The aim of the ontology alignment problem is to find meaningful correspondences between two ontologies represented as collections of entities. This problem can be modelled as a novel mechanism design problem on an edge-weighted bipartite graph, where each side of the graph holds each agent's private entities, and the objective is to maximise the agents' social welfare. Having studied implementation in dominant strategies with and without payments, we report on findings that for truthful mechanisms, these problems need to be solved optimally. We also study greedy allocation rules with a first-price payment rule, and implementation in pure, mixed and Bayesian Nash equilibria, and have found tight bounds on the price of anarchy and stability.