On combining approval with disapproval voting

In this paper we compare two nonranked voting methods: regular approval voting (RAV) and combined approval voting (CAV). Under RAV a voter has two options with respect to each of the competing alternatives—cast one vote in favor or abstain, whereas under CAV he has three options—cast one vote in favor, abstain, or cast one vote against. Under both voting methods it is assumed that one must elect one out of three or more competing alternatives and that the alternative obtaining the largest (net) number of votes is elected. The comparison between these two voting methods shows: 1. Given that a voter is in a decisive position, he will prefer to operate under CAV than under RAV. No voter under CAV will choose to abstain with respect to any of the alternatives. 2. If all voters employ their dominant strategies the final outcome under both voting systems will be the same. 3. For any number of voters and competing alternatives, a single (nth) voter has a smaller probability of not being decisive under RAV and a larger probability of being strongly decisive under CAV. Consequently, the choice between RAV and CAV should depend on: (i) the magnitude of the difference between the expected utility a voter enjoys when he is strongly decisive and the expected disutility he suffers when he is not decisive; (ii) whether, ceteris paribus, a voter derives any psychological satisfaction from being able to vote against disapproved alternatives (CAV) instead of merely abstaining (RAV).