Book Excerpt
The rules of a game say how to make a move but nothing about making a good move. This property is cherished by adherents of the genre and advertised as "a minute to learn, a lifetime to master." Yet whenever it is your turn: how to find that good move?
The most basic option is to calculate: if Black plays here, White can play there, and Black again can do this.... That is what everyone is forced to do when facing a new game without experience to rely on. The process of taking turns in one's mind is slow and exhausting. For a machine, it is called "brute force" and is computationally expensive. Anyhow, it is this activity which to an extent defines games like Chess and Go. David Parlett calls this specific intellectual exercise "forward visualization" and it goes well with spatial recognition skills. Another common descriptor is "look-ahead" and Go players say that they are "reading lines" in a position.
Clearly, a better player can calculate more lines, with more precision and higher speed. But this accounts only for a fraction of the skill! In fact, Chess masters do not calculate extremely many lines, a huge distinction from machine play. Instead, they are much more selective about which lines to look at, and those lines they will compute far deeper than an amateur.
As players of any game know, it is almost always impossible to see all the way to the end, there are too many choices! When we cannot play perfectly, we have to settle for less: intuition, experience, approximations. This is how heuristics come into play, the methods players resort to other than brute-force computations—they are shortcuts in the planning stage.
Board game heuristics are often identified with rules of thumb. While not wrong, I am going to argue that a more specific description is possible and desirable.
The most basic property of heuristics is that they can be applied to positions new to a player. That is the reason for their existence: human players need heuristics to deal with the overly complex game space. In the Chess literature this is called generalization (D'Ereditá & Mario Ferro) and explicitly stated by Frank Lantz et al:
"In a sense the heuristic operates as a kind of compression function on the game tree—a map that reveals structural features in the underlying tree. A blurry, tattered map, but a map nonetheless."
One reason why board games are fun is that we can transcend the initial stage of mere calculation. Taking our use of heuristics into account, we turn again to the question about how to find decent moves. A coarse flowchart for the generic thought process of "making a move" is this:
The most basic option is to calculate: if Black plays here, White can play there, and Black again can do this.... That is what everyone is forced to do when facing a new game without experience to rely on. The process of taking turns in one's mind is slow and exhausting. For a machine, it is called "brute force" and is computationally expensive. Anyhow, it is this activity which to an extent defines games like Chess and Go. David Parlett calls this specific intellectual exercise "forward visualization" and it goes well with spatial recognition skills. Another common descriptor is "look-ahead" and Go players say that they are "reading lines" in a position.
Clearly, a better player can calculate more lines, with more precision and higher speed. But this accounts only for a fraction of the skill! In fact, Chess masters do not calculate extremely many lines, a huge distinction from machine play. Instead, they are much more selective about which lines to look at, and those lines they will compute far deeper than an amateur.
As players of any game know, it is almost always impossible to see all the way to the end, there are too many choices! When we cannot play perfectly, we have to settle for less: intuition, experience, approximations. This is how heuristics come into play, the methods players resort to other than brute-force computations—they are shortcuts in the planning stage.
Board game heuristics are often identified with rules of thumb. While not wrong, I am going to argue that a more specific description is possible and desirable.
The most basic property of heuristics is that they can be applied to positions new to a player. That is the reason for their existence: human players need heuristics to deal with the overly complex game space. In the Chess literature this is called generalization (D'Ereditá & Mario Ferro) and explicitly stated by Frank Lantz et al:
"In a sense the heuristic operates as a kind of compression function on the game tree—a map that reveals structural features in the underlying tree. A blurry, tattered map, but a map nonetheless."
One reason why board games are fun is that we can transcend the initial stage of mere calculation. Taking our use of heuristics into account, we turn again to the question about how to find decent moves. A coarse flowchart for the generic thought process of "making a move" is this:
Why is this crude? Firstly, at any stage, the thought process can revert. For example, calculation might show that a particular subgoal is out of reach in the current position, forcing the player to backtrack all the way to re-evaluating the position. Secondly, subsequent position evaluations will take previous ones into account, even several turns ago. For example, in tactically hot positions, the first two steps are not needed because evaluation and subgoals are preserved from previous turns. Thirdly, the process will iterate and fork, for example, by following various lines or by pursuing several subgoals.
I will sort heuristics into the following four flavours:
Tactical heuristics are about candidate moves as well as calculation. Patterns occur across the whole spectrum, and this is not surprising: when playing board games, shapes are crucial at every stage. I think it makes sense to say that patterns are the building blocks of (higher) heuristics.
