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?
Though I am not super familiar with the research, it seems that in general more indirect democracy functions better due to the fact that voters have little incentive to cast informed votes, whereas representatives are incentivized to make informed decisions on voters behalf.
I think the book 10% Less Democracy can point you to relevant research on this topic. It was discussed briefly on MR here.
You may also want to check out Caplan's The Myth of the Rational Voter for research along similar lines.
Thanks for your thoughts!
To let voters cast more informed votes, there's a whole movement around deliberation. You can argue that increases costs for voters, but I think the trade-off is unclear and likely context dependent.
The books are great pointers, will have a look at the research referenced!