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I am working on a model that can be defined as a utility optimisation problem but I'm struggling with the enunciation and notation.

The model should describe how the utilities of a set of agents A={1,2,...,n} increase with the availability of a larger set of product types P={1,2,...n}, while the utilities of a supplier S decrease with the size of P. The ideal state for A is Pn and for S is P1 (one product type for each agent v/s a single product type for all). This assuming that each product type can potentially give a certain amount of utilities to each agent, so for A1 utilities might be u(P1) > u(P2), for A2 utilities might be u(P1) < u(P2), and for A3 utilities might be u(P1) = u(P2), etc. The utilities perceived are different for each agent and they are not indifferent to any product, so if P needs to be reduced to e.g. 5 types, the problem of choosing element types is not irrelevant.

Then, the (optimisation-like) problem is to maximise utilities for An finding the best combination of products among the whole 'virtual' set Pn if the 'actual' set P has to have no more than x number of elements.

Is this more or less ok or am I completely lost? How can I express it as a utility maximisation equation? Any help/directions refining this would be great help.

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