How Hard is Bribery in Elections with Randomly Selected Voters

Many research works in computational social choice assume a fixed set of voters in an election and study the resistance of different voting rules against electoral manipulation. In recent years, however, a new technique known as random sample voting has been adopted in many multi-agent systems. One of the most prominent examples is blockchain. Many proof-of-stake based blockchain systems like Algorand will randomly select a subset of participants of the system to form a committee, and only the committee members will be involved in the decision of some important system parameters. This can be viewed as running an election where the voter committee (i.e., the voters whose votes will be counted) is randomly selected. It is generally expected that the introduction of such randomness should make the election more resistant to electoral manipulation, despite the lack of theoretical analysis. In this paper, we present a systematic study on the resistance of an election with a randomly selected voter committee against bribery. Since the committee is randomly generated, by bribing any fixed subset of voters, the designated candidate may or may not win. Consequently, we consider the problem of finding a feasible solution that maximizes the winning probability of the designated candidate. We show that for most voting rules, this problem becomes extremely difficult for the briber as even finding any non-trivial solution with non-zero objective value becomes NP-hard. However, for plurality and veto, there exists a polynomial time approximation scheme that computes a near-optimal solution efficiently. The algorithm builds upon a novel integer programming formulation together with techniques from n-fold integer programming, which may be of a separate interest.