Theory of abstract games
The definition of abstract games is a long debated topic. At Abstrakta 2020 (the Italian symposium for abstract game players) the question of the definition of abstract games came up again. Various speakers talked about it, and there was a presentation by Spartaco Albertarelli on the subject. This question that interests participants in the Abstract Games groups on Facebook, readers of the Fogliaccio degli Astratti, and readers of Abstract Games magazine is the following: What do we have in common in our definitions? What unites us is not interest in the lack of theme. Santorini is themed, but fully falls within our definition. Roulette is not themed, but it is not of interest for abstract gamers. What we have in common is an interest in some kinds of strategy games. Let us try to better define the concept, starting with a mathematical definition.
Definition by Alberto Bertoni (University of Milan)
A combinatorial game is a game that satisfies the following conditions:
Of these points we are only interested in 3 and 4. Point 1 does not interest us because we also deal with games with a number of players other than two. Point 2 is not relevant for us because we are also interested in non-finite games, such as Berlekamp's Entrepreneurial Chess or Fractal Tic-tac-toe.
Definition by Alberto Bertoni (University of Milan)
A combinatorial game is a game that satisfies the following conditions:
- There are two players.
- There is a set (which we will consider finite) of possible positions of the game which we will call states.
- The rules of the game specify, for each state and each player, which possible future states can be reached; a player's move is to choose one of the future legal states. If the rules do not depend on the player, the game is called impartial, otherwise it is called partisan.
- The two players alternate their moves.
- The game ends when there are no more possible moves.
Of these points we are only interested in 3 and 4. Point 1 does not interest us because we also deal with games with a number of players other than two. Point 2 is not relevant for us because we are also interested in non-finite games, such as Berlekamp's Entrepreneurial Chess or Fractal Tic-tac-toe.
A clarification regarding Point 3: in the definition of impartial and partizan, the word "rules" includes the set of possible moves. For example, Nim is impartial, but Chess is partizan. A clarification regarding Point 4: surely there are games in which the alternation of moves is not respected (Arimaa, Progressive Chess, etc.); Point 4 must be interpreted rather in the sense, "The two players never move simultaneously"; in this sense most (but not all) games we deal with have no simultaneous moves. We are not interested in Point 5, as many games end when there are still possible moves: for example, Go with Japanese rules ends when there are still dame (neutral points) that are filled in when the game end; in mancala games the game often ends as soon as a player has more than half of the seeds, even though there are still moves to play. Let us see another mathematical definition.
Definition by Aaron Siegel (University of California, Berkeley)
One of the most recent books on the subject is Combinatorial Game Theory (2013) by Aaron Siegel. His definition is as follows: combinatorial games are two-player games with no hidden information and no elements of randomness. Siegel then analyzes four distinctions:
I would say that we are interested in all these categories, and in addition also the games with a number of players other than two. Both Bertoni and Siegel include Chess and Go in the set of combinatorial games. Combinatorial games in this sense are a subset of the games we deal with. In the following section, we will define pure-strategy games as combinatorial games, and then define abstract games as a superset of the pure-strategy games.
Definition by Aaron Siegel (University of California, Berkeley)
One of the most recent books on the subject is Combinatorial Game Theory (2013) by Aaron Siegel. His definition is as follows: combinatorial games are two-player games with no hidden information and no elements of randomness. Siegel then analyzes four distinctions:
- Impartial or partizan games
- Games with cycles or without cycles
- Finite or transfinite games
- Games with the goal to win or to lose (misère)
I would say that we are interested in all these categories, and in addition also the games with a number of players other than two. Both Bertoni and Siegel include Chess and Go in the set of combinatorial games. Combinatorial games in this sense are a subset of the games we deal with. In the following section, we will define pure-strategy games as combinatorial games, and then define abstract games as a superset of the pure-strategy games.
