Cake cutting: Explicit examples for impossibility results

In this article we suggest a model of computation for the cake cutting problem. In this model the mediator can ask the same queries as in the Robertson–Webb model but he or she can only perform algebraic operations as in the Blum–Shub–Smale model. All existing algorithms described in the Robertson–Webb model can be described in this new model. We show that in this model there exist explicit couples of measures for which no algorithm outputs an equitable fair division with connected parts. We also show that there exist explicit sets of measures for which no algorithm in this model outputs a fair division which maximizes the utilitarian social welfare function. The main tool of our approach is Galois theory. •In this article we suggest a model of computation for the cake cutting problem.•The mediator can ask the same queries as in the Robertson–Webb model.•The mediator can perform algebraic operations as in the Blum–Shub–Smale model.•We give explicit examples of impossible fair divisions in this model.•The main tool is field theory.