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The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player. In static games of complete, perfect information, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of ...
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
For (3)⇒(2), define a game where player 1 attempts to show that the desired element s is inside the least fixed point. Player 1 gradually labels elements including s with rational numbers, intended to correspond to ordinal stages of the monotonic induction (any countable ordinal is embeddable into ). Player 2 plays elements with strictly ...
If β = ω λ m + ω μ n + smaller terms, then β = (ω μ n + smaller terms)(ω λ−μ + 1)m is a product of a smaller ordinal and a prime and a natural number m. Repeating this and factorizing the natural numbers into primes gives the prime factorization of β. So the factorization of the Cantor normal form ordinal ω α 1 n 1 + ⋯ + ω α ...
The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games.
Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
Buchholz's paper in 1987 showed that the canonical correspondence between a hydra and an infinitary well-founded tree (or the corresponding term in the notation system associated to Buchholz's function, which does not necessarily belong to the ordinal notation system ), preserves fundamental sequences of choosing the rightmost leaves and the operation on an infinitary well-founded tree or the ...
The game {0|} is called 1, and the game {|0} is called -1. The game {0|0} is called * (star), and is the first game we find that is not a number. All numbers are positive, negative, or zero, and we say that a game is positive if Left has a winning strategy, negative if Right has a winning strategy, or zero if the second player has a winning ...