Multiwinner Election Rules: Axioms and Applications

We consider multiwinner election rules and discuss some of their properties. In particular we discuss the concept of proportional representation of multiwinner election rules. Informally, proportional representation requires that the extent to which a particular preference or opinion is represented in the outcome of elections should be proportional to the frequency with which this preference or opinion occurs within the population. We extend the principle of proportional representation to rankings, as a formal measure of diversity of a collective ranking. There are numerous applications where such diversity within collective rankings is desirable: from displaying a diversified search results to users of a search engine, through accommodating different types of users in rankings provided by recommendation systems, to supporting decision-making processes under liquid democracy or even finding committees of alternatives with additional underlying structure.