set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, that examines the conditions under which one or the other player of a
game
A game is a structured form of play (activity), play, usually undertaken for enjoyment, entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator s ...
has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness".
The games studied in set theory are usually
Gale
A gale is a strong wind; the word is typically used as a descriptor in nautical contexts. The U.S. National Weather Service defines a gale as sustained surface winds moving at a speed of between 34 and 47 knots (, or ).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 ...
in which the players make an infinite sequence of moves and there are no draws. The field of
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 ...
studies more general kinds of games, including games with draws such as
tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. T ...
,
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 ...
, or
infinite chess
Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a mod ...
, or games with imperfect information such as
poker
Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
.
Basic notions
Games
The first sort of game we shall consider is the two-player game of
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 ...
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s. These games are often called Gale–Stewart games.
In this sort of game there are two players, often named ''I'' and ''II'', who take turns playing natural numbers, with ''I'' going first. They play "forever"; that is, their plays are indexed by the natural numbers. When they're finished, a predetermined condition decides which player won. This condition need not be specified by any definable ''rule''; it may simply be an arbitrary (infinitely long)
lookup table
In computer science, a lookup table (LUT) is an array that replaces runtime computation with a simpler array indexing operation. The process is termed as "direct addressing" and LUTs differ from hash tables in a way that, to retrieve a value v wi ...
saying who has won given a particular sequence of plays.
More formally, consider a subset ''A'' of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
; recall that the latter consists of all ω-sequences of natural numbers. Then in the game G''A'',
''I'' plays a natural number ''a''0, then ''II'' plays ''a''1, then ''I'' plays ''a''2, and so on. Then ''I'' wins the game if and only if
:
and otherwise ''II'' wins. ''A'' is then called the ''payoff set'' of G''A''.
It is assumed that each player can see all moves preceding each of his moves, and also knows the winning condition.
Strategies
Informally, a ''strategy'' for a player is a way of playing in which his plays are entirely determined by the foregoing plays. Again, such a "way" does not have to be capable of being captured by any explicable "rule", but may simply be a lookup table.
More formally, a strategy for player ''I'' (for a game in the sense of the preceding subsection) is a function that accepts as an argument any finite sequence of natural numbers, of even length, and returns a natural number. If ''σ'' is such a strategy and <a0,...,a2n-1>
is a sequence of plays, then ''σ''(<a0,...,a2n-1>) is the next play ''I'' will make, if ''I'' is following the strategy ''σ''. Strategies for ''II'' are just the same, substituting "odd" for "even".
Note that we have said nothing, as yet, about whether a strategy is in any way ''good''. A strategy might direct a player to make aggressively bad moves, and it would still be a strategy. In fact it is not necessary even to know the winning condition for a game, to know what strategies exist for the game.
Winning strategies
A strategy is ''winning'' if the player following it must necessarily win, no matter what his opponent plays. For example, if ''σ'' is a strategy for ''I'', then ''σ'' is a winning strategy for ''I'' in the game G''A'' if, for any sequence of natural numbers to be played by ''II'', say <a1,a3,a5,...>, the sequence of plays produced by ''σ'' when ''II'' plays thus, namely
:
is an element of ''A''.
Determined games
A (class of) game(s) is ''determined'' if for all instances of the game there is a winning strategy for one of the players (not necessarily the same player for each instance). Note that there cannot be a winning strategy for ''both'' players for the same game, for if there were, the two strategies could be played against each other. The resulting outcome would then, by hypothesis, be a win for both players, which is impossible.https://www.math.uni-hamburg.de/Infinite Games, Yurii Khomskii (2010) Infinite Games, Yurii Khomskii (2010)
Determinacy from elementary considerations
All finite games of perfect information in which draws do not occur are determined.
Real-world games of perfect information, such as
tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. T ...
,
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 ...
, or
infinite chess
Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a mod ...
, are always finished in a finite number of moves (in chess-games this assumes the 50-move rule is applied). If such a game is modified so that a particular player wins under any condition where the game would have been called a draw, then it is always determined. The condition that the game is always over (i.e. all possible extensions of the finite position result in a win for the same player) in a finite number of moves corresponds to the topological condition that the set ''A'' giving the winning condition for G''A'' is
clopen
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...
in the
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
.
For example, modifying the rules of chess to make drawn games a win for Black makes chess a determined game. As it happens, chess has a finite number of positions and a draw-by-repetition rules, so with these modified rules, if play continues long enough without White having won, then Black can eventually force a win (due to the modification of draw = win for black).
