Strategic Vote Timing in Online Elections With Public Tallies

TL;DR

This work models sequential online elections with public tallies and costly voting, distinguishing informed voters with a fixed preference from uninformed swing voters who can be swayed by interim results. It proves the existence of equilibria where both early and late voting occur with positive probability and identifies specialized late-voting equilibria (late-bloomer) as well as mixed-timing equilibria where informed voters randomize between timing choices. The analysis derives information-set thresholds that govern when voting early or waiting is optimal, and shows how parameter regimes and arrival probabilities of uninformed voters shape these equilibria. The findings have practical relevance for blockchain governance and online polls, where timing decisions can affect convergence speed, turnout, and the stability of outcomes.

Abstract

We study the effect of public tallies on online elections, in a setting where voting is costly and voters are allowed to strategically time their votes. The strategic importance of choosing \emph{when} to vote arises when votes are public, such as in online event scheduling polls (e.g., Doodle), or in blockchain governance mechanisms. In particular, there is a tension between voting early to influence future votes and waiting to observe interim results and avoid voting costs if the outcome has already been decided. Our study draws on empirical findings showing that "temporal" bandwagon effects occur when interim results are revealed to the electorate: late voters are more likely to vote for leading candidates. To capture this phenomenon, we analyze a novel model where the electorate consists of informed voters who have a preferred candidate, and uninformed swing voters who can be swayed according to the interim outcome at the time of voting. In our main results, we prove the existence of equilibria where both early and late voting occur with a positive probability, and we characterize conditions that lead to the appearance of "last minute" voting behavior, where all informed voters vote late.

Figures (4)

• Figure 1: Schematic depiction of the action space of voters. By waiting, voters view the interim tally and use it when deciding their actions. Informed decision points are denoted by circles, while squares denote uninformed actions that are exogenously determined (i.e., whether to vote early, vote late, or abstain).
• Figure 2: A game tree, where the voting window is two periods long and voting blocs have equal weight. Sloped branches are player moves; cornered branches are other voters' moves. Color indicates interim information sets.
• Figure 3: A schematic depiction of equilibria strategies when ${hyper=false]{uninformed}}_1 = 0$, as given by \ref{['thm:QoZeroStrategiesTurnOne', 'prop:QoZeroStrategiesTurnTwo']}.
• Figure 4: Characterization of parameter regimes where waiting dominates early voting. In the proof of \ref{['thm:WaitingEquilibria']}, we show that the cost threshold that admits such equilibria is lower when uninformed voters have a high probability ${hyper=false]{uninformed}}_1$ of voting early (\ref{['prop:WaitingPtpOne']}), while costs are higher when this probability is low (\ref{['prop:WaitingPtpZero']}, which characterizes two complementary parameter regimes). Intuitively, this is due to the inability of informed voters to influence the uninformed decision. In particular, for the lower cost regime of \ref{['prop:WaitingPtpOne']}, we have that ${hyper=false]{voteProb}_{2,\left\langle0, 0\right\rangle}} = 1$, and for the higher cost regime of \ref{['prop:WaitingPtpZero']} has ${hyper=false]{voteProb}_{2,\left\langle0, 0\right\rangle}} = 0$. Via numeric evaluations, we find that the average threshold cost for the former is $0.57$, and for the latter is $0.76$.

Theorems & Definitions (59)

• Definition 2.1
• Remark 2.2
• Proposition 3.0
• Proposition 3.0
• Proposition 3.0
• Proposition 3.0
• proof
• Proposition 3.0
• proof
• Proposition 4.0