Axioms and setup

In Root's Paper:

Setup in Root

We have the following elements:

Mechanism Properties in Root

The following are some properties that mechanisms might have:

Local Dictatorship

Let \(\bar{C^*}\) be the set of infeasible allocations in which each agent's object is individually given to them in some feasible allocation. Let The blocks of \(\bar{C^*}\), \(E_1,...E_p\) be a partition formed by equivocating any two allocations in \(\bar{C^*}\) which assign the same object to at least one person.

\(f: P^2 \to C\) is called a Local Dictatorship if each block \(E_i\) is assigned a local dictator \(d_i\) so that for any \(\succsim\), if agent one ranks object \(a\) the highest and agent two ranks object \(b\) the highest, then: $$f(\succsim) =\begin{cases} (a,b), & (a,b)\in C\\ (a,\max_{\succsim_2}\{y|(a,y)\in C\}), & (a,b)\in E_k \text{ and } d_k=1 \\ (\max_{\succsim_1}\{x|(x,b)\in C\}, b), & (a,b)\in E_k \text{ and } d_k=2 \\ \end{cases}$$

Theorems in Root
Root's GS proof

Look at marginal mechanisms, show that there is some pair of agents and profile of preferences such that the marginal mechanism has at least three allocations in its option set.

Next, in this marginal mechanism, we have a dictator.

In Standard Gibbard-Saitherwaite:

There is a set of social outcomes, \(O\), and a set of individuals \(N\), with individual \(i\) having ordinal preferences \(\succsim_i\in P\) over these outcomes. A social choice function is a map from the profiles \(\succsim \equiv (\succsim_1, ... \succsim_{|N|})\) of everyone's preferences onto specific social outcomes: \(f: P^{|N|} \to O \)

The theorem states that if \(|O| > 2 \), the domain of \(f\) is unrestricted, and \(f\) is onto and strategy proof, then \(f\) is dictatorial .

GS Compared to Root's Setup

In both setups, each agent has preferences \(\succsim_i\in P\) over \(O\). And the social mechanism \(f\) has domain \(P^{|N|}\). In both theorems, \(|O|\geq 3\)

The mechanism selects an option from \(O\)The mechanism selects an option from \(O^{|N|}\)
The codomain of \(f\) is \(O\).The codomain of \(f\) is \( O^{|N|}\).
The image of \(f\) is the entire set of social options \(O\).The image of \(f\) is some feasible subset of the set of profiles of societal options \(C \subset O^{|N|}\). In this case, \(C\) is the diagonals.
Requires that \(|O|>2\) and \(f\) surjective.Technically only requires that the image of \(f\) has more than two elements.
strategy-proofness: \[f(\succsim) \succsim_i f(\succsim^\prime_i, \succsim_{-i})\]strategy-proofness: \[f_i(\succsim) \succsim_i f_i(\succsim^\prime_i, \succsim_{-i})\] (Agent only cares about the part of the social allocation given to them)
DictatorialConcerned more generally with mechanisms which are serial dictatorships. In the diagnonals, the first dictator happens to be a full dictator.

Standard Arrow

Modify Arrow for Root's \(C\):

So standard social choice has a mapping from preference profiles to a single object. Root maps profiles of preferences to a profile of outcomes and says social choice is the specific case where only diagonals are feasible.

Standard Arrow maps prefence profiles to a social preference ordering. To apply the same generalization, we need to map preference profiles to a profile of preference profiles???