In an election, votes are often given as ordered lists over candidates. A common way of determining the winner is then to apply some scoring system, where each position is associated with a specific score. This setting is also transferable to other situations, such as sports tournaments. The design of such systems, i.e.,  the choice of the score values, may have a crucial influence on the outcome. We study the computational complexity of two related decision problems. In addition, we provide a case study of data from Formula 1 using ILP formulations. Our results show that under some mild conditions there are cases where the actual scoring system has no influence, whereas in other cases very small changes may lead to a different winner. This may be seen as a measure of robustness of the winning candidate.

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The bribery problem in elections asks whether an external agent can make some distinguished candidate win or prevent her from winning, by bribing some of the voters. This problem was studied with respect to the weighted swap distance between two votes by Elkind et al. (2009). We generalize this definition by introducing a bound on the distance between the original and the bribed votes. The distance measures we consider include a restriction of the weighted swap distance and variants of the footrule distance, which capture some realworld models of influence an external agent may have on the voters. We study constructive and destructive variants of distance bribery for scoring rules and obtain polynomial-time algorithms as well as NP-hardness results. For the case of element-weighted swap and element-weighted footrule distances, we give a complete dichotomy result for the class of pure scoring rules.

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