Abstract strategy games admit a number of definitions which distinguish these from

strategy game A strategy game or strategic game is a game (e.g. a board game) in which the players' uncoerced, and often autonomous, decision-making skills have a high significance in determining the outcome. Almost all strategy games require internal decisio ...

s in general, mostly involving no or minimal narrative theme, outcomes determined only by player choice (with no

randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...

), and

perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...

. For example, Go is a pure abstract strategy game since it fulfills all three criteria;

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...

and related games are nearly so but feature a recognizable theme of ancient warfare; and

Stratego ''Stratego'' ( ) is a strategy board game for two players on a board of 10×10 squares. Each player controls 40 pieces representing individual officer and soldier ranks in an army. The pieces have Napoleonic insignia. The objective of the game ...

is borderline since it is deterministic, loosely based on 19th-century

Napoleonic warfare The Napoleonic Wars (1803–1815) were a series of major global conflicts pitting the French Empire and its allies, led by Napoleon I, against a fluctuating array of European states formed into various coalitions. It produced a period of Fren ...

, and features concealed information.

Definition

Combinatorial games have no randomizers such as dice, no simultaneous movement, nor hidden information. Some games that do have these elements are sometimes classified as abstract strategy games. (Games such as '' Continuo'', Octiles, '' Can't Stop'', and

Sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

, could be considered abstract strategy games, despite having a luck or bluffing element.) A smaller category of abstract strategy games manages to incorporate hidden information without using any random elements; the best known example is ''

''. Traditional abstract strategy games are often treated as a separate game category, hence the term 'abstract games' is often used for competitions that exclude them and can be thought of as referring to ''modern'' abstract strategy games. Two examples are the IAGO World Tour (2007–2010) and the Abstract Games World Championship held annually since 2008 as part of the

Mind Sports Olympiad The Mind Sports Olympiad (MSO) is an annual international multi-disciplined competition and festival for games of mental skill and mind sports. The inaugural event was held in 1997 in London with £100,000 prize fund''Mind Sports Olympiad Supp ...

. Some abstract strategy games have multiple starting positions of which it is required that one be randomly determined. For a game to be one of skill, a starting position needs to be chosen by impartial means. Some games, such as ''

Arimaa Arimaa () is a two-player strategy board game that was designed to be playable with a standard chess set and difficult for computers while still being easy to learn and fun to play for humans. It was invented in 2003 by Omar Syed, an Indian-Am ...

'' and ''

DVONN ''DVONN'' is a two-player strategy board game in which the objective is to accumulate pieces in stacks. It was released in 2001 by Kris Burm as the fourth game of the ''GIPF'' Project. ''DVONN'' won the 2002 International Gamers Award and the '' ...

'', have the players build the starting position in a separate initial phase which itself conforms strictly to combinatorial game principles. Most players, however, would consider that although one is then starting each game from a different position, the game itself contains no luck element. Indeed,

Bobby Fischer Robert James Fischer (March 9, 1943January 17, 2008) was an American chess grandmaster and the eleventh World Chess Champion. A chess prodigy, he won his first of a record eight US Championships at the age of 14. In 1964, he won with an 11� ...

promoted randomization of the starting position in chess in order to ''increase'' player dependence on thinking at the board. As J. Mark Thompson wrote in his article "Defining the Abstract", play is sometimes said to resemble a series of

puzzle A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together (Disentanglement puzzle, or take them apart) in a logical way, in order to arrive at th ...

s the players pose to each other:

There is an intimate relationship between such games and puzzles: every board position presents the player with the puzzle, What is the best move?, which in theory could be solved by logic alone. A good abstract game can therefore be thought of as a "family" of potentially interesting logic puzzles, and the play consists of each player posing such a puzzle to the other. Good players are the ones who find the most difficult puzzles to present to their opponents.

Many abstract strategy games also happen to be "

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

"; i.e., there is no hidden information, no non-deterministic elements (such as shuffled cards or dice rolls), no simultaneous or hidden movement or setup, and (usually) two players or teams take a finite number of alternating . Many games which are abstract in nature historically might have developed from thematic games, such as representation of military tactics. In turn, it is common to see thematic version of such games; for example, chess is considered an abstract game, but many thematic versions, such as

Star Wars ''Star Wars'' is an American epic film, epic space opera multimedia franchise created by George Lucas, which began with the Star Wars (film), eponymous 1977 film and quickly became a worldwide popular culture, pop-culture Cultural impact of S ...

-themed chess, exist.

History

Akhilleus Aias Staatliche Antikensammlungen 1417

Mancala The mancala games are a family of two-player turn-based strategy board games played with small stones, beans, or seeds and rows of holes or pits in the earth, a board or other playing surface. The objective is usually to capture all or some ...

is among the oldest known games to still be widely played today.

Chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...

is believed to have originated in northwest

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

, in the

Gupta Empire The Gupta Empire was an ancient Indian empire which existed from the early 4th century CE to late 6th century CE. At its zenith, from approximately 319 to 467 CE, it covered much of the Indian subcontinent. This period is considered as the Gol ...