Our definitions
Here I present definitions summarized from comments that lasted several weeks in the Abstract Games group on Facebook in 2018-2019. I re-analyzed these comments after the Abstrakta symposium, and we adopted the conclusions for the abstract games team tournament, NonSoloNumeri.
Here I present definitions summarized from comments that lasted several weeks in the Abstract Games group on Facebook in 2018-2019. I re-analyzed these comments after the Abstrakta symposium, and we adopted the conclusions for the abstract games team tournament, NonSoloNumeri.
By combinatorial games (or pure-strategy games or pure abstract games) we mean games in which, given a game situation and sufficient calculation or reflection time, a player or a fairly powerful calculator can analyze the tree of possible games (up to a defined depth of analysis and with respect to a given evaluation function) and identify the best move (or the set of best moves with equal merit). These are games devoid of hidden information, elements of randomness, simultaneous moves and possible alliances. Chess and Go are the most famous and obvious examples of this kind of game. This definition aims to be similar to that of Combinatorial Game Theory, above, but it is intended more for players than for mathematicians. Please note that some scholars add a restriction to the definitions seen so far and consider "combinatorial games" only those (like Nim or Domineering) in which who moves last wins (called "normal play") or loses (called "misère play"); according to them, this is only a subset of "pure-strategy games" (or "pure abstract games") like Chess or Go.
By abstract games (or more precisely abstract-strategy games), we mean a superset of the pure-strategy games, which also includes games that are very close to pure-strategy games, although not exactly within the definition. The abstract games include Backgammon (which has chance), Chinese Checkers with more than 2 players (which has possible alliances), 55stones (which has simultaneous moves), or Stratego (which has hidden information). We now present some clarifications.
A. Number of players
A1. Games for more than two players.
Some abstract games are for more than two players. For example, Chinese Checkers can be played with three, four, five or six players, all against all. Such games are not in the mathematical interest as, even without considering possible alliances, the "kingmaker effect" can occur; this is the effect whereby a player who can no longer win must choose between move A and move B (that are equivalent for him/her); with A one player wins, but with B another player wins. Game analysis does not make sense after this point. Only in two-player games is it possible to really ascertain the best player. For this reason, the mathematical definitions of combinatorial games include that the game must be for two players. But in our definition of abstract game, it would make no sense to exclude games for more than two players.
By abstract games (or more precisely abstract-strategy games), we mean a superset of the pure-strategy games, which also includes games that are very close to pure-strategy games, although not exactly within the definition. The abstract games include Backgammon (which has chance), Chinese Checkers with more than 2 players (which has possible alliances), 55stones (which has simultaneous moves), or Stratego (which has hidden information). We now present some clarifications.
A. Number of players
A1. Games for more than two players.
Some abstract games are for more than two players. For example, Chinese Checkers can be played with three, four, five or six players, all against all. Such games are not in the mathematical interest as, even without considering possible alliances, the "kingmaker effect" can occur; this is the effect whereby a player who can no longer win must choose between move A and move B (that are equivalent for him/her); with A one player wins, but with B another player wins. Game analysis does not make sense after this point. Only in two-player games is it possible to really ascertain the best player. For this reason, the mathematical definitions of combinatorial games include that the game must be for two players. But in our definition of abstract game, it would make no sense to exclude games for more than two players.
A2. Team games
Furthermore, games for more than two players can also be for teams of two or more players. Examples include Bughouse Chess, Chinese Checkers in teams, Rengo, or any other game played with N teams, in which the players of the same team alternate moves, in general without the possibility of communicating, or without agreeing on moves even if they can communicate. Such games can be analyzed in a similar way to N-player games, especially if the players of a team can communicate in secret. Team games also fully belong to our area of interest, and therefore should be included in our definition of abstract game.
A3. Games for less than two players.
Concerning games for one player (such as Solitaire), I would say that they also fall within our field of interest, although to a lesser extent, and therefore it would not make sense to exclude them from our definition. If we want to add an extreme example, we could also include games for zero players [6], such as Conway's Game of Life.
