Abstract
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.
Highlights
Since Austen-Smith and Banks [1] pointed out that sincere voting may not constitute aNash Equilibrium of a voting game with incomplete information, a series of papers and startling results based on their basic model have followed
The main critique of these authors is the blunt application of game theory to political science: Since these results rely on the pivotality condition, the results so obtained are, put, unrealistic and too surprising to be useful to predict real-world behaviour
The idea that voters determine their votes only by conditioning on them being pivotal, i.e., in the event that they affect the voting outcome, is considered to be too reductive of what drives voting behaviour. This is not as heroic as it is accused of being as an equilibrium condition in a normative sense when strategic voting akin to that defined by Austen-Smith and Banks [1] is considered
Summary
Since Austen-Smith and Banks [1] pointed out that sincere voting may not constitute a. X−i = { x−i = (x j ) j∈I/{i} ∣ x−i ∶ Θ × T−i → {0, 1} } is the set of all vectors of the random piv variables, which is defined over the pivotal voting profiles of all voters except i. { x−i ∣ x−i ∉ X−i and f (Λk ; x−i ) = 1 } is the set of all vectors of the random variables that indicates the correct selected alternative, defined over the non-pivotal voting profiles of the I/{i} voters. I denote by pivi the event that voter i is pivotal, which is the case when t−i ∈ T−i given strategy profile σ−i.
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