Our work proposes a distinct 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 “important”, which we outline as groups that might kind coalitions that result in a group within the coalition being the clear winner.
We classify every doable event end result into one in every of 5 buckets relying on how shut it’s to having a group win all their video games. If the event has a group that wins all their video games, we declare that group the winner. If the event is way from having a group that wins all their video games, we select a winner uniformly at random. For tournaments which can be “shut” to having a group that wins all their video games, we establish these groups that might considerably acquire from manipulating the event. We show that there are usually not many such groups, and for every “shut” event, we assign particular chances to every group 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 all the time beats B, and this will not align with actual tournaments. In spite of everything, underdogs all the time have an opportunity! Do our outcomes maintain if we enable 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 cases are these with deterministic outcomes. Associated work reveals that if we prohibit the win chance of all video games to, say, the 60–40% interval, then we will anticipate to lower the positive factors from manipulation because the event turns into extra aggressive.
In one other try to beat the impossibility end result we introduced earlier than, we introduce a new mannequin for figuring out which manipulations are useful. Within the mannequin outlined so far, groups within the manipulation coalitions deal with their joint chance of successful as a uniform mass. That’s, a group within the coalition doesn’t care which different group’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 not less than a bit egocentric. To mannequin the belief that groups in manipulating coalitions are nonetheless a bit egocentric, we introduce weights to the manipulation calculations to mirror this.
We noticed that if every group 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 could certainly exist guidelines that fulfill properties 1 and three precisely. We additionally present that for a number of in style guidelines, a big weight is required for the rule to fulfill properties 1 and three.