B. Finiteness
Some games are finite, because they have a finite number of possible game states and a finite number of possible games that can be played. Others are non-finite: with infinite moves, with cycles, discrete transfinite or continuous transfinite. These are all generally considered abstract games.
B1. Infinite games
Bao is an example of a game that can lead to an infinite number of moves [13]. In other words, there are infinite sowings not foreseen by the traditional rules. These infinite sowings are now considered illegal moves, so if a player finds himself in an infinite move, that player loses the game.
B2. Games with cycles
Games with cycles are games with a finite number of states, but an infinite tree of possible matches. In this case, there may be cyclical repetitions not well managed by the rules. The triple ko in Go is an example, but also Awele has recently discovered doubtful cases [7].
Furthermore, games for more than two players can also be for teams of two or more players. Examples include Bughouse Chess, Chinese Checkers in teams, Rengo, or any other game played with N teams, in which the players of the same team alternate moves, in general without the possibility of communicating, or without agreeing on moves even if they can communicate. Such games can be analyzed in a similar way to N-player games, especially if the players of a team can communicate in secret. Team games also fully belong to our area of interest, and therefore should be included in our definition of abstract game.
A3. Games for less than two players.
Concerning games for one player (such as Solitaire), I would say that they also fall within our field of interest, although to a lesser extent, and therefore it would not make sense to exclude them from our definition. If we want to add an extreme example, we could also include games for zero players [6], such as Conway's Game of Life.
B. Finiteness
Some games are finite, because they have a finite number of possible game states and a finite number of possible games that can be played. Others are non-finite: with infinite moves, with cycles, discrete transfinite or continuous transfinite. These are all generally considered abstract games.
B1. Infinite games
Bao is an example of a game that can lead to an infinite number of moves [13]. In other words, there are infinite sowings not foreseen by the traditional rules. These infinite sowings are now considered illegal moves, so if a player finds himself in an infinite move, that player loses the game.
B2. Games with cycles
Games with cycles are games with a finite number of states, but an infinite tree of possible matches. In this case, there may be cyclical repetitions not well managed by the rules. The triple ko in Go is an example, but also Awele has recently discovered doubtful cases [7].
B3. Discrete transfinite games
Discrete transfinite games are games with an infinite number of countable states. We have examples such as Berlekamp's Entrepreneurial Chess or Fractal Tic-tac-toe, the latter purposely constructed as transfinite, to represent the recursion of fractals. [17]
B4. Continuous transfinite games
Other games are continuous transfinite and have an uncountable infinite number of states. For example, in Tamsk the pieces are hourglasses and when the time of an hourglass runs out, that piece can no longer move. So even in the absence of moves, the set of possible moves changes over time! It is a game that involves not only discrete-time (for the succession of moves) but also continuous-time, and to describe a game state one would have to indicate the remaining time of each hourglass. In classical physics, time is considered continuous, so the number of states is infinite and uncountable (unless we consider quantized time). Similarly, continuous Go, like many wargames, is played on a space without lines, and, since space is considered continuous in classical physics, the number of states is infinite and uncountable (unless we consider quantized space).
Discrete transfinite games are games with an infinite number of countable states. We have examples such as Berlekamp's Entrepreneurial Chess or Fractal Tic-tac-toe, the latter purposely constructed as transfinite, to represent the recursion of fractals. [17]
B4. Continuous transfinite games
Other games are continuous transfinite and have an uncountable infinite number of states. For example, in Tamsk the pieces are hourglasses and when the time of an hourglass runs out, that piece can no longer move. So even in the absence of moves, the set of possible moves changes over time! It is a game that involves not only discrete-time (for the succession of moves) but also continuous-time, and to describe a game state one would have to indicate the remaining time of each hourglass. In classical physics, time is considered continuous, so the number of states is infinite and uncountable (unless we consider quantized time). Similarly, continuous Go, like many wargames, is played on a space without lines, and, since space is considered continuous in classical physics, the number of states is infinite and uncountable (unless we consider quantized space).
