Paradoxes in Logic, Mathematics and Computer Science

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Lecture 3

• The Arithmetic of Games

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What games really are

• Starting from a game position, the player moving
  next chooses among several options.
• Each option is itself another game.
• Removing all irrelevant details, a game is
  uniquely determined by the possible options for
  each of the players Left and Right. Thus
• Definition A game G is a pair (L,R), where
• L is the set of all options for Left, and
• R is the set of all options for Right.
• Since options are games, this definition is
  recursive.

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A Recursive Definition!

• Definition A game G is a pair (L,R), where
  L and R are sets of games denoting the options
  for Left and Right, respectively.
• If Left plays next and L , then Right wins.
  (Here ? denotes the empty set.)
• If Right plays next and R , then Left wins.
• Notes
• The definition of the game does not include the
  information of whos moving next.
• Losing is defined by running out of moves.

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What could a recursive definition do?

• Definition A game G is a pair (L,R), where L and
  R are sets of games.
• Though the above definition is recursive, we get
  the following examples
• The first game constructed is 0 (,) (name
  to be later justified). In 0 whoever moves first
  loses.
• We can also form the games
• 1 (0,) L has one move, R has none. L wins.
• ?1 (,0) R has one move, L has none. R
  wins.
• ? (0,0) Whoever moves first wins.

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More Examples

• 0 (,) Whoever moves first loses.
• 1 (0,) L has one move, R has none. L wins.
• ?1 (,0) R has one move, L has none. R
  wins.
• ? (0,0) Whoever moves first wins.
• 2 (1,) Two moves for L. L wins.
• 3 (2,) Three moves for L. L wins. Etc..
• Note 3 ((((,),),),). Also,
• ?2 (, ?1) Two moves for R. R wins.
• ?3 (, ?2) Three moves for R. R wins. Etc..

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A Simpler Notation

• If G (GL1,GL2,,GR1,GR2,), we can simply
  denote it by G GL1,GL2,GR1,GR2,, as if G
  is just a set with left and right elements. Thus
  
• 0
• 1 0
• 2 1
• 3 2, etc..
• ?1 0
• ?2 ?1
• ?3 ?2, etc,..

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Addition of Games

• If G GL1,GR1, and H HL1,HR1, are
  games, we define
• GH GL1H,,GHL1,GR1H,,GHR1,
• Meaning GH is both G and H played in parallel.
  Each player chooses to move in any one of them,
  leaving the other unchanged.
• Notes
• This is why in the definition of the game we have
  not specified whos next.
• The above definition is again recursive!

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Examples of Game Sums

• If G GL1,GR1, and H HL1,HR1, are
  games, we define
• GH GL1H,,GHL1,GR1H,,GHR1,
• Examples
• 1 1 0 0 01,10 1 2
• 2 1 1 0 11,20 2 3,
  etc..
• 1 (?1) 0 0 0(?1)10 ?11,
  which behaves like 0 (whoever moves loses).
  
• Exercises Show that G 0 G G
• G H H G and (G H) F G (H F)

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The Negation of Games

• If G GL1,GR1,, we define
• ?G ?GR1, ?GL1,
• Meaning ?G is the same as G when the players
  interchange roles.
• The definition is again recursive!
• Note This fits well in the definitions of the
  games ?1, ?2, ?3, etc..
• E.g. ?2 ?1 ?1. Thus all number names
  of the games are justified.
• Also, ?0 ? 0.

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Zero Games

• Since in the game 0 , whoever moves first
  loses, we define
• Definition A game G is called a zero game iff
  whoever moves first loses, i.e. the second player
  has a winning strategy.
• We then simply write G 0. Are we allowed to do
  that?
• Example 1 (?1) 0
• Exercise Show that in general G (?G) 0.
• Note You can beat a chess master by playing
  Chess (?Chess) and letting him start.

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Exercise

• Recall that the integer number games are defined
  recursively as follows
• 0 ,
• n1 n,
• ?(n1) ?n
• Show that
• 1) If n is a positive natural number, then n
  (?n) 0.
• 2) If m and n are two positive natural numbers,
  then ?mn 0.

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• Thank you for listening.
• Wafik