Computing the Best Policy That Survives a Vote
Abstract
An assembly of $n$ voters needs to decide on $t$ independent binary issues. Each voter has opinions about the issues, given by a $t$bit vector. Anscombe's paradox shows that a policy following the majority opinion in each issue may not survive a vote by the very same set of $n$ voters, i.e., more voters may feel unrepresented by such a majoritydriven policy than represented. A natural resolution is to come up with a policy that deviates a bit from the majority policy but no longer gets more opposition than support from the electorate. We show that a Hamming distance to the majority policy of at most $\lfloor (t  1) / 2 \rfloor$ can always be guaranteed, by giving a new probabilistic argument relying on structurepreserving symmetries of the space of potential policies. Unless the electorate is evenly divided between the two options on all issues, we in fact show that a policy strictly winning the vote exists within this distance bound. Our approach also leads to a deterministic polynomialtime algorithm for finding policies with the stated guarantees, answering an open problem of previous work. For odd $t$, unless we are in the pathological case described above, we also give a simpler and more efficient algorithm running in expected polynomial time with the same guarantees. We further show that checking whether distance strictly less than $\lfloor (t  1) /2 \rfloor$ can be achieved is NPhard, and that checking for distance at most some input $k$ is FPT with respect to several natural parameters.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.00660
 arXiv:
 arXiv:2303.00660
 Bibcode:
 2023arXiv230300660C
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 Accepted by AAMAS'23