In finite games without cycles (called short games) it is possible to analyze the whole tree and find perfect play. Also in finite games with cycles you can find perfect play, even if it may end in a cycle. In discrete transfinite games one can only establish a number N of levels of analysis and obtain evaluations on possible moves, but these evaluations can be improved by increasing N and there may not be a definitive analysis. In continuous transfinite games, the number of states is already infinite at the first level of analysis.
C. Perfect and complete information
In the literature there is no consensus on these definitions, indeed there are often uses that are not entirely congruent. I will limit myself to a short overview, with respect to the complexity of the theme.
C1. Perfect information games
In general, by perfect information we mean that each player, when they have to make their move, knows perfectly the situation of the game. Therefore, games with hidden elements, like many card games and games with simultaneous moves, are games with imperfect information.
C2. Complete information games
On the other hand, complete information means knowledge of the goals of the players. But be careful, not in the sense of the objectives of Risk, for example. In Risk everyone knows that there will be only one winner, and in general the aim of each player is to become the winner, while arriving second or third does not count. On the other hand, in games with incomplete information, a player may want to get the second position, or another position. For example, in a prize game with various competitors, where the first prize is an evening with a famous person and the second prize is a book, a player might want to be second, because the player either does not know the famous person or dislikes them. If this preference is not known to other players, then there is incomplete information.
To give an example among the abstracts, let us consider Chinese Checkers with more than two players. For some X-type players it may be important to just finish first, and they make no distinction between finishing second or last. Other Y-type players, on the other hand, may prefer to consolidate a second place position rather than try all-out to finish first, with the risk of losing the second position and finishing third or fourth. Suppose the current player, G1, is analyzing the possible next moves of his opponents to decide what move to make, and may come to a point where they do not know whether player G2 is X-type, Y-type, or something else. In that case, his analysis will be incomplete, and therefore the game is said to be with incomplete information.
If, on the other hand, such situations never occur, and all the players know the "utility functions" of each opponent (i.e., the criteria according to which the opponent chooses moves), then the game is said to be with complete information.
C3. Distinction between imperfect information and incomplete information games
In summary, there is imperfect information when there are hidden elements due to the game mechanisms, for example, with hidden cards or simultaneous moves, but not with chance only. There is incomplete information when there are hidden elements due to player preferences. Furthermore, under all definitions of perfect and complete information, all pure-strategy games have perfect and complete information, as are abstract non-pure-strategy games with chance, like Backgammon. Chance alone is not considered a hidden element, since the best move can still be found from a probabilistic point of view.
C. Perfect and complete information
In the literature there is no consensus on these definitions, indeed there are often uses that are not entirely congruent. I will limit myself to a short overview, with respect to the complexity of the theme.
C1. Perfect information games
In general, by perfect information we mean that each player, when they have to make their move, knows perfectly the situation of the game. Therefore, games with hidden elements, like many card games and games with simultaneous moves, are games with imperfect information.
C2. Complete information games
On the other hand, complete information means knowledge of the goals of the players. But be careful, not in the sense of the objectives of Risk, for example. In Risk everyone knows that there will be only one winner, and in general the aim of each player is to become the winner, while arriving second or third does not count. On the other hand, in games with incomplete information, a player may want to get the second position, or another position. For example, in a prize game with various competitors, where the first prize is an evening with a famous person and the second prize is a book, a player might want to be second, because the player either does not know the famous person or dislikes them. If this preference is not known to other players, then there is incomplete information.
To give an example among the abstracts, let us consider Chinese Checkers with more than two players. For some X-type players it may be important to just finish first, and they make no distinction between finishing second or last. Other Y-type players, on the other hand, may prefer to consolidate a second place position rather than try all-out to finish first, with the risk of losing the second position and finishing third or fourth. Suppose the current player, G1, is analyzing the possible next moves of his opponents to decide what move to make, and may come to a point where they do not know whether player G2 is X-type, Y-type, or something else. In that case, his analysis will be incomplete, and therefore the game is said to be with incomplete information.
