Approximately strategyproof tournament rules: On large manipulating sets and cover-consistence

We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all ⁿ₂ matches. Prior work defines a tournament rule to be k-SNM-α if no set of ≤ k teams can fix the ≤ ^k₂ matches among them to increase their probability of winning by > α and asks: for each k, what is the minimum α(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-α(k) tournament rule exists? A simple example witnesses that α(k) ≥ ₂^k_k⁻₋¹₁ for all k, and [22] conjectures that this is tight (and prove it is tight for k = 2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(< 1) for all k). Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [22], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1).