Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game 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.
The games studied in set theory are usually Gale –Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker.
The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. 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 saying who has won given a particular sequence of plays.
More formally, consider a subset A of Baire space; recall that the latter consists of all ω-sequences of natural numbers. Then in the game GA, I plays a natural number a0, then II plays a1, then I plays a2, and so on. Then I wins the game if and only if
and otherwise II wins. A is then called the payoff set of GA.
It is assumed that each player can see all moves preceding each of his moves, and also knows the winning condition.
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.
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 GA 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.
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.
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, chess, or infinite chess, 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 GA is clopen in the topology of Baire space.
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 afterI's move. If player Icannot do this, then it means player II had a winning strategy from the beginning. On the other hand, if player Ican 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 and F. M. Stewart proved the open and closed games are determined. Determinacy for second level of the Borel hierarchy 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.[ specify ]
In 1975, Donald A. Martin proved that all Borel games are determined; that is, if A is a Borel subset of Baire space, then GA is determined. This result, known as Borel determinacy, 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 showed that any proof of Borel determinacy must use the axiom of replacement in an essential way, in order to iterate the powerset axiom transfinitely 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.
For every integer n, ZFC\P proves determinacy in the nth level of the difference hierarchy of sets, but ZFC\P does not prove that for every integer nnth level of the difference hierarchy of sets is determined. See reverse mathematics for other relations between determinacy and subsystems of second-order arithmetic.
There is an intimate relationship between determinacy and large cardinals. In general, stronger large cardinal axioms prove the determinacy of larger pointclasses, higher in the Wadge hierarchy, and the determinacy of such pointclasses, in turn, proves the existence of inner models of slightly weaker large cardinal axioms than those used to prove the determinacy of the pointclass in the first place.
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 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.
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
Recall that Kleene–Brouwer order is like lexicographical order except that if s properly extends t then s<t. It is a well-ordering iff the tree is well-founded.
The auxiliary game is open. Proof: If player 2 does not lose at a finite stage, then the union of all (which is the tree that corresponds to the play) is well-founded, and so the result of the non-auxiliary play is not in A.
Thus, the auxiliary game is determined. Proof: By transfinite induction, for each ordinal α compute the set of positions where player 1 can force a win in α steps, where a position with player 2 to move is losing (for player 2) in α steps iff for every move the resulting position is losing in less than α steps. One strategy for player 1 is to reduce α with each position (say picking the least α and breaking ties by picking the least move), and one strategy for player 2 is to pick the least (actually any would work) move that does not lead to a position with an α assigned. Note that L(r) contains the set of winning positions as well as the winning strategies given above.
A winning strategy for player 2 in the original game leads to winning strategy in the auxiliary game: The subtree of T corresponding to the winning strategy is well-founded, so player 2 can pick ordinals based on the Kleene–Brouwer order of the tree. Also, trivially, a winning strategy for player 2 in the auxiliary game gives a winning strategy for player 2 in original game.
It remains to show that using r#, the above-mentioned winning strategy for player 1 in the auxiliary game can be converted into a winning strategy in the original game. r# gives a proper class I of (L(r),∈,r) indiscernible 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.
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 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), L[x] satisfies OD-determinacy (that is determinacy of games on integers of length ω and ordinal-definable payoff), and in HODL[x] is a Woodin cardinal.
If there are infinitely many Woodin cardinals, then projective determinacy holds; that is, every game whose winning condition is a projective set 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.
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 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) is a model of ZF that satisfies AD.
If A is a subset of Baire space such that the Banach–Mazur game 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 on some open neighborhood .
This does not quite imply that A has the property of Baire, but it comes close: A simple modification of the argument shows that if Γ is an adequate pointclass 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 Π1n determinacy implies that every Σ1n+1 set of reals has the property of Baire, is Lebesgue measurable (in fact universally measurable) and has the perfect set property.
In 1969, Michael O. Rabin proved that the second-order theory of n successors is decidable.A key component of the proof requires showing determinacy of parity games, which lie in the third level of the Borel hierarchy.
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Wadge determinacy is the statement that for all pairs A, B of subsets of Baire space, the Wadge game G(A ,B) is determined. Similarly for a pointclass 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 general, Γ Wadge determinacy is a consequence of the determinacy of Boolean combinations of sets in Γ. In the projective hierarchy, Π11 Wadge determinacy is equivalent to Π11 determinacy, as proved by Leo Harrington. 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.
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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.
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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.
In any interesting game with imperfect information, a winning strategy will be a mixed strategy: 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). But the probability distribution 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 (over possible counter-strategies) that exceeds a given value. Against this definition, all finite 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 proved that some "infinite games with imperfect information" (now called "Blackwell games") are determined, and in 1998 Donald A. Martin 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.
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In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.
In the mathematical discipline of set theory, 0# is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers, or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscovered by Solovay, who considered it as a subset of the natural numbers and introduced the notation O#.
In set theory, a Woodin cardinal is a cardinal number λ such that for all functions
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.
In the mathematical field of descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by Luzin (1917) and his student Souslin (1917).
In mathematics, the axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.
In set theory, a subset of a Polish space is ∞-Borel if it can be obtained by starting with the open subsets of , and transfinitely iterating the operations of complementation and wellordered union. Note that the set of ∞-Borel sets may not actually be closed under wellordered union; see below.
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.
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.
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably covering properties. Intuitively, the core model is "the largest canonical inner model there is" and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers to the definable inner model that exhibits the special properties under the assumption that there does not exist a cardinal satisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models below them.
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. Wadge.
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 the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.
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 some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass.
In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure . Just as the axiom of projective determinacy yields a canonical theory of , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.
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.
This is a glossary of set theory.