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## Logical preference representation and combinatorial vote, (2004)

Venue: | Annals of Mathematics and Artificial Intelligence |

Citations: | 96 - 16 self |

### Citations

2825 | Computational Complexity - Papadimitriou - 1994 |

626 | Nonmonotonic reasoning, preferential models and cumulative logics.
- Kraus, Lehmann, et al.
- 1990
(Show Context)
Citation Context .... We list here several topics related to this paper and point to possible directions for further research. 5.1. Compact languages for preference representation: going further The role of the first half of the paper was to give a survey of logical representation of preference, especially from a computational point of view. I do not claim that it is exhaustive, but I tried to refer to most significant approaches. At least three kinds of languages have not been considered, and I’ll try to explain here why: – So-called “preferential logics”, developed either in a nonmonotonic reasoning framework ([32,48] and subsequent works), or more recently in a logic programming framework, are not considered because they do not really deal with (decisiontheoretic) preference representation. The terminology “preference” in the latter approaches is rather technical (“preference” is used as a synonym of “[weak/strict] order”) and not especially connected to decision-theoretic issues. 16 Actually, they differ in the condition required: for the Simpson rule we required that COMPARISON is polynomial whereas for plurality we required the weaker condition that NON-DOMINANCE is in NP or in coNP. 56 J. Lang / Logic... |

582 | Improved algorithms for optimal winner determination in combinatorial auctions and generalizations.
- Sandholm, Suri
- 2000
(Show Context)
Citation Context ... years, AI researchers have been concerned with interaction, cooperation or negotiation within agent societies. For these problems, it often occurs that the set of all feasible states has a very large size, due to its combinatorial nature. For this reason, research has been done so as to develop representation languages aiming at enabling a succinct representation of the description of the problem, without having to enumerate a prohibitive number of states. Languages based on propositional logic have been proposed recently for some multi-agent problems, for instance for combinatorial auctions [8,43,46] and automated negotiation [52]. In this paper we focus on combinatorial vote. Combinatorial vote is located within the larger class of group decision making problems. Each one of a set of agents (called voters) initially expresses her preferences on a set of alternatives (called candidates); these preferences are then aggregated so as to identify (or elect) an acceptable common alternative in an automated way (without negotiation). Formulated as such, this can be identified as a vote problem. Vote problems have been investigated by researchers in social choice theory (see for instance [40] fo... |

362 | The computational complexity of propositional STRIPS planning
- Bylander
- 1994
(Show Context)
Citation Context ...andidates such that x |= ¬p ∧ ¬q and y |= ¬p ∧ q. We have κK,D(x) = κKB(ϕ ∧ ¬ψ) and κK,D(y) = κKB(ϕ), therefore x cond,ZK,D y if and only if κKB(ϕ ∧ ¬ψ) κKB(ϕ), i.e., if and only if KB does not entail D(ψ |ϕ). Proposition 3. The complexity of COMPARISON, NON-DOMINANCE and CAND-OPTSAT for ceteris paribus desires is reported in table 5. We start by proving the most difficult result. 1. COMPARISON for ceteris paribus preferences is PSPACE-complete. Proof. The membership proof is the same as the PSPACE-membership proof for plan existence with deterministic actions represented in STRIPS [11]: the key points are that (i) the length of a dominance path from x to y, if such a dominance path, is bounded by 2|VAR|−1 19 and (ii) for any two candidates x1, x2, checking whether x1 >C:G>G′[V ] x2 can be done in polynomial time (it is sufficient to check that x1 |= C ∧ G ∧ ¬G′, that x2 |= C ∧ ¬G ∧ G′, and that x1 and x2 coincide in all variables not in V ). The hardness proof is much more complex. It comes from a polynomial reduction from plan existence with deterministic actions represented in STRIPS;20 the latter prob19 As in [11], the reason for this is that there are |Mod(K) ... |

331 | Valued constraint satisfaction problems: Hard and easy problems
- Schiex, Fargier, et al.
- 1995
(Show Context)
Citation Context ... course”, “I prefer white wine if one of the courses is fish and none is meat, red wine if one of the courses is meat and none is fish, and in the remaining cases I would like equally red or white wine”, etc. Since the preference structure of each voter cannot be expressed explicitly by listing all candidates, what is needed is a compact preference representation language. Such preference representation languages have been developed within the AI community; they are often build upon propositional logic, but not always (see for instance utility networks [1,33] or valued constraint satisfaction [47] – however in this paper we restrict the study to logical approaches); they enable a much more concise representation of the preference structure, while preserving a good readability (and hence a proximity with the way agents express their preferences in natural language). Therefore, the first parameter to be fixed, for a combinatorial vote problem, is the language for representing the preferences of the voters. In section 3, we recall different logical representation languages proposed in the literature, we discuss their relevance to combinatorial vote, and discuss their computational complex... |