In Characteristics of Games, the authors distinguish between "state heuristics" (reading the game state, called evaluations above) and "directional heuristics" (indicating the next action, split up into tactics and strategies above). Again, patterns belong to both kinds.
First examples from Chess
Let us start with one of the most studied games, brimming with established knowledge. Every beginner picks up quickly that the Queen is the strongest force in the entire army, so it ought to be protected. But there are many marvellous winning combinations enabled by Queen sacrifices, like the one shown here.
I will sort heuristics into the following four flavours:
- Evaluations: Assessing a position, locally and globally.
- Strategies: Global methods and formulation of subgoals.
- Tactics: Local methods, generally small scale and short term.
- Patterns: Specially denoted moves or structures on the board.
Tactical heuristics are about candidate moves as well as calculation. Patterns occur across the whole spectrum, and this is not surprising: when playing board games, shapes are crucial at every stage. I think it makes sense to say that patterns are the building blocks of (higher) heuristics.
In Characteristics of Games, the authors distinguish between "state heuristics" (reading the game state, called evaluations above) and "directional heuristics" (indicating the next action, split up into tactics and strategies above). Again, patterns belong to both kinds.
First examples from Chess
Let us start with one of the most studied games, brimming with established knowledge. Every beginner picks up quickly that the Queen is the strongest force in the entire army, so it ought to be protected. But there are many marvellous winning combinations enabled by Queen sacrifices, like the one shown here.
These combinations are exciting and beautiful because they run counter to the basic heuristic. On the other hand, such a move is not surprising to a purely calculating human or device.
Protecting the Queen is part of a larger guideline learned early: point values assigned to pieces, usually Pawn = 1, Knight = 3, Bishop = 3, Rook = 5, Queen = 9. This provides decision guidance when contemplating the trade of a Rook for two Bishops, for example. The scheme is imperfect but easy to use.
Here we get a glimpse at how heuristics work: novices memorize the numbers; soon enough, players have the material values internalized; at some point, they learn about nuances, such as Knights being more valuable than Bishops in closed positions but worse in the late game; finally, as with all heuristics, players become aware when to ignore them—such as sacrificing material for a combination, position, or tempo.
Concerning the heuristic types, piece values belong to strategy and to evaluation. This is because position assessment comprises material comparison at its core, later refined by concepts like piece development. It is strategic because aiming for an endgame with a material advantage can easily translate to a win, so it a reasonable subgoal.
Many patterns are tactical heuristics: the position below shows linked Rooks (generally good) and an edge Knight (bad). Other standard patterns are forks and pins (both good).
Pawns are very important, and their asymmetric behaviour triggers interesting gameplay. Many heuristics relate to Pawn patterns; stable Pawn structures have strategic significance. The position shows chained Pawns (good), a free and advanced Pawn (very good), an isolated Pawn (bad), doubled Pawns (bad). A free Pawn (meaning that its further forward movement is not impeded by opposing Pawns) can form a valuable subgoal, possibly accruing strategic significance—in this game, White sacrificed material to gain a free Pawn and won with it.
Protecting the Queen is part of a larger guideline learned early: point values assigned to pieces, usually Pawn = 1, Knight = 3, Bishop = 3, Rook = 5, Queen = 9. This provides decision guidance when contemplating the trade of a Rook for two Bishops, for example. The scheme is imperfect but easy to use.
Here we get a glimpse at how heuristics work: novices memorize the numbers; soon enough, players have the material values internalized; at some point, they learn about nuances, such as Knights being more valuable than Bishops in closed positions but worse in the late game; finally, as with all heuristics, players become aware when to ignore them—such as sacrificing material for a combination, position, or tempo.
Concerning the heuristic types, piece values belong to strategy and to evaluation. This is because position assessment comprises material comparison at its core, later refined by concepts like piece development. It is strategic because aiming for an endgame with a material advantage can easily translate to a win, so it a reasonable subgoal.
Many patterns are tactical heuristics: the position below shows linked Rooks (generally good) and an edge Knight (bad). Other standard patterns are forks and pins (both good).
Pawns are very important, and their asymmetric behaviour triggers interesting gameplay. Many heuristics relate to Pawn patterns; stable Pawn structures have strategic significance. The position shows chained Pawns (good), a free and advanced Pawn (very good), an isolated Pawn (bad), doubled Pawns (bad). A free Pawn (meaning that its further forward movement is not impeded by opposing Pawns) can form a valuable subgoal, possibly accruing strategic significance—in this game, White sacrificed material to gain a free Pawn and won with it.
