# Determinacy

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In set theory, a branch of mathematics, **determinacy** is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.

## Basic notions

### Games

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.

In this sort of game we consider 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 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 <a_{0},…,a_{2n-1}>
is a sequence of plays, then σ(<a_{0},…,a_{2n-1}>) is the next play *I* will make, if he 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 <a_{1},a_{3},a_{5},…>, 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 instance 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.

## Determinacy from elementary considerations

All finite games of perfect information in which draws do not occur are determined.

Familiar real-world games of perfect information, such as chess or tic-tac-toe, are always finished in a finite number of moves. 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 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).

It is an instructive exercise to figure out how to represent such games as games in the context of this article.

The proof that such games are determined is rather simple: Player *I* simply plays *not to lose*; that is, he plays to make sure that player *II* does not have a winning strategy *after* *I'*s 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 he must win, because the game will be over after some finite number of moves, and he 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.)

## Determinacy from ZFC

Gale and 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.Template: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 G_{A} 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.

## Determinacy and large cardinals

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.

### Measurable cardinals

It follows from the existence of a measurable cardinal that every analytic game (also called a **Σ**^{1}_{1} game) is determined, or equivalently that every coanalytic (or **Π**^{1}_{1} ) 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 ω^{2} level, i.e. ω·n-**Π**^{1}_{1} determinacy for every .

From a measurable cardinal we can improve this very slightly to ω^{2}-**Π**^{1}_{1} determinacy. From the existence of more measurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over **Π**^{1}_{1}.

### Woodin cardinals

If there is a Woodin cardinal with a measurable cardinal above it, then **Π**^{1}_{2} determinacy holds. More generally, if there are *n* Woodin cardinals with a measurable cardinal above them all, then **Π**^{1}_{n+1} determinacy holds. From **Π**^{1}_{n+1} determinacy, it follows that there is a transitive inner model containing *n* Woodin cardinals.

### Projective determinacy

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 which 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 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.

## Consequences of determinacy

### Regularity properties for sets of reals

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^{[1]}.

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 **Π**^{1}_{1} determinacy, which in turn implies that every **Σ**^{1}_{2} set of reals has the property of Baire.

By considering other games, we can show that **Π**^{1}_{n} determinacy implies that every **Σ**^{1}_{n+1} set of reals has the property of Baire, is Lebesgue measurable (in fact universally measurable) and has the perfect set property.

### Periodicity theorems

- The
**first periodicity theorem**implies that, for every natural number*n*, if**Δ**^{1}_{2n+1}determinacy holds, then**Π**^{1}_{2n+1}and**Σ**^{1}_{2n+2}have the prewellordering property (and that**Σ**^{1}_{2n+1}and**Π**^{1}_{2n+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**Δ**^{1}_{2n+1}determinacy holds, then**Π**^{1}_{2n+1}and**Σ**^{1}_{2n}have the scale property.^{[1]}In particular, if projective determinacy holds, then every projective relation 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 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.

### Wadge determinacy

**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, **Π**^{1}_{1} Wadge determinacy is equivalent to **Π**^{1}_{1} determinacy, as proved by Harrington. This result was extendend by Hjorth to prove that **Π**^{1}_{2} Wadge determinacy (and in fact the semilinear ordering principle for **Π**^{1}_{2}) already implies **Π**^{1}_{2} determinacy.

*This subsection is still incomplete*

## More general games

*This section is still to be written*

### Games in which the objects played are not natural numbers

*This subsection is still to be written*

### Games played on trees

*This subsection is still to be written*

### Long games

*This subsection is still to be written*

### Games of imperfect information

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.^{[2]}

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.^{[3]}
^{[4]} 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.^{[5]}

## Quasistrategies and quasideterminacy

*This section is still to be written*

## Footnotes

- ↑ http://web.mit.edu/dmytro/www/DeterminacyMaximum.htm
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**^**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.

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