If, on the other hand, such situations never occur, and all the players know the "utility functions" of each opponent (i.e., the criteria according to which the opponent chooses moves), then the game is said to be with complete information.
C3. Distinction between imperfect information and incomplete information games
In summary, there is imperfect information when there are hidden elements due to the game mechanisms, for example, with hidden cards or simultaneous moves, but not with chance only. There is incomplete information when there are hidden elements due to player preferences. Furthermore, under all definitions of perfect and complete information, all pure-strategy games have perfect and complete information, as are abstract non-pure-strategy games with chance, like Backgammon. Chance alone is not considered a hidden element, since the best move can still be found from a probabilistic point of view.
On the other hand, in card games, chance is often combined with information known to a single player, who can use it to make his choices. For example, a player's hand of cards is both random and unknown to the opponent, which completely changes the probabilistic analysis concerning the best move, as bluffing is possible in these games.
D. Cooperative games
Some rare abstract games are cooperative, for example Maze (AG22). Maze is a game without chance, even if the initial setup is decided randomly. Being a member of the jury of COGITA (Concorso di Giochi Inediti da Tavolo Astratti, meaning Contest for Unpublished Abstract Board Games), I proposed the theme "cooperative abstract" for the 2018 competition, and we received a dozen prototypes, some of them very interesting. In these games the players (usually two) play against the game itself. Cooperative abstracts can be likened to solitaire games, in which players are generally not permitted to communicate. The interest is precisely to see if the players can defeat the game by understanding the intentions of the other player only through the analysis of the moves. If the game is difficult enough, communicating can also be interesting. Some cooperative abstract games, like Maze, are pure-strategy games.
D. Cooperative games
Some rare abstract games are cooperative, for example Maze (AG22). Maze is a game without chance, even if the initial setup is decided randomly. Being a member of the jury of COGITA (Concorso di Giochi Inediti da Tavolo Astratti, meaning Contest for Unpublished Abstract Board Games), I proposed the theme "cooperative abstract" for the 2018 competition, and we received a dozen prototypes, some of them very interesting. In these games the players (usually two) play against the game itself. Cooperative abstracts can be likened to solitaire games, in which players are generally not permitted to communicate. The interest is precisely to see if the players can defeat the game by understanding the intentions of the other player only through the analysis of the moves. If the game is difficult enough, communicating can also be interesting. Some cooperative abstract games, like Maze, are pure-strategy games.
E. "Degree of strategy" on a scale
Above, we defined "pure-strategy game" to correspond to the definition of combinatorial game. The abstract games, as we indicated, are a superset of the pure-strategy games. The degree of abstractness, therefore, corresponds to the degree of strategy, where "strategy" is used in this special way, rather than in the usual distinction of strategy versus tactics.
We may consider that the definition of strategy games, unlike that of combinatorial games, must be understood not in a dichotomous but in a continuous way. Therefore, the adjectives "abstract" and "strategic" may be understood as gradable (continuously variable), i.e., they can have degrees of intensity and comparisons (such as "beautiful," which can become "very beautiful," "more beautiful," etc.), differently from non-gradable adjectives (such as "Spanish," "postal," "triangular," etc.). Under this interpretation, games may be compared on their degree of strategy.
Various categories can be chosen, then coefficients of importance can be assigned to each category, in order to make a weighted average of these values and in this way the degree of strategy/abstractness of the game could be defined. Here a proposal of categories is presented.
E1. Chance
How much do random elements (such as dice) affect the game: in Chess there is no chance and in Backgammon there is some chance, while Roulette is 100% chance.
E2. Perfect and complete information
How much does hidden information affect the game: in Chess there is none, in Stratego it is important, and in Gobblet it depends on memory. Please note that there are also card games with perfect and complete information, such as the Russian game свои козыри (Svoi Kozyri = one's trumps), and some games invented by David Parlett.
