Weighted partial order oriented three-way decisions under score-based common voting rules

Recently, with the trisecting and acting models of human cognitive behaviors, three-way decisions are introduced to deal with uncertain decisions in many applications. To define and conduct the three-way decisions on the possible or necessary loser or co-loser and/or winner or co-winner, we first define the score-based common voting rules, where a score-based function is introduced to evaluate certain utility, including positional scoring rules, maximin, Bucklin, and so on. Then on one hand, it has been proved that under certain conditions three-way decisions on the possible loser or co-loser and/or winner or co-winner are NP-complete with positional scoring rules, maximin, and Bucklin, respectively. On the other hand, we have presented algorithms whose time complexity is polynomial to conduct three-way decisions on the necessary winner or co-winner and/or necessary loser or co-loser with positional scoring rules, maximin, and Bucklin in time O(nm²), O(nm³), and O(nm²), respectively, which has been validated experimentally.