Talk:Determinacy

Rationale for this page

I can easily imagine someone stumbling across this page and thinking it's an unnecessary duplication of Axiom of determinacy, an article of long standing, and ask why I don't simply try to improve that page.

Well, that was my thought originally, but I was running into all sorts of difficulties figuring out where things should go. Axiom of determinacy didn't have a good definition of "game", "strategy", or "winning strategy", and it wasn't clear where to put them. There's a winning strategy article, but it wants to be useful to game theorists as well, which was too much of a constraint.

Moreover, lots of times one wants to talk about determinacy but is not, in context, particularly interested in the axiom of determinacy. In particular, several pages had links that looked like [[Axiom of determinacy|determinacy]], when AD was perhaps not the main point of interest.

So it occurred to me that there should be a central article about determinacy in general. This is it, or rather the start of it. My vision is that this article will give the basic notions and say something about how they stand in relation to each other. Then there will be links out to more detailed articles on particular topics, such as axiom of determinacy, axiom of real determinacy, projective determinacy, long games, Blackwell games, axiom of quasideterminacy.

This article should be the central reference point for the new Category:Determinacy. If a major aspect of determinacy theory has been left out, then ideally in addition to writing an article on that particular topic, a new section or subsection should be added to this article, describing the topic briefly and saying how it relates to the other determinacy topics. --Trovatore 05:46, 2 September 2005 (UTC)[reply]

Too technical

This page needs some help to make it more accessible. I have a semi-math background (CS + statistics) and I have no idea what is going on. This needs a bit of a gentler intro (like the one for NP vs. P article which is very good ) novacatz 13:17, 13 January 2006 (UTC)[reply]

Well, that's what I had in mind for the Determinacy#Basic notions section. That strikes me as being of roughly the same level of technicality as the lead section of Complexity classes P and NP. Would you look over the "Basic notions" section again and see if you agree, and if not, let me know where it is you get lost? (As for the rest of the article, I wouldn't really expect it to be accessible to anyone without a set theory background; it's an inherently technical subject.) Thanks, Trovatore 00:13, 14 January 2006 (UTC)[reply]

Currently, the basic notions section reads like a math textbook (cf. 'firstly we shall consider...' and 'More formally, ... , recall that the latter consists of...'). If you look at the gentle introduction of NP vs. P article intro, you will see they spend a lot of time explaining what is the problem using layman English (cf. the 'time space tradeoff' has an explanation what 'time' mean and what 'space' means. You can see that the page has a lot of somewhat loose descriptions that are useful on helping people grasp why the area is interesting (eg. 'NP Complete problems can be loosely described as...'). For determinacy - the intro is a bit weak on why the field is important (some examples might help) and starts getting into the results without explaning why they are significant (eg. determinacy from ZFC). Someone with some elementary set theory knowledge (me) can appeciate this result a lot more if some explaination is provided on interepetations of the result. novacatz 04:58, 14 January 2006 (UTC)[reply]

Query

What does this phrase mean? (for example, if we say that Black wins draws at chess) Stephen B Streater 07:54, 1 April 2006 (UTC)[reply]

It means, if we modify the rules of chess, keeping everything the same except that if the play ends in a draw according to traditional chess, we'll say that Black has won. It hadn't occurred to me that this wasn't clear. Can you think of better, but still hopefully concise, language? --Trovatore 15:28, 1 April 2006 (UTC)[reply]

I misread "draws" as a verb like "wins". It makes sense as a noun. How about "wins all draws"? Stephen B Streater 16:37, 1 April 2006 (UTC)[reply]

Or "wins all drawn positions". Stephen B Streater 16:37, 1 April 2006 (UTC)[reply]

Go for it. You might also mention that Black should win if play continues forever (that's optional because it doesn't really matter; in any play that continues forever there are opportunities for Black to claim a draw, and therefore a win). --Trovatore 16:40, 1 April 2006 (UTC)[reply]

I've made the first change. Is your second point about play continuing for ever referring only to the modified chess (ie repeated positions leading to a draw)? Presumably there are games which can go on for ever without leading to a draw eg two sides take turns to write down numbers and the one with the biggest number written down wins. I'll have to think this game through to see if it counts as a game in this article, as the game may never finish but it never be obvious this is going to happen so never be a draw either. Stephen B Streater 18:37, 1 April 2006 (UTC)[reply]

Oh yes, that point was specific to chess. In most of the interesting cases studied, play can go on forever without any finite part of the play determining the winner. (However I don't really follow your "biggest number" example; why should the set of numbers written down have a greatest element?) --Trovatore 18:40, 1 April 2006 (UTC)[reply]

Thanks - I'll put something in, though the chess example is getting quite long, particularly as we can't necessarily assume that everyone understands the draw-by-repetition rule - I might put all the chess bits together and not in brackets. The game I described may not have a biggest element, but it may do. But if it did have a biggest element, I was thinking you wouldn't know in finite time, so couldn't end the game. But then perhaps you could find the maximum element in an infinite set. Stephen B Streater 19:20, 1 April 2006 (UTC)[reply]

An infinite set of naturals cannot have a maximal element. However, you could make rules like:

If there is a greatest number played, then the player who played it, wins. If both players have played that number, then the first one to play it wins.
If there is no greatest number played, then the second player wins.

This game is trivially a win for the second player, who can guarantee that no greatest number is played, just by playing ever-bigger numbers on subsequent plays. But it is an example of a game whose winning condition is not determined by any finite part of the play (after any finitely many moves, it is still possible for either player to win; while the second player can always force a win if he wants to, there's no guarantee that he will). --Trovatore 19:26, 1 April 2006 (UTC)[reply]