Disclaimer: I'm not a political science expert and also appreciate input which sharpens the question or links me to other experts.
In the past few years, there have been many awesome mechanism design developments (e.g. quadratic voting, quadratic funding, product nash rule). These mechanisms are proven optimal for a variety of things like efficient preference expression, incentive-compatibility, decomposabiltiy and robustness. However, the research often leaves open the question whether the outcome of the mechanism should be to choose a representative or to implement an actual change in legislation/budgeting.
For these mechanisms to be implemented in current governments, it would help if they can be shown to be superior to some form of indirect democracy (where the public votes for representatives, and perhaps sets some additional constraints).
Do you know research which compared direct and indirect democracy? Given that the decisions of a free-thinking political representative may be hard to model, some related empirical questions may be easier to answer:
(1) how often do the decisions of representatives diverge from those of their constituents?
(2) does direct democracy score better than indirect democracy according to some objective measure (GDP growth), preferably in a very small/controlled setting.
(3) how do people rate the outcomes from participatory budgeting to centralized budgets?