game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

, a finite game (sometimes called a founded game or a well-founded game) is a

two-player game A two-player game is a multiplayer game that is played by precisely two players. This is distinct from a solitaire game, which is played by only one player. Examples The following are some examples of two-player games. This list is not intended ...

that is assured to end after a

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

number of moves. Finite games may have an infinite number of possibilities or even an unbounded number of moves, so long as they are guaranteed to end in a finite number of turns.

Formal definition

William Zwicker defined a game, ''G'', to be ''totally finite'' if it met the following five conditions: # Two players, I and II, move alternately, I going first. Each has complete knowledge of the other's moves. # There is no chance involved. # There are no ties (when a play of ''G'' is complete, there is one winner). # Every play ends after finitely many moves. # At any point in a play of ''G'', there are but finitely many legal possibilities for the next move.

Examples

Tic Tac Toe Tic-tac-toe (American English), noughts and crosses (English in the Commonwealth of Nations, Commonwealth English), or Xs and Os (Canadian English, Canadian or Hiberno-English, Irish English) is a paper-and-pencil game for two players who ta ...

Chess Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...

Checkers Checkers (American English), also known as draughts (; English in the Commonwealth of Nations, Commonwealth English), is a group of Abstract strategy game, strategy board games for two players which involve forward movements of uniform game ...

Poker Poker is a family of Card game#Comparing games, comparing card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules. It is played worldwide, with varying rules i ...

* The game where player one chooses any number and immediately wins (this is an example of a finite game with infinite possibilities) * The game where player one names any number N, then N moves pass with nothing happening before player one wins (this is an example of a finite game with an unbounded number of moves)

Supergame

A supergame is a variant of the finite game invented by William Zwicker. Zwicker defined a supergame to have the following rules: "On the first move, I name any totally finite game ''G'' (called the subgame). The players then proceed to play ''G'', with II playing the role of I while ''G'' is being played. The winner of the play of the subgame is declared to be the winner of the play of the supergame." Zwicker notes that a supergame satisfies properties 1-4 of a totally finite game, but not property 5. He defines games of this type to be ''somewhat finite.''

Hypergame paradox

A hypergame has the same rules as a super game except that I may name any somewhat finite game on the first move. The hypergame is closely related to the "hypergame paradox" a self-referential, set-theoretic paradox like

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains ...

and

Cantor's paradox In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" i ...

. The hypergame paradox arises from trying to answer the question ''"Is a hypergame somewhat finite?"'' The paradox, as Zwicker note, satisfies conditions 1- 4 making it somewhat finite in the same way a supergame was. However, if hypergame is a somewhat finite game, then play can proceed infinitely with both players choosing hypergame as their subgame forever. This infinite would appear to violate property 4, making the hypergame not somewhat finite. Thus, the paradox.

References

{{Reflist Logic Game theory game classes Game theory