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[BLANK_AUDIO]

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Having discussed single player games,
let's turn now to multiple player

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games.
Such as chess and Othello, and Go,

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and Chinese checkers, and so forth.

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In most cases, the other players in

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these games are general game playing
programs themselves.

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Or in some cases, humans.

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However in some cases, the other players
represent uncertainty

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in the game.

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For example, it's common to model some
card games by representing a

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randomly shuffled deck of cards as an
additional player in the game.

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One that deals, or reveals, cards as the
game progresses.

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Before proceeding, it's worth emphasizing
that multiple

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player games need not be fixed sum.

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In a fixed sum game, the total number of
points is fixed.

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When this number is zero, such games are
often said to be zero sum.

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In order for one player to get

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more points, some other player must lose
points.

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For this reason fixed sum games are
necessarily competitive.

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In general game playing there's no such
restriction.

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Some games are competitive, most games are
competitive, but some are cooperative.

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It may be that the only way for one player
to get a

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higher reward is to help the other players
to get higher rewards as well.

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This is sometimes called variable sum
games.

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In this lesson, as in the preceding
lesson, we look at settings

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in which there's sufficient time for
players to search the entire game tree.

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That said, as in single player games it's
sometimes possible

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to find optimal actions even without
searching the entire game tree.

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Now the small size does rule out some
interesting

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games like chess and Othello, which
require more complicated techniques.

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And we'll discuss some of those later.
However, again, the more

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complicated techniques can be viewed as

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variations on the simple techniques
described here.

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Begin this lesson with a procedure called
minimax,

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and a more efficient version called
bounded minimax.

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We then turn it to an even more efficient
procedure called alpha-beta search.

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Which produces the same results, but
eliminates much

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of the needless computation of minimax and
bounded minimax.

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There's also an analog to sequential
planning called conditional planning.

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However it's a little complicated and it's
not used

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all that often, so we'll bypass that for
now.