Above, we defined "pure-strategy game" to correspond to the definition of combinatorial game. The abstract games, as we indicated, are a superset of the pure-strategy games. The degree of abstractness, therefore, corresponds to the degree of strategy, where "strategy" is used in this special way, rather than in the usual distinction of strategy versus tactics.
We may consider that the definition of strategy games, unlike that of combinatorial games, must be understood not in a dichotomous but in a continuous way. Therefore, the adjectives "abstract" and "strategic" may be understood as gradable (continuously variable), i.e., they can have degrees of intensity and comparisons (such as "beautiful," which can become "very beautiful," "more beautiful," etc.), differently from non-gradable adjectives (such as "Spanish," "postal," "triangular," etc.). Under this interpretation, games may be compared on their degree of strategy.
Various categories can be chosen, then coefficients of importance can be assigned to each category, in order to make a weighted average of these values and in this way the degree of strategy/abstractness of the game could be defined. Here a proposal of categories is presented.
E1. Chance
How much do random elements (such as dice) affect the game: in Chess there is no chance and in Backgammon there is some chance, while Roulette is 100% chance.
E2. Perfect and complete information
How much does hidden information affect the game: in Chess there is none, in Stratego it is important, and in Gobblet it depends on memory. Please note that there are also card games with perfect and complete information, such as the Russian game свои козыри (Svoi Kozyri = one's trumps), and some games invented by David Parlett.
E3. Simultaneity of the moves
In Chess simultaneous moves are absent, but in Morra, Diplomacy, or Assembly Line simultaneous moves are essentially part of the game. There are also variants of abstract games with simultaneous moves (such as Simultaneous Chess, Simultaneous Connect 4, the modern mancala 55Stones). Traditional mancala with simultaneous moves are Agsinnoninka, Sungka, and Baré— in the last two games, only the first move is simultaneous. Some of these games, even if strictly speaking they do not respect the definition of pure-strategy games, have a very high degree of strategy.
In Chess simultaneous moves are absent, but in Morra, Diplomacy, or Assembly Line simultaneous moves are essentially part of the game. There are also variants of abstract games with simultaneous moves (such as Simultaneous Chess, Simultaneous Connect 4, the modern mancala 55Stones). Traditional mancala with simultaneous moves are Agsinnoninka, Sungka, and Baré— in the last two games, only the first move is simultaneous. Some of these games, even if strictly speaking they do not respect the definition of pure-strategy games, have a very high degree of strategy.
E4. Two-players
In two-player competitive games there are no alliances, in Chinese checkers with more than two players there may be, in Diplomacy they are essential. Moreover, with more than two players there is always the risk of the Kingmaker effect, previously discussed.
E5. Human calculability
In mancala games with multiple sowing, such as Bao, it is more difficult to calculate a large number of moves in advance, while in Go it is easier. One could invent a game so complex that the current move is humanly incalculable, and this would make them very uninteresting, because the analysis that can be carried out is negligible compared to the complexity of the game. The game Went goes in this direction, with the aim of testing human-machine cooperation.
One might consider also other categories, such as cooperativity and finiteness. Cooperative games are a bit less "strategic" than competitive games, because victory depends also on the affinity with the other player: a pair of players with great affinity and similar average strength might play much better than a pair of very strong players with no affinity, that might follow different ideas.
Finiteness and more generally depth of the tree of moves also have an influence, because if the tree is too short (as in Tic-tac-toe) the degree of strategy is very low, and if the tree is infinite (as in transfinite games) the degree of strategy is more difficult to define.
But I will not consider these categories here because they have a significant influence only on rare games. Proposals for other meaningful categories are welcome. Please contact me (e.g. on Facebook, in the group Abstract Nation) and we can discuss them.
