Question 13.61

31. Here are the steps in the Selfridge-Conway procedure for three players.

  • Stage 1. The initial division

    • Step 1. Player 1 cuts the cake into, what in his view, is three equal pieces.
    • Step 2. Player 2, if he thinks one piece is largest, trims from that piece to create what he believes is a two-way tie for largest piece. The trimmings are set aside. If Player 2 thinks that the original split was fair, he does nothing.
    • Step 3. Player 3 may choose any piece.
    • Step 4. Player 2 chooses a piece. If the trimmed piece remains, he must choose it. If not, he chooses the one he feels is tied with the trimmed piece for largest.
    • Step 5. Player 1 gets the remaining piece.

  • Stage 2. Dividing the trimmings

    Assume that Player 3 received the trimmed piece in Stage 1.

    • Step 6. Player 2 divides the trimmings into what he considers three equal parts.
    • Step 7. Player 3 chooses one part of the trimmings.
    • Step 8. Player 1 chooses a piece of the trimmings.
    • Step 9. Player 2 receives the remaining trimmings.
      1. Explain why Player 1 is envy-free after Stage 1.
      2. Explain why Player 2 is envy-free after Stage 1.
      3. Explain why Player 3 is envy-free after Stage 1.
      4. Explain why Player 1 is envy-free after Stage 2.
      5. Explain why Player 2 is envy-free after Stage 2.
      6. Explain why Player 3 is envy-free after Stage 2.

(Thanks to Michael Rosenthal for suggesting this exercise topic.)

31.

(a) This is because Player 1 views all three pieces as being of equal size or value.

(b) Player 2 created a two-way tie for largest or most valuable piece. Player 3 may take one of these, but at least one will still be available for Player 2 to choose.

(c) This is because Player 3 gets to choose first.

(d) Player 1 will not envy Player 3 because Player 3 received the trimmed piece, and so Player I’s piece is as large in his eyes as Player 3’s piece plus all of the trimmings. Player 1 will not envy Player 2 because he is choosing before Player 2.

(e) This is because Player 2 views all three pieces of the trimmings as being of equal size or value.

(f) This is because Player 3 gets to choose first.