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From: ark@rabbit.UUCP (Andrew Koenig)
Newsgroups: net.chess
Subject: Re: Rule variations
Message-ID: <2715@rabbit.UUCP>
Date: Sun, 15-Apr-84 10:36:37 EST
Article-I.D.: rabbit.2715
Posted: Sun Apr 15 10:36:37 1984
Date-Received: Sun, 15-Apr-84 23:35:34 EST
References: <256@heurikon.UUCP>
Organization: AT&T Bell Laboratories, Murray Hill
Lines: 38

How about using a doubling cube, as in backgammon?

For those who don't play backgammon, I'll explain.  A doubling cube is
a die whose sides contain 2, 4, 8, 16, 32, and 64.  You start the game
with the cube in the middle of the table, signifying that no one
'owns' it yet.

If your opponent does not own the cube, then any time it is your turn to
move, you can offer a double BEFORE you move.  This means that you take
the cube, turn it so the next higher power of 2 shows on top (you use
2 the first time, of course), and present it to your opponent.  Your
opponent has the following options:

	1.  Refuse the double by resigning.  The opponent
	loses the old (i. e. undoubled) stake.

	2.  Accept the double.  The opponent puts the cube on
	his side of the board, and play continues at twice the
	previous stake.  Your opponent now owns the cube, and
	you therefore cannot double again until you have accepted
	the next double.

	3.  Redouble.  This is the same as accepting the double, but
	your opponent gives the cube a second twist before accepting
	it and the game continues at four times the stake.  (some
	backgammon players do not allow redoubles)

You might think that it is never right to accept a properly offered
double, but this is not tue.  Suppose you consider the odds to be
about 2:1 that you will win.  Then your expectation if you do not
double is 2/3 if you win minus 1/3 if you lose, or 1/3.  If you double,
your expectation is 1 if the double is refused and 2/3 if it is accepted,
so you certainly should double.  Further, your opponent should accept
the double for exactly the same reason:  his expectation is -1 if he
refuses and -2/3 if he accepts.

This overly simplistic analysis does not take cube ownership or redoubling
into account.