Proxy selection in transitive proxy voting

One might worry that there is a prohibitively large effort involved in submitting \(S_{i}\) and \(D_{i}\). To defend the cost of \(D_{i}\), note that the default vote can be thought of as little more than a placeholder, a device which serves some practical purpose but has little ideological significance (e.g. each voter extends \(P_{i}\) at random, or chooses the lexicographically earliest extension of \(P_{i}\)). To defend \(S_{i}\), note that any real world version of transitive proxy voting will be situated within a dynamic environment. If we assume that the model is a static representation of a process which is inherently dynamic, then we might interpret the situation described by the model as follows. A voter i submits \(P_{i}\). She then calculates \(g({\varvec{P}},i)\), the set of proxies she feels represent her interests, using the preferences submitted by the other voters. If \(g({\varvec{P}},i)=\emptyset\), she submits \(D_{i}\supset P_{i}\), her default preference. If \(g({\varvec{P}},i)\ne \emptyset\), she picks some \(j \in g({\varvec{P}},i)\) to be her proxy. So, rather than \(S_{i}\), the voter i is really only required to specify the name of a proxy in \(g({\varvec{P}},i)\). It’s true that the calculation of \(g({\varvec{P}},i)\) will require some computation, but this should not surprise us: the process of choosing a proxy does take some effort from the voter!

It should be clear that we could modify the proof to accommodate irresolute rules.

IIA-manipulability can be thought of as a much weaker condition than ‘one-way monotonicity’ (Sanver and Zwicker 2009), which features in the preference reversal paradox (Dominik 2017). In effect, one-way monotonicity says that every example of GS-manipulability is an example of IIA-manipulability.