What exactly is a game heuristic?
Heuristics occur in all areas of human endeavour and in this generality are completely outside the scope of the article. Let me stress that the following definition is only about heuristics for playing of board games. It is still vague—necessarily so because the concept is very broad—but yet more specific than mere rules of thumb. A working definition is as follows: A heuristic attaches a label to a collection of [sequences of] [partial] positions; heuristics often, but not always, include an assessment (good/bad).
The label is the players' handle for thinking and talking about the heuristic. Specialized jargon is common although different communities may have divergent terminology. It is reasonable to include negative heuristics, in other words, attach names to patterns, tactics and strategies which are generally inferior. Not only does this allow us to avoid them on purpose, it also makes it possible to discuss when such a heuristic does work. In fact, Nick Bentley argues that the availability of heuristics which seem good but are not—i.e., which ought to be superseded by better ones—is a hallmark of a good game; he calls this property "speciousness." I want to underline the importance of having a name by quoting John Fairbairn:
"Difficult concepts or positions are usually difficult for most of us to understand without a handy label. Imagine where we would be without the label 'good shape.' Would we then be aware that such a thing even existed?"
In the simplest case, a heuristic associates a true/false value or a number to each legal full-board position. Is a Symple position already cold? The material values for each side in Chess. The number of clumps in Lines of Action—a bad approach, by the way. Recall that heuristics can go awry; in fact, the way to better heuristics is paved with worse ones.
Many evaluation heuristics are global (about full board positions) and static (a single position, not a sequence of moves). Reversi has a primitive global heuristic assigning values to each square, with corners getting the best ratings. As with piece values in Chess, the numbers are not hard and fast; beginners learn first that corner squares are extremely valuable.
Heuristics occur in all areas of human endeavour and in this generality are completely outside the scope of the article. Let me stress that the following definition is only about heuristics for playing of board games. It is still vague—necessarily so because the concept is very broad—but yet more specific than mere rules of thumb. A working definition is as follows: A heuristic attaches a label to a collection of [sequences of] [partial] positions; heuristics often, but not always, include an assessment (good/bad).
The label is the players' handle for thinking and talking about the heuristic. Specialized jargon is common although different communities may have divergent terminology. It is reasonable to include negative heuristics, in other words, attach names to patterns, tactics and strategies which are generally inferior. Not only does this allow us to avoid them on purpose, it also makes it possible to discuss when such a heuristic does work. In fact, Nick Bentley argues that the availability of heuristics which seem good but are not—i.e., which ought to be superseded by better ones—is a hallmark of a good game; he calls this property "speciousness." I want to underline the importance of having a name by quoting John Fairbairn:
"Difficult concepts or positions are usually difficult for most of us to understand without a handy label. Imagine where we would be without the label 'good shape.' Would we then be aware that such a thing even existed?"
In the simplest case, a heuristic associates a true/false value or a number to each legal full-board position. Is a Symple position already cold? The material values for each side in Chess. The number of clumps in Lines of Action—a bad approach, by the way. Recall that heuristics can go awry; in fact, the way to better heuristics is paved with worse ones.
Many evaluation heuristics are global (about full board positions) and static (a single position, not a sequence of moves). Reversi has a primitive global heuristic assigning values to each square, with corners getting the best ratings. As with piece values in Chess, the numbers are not hard and fast; beginners learn first that corner squares are extremely valuable.
A little more situational: The minimal number of placements needed to finish a Havannah frame. These can only be counted once a ring, fork, or bridge frame has been established, so they are not available for all positions. Of course, before that, players can (and will) count the numbers of moves needed to turn partial frames into proper ones.
Many heuristics are local, in that they refer to a part of the board, sometimes very local; this is what is meant by "partial position" above. Common are patterns, that is piece configurations of importance. Various Chess patterns have been mentioned before. Needless to say, Go—as the primordial placement game—has a myriad of patterns, including the empty triangle (a bad shape), bamboo joints, snapback, and many more. Other examples are the chariot of Murus Gallicus, the bridge of Hex, the open three and four of Connect6. Standard Backgammon heuristics are: pip count (global, evaluation), points (pattern, tactical), prime (pattern, strategic); all of these are static.
