There’s hardly sufficient coverage of the evolution of ‘play out’ in smaller dictionaries. So here’s my contribution of how I read it by consulting the biggies.

According to the OED (from which most of the following examples are drawn), ‘play out’ comes from the theatre: to perform to the end.
1596 Shakes. 1 Hen. IV, ii. iv. 531 Out you Rogue, play out the Play.

We then find it used in the passive with the meaning of being performed to the end, and then with the meaning of becoming exhausted, worn out.
1864 (Burton) The drama is not yet entirely played out.
1867 (H. Conybeare) The classical and pointed styles each ran their course from prime to decadence--in fact, ‘played themselves out’ completely.
1887 (Westm. Rev.) About twelve or fifteen years ago he was decidedly of opinion that Mr. Gladstone was played out.
1888 (Lees & Clutterbuck B.C.) Today they had made forty miles over this awful trail, and their horses were not unnaturally quite played out.

The reflexive and intransitive forms are then used with the same meaning:
1884 (J. Quincy) This burlesque... gradually played itself out, and came to an end.
1964 (Mrs. L. B. Johnson, White House Diary) A little past one my enthusiasm played out and I put my head in the pillow.

However, the passive and reflexive forms have for some time been used with the simpler meaning of ‘be enacted, be performed, develop, unfold over time’. No emphasis on exhaustion of any kind.
1854 (S. Austin) The great heroic tragedy which was now being played out on the world's stage.
(Collins Cobuild) ...a political power struggle being played out in Cambodia... The film has eerie parallels with the drama being played out in real life.

According to Cambridge International:
If a situation plays itself out, it develops until nothing more can happen and it is no longer very important: The best thing to do is to stand back and let the crisis play itself out.
(esp. Am.) If a situation plays out, it happens and develops: The debate will play out in the meetings and in the media over the next week or two.
[Note how Cambridge omit the reflexive use and believe the latter to be an American usage.]

The Americans (Webster’s) probably see a different origin to the meaning of ‘unfolding over time’:

- transitive verb
1 a : to perform to the end: played out the guilty role assigned to them.
b : to bring to an end : use up : FINISH: the pie's played out | that graft's played out.
2 : *UNREEL, UNFOLD: played out a length of line*
- intransitive verb
1 : to become spent or exhausted: the twister raged along for about 400 feet before it played out | the pony played out twice.
2 : *to unreel or unfold to a considerable length: the possibility of a snag when hose is playing out.*

In spite of the meagre coverage by the dictionaries, there are thousands of web examples of the reflexive use (play itself out) with the meaning of gradually unfolding over time. The idea of a drama unfolding on stage is behind most of them.

(Daily Telegraph): In this third act of the Blair era, we see the very familiar feuds and rivalries of New Labour now *playing themselves out* at geopolitical level.
(BBC): ... and yet perhaps early advocates of Copernicanism would recognize the debates and contests of ideologies that are *playing themselves out* today.
(From an introduction to Pride and Prejudice): And while the great events and philosophical movements of history *play themselves out* around us, it is our own nature and actions, and the nature and actions of the people around us, that most influence our lives.
(Guardian): A parallel dystopia is currently playing itself out in Britain.

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Note added at 1 day 10 hrs 2 mins (2005-10-29 14:40:11 GMT)
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Re your note ("come to its logical end").
Even without reading the article (http://www.untimely-thoughts.com/?cat=4&type=3&art=1865), I could say that the very structure of the sentence (It is on the basis of this breadth and depth of support for the reunion that the higher elite games play themselves out) points towards the meaning of 'unfold', or simply 'are played' as Tania says. The breadth and depth of support for the reunion feeds these higher elite games. If you say 'come to their logical end', it would mean that support for the reunion is what is putting an end to these games (and that is not what it means, IMO).