Here below I present a Venn diagram of games, conceived and created by Maurizio De Leo according to four categories: two-player, sequential, no hidden information, and deterministic. The categories are here considered dichotomously and not continuously. In each subset the games in the top line have no theme, and those in the bottom line have a theme. Question marks indicate that we could not find any examples. Games of perfect information are in the intersection of sequential and no hidden information. The central subset (with Go, Hex, and Hive) represents the pure-strategy games and they are all considered abstract games. Let us now analyze other subsets.
The number of players is probably the least important category in considering whether a game is abstract: generally people do not stop to consider whether a game is still "abstract" when adding players—for example, Chinese Checkers for more than two players is generally still considered an abstract game. Among the other three categories, when at least two of them are missing, the games of that subset are generally considered not abstract: for example, the Claustrophobia (discrete-space wargame), Marrakesh (1978) and GOPS all miss two of the three categories. When only one of the three categories is missing, some games of the subset are usually considered abstract and interesting for abstract-game players—for example, Backgammon (but not Yahtzee), Stratego, Chess with simultaneous moves (but not Rock Paper Scissors). We might call them "abstractish."
In two-player competitive games there are no alliances, in Chinese checkers with more than two players there may be, in Diplomacy they are essential. Moreover, with more than two players there is always the risk of the Kingmaker effect, previously discussed.
E5. Human calculability
In mancala games with multiple sowing, such as Bao, it is more difficult to calculate a large number of moves in advance, while in Go it is easier. One could invent a game so complex that the current move is humanly incalculable, and this would make them very uninteresting, because the analysis that can be carried out is negligible compared to the complexity of the game. The game Went goes in this direction, with the aim of testing human-machine cooperation.
One might consider also other categories, such as cooperativity and finiteness. Cooperative games are a bit less "strategic" than competitive games, because victory depends also on the affinity with the other player: a pair of players with great affinity and similar average strength might play much better than a pair of very strong players with no affinity, that might follow different ideas.
Finiteness and more generally depth of the tree of moves also have an influence, because if the tree is too short (as in Tic-tac-toe) the degree of strategy is very low, and if the tree is infinite (as in transfinite games) the degree of strategy is more difficult to define.
But I will not consider these categories here because they have a significant influence only on rare games. Proposals for other meaningful categories are welcome. Please contact me (e.g. on Facebook, in the group Abstract Nation) and we can discuss them.
Here below I present a Venn diagram of games, conceived and created by Maurizio De Leo according to four categories: two-player, sequential, no hidden information, and deterministic. The categories are here considered dichotomously and not continuously. In each subset the games in the top line have no theme, and those in the bottom line have a theme. Question marks indicate that we could not find any examples. Games of perfect information are in the intersection of sequential and no hidden information. The central subset (with Go, Hex, and Hive) represents the pure-strategy games and they are all considered abstract games. Let us now analyze other subsets.
The number of players is probably the least important category in considering whether a game is abstract: generally people do not stop to consider whether a game is still "abstract" when adding players—for example, Chinese Checkers for more than two players is generally still considered an abstract game. Among the other three categories, when at least two of them are missing, the games of that subset are generally considered not abstract: for example, the Claustrophobia (discrete-space wargame), Marrakesh (1978) and GOPS all miss two of the three categories. When only one of the three categories is missing, some games of the subset are usually considered abstract and interesting for abstract-game players—for example, Backgammon (but not Yahtzee), Stratego, Chess with simultaneous moves (but not Rock Paper Scissors). We might call them "abstractish."
In the table below I present an example of estimation of the degree of strategy, or the degree of abstractness, of some games, in other words, how close the game is to pure-strategy, also taking into account human calculability. The categories are here considered to be continuous and not discrete. The pure-strategy games (in dark yellow) are those that have the maximum score in all categories, neglecting human calculability. The games usually considered abstract are all the more strategic ones up to Stratego, minus Svoi Kozyri, plus Went. Since it is difficult to find objectivity in this area, this table is just meant to be food for thought.