More advanced heuristics are dynamic, thus containing (usually varying) follow-up moves. Such heuristics refer to a collection of move sequences. Typical instances are opening formulas, like the Sicilian Defense in Chess or the many joseki of Go. In the tactical realm, many sequences are grouped under a particular label, such as ladders (standing for a particular position whose development is implied).
The distinction whether a heuristic is static or dynamic can be blurry. For example, Hex players are familiar with templates, configurations ensuring a partial connection (often towards an edge). The template itself is static, but the execution against an opponent's intrusion inside it is dynamic. Likewise, while the assessment of the status of a Go group (as alive, dead, depending on ko or tempo) is a snapshot, whereas the process of killing or making life is dynamic.
Proverbs are the other classic method to store knowledge, besides game-specific terminology. The classics have plenty of them. Here are two examples from Go: "There is death in hane" (tactics); "A ponnuki is worth 40 points" (evaluation). Most Chess players will have heard early on that "A Knight on the rim is grim." Japanese Shogi also has lots of interesting proverbs, including "Early escape by the King is worth eight moves." I believe it is a sign of maturity and greatness when players start inventing such phrases for their game.
The life cycle of heuristics
Heuristics are not intrinsic properties of the game as a system of rules. Instead they are developed and preserved by the playing community. It is fun but far outside the scope of this text to speculate whether programs based on neural networks build their own heuristics or do something else. Older game-playing programs implemented human heuristics, especially drawing on expert knowledge in their evaluation functions.
A heuristic is proved wrong when a new, better heuristic establishes its superiority. The classical games have histories long enough for us to be aware of many heuristics that have fallen by the wayside.
A current example from Go is the 3-3 invasion. It has always been clear that a single stone on the star point does not secure the corner territory. By invading, the opponent gains a small corner territory at the expense of thickness. The question becomes at which stage of a game the 3-3 invasion is appropriate. Accepted wisdom, laid out in books, was to delay the 3-3 invasion to the midgame. Generally, it was thought, the outside influence is too valuable early on. Imagine the shock when computer Go started playing the invasion at turn 5—with success! This was an extreme departure from a well-established heuristic. By now, turn 5 corner invasions are common in professional games, too. Following AI lead, the joseki (corner formulas) have been changed so as to reduce the outside influence considerably.
Many heuristics are local, in that they refer to a part of the board, sometimes very local; this is what is meant by "partial position" above. Common are patterns, that is piece configurations of importance. Various Chess patterns have been mentioned before. Needless to say, Go—as the primordial placement game—has a myriad of patterns, including the empty triangle (a bad shape), bamboo joints, snapback, and many more. Other examples are the chariot of Murus Gallicus, the bridge of Hex, the open three and four of Connect6. Standard Backgammon heuristics are: pip count (global, evaluation), points (pattern, tactical), prime (pattern, strategic); all of these are static.
More advanced heuristics are dynamic, thus containing (usually varying) follow-up moves. Such heuristics refer to a collection of move sequences. Typical instances are opening formulas, like the Sicilian Defense in Chess or the many joseki of Go. In the tactical realm, many sequences are grouped under a particular label, such as ladders (standing for a particular position whose development is implied).
The distinction whether a heuristic is static or dynamic can be blurry. For example, Hex players are familiar with templates, configurations ensuring a partial connection (often towards an edge). The template itself is static, but the execution against an opponent's intrusion inside it is dynamic. Likewise, while the assessment of the status of a Go group (as alive, dead, depending on ko or tempo) is a snapshot, whereas the process of killing or making life is dynamic.
Proverbs are the other classic method to store knowledge, besides game-specific terminology. The classics have plenty of them. Here are two examples from Go: "There is death in hane" (tactics); "A ponnuki is worth 40 points" (evaluation). Most Chess players will have heard early on that "A Knight on the rim is grim." Japanese Shogi also has lots of interesting proverbs, including "Early escape by the King is worth eight moves." I believe it is a sign of maturity and greatness when players start inventing such phrases for their game.
The life cycle of heuristics
Heuristics are not intrinsic properties of the game as a system of rules. Instead they are developed and preserved by the playing community. It is fun but far outside the scope of this text to speculate whether programs based on neural networks build their own heuristics or do something else. Older game-playing programs implemented human heuristics, especially drawing on expert knowledge in their evaluation functions.
A heuristic is proved wrong when a new, better heuristic establishes its superiority. The classical games have histories long enough for us to be aware of many heuristics that have fallen by the wayside.