305 |
Axioms of Cooperative Decision Making,
- Moulin
- 1988
(Show Context)
Citation Context ...,43,46] and automated negotiation [52]. In this paper we focus on combinatorial vote. Combinatorial vote is located within the larger class of group decision making problems. Each one of a set of agents (called voters) initially expresses her preferences on a set of alternatives (called candidates); these preferences are then aggregated so as to identify (or elect) an acceptable common alternative in an automated way (without negotiation). Formulated as such, this can be identified as a vote problem. Vote problems have been investigated by researchers in social choice theory (see for instance [40] for an overview) who have studied extensively all properties of various families of vote rules, up to an important detail: candidates are supposed to be listed explicitly (typically, they are individuals or lists of individuals, as in political elections), which assumes that they should not be too numerous. ∗ This paper is a revised and extended version of the paper entitled “From preference representation to combinatorial vote”, Proceedings of the Eighth International Conference on Principles of Knowledge Representation and Reasoning (KR2002) (Morgan Kaufmann, 2002) pp. 277–288. 38 J. Lang /... |

274 | Bidding and allocation in combinatorial auctions.
- Nisan
- 2000
(Show Context)
Citation Context ... years, AI researchers have been concerned with interaction, cooperation or negotiation within agent societies. For these problems, it often occurs that the set of all feasible states has a very large size, due to its combinatorial nature. For this reason, research has been done so as to develop representation languages aiming at enabling a succinct representation of the description of the problem, without having to enumerate a prohibitive number of states. Languages based on propositional logic have been proposed recently for some multi-agent problems, for instance for combinatorial auctions [8,43,46] and automated negotiation [52]. In this paper we focus on combinatorial vote. Combinatorial vote is located within the larger class of group decision making problems. Each one of a set of agents (called voters) initially expresses her preferences on a set of alternatives (called candidates); these preferences are then aggregated so as to identify (or elect) an acceptable common alternative in an automated way (without negotiation). Formulated as such, this can be identified as a vote problem. Vote problems have been investigated by researchers in social choice theory (see for instance [40] fo... |

254 | Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence. - Shoham - 1988 |

217 | Toward a logic for qualitative decision theory.
- Boutilier
- 1994
(Show Context)
Citation Context ...oritized goals, the preference relation then being based on the leximin ordering (however the transformation is not polynomial). Therefore the problems associated with RQCL are at least as hard as with Rleximin and possibly above (we did not investigate them). 3.3.4. Conditional logics Each goal Gi is now attached to a context Ci: GB = {C1 : G1, . . . , Cn : Gn}, and C : G is interpreted as C ⇒ G in the simplest conditional logics: being a complete weak order on candidates, we say that C : G is satisfied by if and only if Max(Mod(C),) ⊆ Mod(G); this can be interpreted as “ideally G if C” [5]. This constraint does not fully determine the preference relation induced from D. Several possibilities exist: 3.3.4.1. Standard preference relation. Rcond,S consists in considering that a candidate is at least as good as another one if and only if this holds in all models of GB. Formally: x cond,SGB y if and only if for any satisfying GB we have x y. Note that GB is only a partial preoder which is generally very weak, often much too weak (i.e., does not enable enough comparisons) to be a good candidate for preference representation, as it can be seen on the following example: let u... |

215 | On the complexity of propositional knowledge base revision, updates, and counterfactuals.
- Eiter, Gottlob
- 1992
(Show Context)
Citation Context ...elation is complete). We omit its definition. It is not hard to show that as soon as x GB y can be decided in polynomial time from , which is the case for R⊆, Rdiscrimin and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 preferred to ¬ϕ1 ∧ϕ2 preferred to ¬ϕ1 ∧ ¬ϕ2 ∧ ϕ3, . . . . Formulas expressed in this language can be translated into a set of prioritized goals, the preference relation then being based on the leximin ordering (however the tr... |

