Democracy as voting can be conceptualized as a social information processor, composed of a set of individual processors (the voters) and an integrator (the voting system) that together produce global decisions.
Can one speak precisely about the quality of these decisions? Can the concept of performance and error be meaningfully defined?
Let’s restrict the analysis to the simplest case, a direct democracy Yes/No choice. Assume the concept of correct vote is well defined for the individual information processors, which we can call voting agents. If all the voting agents correctly emit Yes, but the global decision is No, then it seems reasonable to say that the global decision is incorrect. This is just a limiting case of the principle of majority vote, which we can restate as
1. The globally correct decision is that for which there is a highest number of votes that match individual correctness 
This establishes, pending an individual definition of correctness, a binary definition of correctness for the global processor. Furthermore, if voting is exercised repeatedly for a sequence of decisions, we can define a success rate for the global processor as
2. The success rate is the fraction of correct results that the processor emits
Let’s define the vote for an agent a on an issue i as V(a, i), and the correct vote as C(a, i). So
3. An agent’s a vote on an issue i is V(a,i)
4. An agent a votes correctly on an issue i if and only if V(a, i) = C(a, i)
What 4 says is that there exists a correct vote for an agent on an issue, and that the agent may or may not emit that vote. This definition does not require an interpretation of what V and C are, merely that they exist. An interpretation can be made in the framework of decision theory as we will see in the next post. But whatever the interpretation, one can ask, given a fixed C and V how does the performance of the system vary as a function of integrator design?
Say for example, comparing direct to representative democracy. It may be that in representative democracy representatives have greater expertise and emit votes that match C for a greater fraction of voters than if these voters had voted directly. This would speak in favor of representative democracy as a better processor. Conversely, one could say that the representatives are unaligned with the voters’ preferences, yielding worse performance.
Let’s extend 1–4 to allow for representative and liquid democracy.
In representative democracy there is no possibility of direct vote. It can be characterized as a low frequency delegatation-vote-only integrator. This adds to our previous analysis the notion of a delegation vote which is distinct from a standard vote (as described by V and C). A delegation vote is a voting agent’s choice of delegate.
Define the delegation vote for an agent a as D(a), which points to a delegate agent d. Thus
3.1 (representative) A voting agent a’s vote is given by V(D(a), i)
In a liquid democracy agents can choose to vote directly or delegate their vote
3.2 (liquid) A voting agent a’s vote is given by V(a, i) or V(D(a), i)
and the delegation transitivity of liquid democracy yields
3.3 (liquid) A delegate agent d’s vote is V(d, i) or V(D(d), i)
having extended V and D to apply also to delegate agents.
Getting back to the big picture, here’s what we have. We have suggested simple definitions for performance and error in global decisions as a function of individual correctness, which remains unspecified. We need to provide interpretations and specifications for individual voting, delegation and individual correctness (V, D and C). These specifications may yield a model that is operational, whose resultant dynamics can be observed.
Can democratic performance be investigated with such an agent based model? And more specifically, can such a model reveal or justify the hidden assumptions present when asserting superior performance of direct vs representative vs liquid democracy integrators?
(Continued in further posts)
 An alternative definition could incorporate degrees, for example depending on the exact numbers involved.
 The voting agent / delegate agent distinction is not obligatory, but is required for private voting systems
 Let’s say the agent has some preferences, or a utility function, about the state of the world. Given a voting decision, it must judge which outcome results in a higher expected utility. This rationally ideal choice is C. The agent, constrained by limited knowledge, limited computing resources, and cognitive errors, is modeled by V.
 The definition of correctness for a delegation vote should be defined entirely in terms of the correctness of the delegate’s vote. As an extreme case, if the delegate’s votes are all correct from the point of view of the voter, then that delegate vote was correct. Conversely, the vote is incorrect if none of the delegate’s vote were correct (again from the point of view of the voter). So judging from these extremes it seems that the correctness of a delegated vote admits degrees and is similar to the success rate of 2:
5. An agent a has a delegation success rate equal to the fraction of issues i where V(D(a), i) = C(a, i)
 The wording is somewhat convoluted to account for unintuitive scenarios. It is possible, for example, for almost every voter to individually vote incorrectly but that the resulting global decision is correct (due to errors cancelling).