We provide necessary and sufficient conditions on the syntactic type of integrity constraint such that the domain it describes admits a nondictator aggregator. We show that option integrity constraints are easily identifiable and provide algorithms that, given a domain D ⊆ {0, 1}m, check a time polynomial in its size for a nondictator aggregator admits, and indeed produce an option integrity constraint that describes the case that .
Introduction
In addition to the problem of choosing the winning candidate in the election, social choice theory has its origins in the normative analysis of welfare economics [31], a branch of modern economics that evaluates economic policies based on the welfare of society in general. Non-dictatorship: requires that there be no individual in society whose, for any domain of the social welfare function, the collective preference is the same as that individual's (ie, the dictator's) preference.
Logic based judgment aggregation
Given a problem of judgment aggregation, individuals express their views on the issues on the agenda. A judgment set provides information about which proposals/agenda items are rejected and which are accepted.
Abstract judgment aggregation
Taking the new variable Z to be false, we thus obtain that a does not satisfy φm, or equivalently, xm = 0. These differences may concern the interpretation as well as the syntax of the formulas that constitute the problems.
Property based judgment aggregation
Condition 3 is a separation condition in the sense that different scores differ on at least one property. This allows us to identify (E,H) with XH ⊆ {0, 1}m and therefore the clustering problem in a feature-based framework can always be expressed in an abstract framework.
Integrity constraint based judgment aggregation
As mentioned earlier, the set of feasible voting patterns X ⊆ {0, 1}m is a domain of possibilities if it accepts a non-dictatorial aggregator of certain arity. So X admits a nondictatorial projection aggregator, which in turn means that any set of feasible voting patterns that is a Cartesian product is a domain of possibilities.
Syntactic characterization of integrity constraints: Conjectures and
The following lemma is an immediate consequence of the fact that an aggregator is by definition supportive. Recall that a binary non-projection aggregator f = (f1, . . . ,fm) can consist of any combination of the operators. The latter is an immediate consequence of the fact that Dadmits the aggregator f = (f1, . . . ,fm), where.
Intuitively, the goal is to obtain a formula φ with a "Horn part" consisting of the variables φ corresponding to the ∧components of the aggregator such that Mod(φ) = D. Characterization of possibility domains via integrity constraints Z In the above discussion, the fact that Assumption3 does not contradict to none of Counterexamples 1 or 2, should come as no surprise. So we have proved that whenever Dad allows a binary nondictator aggregator, there exists a formula φ of the form described in condition 2 such that Mod(φ) = D.
Sets V1 and V2 were intended to include variables corresponding to the pr21 and pr22 components of the aggregator, respectively. Condition (c), as we have seen, conforms to the requirement of maximality of the symmetric components of the adumbrated aggregator.
Syntactic characterization of integrity constraints
Note that any Horn, renamable horn, or partially Horn formula is trivially renamable, partially Horn. Let φ be a renamable partially Horn formula, and let φ∗ be a partially Horn formula obtained by renaming some of the variables of φ, with V0 being the admissible set of variables. Consider the partially renamable Horn formula of the previous example, defined over the variable set V={x1,x2,x3,x4},.
We will prove that option integrity constraints are the answer to the problem of the syntactic characterization of option domains. By Lemma4, the renamable partial Horn formula for D can be obtained by renaming inφvariablesxi such that, ati∈ J. Finally, we are ready to prove that possibility integrity constraints are the answer to the problem of the syntactic characterization of possibility domains.
Also, letφ be the partial Horn formula obtained by renaming the variables of a subset V∗ ⊆ V0. Other forms of non-dictatorial aggregation- Characterizations of the corresponding domains Since abc,a0b0c0,a00b00c00satisfyφ, it holds thatc,c0andc00satisfyC0q.
Generalized Dictatorships
Interestingly, φ13 thus describes a domain that admits an aggregator which is not a generalized dictatorship, even though it is not the aggregator that "corresponds to" the formula. This means that the decision whether an aggregator is a generalized dictatorship depends on the domain in question. Assume that D = {x,y}, where x and y are distinct, and that f is an n-ary aggregator for D that is not a generalized dictatorship for D.
A domain D⊆ {0, 1}m with at least three elements admits an aggregator that is not a generalized dictatorship if and only if it is a domain of possibilities. The forward direction is achieved by the trivial fact that an aggregator that is not a generalized dictatorship is also not dictatorial. Then there exists a binary symmetric aggregator g = (g1, . . . ,gm) for D (g may be different from f ) that is not a generalized dictatorship for D.
Our goal is to show that D∗ admits a symmetric aggregator that is not a generalized dictatorship. A domain D⊆ {0, 1}m, with at least three elements, admits an aggregator that is not a generalized dictatorship if and only if there exists a possible integrity constraint whose set of models is equal to D
Anonymous, Monotone and StrongDem Aggregators
Characterizations for domains admitting anonymous, mono-
In the previous section, we studied three forms of nondictatorial gatherers with attractive properties. In particular, we show that domains accepting anonymous aggregates are described by local possibility integrity constraints, while domains accepting monotone non-dictatorial aggregates by separable or partially renamable Horn formulas and, finally, that domains accepting StrongDem aggregates are described from a subclass of local opportunity integrity constraints. A domain D ⊆ {0, 1}m admits a locally nondictatorial monotone accumulator if and only if it admits an anonymous monotone.
