Strategyproof Tournament Rules for Teams with a Constant Degree of Selfishness

We revisit the well-studied problem of designing fair and manipulation-resistant tournament rules. In this problem, we seek a mechanism that (probabilistically) identifies the winner of a tournament after observing round-robin play among $n$ teams in a league. Such a mechanism should satisfy the natural properties of monotonicity and Condorcet consistency. Moreover, from the league's perspective, the winner-determination tournament rule should be strategyproof, meaning that no team can do better by losing a game on purpose. Past work considered settings in which each team is fully selfish, caring only about its own probability of winning, and settings in which each team is fully selfless, caring only about the total winning probability of itself and the team to which it deliberately loses. More recently, researchers considered a mixture of these two settings with a parameter $\lambda$. Intermediate selfishness $\lambda$ means that a team will not lose on purpose unless its pair gains at least $\lambda s$ winning probability, where $s$ is the individual team's sacrifice from its own winning probability. All of the dozens of previously known tournament rules require $\lambda = \Omega(n)$ to be strategyproof, and it has been an open problem to find such a rule with the smallest $\lambda$. In this work, we make significant progress by designing a tournament rule that is strategyproof with $\lambda = 11$. Along the way, we propose a new notion of multiplicative pairwise non-manipulability that ensures that two teams cannot manipulate the outcome of a game to increase the sum of their winning probabilities by more than a multiplicative factor $\delta$ and provide a rule which is multiplicatively pairwise non-manipulable for $\delta = 3.5$.