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.
Strategy-proof location on a network.Journal of Economic Theory, 104(2):405–428, 2002
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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.