What is a Proxy Mechanism?

A natural question concerns the nature of the proxy mechanism g. Recall that a proxy mechanism takes into account the partial preferences over the set of alternatives submitted by the voters and assigns to each voter a set of permitted proxies, the voters who she is allowed to delegate her vote to. I have suggested that we should interpret this set of permitted proxies as the set of delegators who would represent the voter’s interests (based on the preferences they have submitted). But what is the proxy mechanism itself? How does it decide which voters are capable of representing which others?

The first point to make is that there are a lot of possible proxy mecha- nisms, just as there are a lot of social choice functions. I postpone examining proxy mechanisms from an axiomatic perspective until the next chapter. But it is worth giving some examples of simple proxy mechanisms here.

Definition 2.13. (TRIV) TRIV(P, i) =      N\{i} ifPi=∅ {i} ifPi∈ L(A) ∅ otherwise Definition 2.14. (SUBSET) SUBSET(P, i) =      N\{i} ifPi=∅ {i} ifPi∈ L(A) {j∈N\{i} |Pi ⊆Pj} otherwise Definition 2.15. (STRICT-SUBSET) STRICT-SUBSET(P, i) =      N\{i} ifPi=∅ {i} ifPi∈ L(A) {j∈N\{i} |Pi ⊂Pj} otherwise Definition 2.16. (UNIV) UNIV(P, i) =      N\{i} ifPi=∅ {i} ifPi∈ L(A) N\{i} otherwise

Definition 2.17. (DICTATOR) For eachi∈N, fix somej∈N\{i}(to make this concrete, we could, for example, pick the lexicographically earliest voter inN\{i}). Then DICTATOR(P, i) =      N\{i} ifPi =∅ {i} ifPi ∈ L(A) {j} otherwise

Definition 2.18. (HYBRID) Fix somej∈N. Then HYBRID(P, i) =            N\{i} ifPi =∅ {i} ifPi ∈ L(A) N\{i} ifi=j and Pi∈ L/ (A) ∅ otherwise

For most of the remainder of this thesis, I’ll focus on theSUBSETmecha- nism, as I think it is a natural interpretation of what it means for a delegate to represent a voter. In the next chapter, I’ll also show that it is the unique mechanism satisfying certain desirable properties.

For now, though, note that the choice of proxy mechanismg has a large effect on the proxy vote setting.

If g = TRIV, then every agent will cast her own vote, unless she has no preferences at all over the alternatives (in which case she will delegate). So we are close to a classical vote; proxy voting plays little role here. In particular, the proxy choice profileSis often fairly irrelevant to the outcome of the election, although the default vote profileD can be highly relevant.

By contrast, if g= UNIV, then every agent who has not made her mind up fully can delegate her vote to any other agent, regardless of what she thinks on the issues she has made her mind up on. In effect, this is the formal set up of many of the accounts we discussed in the previous chapter; the strictly partial components of the preference profile P are irrelevant to the outcome of the vote, as is the default vote profile D. Instead, it is the proxy choice S that plays a large role in determining the outcome of the election.

If g = DICTATOR, then each agent i has a unique dictator j; when i

submits some but not all pairwise comparisons, then she must delegate her vote to j. Similarly, if g= HYBRID, then the mechanism acts like UNIV for some distinguished j ∈ N, and acts like TRIV for every i ∈N\{j}. These are not intended as real suggestion for a proxy mechanisms, but should serve to indicate the sheer range of available options.

Having sketched some example of proxy mechanisms and their effects on the voting system, it is time to turn to the question at hand. What actually is a proxy mechanism? I see at least two interpretations of proxy mechanisms.

Firstly, there is a descriptive interpretation of proxy mechanisms. On this interpretation, a proxy mechanism describes the behaviour of voters (assuming they act in their own interest). Voters will only choose dele- gates who represent their interests, and the proxy mechanism makes this constraint explicit. Note that proxy mechanisms allow for different voters to have different interpretations of what it takes for a delegate to represent their interests (since they take the name of the voter as an input). Different proxy mechanisms correspond to different constraints on the judgement of

voters. I think the descriptive interpretation best corresponds to the ac- count of proxy choice I offered at the beginning of this chapter (consider, for example, the Brexit thought experiment).

There is also aprescriptive interpretation of proxy mechanisms. On this interpretation, a proxy mechanism is something external to a voter. It could be some aspect of a centralised voting system, or some rule which a voter is required to obey.

When it comes to the classical picture of liquid democracy as a happy medium between direct and representative democracies, I think it is clear that we ought to prefer the descriptive interpretation of proxy mechanisms. When it comes to transitive proxy voting more broadly, though, I think the idea of a centralised proxy mechanism becomes less strange. Suppose, for example, that agents are autonomous software agents making decisions over a large number of alternatives. At this point, one might plausibly want to constrain the sort of delegations that are allowed, to have some means to predict the behaviour of the system.