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.

Contents

Basic notions [link]

Games [link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle a_0,a_1,a_2,\ldots\rangle\in A

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.

Strategies [link]

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.

Winning strategies [link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\sigma(\langle\rangle),a_1,\sigma(\langle\sigma(\langle\rangle),a_1\rangle),a_3,\ldots\rangle

is an element of A.

Determined games [link]

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 [link]

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.)

[edit] 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.[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 [link]

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.

[edit] Measurable cardinals

It follows from the existence of a measurable cardinal that every analytic game (also called a Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma^1_1

game) is determined, or equivalently that every coanalytic (or Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi_1^1
) 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi^1_1

is determined.  This is only apparently stronger; ω2-Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi_1^1
determinacy turns out to be equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi_1^1
determinacy.

From the existence of more measurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi_1^1 .

[edit] Woodin cardinals

If there is a Woodin cardinal with a measurable cardinal above it, then ΠFailed to parse (Missing texvc executable; please see math/README to configure.): ^1_2

determinacy holds.  More generally, if there are n Woodin cardinals with a measurable cardinal above them all, then 

ΠFailed to parse (Missing texvc executable; please see math/README to configure.): ^1_{n+1}

  determinacy holds.  From ΠFailed to parse (Missing texvc executable; please see math/README to configure.): ^1_{n+1}
determinacy, it follows that there is a transitive inner model containing n Woodin cardinals.

[edit] 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.

[edit] 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 [link]

Regularity properties for sets of reals [link]

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 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.

Periodicity theorems [link]

  • 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.[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 [link]

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 [link]

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 [link]

This section is still to be written

Games in which the objects played are not natural numbers [link]

This subsection is still to be written

[edit] Games played on trees

This subsection is still to be written

Long games [link]

This subsection is still to be written

Games of imperfect information [link]

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 [link]

This section is still to be written

Footnotes [link]

  1. ^ https://web.mit.edu/dmytro/www/DeterminacyMaximum.htm
  2. ^ Vervoort, M. R. (1996). "Blackwell Games". Statistics, Probability and Game Theory 30: 4 & 5. https://staff.science.uva.nl/~vervoort/blackwell-article.pdf. 
  3. ^ Martin, D. A. (December 1998). "The determinacy of Blackwell games". Journal of Symbolic Logic 63 (4): 1565. 
  4. ^ Shmaya, E. (2009). The determinacy of infinite games with eventual perfect monitoring. 30. arXiv:0902.2254. 
  5. ^ Löwe, Benedikt (2006). "SET THEORY OF INFINITE IMPERFECT INFORMATION". CiteSeerX. https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.7976. Retrieved 2010-06-06. 
  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 [link]

  • 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 (4): 685–693. DOI:10.2307/2273508. JSTOR 2273508. 
  • 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. DOI:10.2307/1971035. 
  • Martin, Donald A. and John R. Steel (Jan., 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical Society 2 (1): 71–125. DOI:10.2307/1990913. JSTOR 1990913. 
  • 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. DOI:10.1073/pnas.85.18.6587. PMC 282022. PMID 16593979. //www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=282022. 
  • 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. Math. 5: Supplement I:841–847. 

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Determined (song)

"Determined" is the first single from American band Mudvayne's third studio album, Lost and Found.

Music

The song contains elements of thrash metal and hardcore punk.

Music video

The music video for "Determined" shows the band playing the song in front of a large group of moshing fans. It was recorded in New York City.

Appearances in other media

The song is featured in the soundtrack to Need for Speed: Underground 2.

Live performance

Mudvayne performed the song live for the first time in early 2005 during a small clubs tour. It has since become a regular part of the band's setlist.

Reception

Johnny Loftus of Allmusic called the song "one of Mudvayne's all-time strongest tracks". It also received praise from the Baltimore Sun.

The song was nominated at the 2006 Grammy Awards for Best Metal Performance but lost to Slipknot's "Before I Forget".

Track listing


Sources

External links

  • Official music video

  • Words (Bee Gees song)

    "Words" is a song by the Bee Gees, written by Barry, Robin & Maurice Gibb. The song reached No. 1 in Germany, Switzerland, Netherlands and China.

    "Words" was the Bee Gees third UK top 10 hit, reaching number 8, and in a UK television special on ITV in December 2011 it was voted fourth in "The Nation's Favourite Bee Gees Song". The song has been recorded by many other artists., including hit versions by Rita Coolidge in 1978 and Boyzone in 1996. This was Boyzone's fifth single and their first number one hit in the UK.

    Writing

    Barry Gibb explains:

    Robin Gibb: "'Words' reflects a mood, It was written after an argument. Barry had been arguing with someone, I had been arguing with someone, and happened to be in the same mood. [The arguments were] about absolutely nothing. They were just words. That is what the song is all about; words can make you happy or words can make you sad".

    Barry said in 1996 on the VH1 Storytellers television show that it was written for their manager, Robert Stigwood.

    Words (F. R. David song)

    "Words" is a 1982 song by F. R. David, which sold eight million copies worldwide and peaked at number two on the British charts in spring of 1983. The song was originally released only in France and Monaco in the winter of 1981, later it was released in the rest of Europe. In 1983, it finally was released in America and the UK. It was a huge European hit, peaking at number one in Germany, Switzerland, Sweden, Austria and Norway. It also went to number one in South Africa in late 1982 and spent 25 weeks on the charts. The photography of the 7" vinyl was made by Vassili Ulrich.

    Initial copies of the recording on both LP and single credit the composition of "Words" solely to Robert Fitoussi, which is the real name of F. R. David. All later reissues of F. R. David's original recording of "Words", as well as all re-recordings, credit the composition of the song to Fitoussi (music), and Marty Kupersmith & Louis S. Yaguda (lyrics).

    In the 2000s, David released a French language duo version of the song with the singer Winda entitled "Words, j'aime ces mots". F. R. David and Winda included also an English version as a duo.

    Words (Anthony David song)

    "Words" is a song by American R&B singer-songwriter Anthony David, from his third studio album Acey Duecy. It features fellow contemporary R&B singer-songwriter India.Arie. The song peaked at #53 on the Billboard Hot R&B/Hip-Hop Songs chart, since its release. The song was nominated for a Grammy for Best R&B Performance by a Duo or Group with Vocals in 2009.

    Charts

    References

    External links

  • Lyrics of this song at MetroLyrics
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