Our work proposes a unique rule that barely, however nonetheless not optimally, reduces the potential positive factors from manipulation. The theoretical rule we suggest identifies a small set of groups as “vital”, which we outline as groups that might type coalitions that result in a staff within the coalition being the clear winner.
We classify every attainable match final result into one in every of 5 buckets relying on how shut it’s to having a staff win all their video games. If the match has a staff that wins all their video games, we declare that staff the winner. If the match is much from having a staff that wins all their video games, we select a winner uniformly at random. For tournaments which are “shut” to having a staff that wins all their video games, we determine these groups that might considerably achieve from manipulating the match. We show that there usually are not many such groups, and for every “shut” match, we assign particular chances to every staff in order that the positive factors from manipulation for teams of measurement three are 50% at most (and nonetheless the optimum 33% for teams of measurement two).
One frequent concern about our mannequin is that we assume all outcomes of the bottom reality are deterministic, e.g., A at all times beats B, and this will not align with actual tournaments. In any case, underdogs at all times have an opportunity! Do our outcomes maintain if we permit for randomized outcomes, e.g., A beats B 80% of the time? It seems that, because of prior work, the reply to this query is “sure” as a result of the worst case situations are these with deterministic outcomes. Associated work reveals that if we limit the win likelihood of all video games to, say, the 60–40% interval, then we will anticipate to lower the positive factors from manipulation because the match turns into extra aggressive.
In one other try to beat the impossibility end result we offered earlier than, we introduce a new mannequin for figuring out which manipulations are useful. Within the mannequin outlined to date, groups within the manipulation coalitions deal with their joint likelihood of successful as a uniform mass. That’s, a staff within the coalition doesn’t care which different staff’s probabilities of successful go up or down, even whether it is their very own — an assumption that’s unlikely since groups naturally care about their very own likelihood of successful, or are at the least a little bit egocentric. To mannequin the idea that groups in manipulating coalitions are nonetheless a little bit egocentric, we introduce weights to the manipulation calculations to replicate this.
We noticed that if every staff weights their very own probabilities of successful twice as a lot as that of the opposite groups within the manipulating coalition, there exist guidelines that fulfill properties 1 and three for tournaments with at most six groups. We conjecture that, below this mannequin, there might certainly exist guidelines that fulfill properties 1 and three precisely. We additionally present that for a number of standard guidelines, a big weight is required for the rule to fulfill properties 1 and three.