Deterministic Impartial Selection with Weights

Javier Cembrano, Svenja M. Griesbach, Maximilian J. Stahlberg

Abstract: In the impartial selection problem, a subset of agents up to a fixed size k among a group of n is to be chosen based on votes cast by the agents themselves. A selection mechanism is impartial if no agent can influence its own chance of being selected by changing its vote. It is \(\alpha \)-optimal if, for every instance, the ratio between the votes received by the selected subset is at least a fraction of \(\alpha \) of the votes received by the subset of size k with the highest number of votes. We study deterministic impartial mechanisms in a more general setting with arbitrarily weighted votes and provide the first approximation guarantee, roughly \(1/\lceil 2n/k\rceil \). When the number of agents to select is large enough compared to the total number of agents, this yields an improvement on the previously best known approximation ratio of 1/k for the unweighted setting. We further show that our mechanism can be adapted to the impartial assignment problem, in which multiple sets of up to k agents are to be selected, with a loss in the approximation ratio of 1/2.