An Elementary Cross-Impact Model

Cross-impact analysis is a method for revising estimated probabilities of future events in terms of estimated interactions among those events. This report presents an elementary cross-impact model where the cross-impacts are formulated as relative probabilities. Conditions are derived for the consistency of the matrix of relative probabilities of n events. An extension also provides a necessary condition for the vector of absolute probabilities to be consistent with the relative probability matrix. An averaging technique is formulated for resolving inconsistencies in the matrix, and a nearest-point computation is derived for resolving inconsistencies between the set of absolute probabilities and the matrix. Although elementary, the present model clarifies some of the conceptual problems associated with cross-impact analysis, and supplies a relatively sound basis for revising probability estimates in the limited case where interactions can be approximated by relative probabilities.