Complements and substitutes in multilateral assignment markets

I prove that, in assignment markets with more than two sides, agents of different sides (or sectors) need not be complements, whereas agents of the same side need not be substitutes. Shapley (1962) showed that this cannot happen when assignment markets are bilateral. Nevertheless, I found sufficient conditions, that always hold for bilateral markets, that guarantee substitutability and a extended notion of complementarity among agents in arbitrary multilateral assignment markets. I also prove that Shapleys (1962) result always holds regardless the number of sectors of the market when goods in the market are homogeneous.complementarity, homogeneous goods, substitutability, assignment problem, multi-sided assignment markets