Negotiation for Calculating Causal Effects in Bi-Agent Causal Models

In this paper we introduce the paradigm of multi-agent causal models (MACM), which are an extension of causal graphical models to a setting where there is no longer one single computational entity (agent) observing or not observing all the domain variables V. Instead there are several agents each having access to non-disjoint subsets of V. The incentive for introducing cooperative multiagent modeling is, as domains become larger, more complex, open and inherently distributed, building and maintaining a model by a single agent would prove to be very costly or even impossible. In our approach every agent has a semi-Markovian causal model over its local domain variables, determined by an acyclic causal diagram with directed and bi-directed edges and a joint probability distribution over its observed variables. After defining MACMs we introduce an algorithm for calculating the causal effect of variable X on another variable Y from purely observational data if possible in a bi-agent model. This is the effect of manipulating variable X on variable Y. In the algorithm the only communication between agents concerns variables they share in their models or variables X and Y and thus protects the privacy of the individual agent models. Our algorithm is an extension of a single agent algorithm due to Tian and Pearl. As it can happen that the set variables that two agents share are not sufficient to answer some multiagent causal queries, we have developed a negotiation algorithm for agents to cooperatively extend the set of variables they share, while maximally preserving the privacy of the individual agent models.