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 imaginatively 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 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 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,…,a_2n-1> is a sequence of plays, then σ(<a₀,…,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 <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 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.

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

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 is the best result possible using ZFC alone, in the sense that the determinacy of the next higher Wadge class is not provable in ZFC.

Martin's proof uses the powerset axiom in an essential way. There is a level-by-level result detailing what fragment of the powerset axiom is necessary to guarantee determinacy through what level of the Borel hierarchy.

Historically, determinacy for second level of the Borel hierarchy games was shown by Wolfe in 1955.

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 Σ¹₁ game) is determined, or equivalently that every coanalytic (or Π¹₁) game is determined. (See Projective hierarchy for definitions.)

Actually an apparently stronger result follows: If there is a measurable cardinal, then every game in the first ω² levels of the difference hierarchy over Π¹₁ is determined. This is only apparently stronger; ω²-Π¹₁ determinacy turns out to be equivalent to Π¹₁ determinacy.

From the existence of more measurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over Π¹₁.

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 Π¹_n+1 determinacy holds. From Π¹_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.

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 Π¹_n determinacy implies that every Σ¹_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 Δ¹_2n+1 determinacy holds, then Π¹_2n+1 and Σ¹_2n+2 have the prewellordering property (and that Σ¹_2n+1 and Π¹_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 Δ¹_2n+1 determinacy holds, then Π¹_2n+1 and Σ¹_2n+2 have the scale property. 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, Π¹₁ Wadge determinacy is equivalent to Π¹₁ determinacy, as proved by Harrington. This result was extendend by Hjorth to prove that Π¹₂ Wadge determinacy (and in fact the semilinear ordering principle for Π¹₂) already implies Π¹₂ 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 (Blackwell games)

This subsection is still to be written

Article about blackwell games

Quasistrategies and quasideterminacy

This section is still to be written

Footnotes

1. 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.

References

• Gale, D. and F. M. Stewart (1953). "Infinite games with perfect information". Ann. Math. Studies 28 245–266.
• Harrington, Leo (Jan., 1978). "Analytic determinacy and 0#". The Journal of Symbolic Logic 43 685–693.
• Hjorth, Greg (Jan., 1996). "Π¹₂ Wadge degrees". Annals of Pure and Applied Logic 77 53–74.
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.
• Martin, Donald A. (1975). "Borel determinacy". Annals of Mathematics. Second Series 102 (2): 363–371.
• Martin, Donald A. and John R. Steel (Jan., 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical Society 2 (1): 71–125.
• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.
• Woodin, W. Hugh (1988). "Supercompact cardinals, sets of reals, and weakly homogeneous trees". Proceedings of the National Academy of Sciences of the United States of America 85 (18): 6587–6591.
• Martin, Donald A. (2003). "A simple proof that determinacy implies Lebesgue measurability". Rend. Sem. Mat. Univ. Pol. Torino 61 (4): 393–399. (PDF)
• Wolfe, P. (1955). "The strict determinateness of certain infinite games". Pacific J. of Math. 5 Supplement I:841–847.