Coalition strategy-proof anonymous binary social choice in a countably infinite society

Abstract

We study anonymous binary social choice functions on the universal weak-preference domain when the society of voters is countably infinite. Voters may strictly support one of the two alternatives or be indifferent, and non-manipulability is required in the form of coalitional strategy-proofness. Full anonymity reduces the relevant information about a coalition not to its cardinality alone, but to its cardinality signature: finite of size \(k\), infinite with infinite complement, or cofinite with complement of size \(k\). Recording the signatures of the two strict-support coalitions yields an infinite triangular grid, ordered by increasing support for one alternative and decreasing support for the other. We prove that anonymous coalition strategy-proof binary social choice functions are exactly those obtained from super order closed subsets, i.e. upper sets, of this grid. We then give a geometric classification of these subsets. Unlike in the finite-voter case, they need not be generated by their minimal elements: countable infinity creates missing-boundary regions corresponding to finite opposition or cofinite strict support. We prove an exact representation of these subsets by genuine or missing-boundary corners, and derive a nonredundant parametrization by an antichain and, when necessary, one exceptional missing-boundary region. This yields four generalized zigzag forms and completes the infinite-society counterpart of the known geometric representation for finite electorates.