Most democracies have an Executive Officer, such as the President of the United States, typically elected either directly or indirectly by the people.
In a DAO Democracy, assuming we retain the Executive Branch in more or less its current form, the simplest approach would be to appoint, as President, that person the prediction market said had the highest positive impact on the collective welfare if appointed as President. The President would serve until the appointment of some other person had a higher positive impact on the collective welfare if appointed as President. An alternative would be to have a special election for the President, using a custom presidential prediction market to select among the candidates.
Ideally, in an election for President we’d like to ask the citizens in the different futures created by electing the different candidates what they thought of them after their terms were over. If A, B and C were candidates, we’d like to hear the historical judgment from (say) a year after the end of their term. If A ranked 60, B ranked 50 and C ranked 40, then we’d know we should elect A, as A was ranked highest by the prediction market just before he was chosen.
Normally, this isn’t possible because we can’t, as in some science fiction movies, examine counterfactual histories by letting the universe follow multiple different paths and see what happens if some historical event that never occurred did, in fact, occur. Using conditional prediction markets, we can do just that. The prediction market will tell us how A would have ranked as President if elected, how B would have ranked as President if elected, and how C would have ranked as President if elected. We can then pick the candidate who would have ranked highest. This system is closest to the existing system, in the sense that there is a fixed point in time when we choose a President, and the selection process is one in which citizens are asked, directly, what they think of the President. It’s a bit unusual, in that it still uses prediction markets as the core mechanism for making the key choices, and the citizens are asked what they thought of the President who was actually chosen after he has finished his term, and their choice is used to make (likely rather significant) payments to those investors who accurately predicted how people would evaluate this President after his term was finished.
Note the key financial incentive: large payments flow to those who can accurately forecast how the bulk of the citizens will evaluate the President five years in the future. The evaluation is carried out a year after the President’s term is finished, and the President’s term normally runs four years. Those people who accurately forecast what the vast majority of citizens would think of the President after his term of office had been over for a year will be given a significant financial payoff. There are many variations on this approach which could be used to elect the President, or any other officer. Which of these variations is best we leave to the prediction market underlying the DAO Democracy to figure out.