In social choice there often arises a conflict between the majority principle (the search for a candidate that is as good as possible for as many voters as possible), and the protection of minority rights (choosing a candidate that is not overly bad for particular individuals or groups). In a context where the latter is our main concern, veto-based rules -- giving individuals or groups the ability to strike off certain candidates from the list -- are a natural and effective way of ensuring that no minority is left with an outcome they find untenable. However, such rules often fail to be anonymous, or impose specific restrictions on the number of voters and candidates. These issues can be addressed by considering the proportional veto core -- the solution to a cooperative game where every coalition is given the power to veto a number of candidates proportional to its size. However, the naive algorithm for the veto core is exponential, and the only known rules for selecting from the veto core, with an arbitrary number of voters, violate either anonymity or neutrality. In this paper we present a polynomial time algorithm for computing the veto core and present a neutral and anonymous algorithm for selecting a candidate from it. We also show that a pessimist can manipulate the veto core in polynomial time.

Abstract We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined by R≤S⇔ ∃f∀x, y[x R y↔f(x)S f(y)]. Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a ${\rm{\Pi }}_1^0$ -complete equivalence relation, but no ${\rm{\Pi }}_k^0$ -complete fork≥ 2. We show that ${\rm{\Sigma }}_k^0$ preorders arising naturally in the above-mentioned areas are ${\rm{\Sigma }}_k^0$ -complete. This includes polynomial timem-reducibility on exponential time sets, which is ${\rm{\Sigma }}_2^0$ , almost inclusion on r.e. sets, which is ${\rm{\Sigma }}_3^0$ , and Turing reducibility on r.e. sets, which is ${\rm{\Sigma }}_4^0$ .

Slinko and White, (2008) have recently introduced a new model of coalitional manipulation of voting rules under limited communication, which they call safe strategic voting. The computational aspects of this model were first studied by Hazon and Elkind, (2010), who provide polynomial-time algorithms for finding a safe strategic vote under k-approval and the Bucklin rule. In this paper, we answer an open question of Hazon and Elkind, (2010) by presenting a polynomial-time algorithm for finding a safe strategic vote under the Borda rule. Our results for Borda generalize to several interesting classes of scoring rules.