A current example from Go is the 3-3 invasion. It has always been clear that a single stone on the star point does not secure the corner territory. By invading, the opponent gains a small corner territory at the expense of thickness. The question becomes at which stage of a game the 3-3 invasion is appropriate. Accepted wisdom, laid out in books, was to delay the 3-3 invasion to the midgame. Generally, it was thought, the outside influence is too valuable early on. Imagine the shock when computer Go started playing the invasion at turn 5—with success! This was an extreme departure from a well-established heuristic. By now, turn 5 corner invasions are common in professional games, too. Following AI lead, the joseki (corner formulas) have been changed so as to reduce the outside influence considerably.
As with the 3-3 invasion, heuristics usually are not flat out wrong but have to be refined, for example by applying them more conditionally. As knowledge about the game accumulates, it is standard routine to adapt existing heuristics.
This process has similarities with biological evolution: It does happen that a heuristic has to be entirely discarded (for example refuted openings in Chess or Go). It also happens that genuinely new ideas are played successfully, and become the seeds of a novel heuristic. Most often, heuristics mutate. Thus heuristics compete with each other, and a new heuristic may remove or redress or restrict a previously dominating one.
The history of Chess exemplifies this: the Romantic school, stressing offence and combinations and represented by Adolf Anderssen or Paul Morphy, was superseded by the positional and defensive approach of the Modern (or Scientific or Classic) school, started by Wilhelm Steinitz. A later paradigm shift came with the Hypermodern school, which rejected the importance of occupying the centre.
One would hope that in this way all heuristics together converge to perfect play. This is an idle hope! One reason is that heuristics can be ever more fine-tuned: cutting away false positives and spinning off sub-heuristics for special cases. By doing so, the heuristics get better and better, but they also become more and more complicated. Which makes them harder to invent, to apply, to store, and to spread, ultimately defeating the point of heuristics. Thus, the most refined heuristics are relevant only to the most dedicated players.
There may be potential in assessing heuristics through programs playing at superhuman level; an option now in principle available for every game. For a long time, AI play felt alien and far removed from heuristics. Thankfully, the latest, neural network-driven generation of programs has changed that: computer moves look much more natural now and are closer to our heuristics. For Chess, this is nicely explained by the authors in their book Game Changer.
That said, it still takes effort by human players to translate computer moves into heuristics. And even if computer play feels more familiar, there are certainly surprises like the early 3-3 invasions mentioned above.
The statistical point of view of heuristics
When studying a particular abstract board game, there are two different views one can take. On the one hand, the game is a huge combinatorial exercise, and assuming a finite game tree (for example, if repeated positions are impossible or forbidden), each position (i.e., node in the game space) has a win/draw/loss value under perfect play, including the starting position.
On the other hand, all interesting games are too large for the combinatorial approach to work. Therefore players approach a game statistically: being unable to precisely calculate the win/draw/loss state of a given position, we are constrained to estimates. Now, some positions are more likely to yield a win than others. It is possible to quantify this statement, at least in principle: count the number of winning moves and the lengths of the corresponding fastest winning lines. For instance, contrast a position in which five out of twenty moves win and each does so in at most six turns with another, where the player has to make the uniquely correct move for ten turns.
A player aims to make moves gravitating towards clearer "win" values. One job of heuristics is to suggest such moves. In other words, heuristics are designed as attractors in the game space towards regions with many (and clearly) won positions, and as repulsors away from lost positions. In the words of Lantz et al., and emphasizing their use of "statistically":
"Heuristics take advantage of regularities in the game tree to guide the player towards areas of state space where winning paths are statistically more dense."
This approach also makes it clear why negative heuristics (such as bad shapes) are so useful. When calculating lines in a particular position, negative heuristics function as warning signs, telling us to search for good moves somewhere else.
Heuristics for new games?
You can argue that invoking shelves full of Chess and Go literature is pointless: sure, these books contain loads of heuristics for these two games, but how would that help with a new design? That is a valid argument, and it is entirely true that many new games are born naked—without heuristics to rely on.
We can import heuristics for offspring of a familiar design, such as Chess or Draughts variants. There are some meta-heuristics that apply very broadly, for example to placement games with the connection goal. One of the them is the broadside, originating in Hex but applicable to many connection games: it refers to building a structure perpendicular to the intended direction of the connection; contrary to what one might believe at first, broadsides are regularly sources of strategic flexibility.
