We study online temporal voting, where a group of voters submit 0/1 approvals on sets of alternatives that arrive online over multiple rounds and a single alternative is chosen in each round. We introduce online variants of two well-known game theoretic properties, strategyproofness (SP) and independence of irrelevant alternatives. We show that online independence of irrelevant alternatives (OIIA) is a sufficient condition for online strategyproofness (OSP), and that several known online voting rules satisfy OIIA and thus OSP, but that they are not SP. In particular, we show that Perpetual Phragmen, the only known online voting rule to satisfy PJR, satisfies OSP. The Method of Equal Shares (MES), a semi-online voting rule knwon to satisfy wEJR, also satisfies OSP. We then introduce the price of manipulability, which quantifies the effect of strategic behaviour on proportional representation guarantees. Finally, we introduce asymptotic satisfaction of proportional representation and show that an online voting rule, Serial Dictator, is fully strategyproof and satisfies proportional justified representation (PJR) up to an additive constant.

翻译：我们研究了在线时序投票，其中一组投票者在多轮在线到达的备选方案集合上提交0/1批准票，每轮选择一个备选方案。我们引入了两个经典博弈论性质——策略防护性（SP）和无关备选方案独立性——的在线变体。研究表明，在线无关备选方案独立性（OIIA）是在线策略防护性（OSP）的充分条件，且若干已知在线投票规则满足OIIA因此也满足OSP，但并不满足SP。特别地，唯一已知满足PJR的在线投票规则Perpetual Phragmen满足OSP，而满足wEJR的半在线投票规则等额份额法（MES）同样满足OSP。我们随后引入可操纵性代价，量化策略行为对比例代表性保障的影响。最后，我们提出比例代表性的渐近满足性，并证明在线投票规则序列独裁者具有完全策略防护性，且以加性常数误差满足比例正当代表性（PJR）。