A note on pivotality

This note provides simple derivations of the equilibrium conditions for different voting games with incomplete information. In the standard voting game à la Austen-Smith and Banks (1996), voters update their beliefs, and, conditional on their being pivotal, cast their votes. However, in voting games such as those of Ellis (2016) and Fabrizi, Lippert, Pan, and Ryan (2019), given a closed and convex set of priors, ambiguity-averse voters would select a prior from this set in a strategy-contingent manner. As a consequence, both the pivotal and non-pivotal events matter to voters when deciding their votes. In this note, I demonstrate that for ambiguous voting games the conditional probability of being pivotal alone is no longer sufficient to determine voters' best responses.