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Cake Division BanachKnaster Procedure

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However, if C trimmed a piece in step 3, he must select that piece if it is ... Now trimmings must be divided using same procedure (apply the trimming procedure ... – PowerPoint PPT presentation

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Title: Cake Division BanachKnaster Procedure


1
Cake Division Banach-Knaster Procedure
  • A cuts ¼ of cake and hands it to B
  • If B feels piece gt ¼, B trims piece placing
    trimmings with remainder of cake and passes cut
    piece to C. If B feels piece is lt or to ¼, then
    B passes piece to C without trimming it.
  • With the piece given to him by B, C does same
    procedure as B, passing the trimmed or untrimmed
    piece onto D.
  • D does same as other players, but D is the last
    player, so if D trims the piece, D keeps it and
    exits the game. Otherwise, the piece goes back to
    the last player who trimmed it or to A if no one
    trimmed it and that player exits the game.
  • Process starts over by cutting another piece of
    approximately ¼ of the original from the
    remaining cake.
  • When down to 2 players, use divide and choose
    scheme. 

2
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3
Cake Division Selfridge-Conway Procedure
  • Independently discovered by Selfridge and Conway
    circa 1960.
  • Player A divides cake into what he considers to
    be 3 pieces.
  • A passes the pieces to player B who trims at most
    1 of the 3 pieces to create a tie for the largest
    piece. B sets aside the trimmings and passes the
    3 pieces to player C.
  • Player C chooses the piece he considers tied for
    largest.

4
Cake Division Selfridge-Conway Procedure
  • Player B must now choose a piece from the
    remaining 2 pieces that he considers tied for
    largest. However, if B trimmed a piece in step 3,
    he must select that piece if it is still
    available.
  • Player A gets the remaining piece.
  • Now trimmings must be divided as follows
  • Player B cuts the trimmings into 3 pieces he
    considers to be the same size.
  • Players choose which of the three pieces they
    want in the following order Player C, A, and
    finally B.

5
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6
Cake Division 1992 Envy-Free Procedure
  • Player A divides cake into what he considers to
    be 5 pieces.
  • A passes the pieces to player B who trims at most
    2 of the 5 pieces to create a 3 way tie for the
    largest piece. B sets aside the trimmings and
    passes the 5 pieces to player C.
  • Player C trims at most 1 of the 5 pieces to
    create at least a 2 way tie for the largest
    piece. C places the trimmings with the other
    trimmings and passes the 5 pieces to player D.
  • Player D chooses the piece he considers tied for
    largest.

7
Cake Division 1992 Envy-Free Procedure
  • Player C must now choose a piece from the
    remaining 4 pieces that he considers tied for
    largest. However, if C trimmed a piece in step
    3, he must select that piece if it is still
    available.
  • Player B must now choose a piece from the
    remaining 3 pieces that he considers tied for
    largest. However, if B trimmed a piece in step
    2, he must select that piece if it is still
    available.
  • Player A now selects from the remaining 2 pieces.
  • Now trimmings must be divided using same
    procedure (apply the trimming procedure an
    infinite scheme) in a finite manner, i.e., keep
    trimming until trimmings are so small no one
    cares.

8
Credits
  • COMAP, For All Practical Purposes, 5th ed
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