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In a proportional cake-cutting, each person receives a piece that he values as at least 1/n of the value of the entire cake. In the example cake, a proportional division can be achieved by giving all the vanilla and 4/9 of the chocolate to George (for a value of 6.66), and the other 5/9 of the chocolate to Alice (for a value of 5). In symbols:
It was the first example of a continuous procedure in fair division. The knife is passed over the cake from the left end to the right. Any player may say stop when they think / of the cake is to the left of the knife, the cake is cut and the player who spoke gets that piece. Repeat with the remaining cake and players, the last player gets the ...
Divide and choose (also Cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource ("the cake") and two partners who have different preferences over parts of the cake (both want as much of it as possible).
In the fair cake-cutting problem, the partners often have different entitlements. For example, the resource may belong to two shareholders such that Alice holds 8/13 and George holds 5/13.
In the above example, the order would be: Lemon (0), Chocolate (1), Vanilla+Cherries (4). Start at the point where all cake is given to George (0,30). Move each piece-of-cake in order from George to Alice; draw a line whose slope is the corresponding utility-ratio. Finish at the point where all cake is given to Alice (30,0).
Truthful cake-cutting is the design of truthful mechanisms for fair cake-cutting. The currently known algorithms and impossibility results are shown here. The main cases in which it is unknown whether a deterministic truthful fair mechanism exists are: [10]
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Despite the simplicity of the model, many classic cake-cutting algorithms can be described only by these two queries. On the other hand, there are fair cake-cutting problems that provably cannot be solved in the RW model using finitely many queries. The Eval and Cut queries were first described in the book of Jack M. Robertson and William A ...