A graph-theoretic approach to revealed preference

One of the issues in the impossibility theorem of Arrow is the difference between choice behaviour, as considered by Arrow in most of the illustrations for the conditions in his theorem, and binary relations as dealt with in Arrow's theorem. The relations between choice behaviour and binary relations are studied in revealed preference theory, a theory which originates from consumer demand theory. This paper presents a graph-theoretic approach to revealed preference theory. This is done by considering alternatives as vertices, and choice situations as arcs. By means of this method alternative proofs are obtained for some known results. In particular it is shown that many results from literature can be derived from what may be the main result from revealed preference theory, a theorem of Richter (1966). Next a duality approach is sketched, where vertices and arcs are interchanged as done in dual graph theory. Finally some results are given for non-transitive binary relations. For these there is an increasing interest because of Arrow's theorem