The proof that such games are determined is rather simple: Player ''I'' simply plays ''not to lose''; that is, player ''I'' plays to make sure that player ''II'' does not have a winning strategy ''after'' ''Is move. If player ''I'' ''cannot'' do this, then it means player ''II'' had a winning strategy from the beginning. On the other hand, if player ''I'' ''can'' play in this way, then ''I'' must win, because the game will be over after some finite number of moves, and player ''I'' can't have lost at that point.
This proof does not actually require that the game ''always'' be over in a finite number of moves, only that it be over in a finite number of moves whenever ''II'' wins. That condition, topologically, is that the set ''A'' is closed. This fact—that all closed games are determined—is called the ''Gale–Stewart theorem''. Note that by symmetry, all open games are determined as well. (A game is ''open'' if ''I'' can win only by winning in a finite number of moves.)
David Gale
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
and F. M. Stewart proved the open and closed games are determined. Determinacy for second level of the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
games was shown by Wolfe in 1955. Over the following 20 years, additional research using ever-more-complicated arguments established that third and fourth levels of the Borel hierarchy are determined.
In 1975,
Donald A. Martin
Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy.
Education and career
Martin rece ...
proved that all
Borel Borel may refer to:
People
* Borel (author), 18th-century French playwright
* Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance
* Émile Borel (1871 – 1956), a French mathematician known for his founding ...
games are determined; that is, if ''A'' is a Borel subset of Baire space, then G''A'' is determined. This result, known as
Borel determinacy In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. A Gale-Stewart game is a poss ...
, is the best possible determinacy result provable in ZFC, in the sense that the determinacy of the next higher Wadge class is not provable in ZFC.
In 1971, before Martin obtained his proof,
Harvey Friedman
__NOTOC__
Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axi ...
showed that any proof of Borel determinacy must use the
axiom of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
in an essential way, in order to iterate the powerset axiomtransfinitely often. Friedman's work gives a level-by-level result detailing how many iterations of the powerset axiom are necessary to guarantee determinacy at each level of the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
.
For every integer ''n'', ZFC\P proves determinacy in the ''n''th level of the difference hierarchy of sets, but ZFC\P does not prove that for every integer ''n'' ''n''th level of the difference hierarchy of sets is determined. See
reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
for other relations between determinacy and subsystems of
second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
A precurs ...
.
Determinacy and large cardinals
There is an intimate relationship between determinacy and
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
s. In general, stronger large cardinal axioms prove the determinacy of larger
pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
es, higher in the
Wadge hierarchy In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wad ...
, and the determinacy of such pointclasses, in turn, proves the existence of
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
s of slightly weaker large cardinal axioms than those used to prove the determinacy of the pointclass in the first place.
Measurable cardinals
It follows from the existence of a measurable cardinal that every analytic game (also called a Σ11 game) is determined, or equivalently that every coanalytic (or Π11 ) game is determined. (See
Projective hierarchy
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
for definitions.)
Actually a measurable cardinal is more than enough. A weaker principle — the existence of 0# is sufficient to prove coanalytic determinacy, and a little bit more: The precise result is that the existence of 0# is equivalent to the determinacy of all levels of the difference hierarchy below the ω2 level, i.e. ω·n-Π11 determinacy for every .
From a measurable cardinal we can improve this very slightly to ω2-Π11 determinacy. From the existence of more measurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over Π11.
Proof of determinacy from sharps
For every real number ''r'', determinacy is equivalent to existence of ''r''#. To illustrate how large cardinals lead to determinacy, here is a proof of determinacy given existence of ''r''#.
Let ''A'' be a subset of the Baire space. ''A'' = p 'T''for some tree ''T'' (constructible from ''r'') on (ω, ω). (That is x∈A iff from some ''y'', is a path through ''T''.)
Given a partial play ''s'', let be the subtree of ''T'' consistent with ''s'' subject to max(y0,y1,...,ylen(s)-1)<len(s). The additional condition ensures that is finite. Consistency means that every path through is of the form where is an initial segment of ''s''.
