Combinatorial Voting

We study elections that simultaneously decide multiple issues, where voters have independent private values over bundles of issues. The innovation is considering nonseparable preferences, where issues may be complements or substitutes. Voters face a political exposure problem: the optimal vote for a particular issue will depend on the resolution of the other issues. Moreover, the probabilities that the other issues will pass should be conditioned on being pivotal. We first prove equilibrium exists when distributions over values have full support or when issues are complements. We then study limits of symmetric equilibria for large elections. Suppose that, conditioning on being pivotal for an issue, the outcomes of the residual issues are asymptotically certain. Then limit equilibria are determined by ordinal comparisons of bundles. We characterize when this asymptotic conditional certainty occurs. Using these characterizations, we construct a nonempty open set of distributions where the outcome of either issue remains uncertain in all limit equilibria. Thus, predictability of large elections is not a generic feature of independent private values. While the Condorcet winner is not necessarily the outcome of the election, we provides conditions that guarantee the implementation of the Condorcet winner. Finally, we prove results that suggest transitivity and ordinal separability of the majority preference relation are conducive for ordinal efficiency and for predictability.