Here we will use it to obtain the analogous results for domains that admit non-dictatorial monotonic and StrongDem aggregators. A domain D ⊆ {0, 1}m admits a monotonic nondictatorial aggregator of some kind if and only if it admits a binary nondictatorial one. That a domain that admits a binary non-dictatorial aggregator also admits a non-dictatorial monotonic one is obvious, since all binary univariate functions are monotonic.
D admits a nondictatorial monotone n-ary stacker if and only if there exists a partially separable or renamable Horn integrity constraint whose pattern set is equal to D. Other forms of nondictatorial stacker- Characterizations of corresponding domains Simply, if the fandgare aggregators for a domainD, then so is, since it is produced by an offandg overlay.
Systematic Aggregators
In this section, we show that, given a formulaφ, we can determine in time linearly in the length of the formula whether it is an option integrity constraint. Note that to calculate φ0 from φ, one would need squared time in the length of φ. There is an algorithm that, upon entering a formulaφ, stops in time linear in the length of φ and either returns that φ cannot be partially renamed Horn or alternatively produces a subset V∗ ⊆ V such that the formula φ∗ obtained from φ by renaming the letters for variables in V∗ is partially Horn.
Since the algorithm runs in linear time in the number of vertices and edges of G, it is also linear in the length of the input formula φ. We can decide whether each of the sccs is bad or not again in linear time in the length of the input formula. Furthermore, the process described to obtain the assignment is linear in the length of the input formula and provides information on which variables should be renamed.
From the above, we get the following theorem, which states that checking whether a formula is an option integrity constraint can be performed in polynomial time in the size of the formula. There is an algorithm that, upon input of a formulaφ, stops in linear time for the length ofφ and either returns thatφ is not an option integrity constraint.
Local possibility integrity constraints & Local possibility domains
Constraints on the integrity of local possibilities & domains of local possibilities 69 Now be a renamable partial Horn formula, where by renaming the vari-. If V0=∅, then φ is not lpic or there exists a partition (V1,V2)V such that no clause of φ contains variables from V1 and V2. Now, if any clause φ contains more than two variables from V1 or variables from both V1 and V2, then φ is not lpic.
We end this section by showing that, given a domain D, we can efficiently determine whether it is an lpd and construct an lpicφ such that Mod(φ) = Din if it is. The first fact we use is that a prime formula that is logically equivalent to a bijunctive one is also bijunctive [36, Proposition 3]. The second is that given a prime formula describing an affine domain D, we can construct an affine formula describing D in linear time in the size of the input formula.
There is an algorithm that, on input D⊆ {0, 1}m, stops in time O(|D|2m2) and either returns that D is not a local possibility domain, or alternatively performs a local possibility integrity constraintφ , containing O (|D|m)clauses, whose set of satisfying truth assignments is D . We then use the linear algorithm of Theorem17 to check whether φ is partially renamable Horn.
Domains closed under other forms of non-dictatorial aggregators
There is an algorithm that stops at input D⊆ {0, 1}m in time O(|D|2m2) and returns that D does not allow a non-dictatorial monotonic aggregator, or alternatively a separable or renameable, partial Horn formulaφ , which contains O(|D|m)-clauses, whose set of satisfying truth assignments is D. According to Corollary10 we have that a Boolean domain D⊆ {0, 1}m admits an ann-ary non-dictatorial systematic aggregator if and only if there exists an integrity constraint that is either Horn, dual Horn, bijunctive or affine, whose set of satisfactory assignments is equal to D. Furthermore, we examined other forms of non-dictatorial aggregators that have appeared in the literature, and presented syntactic characterizations of the integrity constraints describing the corresponding domains.
In particular, we have shown that domains admitting locally nondictatorial aggregators, called local option domains, and domains admitting anonymous aggregators coincide and are described by local option integrity constraints. Domains admitting aggregators other than generalized dictatorships are denoted as possibility integrity constraint models, while domains admitting monotone nondictatorship aggregators are described by a subclass of such formulas, the separable and renamable partial Horn formulas. Furthermore, we have shown that any integrity constraint of the above types is easily (in linear time in the length of the input formula) identifiable.
We also provided algorithms which, given a domain D ⊆ {0, 1}m, can determine in the time polynomial of the size of the domain whether D admits a non-dictatorial aggregator of each of the types mentioned above, and if it does, construct an integrity constraint describing that whose number of sentences is linear in the size of the domain. Besides knowing whether nondictatorial aggregation is possible for a given domain, these algorithms also provide specific information about how to construct such an aggregator.
A problem with plurality voting
An illustration of Condercet’s paradox
An illustration of the doctrinal paradox
An illustration of the discursive dilemma