To prove that A is determined, define auxiliary game as follows:
In addition to ordinary moves, player 2 must play a mapping of into ordinals (below a sufficiently large ordinal ''κ'') such that
* each new move extends the previous mapping and
* the ordering of the ordinals agrees with the
Kleene–Brouwer order In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order is a linear order on finite sequences over some linearly ordered set (X, <), that differs from the more commonly used
indiscernible
In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. Examples
If ''a'', ''b'', and ''c'' are distinct and is a set ...
ordinals. By indiscernibility, if ''κ'' and the ordinals in the auxiliary response are in ''I'', then the moves by player 1 do not depend on the auxiliary moves (or on ''κ''), and so the strategy can be converted into a strategy for the original game (since player 2 can hold out with indiscernibles for any finite number of steps). Suppose that player 1 loses in the original game. Then, the tree corresponding to a play is well-founded. Therefore, player 2 can win the auxiliary game by using auxiliary moves based on the indiscernibles (since the order type of indiscernibles exceeds the Kleene–Brouwer order of the tree), which contradicts player 1 winning the auxiliary game.
Woodin cardinals
If there is a Woodin cardinal with a measurable cardinal above it, then Π12 determinacy holds. More generally, if there are ''n'' Woodin cardinals with a measurable cardinal above them all, then Π1n+1 determinacy holds. From Π1n+1 determinacy, it follows that there is a transitive
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
containing ''n'' Woodin cardinals.
(lightface) determinacy is equiconsistent with a Woodin cardinal. If determinacy holds, then for a Turing cone of ''x'' (that is for every real ''x'' of sufficiently high
Turing degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.
Overview
The concept of Turing degree is fund ...
), L 'x''satisfies OD-determinacy (that is determinacy of games on integers of length ω and ordinal-definable payoff), and in HODL 'x''/sup> is a Woodin cardinal.
Projective determinacy
If there are infinitely many Woodin cardinals, then projective determinacy holds; that is, every game whose winning condition is a
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
is determined. From projective determinacy it follows that, for every natural number ''n'', there is a transitive inner model that satisfies that there are ''n'' Woodin cardinals.
Axiom of determinacy
The axiom of determinacy, or AD, asserts that ''every'' two-player game of perfect information of length ω, in which the players play naturals, is determined.
AD is provably false from ZFC; using the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
one may prove the existence of a non-determined game. However, if there are infinitely many Woodin cardinals with a measurable above them all, then
L(R)
In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
Construction
It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructi ...
Banach–Mazur game
In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire s ...
for ''A'' is determined, then either ''II'' has a winning strategy, in which case ''A'' is meager, or ''I'' has a winning strategy, in which case ''A'' is
comeager
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
on some open neighborhood.
This does not quite imply that ''A'' has the
property of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such th ...
, but it comes close: A simple modification of the argument shows that if Γ is an
adequate pointclass In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in ...
such that every game in Γ is determined, then every set of reals in Γ has the property of Baire.
In fact this result is not optimal; by considering the unfolded Banach–Mazur game we can show that determinacy of Γ (for Γ with sufficient closure properties) implies that every set of reals that is the ''projection'' of a set in Γ has the property of Baire. So for example the existence of a measurable cardinal implies Π11 determinacy, which in turn implies that every Σ12 set of reals has the property of Baire.
By considering other games, we can show that Π1''n'' determinacy implies that every Σ1''n''+1 set of reals has the property of Baire, is
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
(in fact
universally measurable In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. In particular, a universally measurable set of reals is n ...
) and has the
perfect set property In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
.
Periodicity theorems
* The first periodicity theorem implies that, for every natural number ''n'', if Δ12''n''+1 determinacy holds, then Π12''n''+1 and Σ12''n''+2 have the
prewellordering property
In set theory, a prewellordering on a set X is a preorder \leq on X (a transitive and strongly connected relation on X) that is wellfounded in the sense that the relation x \leq y \land y \nleq x is wellfounded. If \leq is a prewellordering on ...
(and that Σ12''n''+1 and Π12''n''+2 do ''not'' have the prewellordering property, but rather have the separation property).
* The second periodicity theorem implies that, for every natural number ''n'', if Δ12''n''+1 determinacy holds, then Π12''n''+1 and Σ12''n'' have the scale property. In particular, if projective determinacy holds, then every projective
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
has a projective uniformization.
* The third periodicity theorem gives a sufficient condition for a game to have a definable winning strategy.
Applications to decidability of certain second-order theories
In 1969,
Michael O. Rabin
Michael Oser Rabin ( he, מִיכָאֵל עוזר רַבִּין; born September 1, 1931) is an Israeli mathematician and computer scientist and a recipient of the Turing Award.
Biography Early life and education
Rabin was born in 1931 in ...
proved that the monadic second-order theory of ''n'' successors ( S2S for ''n'' = 2) is decidable. A key component of the proof requires showing determinacy of
parity game
A parity game is played on a colored directed graph, where each node has been colored by a priority – one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph. The owne ...
s, which lie in the third level of the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
.