190 | System z: A natural ordering of defaults with tractable applications to default reasoning.
- Pearl
- 1990
(Show Context)
Citation Context ...k, often much too weak (i.e., does not enable enough comparisons) to be a good candidate for preference representation, as it can be seen on the following example: let us consider the propositional language generated by two propositional variables a and b and let GB = { : a}. Then, for any x, y ∈ {(a, b), (a,¬b), (¬a, b), (¬a,¬b)}, x GB y holds if and only if x = y (and therefore x >GB y never holds). 3.3.4.2. Preference relation based on Z-ranking. While Rcond,S considered all models satisfying a set of conditionals, the approach based on the Z-completion of GB, at work in System-Z [45] and similar approaches, selects one model and allows much more consequences to be derived. Given a set GB = D of conditional rules φ : ψ and a set of hard facts, System-Z proceeds by partitioning D into a collection D0 ∪ · · · ∪ Dn; if a conditional rule ϕ : ψ is in Di then its rank is i. We omit to explain how the ranking is computed as this is only derivative for the purposes of the paper. Note that ranks intuitively respect specificity relations between rules, i.e., more specific rules are assigned higher ranks. Now, the ranking function on conditional rules induces a ranking function κ on... |

178 | Inconsistency management and prioritized syntax-based entailment.
- Benferhat, Cayol, et al.
- 1993
(Show Context)
Citation Context ...AND-OPT-SAT is in p2 , and in p 2 when the set {uGB(x) |x ∈ Mod(K)} can be computed in polynomial time (and therefore has a polynomial size); CAND-OPT-SAT is p2 -complete even for the simple representation language Rcard. 3.3.3. Prioritized goals Instead of weighting formulas by numerical weights, several approaches proceed by ordering them with a priority relation. Consider GB = 〈{G1, . . . ,Gn},〉 where is a weak order (called a priority relation) on {1, . . . , n}. Let be the strict order induced by , and let nonsatGB(x) = {1, . . . , n} \ satGB(x). A common choice (see, e.g., [3,29]), called the discrimin ordering when is complete, Rdiscrimin, is the following: x discriminGB y if and only if ∀i ∈ nonsatGB(x) \ nonsatGB(y) ∃j ∈ nonsatGB(y) \ nonsatGB(x) such that j i. Note that Rdiscrimin generalizes R⊆ – namely, it coincided with R⊆ when is chosen to be the relation defined by i j for all i, j . An alternative common way of inducing preference on candidates from priorities is the representation Rleximin based on leximin ordering [3]. It generalizes Rcard (and assumes that the priority relation is complete). We omit its definition. It is not hard to show ... |

153 | Graphical Models for Preference and Utility.
- Bacchus, Grove
- 1995
(Show Context)
Citation Context ...ase I would prefer smoked salmon as first course”, “I prefer white wine if one of the courses is fish and none is meat, red wine if one of the courses is meat and none is fish, and in the remaining cases I would like equally red or white wine”, etc. Since the preference structure of each voter cannot be expressed explicitly by listing all candidates, what is needed is a compact preference representation language. Such preference representation languages have been developed within the AI community; they are often build upon propositional logic, but not always (see for instance utility networks [1,33] or valued constraint satisfaction [47] – however in this paper we restrict the study to logical approaches); they enable a much more concise representation of the preference structure, while preserving a good readability (and hence a proximity with the way agents express their preferences in natural language). Therefore, the first parameter to be fixed, for a combinatorial vote problem, is the language for representing the preferences of the voters. In section 3, we recall different logical representation languages proposed in the literature, we discuss their relevance to combinatorial vote, ... |

137 | Preferential Semantics for Goals.
- Wellman, Doyle
- 1991
(Show Context)
Citation Context ...nditional desire induces an explicit utility loss [36]. This principle is further generalized by introducing numerical strengths and polarities in [38,51]. We did not investigate complexity issues for these approaches. 3.3.5. Ceteris paribus preferences C, G and G′ being three propositional formulas and V being a subset of VAR such that Var(G) ∪ Var(G′) ⊆ V , the ceteris paribus desire C : G > G′[V ] is interpreted by: “all irrelevant things being equal, I prefer G ∧ ¬G′ to G′ ∧ ¬G”, where the “irrelevant things” are the variables that are not in V . The definitions proposed in various places [7,23,24,49] differ somehow. We take as a basis the definition of [23], slightly generalized, in the spirit of [49] but with less complications, by introducing the explicit set of variables V which expresses, in an explicit way, which variables are referred to when saying “all other things being equal” (namely, those not in V ). For natural reasons, and to remain consistent with the origiginal definitions, we impose that Var(G)∪Var(G′) ⊆ V . This modification is simple, it does not affect significantly the computational aspects of the framework and answers (to a certain extent) a criticism addressed to ce... |

