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IN POKER, each
of the players makes a series of strategy decisions upon which the
outcome of the game depends. Poker thus comes within the scope of
the "theory of games" originally developed by John von
Neumann.Unquestionably, one can
be a good poker player without understanding this theory. In fact,
the "game-theoretic" considerations are usually outweighed
by simple psychological considerations when a hand of poker is played
at the table instead of on a mathematician's note pad.
However, as the theory of games
may have significant effects on the play of a hand of poker, and
since an understanding of the basic results of this theory may provide
an insight into the nature of the game, I will describe some of
the basic game-theoretic considerations of poker.Matching
coins. Imagine that A and B want to "match pennies." Each
player has a stack of pennies (say) which he prepares in advance,
the coins being placed in the stack so that either a head or a tail
will appear when it is exposed.
They play a game in which each exposes the coins of his stack one
at a time; if, on comparison, the faces match, A wins those two
coins; if the faces do not match, B wins the two coins.Clearly this
is a fair game, with equal chances for A and B. On any one comparison,
each has an equal chance to win, and each bet is made at even money.
One question we might ask with
relation to this game is the following: Is there any strategy that
A or B might adopt (in the stacking of his coins) so as to place
a limit on his losses, or to guarantee a certain minimum gain in
the long run? The answer to this is yes. Furthermore, the most important
significance of von Neumann's theory is that in any two-person,
choice-of strategy game in which, at each phase, one player wins
what the other loses (and vice versa), such a strategy (called an
optimum strategy) exists for each player. |