F. Linguistic note
Little or no theme is a frequent feature of abstract strategy games, although it is an irrelevant feature for their definition. As if to say, being a native Italian speaker is a frequent feature of Italians, although it is irrelevant in the administrative definition of Italian citizenship—there are Italians who are not native speakers of Italian, and there are non- Italians who are. So the word "abstract" is not necessarily the most suitable adjective for the strategy games described here, but recalls mathematical abstraction, recalls one of their frequent features, the absence of theme, and above all is in fact the most used term in this area. As often happens in languages, a word takes on a different meaning from its original, and there is nothing strange about this. Chinese Checkers is not Chinese and is not a kind of Checkers, yet it is called that.
Conclusion
In conclusion, this is intended to be a descriptive analysis, not a prescriptive one. I also agree that it would be better to have two different words for two different things, I'm an Esperantist! In Esperanto, for example, "geometric point" is called punkto, "point in games" is called poento and "stitching point" is called punto. In my lecture "Relations between languages and mathematics" I talk about these things, among other concepts. It would be better to have a word for "game without theme", and another word for "pure-strategy or almost-pure-strategy game"; unfortunately, the word "abstract," originally referred to the first meaning, is currently the most used term also for the second meaning. The language requires an adjective that is a single word, both for ease of use and for being able to create other words: "abstractist," "abstractism," and so on. So, if we want to coin a new term for the second meaning, Andrea Angiolino proposed "abstrategic." But in any case, it would be difficult to get such a term into use. What do you propose? In the meantime, let us be satisfied with defining the terms in use and using them consistently. ◾️
Acknowledgements
The author of this article, Cesco Reale is a collaborator in the in the Italian Festival of Mathematical Games, Swiss Museum of Games, and the Abstrakta Project).
The author thanks Maurizio De Leo, Riccardo Moschetti, Maurizio Parton, Jorge Nuno Silva, Cameron Browne, and Francesco Salerno for the valuable suggestions.
References
Little or no theme is a frequent feature of abstract strategy games, although it is an irrelevant feature for their definition. As if to say, being a native Italian speaker is a frequent feature of Italians, although it is irrelevant in the administrative definition of Italian citizenship—there are Italians who are not native speakers of Italian, and there are non- Italians who are. So the word "abstract" is not necessarily the most suitable adjective for the strategy games described here, but recalls mathematical abstraction, recalls one of their frequent features, the absence of theme, and above all is in fact the most used term in this area. As often happens in languages, a word takes on a different meaning from its original, and there is nothing strange about this. Chinese Checkers is not Chinese and is not a kind of Checkers, yet it is called that.
Conclusion
In conclusion, this is intended to be a descriptive analysis, not a prescriptive one. I also agree that it would be better to have two different words for two different things, I'm an Esperantist! In Esperanto, for example, "geometric point" is called punkto, "point in games" is called poento and "stitching point" is called punto. In my lecture "Relations between languages and mathematics" I talk about these things, among other concepts. It would be better to have a word for "game without theme", and another word for "pure-strategy or almost-pure-strategy game"; unfortunately, the word "abstract," originally referred to the first meaning, is currently the most used term also for the second meaning. The language requires an adjective that is a single word, both for ease of use and for being able to create other words: "abstractist," "abstractism," and so on. So, if we want to coin a new term for the second meaning, Andrea Angiolino proposed "abstrategic." But in any case, it would be difficult to get such a term into use. What do you propose? In the meantime, let us be satisfied with defining the terms in use and using them consistently. ◾️
Acknowledgements
The author of this article, Cesco Reale is a collaborator in the in the Italian Festival of Mathematical Games, Swiss Museum of Games, and the Abstrakta Project).
The author thanks Maurizio De Leo, Riccardo Moschetti, Maurizio Parton, Jorge Nuno Silva, Cameron Browne, and Francesco Salerno for the valuable suggestions.
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