EXAMPLE 3 The Steinhaus Proportional Procedure for Three Players (Lone Divider)

Given three players—Bob, Carol, and ted—we have Bob divide the cake into three pieces (call them , , and ), each of which he thinks is a size or value of exactly one- third. Let’s speak of Carol as “approving of a piece” if she thinks it is of a size or value of at least one-third. Similarly, we will speak of ted as “approving of a piece” if the same criterion applies. Notice that both Carol and ted must approve of at least one piece.

If there are distinct pieces—say, and —with Carol approving of and ted approving of , then we give the third piece, , to Bob (and, of course, to Carol and to ted), and we are done. the problem case is where both Carol and ted approve of only one piece and it is the same piece.

Let’s assume that Carol and ted approve of only one piece, , and hence (of more importance to us) both disapprove of piece . Let denote the result of putting piece and piece back together to form a single piece. Notice that both Carol and ted think thaf is at least two-thirds of the cake because both disapprove of . thus, we can give to Bob and let Carol and ted use divide-and-choose on . Because half of two-thirds is one-third, both Carol and ted are guaranteed a proportional share (as is Bob, who approved of all three pieces).