# An $ABC$-Problem for location and consensus on graphs

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Martyn Mulder

Econometrisch Instituut, Erasmus Universiteit

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F.R. McMorris

IIT Chicago, and University of Louisville

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Beth Novick

Clemson University

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R.C. Powers

University of Louisville

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
In social choice theory the process of achieving consensus can be modeled by a function $L$ on the set of all finite sequences of vertices of a graph. It returns a nonempty subset of vertices. To guarantee rationality of the process {\em consensus axioms} are imposed on $L$. Three such axioms are Anonymity, Betweenness and Consistency. Functions satisfying these three axioms are called $ABC$-functions. The median function $Med$ on a connected graph returns the set of vertices minimizing the distance sum to the elements of the input sequence. It is a prime example of an $ABC$-function. Recently Mulder \& Novick proved that on median graphs $Med$ is the unique function $ABC$-function. This inspired the {\em $ABC$-Problem} for location and consensus on graphs: $(i)$ On what other classes of graphs is $Med$ the unique $ABC$-function? $(ii)$ If on class $\mathcal{G}$ there are other $ABC$-functions besides $Med$, which additional axioms still characterize $Med$? $(iii)$ Find on $\mathcal{G}$ the other $ABC$-functions, and possibly characterize these.
In this talk we present first non-trivial results for this $ABC$-Problem. Moreover we present some interesting voting procedures on $n$ alternatives.