Electoral competition for the 2+1 electoral rule and the close alternatives

The Anglo-American double-member districts employing plurality-at-large are frequently criticized for giving a large majority premium to a winning party, since the large premium may decrease proportionality of the elected assembly relative to single-member districts. We demonstrate that the premium stems from a limited degree of voters' discrimination associated with only two positive votes on the ballot. To enhance voters' ability to discriminate, we consider alternative electoral rules that give voters more positive and negative votes. We identify strict voting equilibria of several alternative rules in a situation where candidates differ in binary ideology and binary quality, voters' ideology-types are binomially distributed, voters are strategic, and a candidate's policy is more salient than candidate's quality. The most generous rules such as approval voting and combined approval-disapproval voting only replicate the electoral outcomes of plurality-at-large. The best performance in a double-member district is achieved by a rule that assigns two positive votes and one negative vote to each voter (2+1 rule). Under a strict and sincere pure-strategy equilibrium of the 2+1 rule, the second largest group frequently wins the second seat and high-quality candidates gain seats more likely than low-quality candidates. The 2+1 rule increases the scope for a voter's discrimination while avoiding the underdog effects and overstating of preferences associated with an unrestricted number of negative votes.