In our political philosophy seminar we've recently been looking at the question of how 'democracy', understood as a kind of popular control, might be realized through representation. One obvious possibility involves the use of (e.g. election) incentives, sanctions, etc., to encourage political representatives to track the popular will. But this won't extend to citizen's juries or statistically representative assemblies, where the participants are ordinary people selected by lot and so not subject to any such incentives. In Pettit's terms, they act as 'proxies' rather than 'deputies', indicating rather than responding to the will of the general population.
Pettit argues that this still constitutes a form of popular control. After all, the assembly was set up in this way (i.e. by lot) precisely because it may be expected to accurately reflect what the population wants. If it didn't, they would've set things up differently. Still, we need to be clear about the scope of the control implied here.
I think it's not really the case that the population controls the assembly's decision in any strict sense. After setting things up, they have no further influence on the assembly, and so (a fortiori) cannot raise the causal probability of this assembly's decision going as they prefer (as per Pettit's analysis of control). Rather, stepping back to before the assembly was chosen, we see there is some popular control over the decision-making process in the abstract. For, by instituting the assembly, the people may raise the probability that the subsequent decision is made as they would wish (in contrast to, say, letting the richest person decide). At this level of abstraction, the possibility of alternative setups (decision procedures) is part of the comparison class. But this is no longer the case once we start talking more specifically about the assembly's decision. Popular control does not extend to this local specification of the event. It is merely general: control over the initial setup, not over the decision reached by the particular procedure.
Indeed, as I objected in class, it isn't clear that the status of the agent (assembly) as "indicative" of the principal (broader population) is playing any essential role here at all. Instead, I think all the work is being done by the principal's control over the setup, i.e. their ability to select the agent. Although it makes most sense to choose an agent that you expect to mirror your own preferences, there are other options. A perverse principal might select an "anti-indicator" agent who is expected to choose precisely the opposite of what they want! The control exercised by the principal over the ultimate outcome is exactly the same in either case. So, as came out in further discussion, what matters here is not that the agent is an accurate indicator of oneself (as principal), but just that they exhibit some systematic predictability, which the principal can aim at (perhaps perversely) in empowering them.
See also: Jane Mansbridge on selection vs. control.