( 280–550),Murray (1913) where its early form in the 6th century was known as '' chaturaṅga'' (), literally ''four divisions''

f the military F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''. Hist ...

nbsp;–

infantry Infantry is a military specialization which engages in ground combat on foot. Infantry generally consists of light infantry, mountain infantry, motorized infantry & mechanized infantry, airborne infantry, air assault infantry, and marine i ...

cavalry Historically, cavalry (from the French word ''cavalerie'', itself derived from "cheval" meaning "horse") are soldiers or warriors who fight mounted on horseback. Cavalry were the most mobile of the combat arms, operating as light cavalry ...

elephants Elephants are the largest existing land animals. Three living species are currently recognised: the African bush elephant, the African forest elephant, and the Asian elephant. They are the only surviving members of the family Elephantidae and ...

, and

chariot A chariot is a type of cart driven by a charioteer, usually using horses to provide rapid motive power. The oldest known chariots have been found in burials of the Sintashta culture in modern-day Chelyabinsk Oblast, Russia, dated to c. 2000&nbs ...

ry, represented by the pieces that would evolve into the modern pawn, knight, bishop, and rook, respectively. ''Chaturanga'' was played on an 8×8 uncheckered board, called ''

ashtāpada Ashtāpada ( sa, अष्टापद) or Ashtapadi is an Indian board game which predates chess and was mentioned on the list of games that Gautama Buddha would not play. Chaturanga, which could be played on the same , appeared sometime around ...

''.

Shogi , also known as Japanese chess, is a strategy board game for two players. It is one of the most popular board games in Japan and is in the same family of games as Western chess, ''chaturanga, Xiangqi'', Indian chess, and '' janggi''. ''Shōgi'' ...

was the earliest chess variant to allow captured pieces to be returned to the board by the capturing player. This drop rule is speculated to have been invented in the 15th century and possibly connected to the practice of 15th century

mercenaries A mercenary, sometimes also known as a soldier of fortune or hired gun, is a private individual, particularly a soldier, that joins a military conflict for personal profit, is otherwise an outsider to the conflict, and is not a member of any o ...

switching loyalties when captured instead of being killed. A board resembling a

Draughts Checkers (American English), also known as draughts (; British English), is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers ...

board was found in Ur dating from 3000 BC. In the

British Museum The British Museum is a public museum dedicated to human history, art and culture located in the Bloomsbury area of London. Its permanent collection of eight million works is among the largest and most comprehensive in existence. It docum ...

are specimens of ancient Egyptian checkerboards, found with their pieces in burial chambers, and the game was played by Queen Hatasu.

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

mentioned a game, πεττεία or ''petteia'', as being of Egyptian origin, and

Homer Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the ''Iliad'' and the ''Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of the ...

also mentions it. Go was considered one of the four essential arts of the cultured

aristocratic Aristocracy (, ) is a form of government that places strength in the hands of a small, privileged ruling class, the aristocrats. The term derives from the el, αριστοκρατία (), meaning 'rule of the best'. At the time of the word's ...

Chinese scholars in antiquity. The earliest written reference to the game is generally recognized as the historical annal '' Zuo Zhuan'' (c. 4th century BC). Englishmen Lewis Waterman and John W. Mollett both claim to have invented the game of Reversi in 1883, each denouncing the other as a fraud. The game gained considerable popularity in England at the end of the nineteenth century. The game's first reliable mention is in 21 August 1886 edition of '' The Saturday Review''.

Comparison

Analysis of "pure" abstract strategy games is the subject of

combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

. Abstract strategy games with hidden information, bluffing, or simultaneous move elements are better served by Von Neumann–Morgenstern

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, while those with a component of luck may require

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

incorporated into either of the above. As for the qualitative aspects, ranking abstract strategy games according to their interest, complexity, or strategy levels is a daunting task and subject to extreme subjectivity. In terms of measuring how finite a mathematical field each of the three top contenders represents, it is estimated that checkers has a

game-tree complexity Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game comple ...

of 10⁴⁰ possible games, whereas chess has approximately 10¹²³. As for Go, the possible legal game positions range in the magnitude of 10¹⁷⁰.

Champions

The

first held the ''Abstract Games World Championship'' in 2008 to try to find the best abstract strategy games

all-rounder An all-rounder is a cricketer who regularly performs well at both batting and bowling. Although all bowlers must bat and quite a handful of batsmen do bowl occasionally, most players are skilled in only one of the two disciplines and are consi ...

. The MSO event saw a change in format in 2011 restricting the competition to players' five best events, and was renamed to the ''Modern Abstract Games World Championship''. * 2008: David M. Pearce (England) * 2009: David M. Pearce (England) * 2010: David M. Pearce (England) * 2011: David M. Pearce (England) * 2012: Andres Kuusk (Estonia) * 2013: Andres Kuusk (Estonia)

References

External links