DETERMINACIÓN DE LOS COSTOS TOTALES DE INVERSIÓN EN UNA REFINERÍA

The property of unanimity imposes to the outcome of a rule on a profile to agree with the agents if they agree with each other. In goal-based voting we can separate this property into two different axioms. The first focuses on the case where the

3.2. Axiomatics 25

unanimous choice of the agents for a specific issue is respected in the outcome, and it is a straightforward generalization of the unanimity axiom in binary aggregation (Grandi and Endriss,2011). It is also closely related to the unanimous consensus class in voting (Elkind et al.,2010), stating that a candidate should win if it is ranked first by all voters in an election. We formally define unanimity as:

Definition 3.7. A rule F is unanimous (U) if for all profiles Γ and for all j ∈ I, if mx_ij =0 for all i∈ N then F(Γ)_j =1−x for x∈ {0, 1}.

It is fairly easy to see that if all agents accept or reject unanimously an issue the outcomes of EMaj, TrueMaj and 2sMaj will agree with the profile. The Approval rule also satisfies it, since it will choose one of the models of the agents’ goals (which by definition are all unanimous on that issue). For TrSh rules, the axiom is not satisfied for certain (degenerate) choices of quotas and weights for the agents’ goals, as shown by the following example:

Example 3.7. ConsiderΓ for three agents and two issues such that γ1 = γ2 = γ3 =

1∧2, where q1 = q2 = 3 are the quotas, and µ1∧2(11) = 0.3 are the weights for

the agents’ goals. If wi = 1 for all i ∈ N are the weights of the agents, we have

TrSh(Γ) = {(00)}. Similar examples can be found for agents’ weights lower than 1. The second type of unanimity captures the idea that if the agents all agree on some models of their goals, these models should be the outcome. More formally, we define model-unanimity8as follows:

Definition 3.8. A rule F is model-unanimous (MU) if on all profilesΓ we have v∈F(Γ) if and only if v∈Mod(γi) for all i∈ N.

The (MU) axiom can be alternatively formulated as stating that if for all Γ we have Mod(V

i∈N γi) 6= ∅ then we must have that F(Γ) = Mod(Vi∈Nγi). Observe

that in Definition3.8, the right-to-left direction ensures that if agents are unanimous about a model it will be accepted in the outcome (but the outcome may include other models which are not unanimously supported), while the left-to-right direc- tion ensures that all the models in the outcome are unanimously accepted by the agents (but the rule may be excluding from the outcome some models that all agents accept). Model-unanimity is also known as the (IC2) postulate in belief merging (Everaere et al.,2015) and since Approval can be expressed as an IC merging opera- tor (see Section3.1.2) it therefore satisfies model-unanimity. The following example shows that EMaj, TrueMaj and 2sMaj do not satisfy this type of unanimity:

Example 3.8. LetΓ be a profile for three agents and three issues where the agents’ goals are such that Mod(γ1)= {(000),(101),(110)}, Mod(γ2) = {(000),(111),(100)}

and Mod(γ3) = {(000),(101),(110)}. Even though Mod(γ1∧γ2∧γ3) = {(000)},

we have that EMaj(Γ)= TrueMaj(Γ)= 2sMaj(Γ) = {(100)}.

Unanimity does not imply model-unanimity since EMaj, TrueMaj and 2sMaj sat- isfy the first but not the second. Interestingly, the opposite is also not the case: Proposition 3.3. There exists a rule F that is model-unanimous and not unanimous.

8_{The rule Conj}

v defined in previous work (Novaro et al.,2018) captured precisely the intuition

behind this axiom, in that its output was defined as the models of the conjunction of the agents’ goals if they were mutually consistent, and a default option otherwise.

Proof. Let F be defined as such that F(Γ) =Mod(V

i∈N γi) if Mod(Vi∈Nγi) 6=∅, and

{0}m_{otherwise. By definition F is model-unanimous. Consider the profileΓ for two}

agents and three issues where γ1=1∧ ¬2 and γ2 =1∧2. Since Mod(γ1∧γ2) =∅,

we have that F(Γ) = {(000)}. However, m0

i1 =0 for all i∈ N and yet F(Γ)1=0.

Observe that (MU) implies in particular that F(γ, . . . , γ) =Mod(γ), which corre-

sponds to the strongly unanimous consensus class in voting (Elkind et al.,2010). The axioms of unanimity that we just saw require that the outcome agrees with the agents if they all agree (on issues or models). The next axiom that we introduce states that the outcome of a rule should be supported by at least one of the agents in the profile. We formally define groundedness as follows:

Definition 3.9. A rule F is grounded (G) if F(Γ) ⊆Mod(W

i∈Nγi).

By definition, Approval is grounded, since it outputs a model approved by at least one agent in the profile. On the other hand, the same is not true for the majority rules EMaj, TrueMaj and 2sMaj, as shown by the following result:

Proposition 3.4. EMaj, TrueMaj and 2sMaj are not grounded.

Proof. Consider a profileΓ for three agents and issues where Mod(γ1) = {(111)},

Mod(γ2) = {(010)}and Mod(γ3) = {(001)}. The rules EMaj, TrueMaj and 2sMaj all

return{(011)}, and since(011) 6∈ Mod(W

i∈Nγi) groundedness is not satisfied.

Proposition3.4implies that the three majority rules do not guarantee that the col- lective choice will satisfy the goal of at least one agent. In some cases, however, this can be seen as the rules finding a compromise issue-by-issue between the conflicting views of the agents.

Since Mod(V

i∈Nγi) ⊆ Mod(

W

i∈Nγi) we may think that model-unanimity im-

plies groundedness. The following result however shows that the two axioms are not related:

Proposition 3.5. There exists rules F and F0such that F is grounded and not model- unanimous, while F0is model-unanimous and not grounded.

Proof. Consider F such that for all profilesΓ, if Mod(W

i∈Nγi) =Mod(

V

i∈N γi) then

F(Γ) = Mod(W

i∈Nγi) and F(Γ) = Mod(Wi∈Nγi)\Mod(Vi∈Nγi) otherwise. For F0

it suffices to consider the same rule defined in the proof of Proposition3.3.

An analogous notion of groundedness has been previously defined for the ag- gregation of ontologies as well (Porello and Endriss,2014).