Our work proposes a special rule that barely, however nonetheless not optimally, reduces the potential good points from manipulation. The theoretical rule we suggest identifies a small set of groups as “important”, which we outline as groups that might type coalitions that result in a crew within the coalition being the clear winner.
We classify every potential match end result into one among 5 buckets relying on how shut it’s to having a crew win all their video games. If the match has a crew that wins all their video games, we declare that crew the winner. If the match is way from having a crew that wins all their video games, we select a winner uniformly at random. For tournaments which might be “shut” to having a crew that wins all their video games, we determine these groups that might considerably achieve from manipulating the match. We show that there are usually not many such groups, and for every “shut” match, we assign particular possibilities to every crew in order that the good points from manipulation for teams of measurement three are 50% at most (and nonetheless the optimum 33% for teams of measurement two).
One widespread 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 may occasionally not align with actual tournaments. In spite of everything, underdogs at all times 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, due to prior work, the reply to this query is “sure” as a result of the worst case situations are these with deterministic outcomes. Associated work exhibits that if we prohibit the win likelihood of all video games to, say, the 60–40% interval, then we will count on to lower the good points from manipulation because the match turns into extra aggressive.
In one other try to beat the impossibility consequence we introduced earlier than, we introduce a new mannequin for figuring out which manipulations are helpful. Within the mannequin outlined to this point, groups within the manipulation coalitions deal with their joint likelihood of profitable as a uniform mass. That’s, a crew within the coalition doesn’t care which different crew’s probabilities of profitable 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 profitable, or are at the very 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 mirror this.
We noticed that if every crew weights their very own probabilities of profitable 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 fashionable guidelines, a big weight is required for the rule to fulfill properties 1 and three.