Another meta-heuristic for games with capturing mechanisms: if in doubt, capture! In fact, this is so deeply rooted in human behaviour that players have to learn to resist it. In Chess, a fair trade in material can still be bad for positional reasons. Worse, a gain in material might be a tainted gift, a sacrifice that triggers a loss. In Go, players have to distinguish between unimportant stones or groups (whose capture is an endgame matter) and crucial stones, usually cuts.
Players feel lost without instincts. This can be seen as a bad thing—on the other hand, a zero baseline makes it easier to develop the very first heuristics! Many players of modern abstract games argue that they prefer to get into new designs rather than working out the well-established knowledge of a classic.
Core heuristics
I believe that many games have a core heuristic, whose understanding enables gameplay on a much deeper level. In other words, while there are manifold finely layered heuristics, one particular heuristic may provide a huge burst in understanding. Roughly speaking, a core heuristic is one that answers a player's query, "What am I doing here?" in the absence of much game-specific knowledge.
The classic example is the concept of life and death in Go. Try to imagine learning the rules but not hearing about alive groups. There is an immeasurable gain of clarity by just being aware of how groups can achieve eternal life. Accordingly, the two-eyes heuristic is taught extremely early. It is, in my experience, part of the immense appeal of Go. I believe that learning about life and death is so satisfactory because it provides moves with a purpose (to live or to kill), accessible to anyone beginning the journey. In their book Deep Learning and the Game of Go, the authors take the reader on a trip from a Python program playing random moves to one playing at dan level. Apart from the rules, the concept of two eyes is the only coded Go information—having that improves performance as much for a machine as it does for a human player.
For a Reversi player, a crucial lesson is the importance of the corners. Just as with the life of Go groups, this is a permanence property, and in fact stability in Reversi is the real core heuristic—corners are one pattern associated with it. That said, Ted Landau's handbook explains how it is often good to give corners away! By the way, in his guide of more than sixty pages, the square values given above are not used—for advanced play, these numbers are too crude.
In Boom & Zoom (AG21), being aware of the timer changes the understanding of the game completely. All kinds of follow-up heuristics ensue naturally, such as the subgoal of creating backwards opposing singletons.
The chain-scoring games Omega and Multiplicity compute the score as the product of all chain sizes of a player. This feels arbitrary and can become numerically overwhelming—the products can have four digits. Knowing one simple arithmetical fact changes everything: among numbers adding up to a fixed sum, the product is maximized when all factors are three, or as close as possible to three. Knowing this fact provides clarity—this core heuristic allows players to bring in connection and separation heuristics from many other games. For most games, it is undecidable whether the rules came first or the basic heuristic. In this particular case, I was able to ask Néstor Romeral Andrés and Christian Freeling, the designers of these two games, who confided to me that they were aware of the rule-of-three. So here, the core heuristic was conscious, not emergent.
I believe that designers and publishers are well advised to mention core heuristics together with the rule sheets, if possible. This makes it much easier for players to get into the game, increasing chances of further plays. Assuming the game is sufficiently deep, this will not detract from the joy of learning.
Verbalize thoughts when playing
If all fails, I find it useful to think aloud when playing. Unless your opponent is game, only do this in isolation, lest you'll attract psychiatric recommendations! Avoid aimless soliloquy; try to say out loud why you prefer a certain move. For unknown games, the reasons will be bizarre, sometimes arbitrary: perhaps a move reminds us of Chess; maybe it makes a nice symmetric pattern. With a little experience, reasons should become more sensible. Once your sentences start, "I play here because...," you may be following a heuristic. As explained before, heuristics may in fact be poor but that is all right: if the result is underwhelming, at least now you know another kind of move to shun.
Conclusion
There is much more to say about heuristics because they are the raw materials for many follow-up concepts. For example, conceptual depth is about the quantity of heuristics, and especially on their interdependency: we think of a game as deep it if has a many levels of heuristics building on each other. Clarity is about the accessibility of heuristics. But most importantly, I find it exciting to invent a heuristic for a new game and try it out in practice. ◾️
This process has similarities with biological evolution: It does happen that a heuristic has to be entirely discarded (for example refuted openings in Chess or Go). It also happens that genuinely new ideas are played successfully, and become the seeds of a novel heuristic. Most often, heuristics mutate. Thus heuristics compete with each other, and a new heuristic may remove or redress or restrict a previously dominating one.
