Are we done with preference rankings? If we are, then what?

Intransitive, incomplete and discontinuous preferences are not always irrational but may be based on quite reasonable considerations. Hence, we pursue the possibility of building a theory of social choice on an alternative foundation, viz. on individual preference tournaments. Tournaments have been studied for a long time independently of rankings and a number of results are therefore just waiting to be applied in social choice. Our focus is on Slater’s rule. A new interpretation of the rule is provided.