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Round 2.
Now there are N-1 players dividing up the remainder of the cake.

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For example, if Bob knew Carol put the pension as her third choice and she really wasn’t going to pick it right away, he might pick the house first. Then the result could be as follows:

Bob: House Carol: Investments

Bob: Pension Carol: Vehicles

In a sense, Bob played insincerely (because he did not pick according to his real preferences) but by doing this he actually did better according to those preferences: he ended up with his first two choices instead of his first and third as he would have by playing sincerely.

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To use the strategy a player must know the other’s preferences.

If the players provided each other with their preferences before taking turns divvying them up, both could use the strategy simultaneously.

The strategy is as follows for each player:

Complete a table representing choices at each turn by alternating filling in a player’s last choice with the other’s least preference starting with whichever player chooses last and working backwards to whichever goes first.

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A cake-division procedure is considered proportional (for all n players), if each player’s strategy guarantees that player a piece of size or value at least 1/n of the whole in his or her own estimation.

Bob, Carol, and Ted will get pieces X, Y, and Z of cake. If Bob cuts the cake, Carol “approves” of piece Y, and Ted “approves” of piece Z, then there is no problem.If Carol and Ted only approve of piece X, then X and Y are rejoined for Carol and Ted to divide and choose while Bob gets piece Z.

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For each object, the following steps are performed:1) The heirs-independently and simultaneously- submit monetary bids for the object 2) The high bidder is awarded the object, and he or she places all but 1/n of his or her bid in a kitty. So, if there are 4 heirs (n=4), then he or she places all but one-fourth– that is, 3/4th– of his or her bid in the kitty

3) Each of the other heirs withdraws from the kitty 1/n of his or her bid.4) The money remaining in the kitty is divided equally among the n heirs.