The history of Chess exemplifies this: the Romantic school, stressing offence and combinations and represented by Adolf Anderssen or Paul Morphy, was superseded by the positional and defensive approach of the Modern (or Scientific or Classic) school, started by Wilhelm Steinitz. A later paradigm shift came with the Hypermodern school, which rejected the importance of occupying the centre.
One would hope that in this way all heuristics together converge to perfect play. This is an idle hope! One reason is that heuristics can be ever more fine-tuned: cutting away false positives and spinning off sub-heuristics for special cases. By doing so, the heuristics get better and better, but they also become more and more complicated. Which makes them harder to invent, to apply, to store, and to spread, ultimately defeating the point of heuristics. Thus, the most refined heuristics are relevant only to the most dedicated players.
There may be potential in assessing heuristics through programs playing at superhuman level; an option now in principle available for every game. For a long time, AI play felt alien and far removed from heuristics. Thankfully, the latest, neural network-driven generation of programs has changed that: computer moves look much more natural now and are closer to our heuristics. For Chess, this is nicely explained by the authors in their book Game Changer.
That said, it still takes effort by human players to translate computer moves into heuristics. And even if computer play feels more familiar, there are certainly surprises like the early 3-3 invasions mentioned above.
The statistical point of view of heuristics
When studying a particular abstract board game, there are two different views one can take. On the one hand, the game is a huge combinatorial exercise, and assuming a finite game tree (for example, if repeated positions are impossible or forbidden), each position (i.e., node in the game space) has a win/draw/loss value under perfect play, including the starting position.
On the other hand, all interesting games are too large for the combinatorial approach to work. Therefore players approach a game statistically: being unable to precisely calculate the win/draw/loss state of a given position, we are constrained to estimates. Now, some positions are more likely to yield a win than others. It is possible to quantify this statement, at least in principle: count the number of winning moves and the lengths of the corresponding fastest winning lines. For instance, contrast a position in which five out of twenty moves win and each does so in at most six turns with another, where the player has to make the uniquely correct move for ten turns.
A player aims to make moves gravitating towards clearer "win" values. One job of heuristics is to suggest such moves. In other words, heuristics are designed as attractors in the game space towards regions with many (and clearly) won positions, and as repulsors away from lost positions. In the words of Lantz et al., and emphasizing their use of "statistically":
"Heuristics take advantage of regularities in the game tree to guide the player towards areas of state space where winning paths are statistically more dense."
This approach also makes it clear why negative heuristics (such as bad shapes) are so useful. When calculating lines in a particular position, negative heuristics function as warning signs, telling us to search for good moves somewhere else.
Heuristics for new games?
You can argue that invoking shelves full of Chess and Go literature is pointless: sure, these books contain loads of heuristics for these two games, but how would that help with a new design? That is a valid argument, and it is entirely true that many new games are born naked—without heuristics to rely on.
We can import heuristics for offspring of a familiar design, such as Chess or Draughts variants. There are some meta-heuristics that apply very broadly, for example to placement games with the connection goal. One of the them is the broadside, originating in Hex but applicable to many connection games: it refers to building a structure perpendicular to the intended direction of the connection; contrary to what one might believe at first, broadsides are regularly sources of strategic flexibility.
Another meta-heuristic for games with capturing mechanisms: if in doubt, capture! In fact, this is so deeply rooted in human behaviour that players have to learn to resist it. In Chess, a fair trade in material can still be bad for positional reasons. Worse, a gain in material might be a tainted gift, a sacrifice that triggers a loss. In Go, players have to distinguish between unimportant stones or groups (whose capture is an endgame matter) and crucial stones, usually cuts.
Players feel lost without instincts. This can be seen as a bad thing—on the other hand, a zero baseline makes it easier to develop the very first heuristics! Many players of modern abstract games argue that they prefer to get into new designs rather than working out the well-established knowledge of a classic.
Core heuristics
I believe that many games have a core heuristic, whose understanding enables gameplay on a much deeper level. In other words, while there are manifold finely layered heuristics, one particular heuristic may provide a huge burst in understanding. Roughly speaking, a core heuristic is one that answers a player's query, "What am I doing here?" in the absence of much game-specific knowledge.
