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.
- 1 Basic notions
- 2 Determinacy from elementary considerations
- 3 Determinacy from ZFC
- 4 Determinacy and large cardinals
- 5 Consequences of determinacy
- 6 More general games
- 7 Quasistrategies and quasideterminacy
- 8 Footnotes
- 9 References
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 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 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.
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 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 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).
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.)
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.[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.
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.
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.
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.
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.
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 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.
- The first periodicity theorem implies that, for every natural number n, if Δ12n+1 determinacy holds, then Π12n+1 and Σ12n+2 have the prewellordering property (and that Σ12n+1 and Π12n+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 Δ12n+1 determinacy holds, then Π12n+1 and Σ12n 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 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.
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
- This subsection is still to be written
- 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.
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.
Quasistrategies and quasideterminacy
- This section is still to be written
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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.
- Gale, D. and F. M. Stewart (1953). Kuhn, H. W.; Tucker, A. W., eds. "Contributions to the Theory of Games, Volume II". Annals of Mathematics Studies 28. Princeton University Press. pp. 245–266. ISBN 9780691079356.
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- Martin, Donald A. (2003). "A simple proof that determinacy implies Lebesgue measurability". Rend. Sem. Mat. Univ. Pol. Torino 61 (4): 393–399. (PDF)
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