Supervisor: Dr Felix Fischer
Project description:
Social choice theory is the area of economics concerned with the aggregation of individual preferences into a collective decision. From a mathematical perspective it studies functions from preferences to decisions, and both preferences and decisions can typically be expressed in terms of combinatorial objects like sets, vectors, or graphs.
Identifying functions with good properties, or proving that functions with certain properties do not exist, often requires a deep understanding of the underlying combinatorial structures. In the past, this has taken the form of a result for a particular structure that has been proved using ad-hoc techniques. The project will work toward a unified theory of the combinatorial structures underlying social choice and toward general techniques for the characterisation of social choice functions. This will involve bringing techniques and principles from extremal combinatorics such as probabilistic and linear algebraic methods into the realm of social choice.
A particularly intriguing class of functions are those on the discrete hypercube (the graph whose vertices are all 0-1 strings of length d with two vertices adjacent if they differ in exactly one position). The discrete hypercube is an important and much-studied combinatorial object, so mathematically this class is a natural choice. The class also covers many settings of practical interest, for example voting based on preferences given by a stance on each of d binary issues.
Some background in combinatorics and graph theory is highly desirable.
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