Cake Division BanachKnaster Procedure

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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. 

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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.

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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.

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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.

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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.

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Credits

• COMAP, For All Practical Purposes, 5th ed