Wadge determinacy
Wadge determinacy is the statement that for all pairs ''A'', ''B'' of subsets of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
, the Wadge game G(''A'' ,''B'') is determined. Similarly for a
pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
Γ, Γ Wadge determinacy is the statement that for all sets ''A'', ''B'' in Γ, the Wadge game G(''A'', ''B'') is determined.
Wadge determinacy implies the semilinear ordering principle for the Wadge order. Another consequence of Wadge determinacy is the
perfect set property In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
.
In general, Γ Wadge determinacy is a consequence of the determinacy of Boolean combinations of sets in Γ. In the
projective hierarchy
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
, Π11 Wadge determinacy is equivalent to Π11 determinacy, as proved by
Leo Harrington
Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in
recursion theory, model theory, and set theory.
Having retired from being a Mathematician, Professor Leo Harrington is ...
. This result was extended by Hjorth to prove that Π12 Wadge determinacy (and in fact the semilinear ordering principle for Π12) already implies Π12 determinacy.
More general games
Games in which the objects played are not natural numbers
Determinacy of games on ordinals with ordinal definable payoff and length ω implies that for every regular cardinal ''κ''>ω there are no ordinal definable disjoint stationary subsets of ''κ'' made of ordinals of cofinality ω. The consistency strength of the determinacy hypothesis is unknown but is expected to be very high.
Games played on
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
Long games
Existence of ω1 Woodin cardinals implies that for every countable ordinal α, all games on integers of length α and projective payoff are determined. Roughly speaking, α Woodin cardinals corresponds to determinacy of games on reals of length α (with a simple payoff set). Assuming a limit of Woodin cardinals ''κ'' with o(''κ'')=''κ''++ and ω Woodin cardinals above ''κ'', games of variable countable length where the game ends as soon as its length is admissible relative to the line of play and with projective payoff are determined. Assuming that a certain iterability conjecture is provable, existence of a measurable Woodin cardinal implies determinacy of open games of length ω1 and projective payoff. (In these games, a winning condition for the first player is triggered at a countable stage, so the payoff can be coded as a set of reals.)
Relative to a Woodin limit of Woodin cardinals and a measurable above them, it is consistent that every game on integers of length ω1 and ordinal definable payoff is determined. It is conjectured that the determinacy hypothesis is equiconsistent with a Woodin limit of Woodin cardinals. ω1 is maximal in that there are undetermined games on integers of length ω1+ω and ordinal definable payoff.
Games of imperfect information
In any interesting game with
imperfect 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 pri ...
, a winning strategy will be a
mixed strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
: that is, it will give some probability of differing responses to the same situation. If both players' optimal strategies are mixed strategies then the outcome of the game cannot be ''certainly'' determinant (as it can for pure strategies, since these are
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
). But the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of outcomes to opposing mixed strategies can be calculated. A game that requires mixed strategies is defined as ''determined'' if a strategy exists that yields a minimum
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
(over possible counter-strategies) that exceeds a given value. Against this definition, all
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* 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 marke ...
two-player zero-sum games are clearly determined. However, the determinacy of ''infinite'' games of imperfect information (Blackwell games) is less clear.
In 1969
David Blackwell
David Harold Blackwell (April 24, 1919 – July 8, 2010) was an American statistician and mathematician who made significant contributions to game theory, probability theory, information theory, and statistics. He is one of the eponyms of th ...
proved that some "infinite games with imperfect information" (now called "Blackwell games") are determined, and in 1998
Donald A. Martin
Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy.
Education and career
Martin rece ...
proved that ordinary (perfect-information game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass. This, combined with the Borel determinacy theorem of Martin, implies that all Blackwell games with Borel payoff functions are determined.
Martin conjectured that ordinary determinacy and Blackwell determinacy for infinite games are equivalent in a strong sense (i.e. that Blackwell determinacy for a boldface pointclass in turn implies ordinary determinacy for that pointclass), but as of 2010, it has not been proven that Blackwell determinacy implies perfect-information-game determinacy.
Quasistrategies and quasideterminacy
See also
*
ω-automaton
In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of finite automata that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety o ...
*
Solved game
A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.
This concept is usually applied to abstract strategy games, and especially to games with full informa ...
Topological game
In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time ...
Footnotes
# This assumes that ''I'' is trying to get the intersection of neighborhoods played to be a singleton whose unique element is an element of ''A''. Some authors make that the goal instead for player ''II''; that usage requires modifying the above remarks accordingly.
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...