We consider judgment aggregation over multiple independent issues, where the chairperson has her own opinion, and can try to bias the outcome by bundling several issues together. Since for each bundle judges must give a uniform answer on all issues, different partitions of the issues may result in an outcome that significantly differs from the "true," issue-wise, decision. We prove that the bundling problem faced by the chairperson, i.e. trying to bias the outcome towards her own opinion, is computationally difficult in the worst case. Then we study the probability that an effective bundling attack exists as the disparity between the opinions of the judges and the chair varies. We show that if every judge initially agrees with the chair on every issue with probability of at least 1/2, then there is almost always a bundling attack (i.e. a partition) where the opinion of the chair on all issues is approved. Moreover, such a partition can be found efficiently. In contrast, when the probability is lower than 1/2 then the chair cannot force her opinion using bundling even on a single issue.

An important element of social choice theory are impossibility theorems, such as Arrow's theorem [1] and Gibbard-Satterthwaite's theorem [2], [3], which state that under certain natural constraints, social choice mechanisms are impossible to construct. In recent years, beginning in Kalai [4], much work has been done in finding robust versions of these theorems, showing that impossibility remains even when the constraints are almost always satisfied. In this work we present an Algebraic scheme for producing such results. We demonstrate it for a variant of Arrow's theorem, found in Dokow and Holzman [5].