lot in this area, because Katago constantly sacrifices even large groups in almost absurd ways and consequently can devalue the opponent's influence in ways that have historically been difficult for humans. That also explains why strong AIs prefer a 3-3 invasion under a hoshi stone so early - they just are that good at handling the opponent's influence and humans are not. Incidentally, this is also a hallmark of strong Chess AIs. In the match between AlphaZero versus Stockfish , AlphaZero happily sacrificed multiple pawns for long-term strategic gains and won several games by paralyzing Stockfish's position, i.e. taking away Stockfish's options.

Multi-purpose moves, like the famous ear-reddening move , are moves that give you multiple ways of continuing and no way for your opponent to take away all of them at once. Miai moves contain the flexibility in their definition - you must have at least a pair of them and your opponent can never take away both simultaneously. Your group being enclosed is an example of being forced to live locally, whereas before you had the additional option of running away. Probing moves are a perfect example of multi-purpose moves and they also help you figure out which of two options is the right one for you. Answering the probing move on the other hand shows your commitment to one course of action and that is valuable information to your opponent, which is why sometimes strong players don't want to answer probing moves.

In the opening, one big reason why hoshi is popular is that it is symmetric and thus gives you the option to extend to either side (and of course gives the opponent the option to approach from either side) , whereas komoku , takamoku and mokuhazushi already make a commitment towards one direction. More generally, many jōseki choices are hard to understand if you can't see that they leave open as many options as possible. Many non-jōseki sequences are suboptimal simply because they have committed too much towards one choice. Another example from Chess is the idiom that the threat is stronger than the execution. Having the threat means having the option to execute, but not the obligation. As soon as you have chosen to execute the threat, the option to not do so has disappeared.

Losing options costs points

Going back to the abstract view, it should be clear that on average, having more options increases the probability that among those extra options is one that is better than what you already have. Or conversely, losing options increases the probability that your continuation is suboptimal, because the optimal path was among the lost options. That statement is so general that it applies no matter how many options we are talking about. If you have one option, gaining a second one is an improvement, but even if you have 999 options, gaining the 1000th option should be a tiny improvement on average. If you already play at a very high level, gaining even just 1 additional point can be quite impactful and thus losing options is even more problematic for strong players. Of course additional options have a cost: The complexity increases markedly and it can become impossible to keep a complete overview, increasing the chance of mistakes. With limited thinking time, the competing interests of simplicity and having many options have to be kept in balance, but in correspondence games, I would expect the balance to go towards more options and more complexity.

This whole idea became much more apparent to me after watching quite a few games of Katago versus Katago. In these games, both players seem to display an almost ridiculous aversion to commitment, always playing Tenuki and constantly changing their minds about whether to save a group or to sacrifice it. I had already seen something similar in games of strong humans, but Katago carries it to extremes. And given how strong Katago is, this is clearly not just a bad habit, an AI quirk. It must be a necessary component of really strong play. My statement about losing options potentially forcing you to take suboptimal paths probably explains it.

The Go board is big and a Go game is long, meaning even tiny advantages can accumulate over time (e.g. losing just 1 point every 10 moves sums to around 15 points lost during the game, easily one additional handicap stone) and options can have influence over long distances (e.g. ladder breakers and Kō threats) .

Committing to one strategy is a loss of options

Remembering the comparison with Rock Paper Scissors, even a small inclination towards one particular strategy (e.g. a preference for influence) may cost you several points over the course of a game if your opponent is as strong as Katago. And that may mean that a perfect Go player might be forced to choose high-level strategies in the same way a perfect Rock Paper Scissors player chooses strategies: randomly.

As an entirely incidental observation, I want to mention that Buddhism thinks attachment is bad. The conclusion of this article is that attachment to one strategy in Go is bad and both Buddhism and Go seem to be popular in the same geographical area (despite coming from different places) .

I don't really like to do this, but I feel the topic is abstract enough that I have explain a bit more in detail what I mean by "strategy". Strategy in this article means any set of rules or algorithm or oracle or divine inspiration or whatever that allows you to choose one of the legal moves when it's your turn. Always choosing the same move (like rock in Rock Paper Scissors) is a strategy. In Go, starting with the corners, expanding to the sides and finally to the center is also a strategy. Obviously, this strategy isn't complete. But strategies can be combined into more complete strategies. In Go you could have a strategy for Kō fights that will only be used when there is a Kō fight and makes no decision when there is no Kō fight. This can then be combined with a strategy for non-Kō aspects of the game. "Never play aggressively" is a strategy that could be part of a more complex strategy. "Never build a Moyō" and "Always build a Moyō" are strategies. "Avoid empty triangles" is a strategy. And strategies can be combined stochastically, like "with 20% probability choose A, with 80% probability choose B". That means that choosing moves completely randomly is also a strategy. And of course "keep all options open for as long as possible" is also a strategy.