The Best of Creative Computing Volume 2 (published 1977)

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Beating the Game (Game theory compares blackjack systems, teaching a computer to play backgammon, Edward Thorp)
by Deitrick E. Thomsen

graphic of page

Game theory compares blackjack systems and proposes to teach a computer

The man who broke the bank at Monto Carlo is a musical fantasy that grew out of
the avid interest many people have in the things that happen on green baize
tables. The man who. according to folklore. was told not to return to Las Vegas
because he had won too much money there is real, He is Edward Thorp, a professor
of mathematics at the University of California at Irvine.

Lately Thorp has been looking for the best way, in a theoretical and practical
sense, to beat the blackjack table. He has devised a way of comparing the
several blackjack systems against each other and a theoretically best possible
system. At the same time his interest has turned to that ancient, but recently
trendy, game, backgammon. He shared some of his latest insights on these topics
with fellow mathematicians at the recent National Mathematics Meeting at

[Image]© HVAS 23

The blackjack systems depend on counting the cards as they fall. As play
proceeds. the deck is depleted, and with the fall of each card the player's
expectation of success changes. Removal of different denominations from the deck
changes expectations in different amounts. Removal of two nines does not have
the same effect as removal of three fives.

From the way the expectations change, a particular numerical value can be
assigned to each denomination of card. As the cards fall from the deck a running
total of these numbers is kept. Different systems assign different numbers to
different denominations. They also differ in how they use the running value
total and the number of cards remaining in the deck. There are various
combinations of addition and division, There is also a difference in whether the
system keeps a separate count of aces. Aces have two values in blackjack. l and
l l, and therefore some system makers like to tally them separately. The result
of all this arithmetic is used to advise the player how to bet.

The question Thorp set himself was whether there is some method of comparing the
different strategies without doing a massive computer simulation of a million
hands. He finds one and he finds a criterion to compare them with each other and
see how close they come to a theoretically possible optimum system.

First he needs a definition of "more advantageous." It may seem obvious that it
means a greater chance of winning, but the case is complicated because a given
system may give a greater expectation of winning when the play is in a
particular stage, but it may be surpassed by another in a different situation.
The final working definition of advantage is a system that gives at least as
good a chance of winning over as wide a range of situations as an alternative
with at least no more risk to the player.

Thorp finds that he can compare the quality of systems by defining an
expectation function for each one that expresses its relative betterness. The
expectation changes as play proceeds. It depends on the fraction of cards
remaining in the deck, and it varies as they fall.

Graphically the expectations define a surface called a simplex, and the falling
of cards causes motion from point to point on this surface as the expectation
changes. Working with the geometry of the simplex Thorp can compare system to
system, and he finds that he can define a single number, which he designates
with the symbol lambda, that expresses a system's betterness relative to others
and its closeness to a theoretically possible optimum system. Thus he has an
analytic method for ranking blackjack systems and no longer has to simulate a
million hands on a computer to compare them. But he does not tell us which is
the best possible system.

Backgammon is among the most ancient games. A set dating to 2600 B.C. has been
found. From the game theore

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