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Game Theory – Cutting a Cake

Cutting a cake had been one of the most popular games in game theory. Cutting a cake is an example of resource sharing, which is extensively studied in economics. The cake division problem was originally developed by Steinhaus in the 1940s. Here the problem is that there will be N users who wish to share a cake into many pieces so that each user gets the maximum satisfaction with the piece that he/ she gets. There will be differences among the preferences of the users regarding the piece, which each wants to have. The division of the cake will result in such a way that each user gets a portion of the cake. If the division of the cake results in such a way that every user gets a piece which is worth at least one fifth of the value of the cake, as perceived by the user, then such a division is called fair division of the cake.

 

In the game, the cake will be represented as an interval I= [0, 1]. The subinterval of this interval will denote a piece of the cake. All such pieces together will collectively constitute the whole cake. The user will do the valuation of the pieces only. The preferences of each user will be represented by the utility function. Each user does not know about the utility function of others. It is assumed that the division of the cake is done by a super user denoted by S .S has no knowledge about the utility functions of different users. It is also assumed that the super user requests each user to cut the cake into ratios based on the specifications of the super user. The super user will be experienced with many such divisions. Hence, his experience enables him to assess how the division of the cake needs to be done so that each user gets his appropriate amount of content. This is thus a fair division of cake. Here, question that can be interesting is the number of cuts needed to perform fair division of cake given that there are N users. The minimum number of cuts is generally preferred for the ease of computation purposes. The divide and conquer process is supposed to be the best cake cutting algorithm. However, it is very hard to determine the required number of minimum cuts for generating a fair cake division, as shown in many studies.

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