Note on compromise axiom

The concept of compromise has been present in the theory of social choice from the very beginning. The result of social choice functions as such is often called a social compromise. In the last two decades, several functions of social choice dedicated to the concept of compromise, such as Fallback bargaining, Majoritarian compromise, Median voting rule or $p$-measure of compromise rules, have been considered in the literature. Furthermore, compromise axioms were formed in several attempts. However, we believe that the previous formalizations of compromise did not axiomatically describe this feature of the social choice functions. In this paper we will follow the line of thought presented by Chatterji, Sen and Zeng (2016) and form a weak and strong version of a Compromise axiom, one that should capture understanding of compromise based on an ability to elect a winner which is not top-ranked in any preference on a profile. After that we will analyze an interaction of those axioms and established social choice functions. We will show that the division of SCFs in three classes with respect to these axioms fairly reflect relationship between those SCFs and colloquial expectations from notion of compromise. We then compare the defined axioms with the compromise axioms of Börgers and Cailloux. Finally, for SCFs that satisfy the strong compromise axiom, we define a compromise intensity function that numerically expresses the degree of tolerance of the SCF for choosing a compromise candidate.