With k strategic liars out of n clients, Nash equilibria always exist with price of anarchy at most (n+2k)/(n-2k) (nearly tight), and strong equilibria exist on lines with cost at most (n+k)/(n-k) times optimal.
Truthful facility assignment with resource augmentation: An exact analysis of serial dictatorship.Mathematical Programming, 203(1):901–930, 2024
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Strategic Facility Location with Limited Liars
With k strategic liars out of n clients, Nash equilibria always exist with price of anarchy at most (n+2k)/(n-2k) (nearly tight), and strong equilibria exist on lines with cost at most (n+k)/(n-k) times optimal.