In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there are no moves left in either of the two parallel games, at which point (in normal play) the last player to move wins. This operation may be extended to disjunctive sums of any number of games, again by playing the games in parallel and moving in exactly one of the games per turn. It is the fundamental operation that is used in the Sprague–Grundy theorem for impartial games and which led to the field of combinatorial game theory for partisan games.

Application to common games

Disjunctive sums arise in games that naturally break up into components or regions that do not interact except in that each player in turn must choose just one component to play in. Examples of such games are Go,

Nim Nim is a mathematical two player game. Nim or NIM may also refer to: * Nim (programming language) * Nim Chimpsky, a signing chimpanzee Acronyms * Network Installation Manager, an IBM framework * Nuclear Instrumentation Module * Negative index met ...

, Sprouts, Domineering, the Game of the Amazons, and the map-coloring games. In such games, each component may be analyzed separately for simplifications that do not affect its outcome or the outcome of its disjunctive sum with other games. Once this analysis has been performed, the components can be combined by taking the disjunctive sum of two games at a time, combining them into a single game with the same outcome as the original game.

Mathematics

The sum operation was formalized by . It is a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

and associative operation: if two games are combined, the outcome is the same regardless of what order they are combined, and if more than two games are combined, the outcome is the same regardless of how they are grouped. The negation −''G'' of a game ''G'' (the game formed by trading the roles of the two players) forms an

additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...

under disjunctive sums: the game ''G'' + −''G'' is a zero game (won by whomever goes second) using a simple echoing strategy in which the second player repeatedly copies the first player's move in the other game. For any two games ''G'' and ''H'', the game ''H'' + ''G'' + −''G'' has the same outcome as ''H'' itself (although it may have a larger set of available moves). Based on these properties, the class of combinatorial games may be thought of as having the structure of an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

, although with a

proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...

of elements rather than (as is more standard for groups) a set of elements. For an important subclass of the games called the

surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surrea ...

s, there exists a multiplication operator that extends this group to a field. For impartial

misère Misère (French language, French for "destitution"), misere, bettel, betl, or (German language, German for "begging, beggar"; equivalent terms in other languages include , , ) is a bidding, bid in various card games, and the player who bids misè ...

play games, an analogous theory of sums can be developed, but with fewer of these properties: these games form a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

with only one nontrivial invertible element, called

star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...

( *), of order two.

References

*. {{DEFAULTSORT:Disjunctive Sum Combinatorial game theory