The product paradox of voting power

We consider a system made up of two committees, with overlapping membership but with different rules for passing decisions; the passage of a joint resolution requires separate approval by each committee. We show that it is possible for one voter a to be stronger (have greater voting power) than another voter b in both of the two committees, while in the two-committee system (considered as a single composite entity) b is stronger than a. We argue that this apparently paradoxical phenomenon is inherent in the structure of the decision-making system in question, rather than being an artefact of any particular index used to measure voting power.