The classic example is the concept of life and death in Go. Try to imagine learning the rules but not hearing about alive groups. There is an immeasurable gain of clarity by just being aware of how groups can achieve eternal life. Accordingly, the two-eyes heuristic is taught extremely early. It is, in my experience, part of the immense appeal of Go. I believe that learning about life and death is so satisfactory because it provides moves with a purpose (to live or to kill), accessible to anyone beginning the journey. In their book Deep Learning and the Game of Go, the authors take the reader on a trip from a Python program playing random moves to one playing at dan level. Apart from the rules, the concept of two eyes is the only coded Go information—having that improves performance as much for a machine as it does for a human player.
For a Reversi player, a crucial lesson is the importance of the corners. Just as with the life of Go groups, this is a permanence property, and in fact stability in Reversi is the real core heuristic—corners are one pattern associated with it. That said, Ted Landau's handbook explains how it is often good to give corners away! By the way, in his guide of more than sixty pages, the square values given above are not used—for advanced play, these numbers are too crude.
In Boom & Zoom (AG21), being aware of the timer changes the understanding of the game completely. All kinds of follow-up heuristics ensue naturally, such as the subgoal of creating backwards opposing singletons.
The chain-scoring games Omega and Multiplicity compute the score as the product of all chain sizes of a player. This feels arbitrary and can become numerically overwhelming—the products can have four digits. Knowing one simple arithmetical fact changes everything: among numbers adding up to a fixed sum, the product is maximized when all factors are three, or as close as possible to three. Knowing this fact provides clarity—this core heuristic allows players to bring in connection and separation heuristics from many other games. For most games, it is undecidable whether the rules came first or the basic heuristic. In this particular case, I was able to ask Néstor Romeral Andrés and Christian Freeling, the designers of these two games, who confided to me that they were aware of the rule-of-three. So here, the core heuristic was conscious, not emergent.
I believe that designers and publishers are well advised to mention core heuristics together with the rule sheets, if possible. This makes it much easier for players to get into the game, increasing chances of further plays. Assuming the game is sufficiently deep, this will not detract from the joy of learning.
Verbalize thoughts when playing
If all fails, I find it useful to think aloud when playing. Unless your opponent is game, only do this in isolation, lest you'll attract psychiatric recommendations! Avoid aimless soliloquy; try to say out loud why you prefer a certain move. For unknown games, the reasons will be bizarre, sometimes arbitrary: perhaps a move reminds us of Chess; maybe it makes a nice symmetric pattern. With a little experience, reasons should become more sensible. Once your sentences start, "I play here because...," you may be following a heuristic. As explained before, heuristics may in fact be poor but that is all right: if the result is underwhelming, at least now you know another kind of move to shun.
Conclusion
There is much more to say about heuristics because they are the raw materials for many follow-up concepts. For example, conceptual depth is about the quantity of heuristics, and especially on their interdependency: we think of a game as deep it if has a many levels of heuristics building on each other. Clarity is about the accessibility of heuristics. But most importantly, I find it exciting to invent a heuristic for a new game and try it out in practice. ◾️
Acknowledgements
My thanks to Myron Samsin, Christian Freeling, Carlos Luna Mota for very useful comments!
The header image shows a game of Tak in progress, with the Tak University Edition (2018).
The footer image shows a game of Tzaar (2007) in progress.
My thanks to Myron Samsin, Christian Freeling, Carlos Luna Mota for very useful comments!
- David Parlett (1998). The Oxford History of Board Games. Oxford University Press.
- Giuliano D’Ereditá, Mario Ferro (2015). "Generalization In Chess Thinking," PNA, 9(3), pp. 245-259.
- Frank Lantz, Aaron Isaksen, Alexander Jaffe, Andy Nealen, & Julian Togelius (2017). "Depth in strategic games," Association for the Advancement of Artificial Intelligence, 8.
- George Skaff Elias, Richard Garfield, K. Robert Gutschera (2012). Characteristics of Games. MIT Press.
- John Fairbairn (1992). "Liu Zhongfu's Go Secrets," GO World 67 (Spring issue).
- Nick Bentley. "Redefining the Abstract." www.nickbentley.games/redefining-the-abstract
- Matthew Sadler, Natasha Regan: Game Changer: AlphaZero's Groundbreaking Chess Strategies and the Promise of AI. New in Chess (2019).
- Ted Landau: "Othello: Brief & Basic." 63 pages, www.tedlandau.com/files/Othello-B%26B.pdf
- Max Pumperla, Kevin Ferguson (2019). Deep Learning and the Game of Go. Manning Publications.
The header image shows a game of Tak in progress, with the Tak University Edition (2018).
The footer image shows a game of Tzaar (2007) in progress.