Heuristics for deciding collectively rational consumption behavior

We propose a directed graph for testing whether observed household consumption behavior satisfies the Collective Axiom of Revealed Preferences (CARP). More precisely, the data satisfy CARP if the graph allows a node-partitioning into two induced subgraphs that are acyclic. We prove that partitioning the obtained graph into two acyclic subgraphs is NP-complete. Next, we derive a necessary condition for CARP that can be verified in polynomial time, and we present an example to show that our necessary and sufficient conditions do not coincide. We also propose and implement fast heuristics for testing the sufficient condition. These heuristics can be used to check reasonably large data sets for CARP, and can be of particular interest when used prior to computationally demanding approaches. Finally, based on computational results for both real-life and synthetic data, we conclude that our heuristics are effective in testing CARP.