The Best of Creative Computing Volume 2 (published 1977)

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Beating the Game

graphic of page

tician's point of view, Thorp says, the hope of solving the problems that it
presents is quite good. 

The game consists of a ladder of 26 cells. Of these, 24 appear on an actual
backgammon board. For the analysis, Thorp adds, the two "off the board" spaces
at each end into which counters that have successfully completed their journeys
are put. Each player's counters start at one end of the board, and the object is
to get them all across the board and off it, passing the other player's counters
coming in the opposite direction, before he gets his across. Moves are
determined by throwing dice.

There is an important complication. If counters of both players arrive in the
same cell, there are situations where one can be sent back to the beginning of
its trip. This possibility of repeated restarts makes the game in a theoretical
sense potentially infinite. In principle a backgammon match could last forever.
It is "a fact which will impede analysis slightly," Thorp concedes.

The way to analyze the game is to set up partial models that are simplified,
removing complexities of the real game, especially the one that makes it
infinite, and then gradually to add back the complications. In Model I each
player has one counter, and the bounce-back rule is suspended so that the
counters can freely pass each other. When this is properly set up it produces a
game of 167 steps. The first to reach step I67 wins. Computer simulation shows
that when goals are equal, the first player to roll has a slight advantage, but
this declines toward even chance as play proceeds. This is a very crude approach
to a real backgammon game, but it leads to interesting insights, Thorp says.

In Model II one sets up an end game. Again there are two counters, but they have
already passed each other so there is no further chance that they could be sent
back to their starting points. This too is a finite game and is amenable to

One complication of the real game is the doubling cube. As the game proceeds, if
one player gains a certain advantage, he can use the doubling cube to double the
stakes. This changes the consequences for the loser and alters the expectations
and strategy of play. Recursion schemes can be devised to solve both Model I
with the doubling cube and the end game with the doubling cube. (A recursion
scheme is a system for calculating a series of related values. Knowing the First
number in the series and the recursion scheme you can calculate the second.
Putting the second number into the recursion scheme gets you the third. And so

In actual backgammon play it is possible that the game might come down
eventually to Model I or Model II, but
these highly restricted situations are still far from the complexity of a full
game, in which each player has several counters on the board at once and the
bounce-back rule can operate. Still enough has been learned so far for Thorp to
conclude that backgammon "can be played better by computer than by any person."
But suppose the computer refuses to go to Reykjavik? 

Reprinted with permission form Science News. Copyright 1975 by Science Service,

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