137 | On the logic of merging, in:
- Konieczny, Pino-Pérez
- 1998
(Show Context)
Citation Context ...f the comparison and the non-dominance problems for ceteris paribus desires under various assumptions is a promising topic. Some significant results have been obtained in the resticted case of CP-nets in [20,21], especially tractable cases (however, the general comparison problem for the restriction of ceteris paribus desires corresponding to CP-nets is still open). 3.3.6. Distances Let d be a (pseudo-)distance on X , i.e., a function from Mod(K) × Mod(K) to N such that (i) d(x, y) = 0 if and only if x = y and (ii) d(x, y) = d(y, x). Distancebased logical representations of preference [31,34,35], denoted by Rd , are based on the 8 This would however be the case with variants of the framework, that we do not consider here for the sake of brevity. J. Lang / Logical preference representation and combinatorial vote 47 intuitive idea that, when expressing a goal G, then ideally, x must satisfy G, and when it is no longer the case, then, the “further” x is from G, the less preferred x. Formally, a pair 〈{G}, d〉, where G is a propositional formula and d a pseudodistance, induces the utility function uGB (x) = −d(x,G) = − min y|=G d (x, y). When d is computable in polynomial time, on... |

118 | Belief revision and default reasoning: Syntax-based approaches, in:
- Nebel
- 1991
(Show Context)
Citation Context ...elation is complete). We omit its definition. It is not hard to show that as soon as x GB y can be decided in polynomial time from , which is the case for R⊆, Rdiscrimin and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 preferred to ¬ϕ1 ∧ϕ2 preferred to ¬ϕ1 ∧ ¬ϕ2 ∧ ϕ3, . . . . Formulas expressed in this language can be translated into a set of prioritized goals, the preference relation then being based on the leximin ordering (however the tr... |

117 | R.: UCPnetworks: a directed graphical representation of conditional utilities. In:
- Boutilier, Bacchus, et al.
- 2001
(Show Context)
Citation Context ...R, generating a profile P = InduceR(B), i.e., depending whether preferences are cardinal or ordinal: InduceR(B) = {uGB1, . . . , uGBp} or InduceR(B) = {GB1, . . . ,GBp}. We will now look at several vote rules that are well-known in the social choice community, and discuss these with respect to two criteria: 1. Relevance for combinatorial vote (i.e., does the vote rule still “mean” something when the set of candidates has a combinatorial structure?). 2. Computational complexity. Let V be a vote rule and R a representation language; we consider the following decision problems. 10 However, see [6] for an introduction of numerical utilities in ceteris paribus networks that remedies this problem, but on the other hand reintroduces the cognitive problem inherent to numerical utilities. 11 For the sake of simplicity we will not distinguish between what is usually called a vote rule, which selects a unique candidate, and a vote correspondance, which selects a subset of candidates; we will use the terminology “vote rule” in all cases. The rules that we will consider in the rest of the paper are strictly speaking correspondances, from which a standard rule can be defined by a tie-breaking rul... |

97 | Bidding languages for combinatorial auctions.
- Boutlier, Hoos
- 2001
(Show Context)
Citation Context ... years, AI researchers have been concerned with interaction, cooperation or negotiation within agent societies. For these problems, it often occurs that the set of all feasible states has a very large size, due to its combinatorial nature. For this reason, research has been done so as to develop representation languages aiming at enabling a succinct representation of the description of the problem, without having to enumerate a prohibitive number of states. Languages based on propositional logic have been proposed recently for some multi-agent problems, for instance for combinatorial auctions [8,43,46] and automated negotiation [52]. In this paper we focus on combinatorial vote. Combinatorial vote is located within the larger class of group decision making problems. Each one of a set of agents (called voters) initially expresses her preferences on a set of alternatives (called candidates); these preferences are then aggregated so as to identify (or elect) an acceptable common alternative in an automated way (without negotiation). Formulated as such, this can be identified as a vote problem. Vote problems have been investigated by researchers in social choice theory (see for instance [40] fo... |

96 |
Default Reasoning: Causal and Conditional theories
- Geffner
- 1992
(Show Context)
Citation Context ...AND-OPT-SAT is in p2 , and in p 2 when the set {uGB(x) |x ∈ Mod(K)} can be computed in polynomial time (and therefore has a polynomial size); CAND-OPT-SAT is p2 -complete even for the simple representation language Rcard. 3.3.3. Prioritized goals Instead of weighting formulas by numerical weights, several approaches proceed by ordering them with a priority relation. Consider GB = 〈{G1, . . . ,Gn},〉 where is a weak order (called a priority relation) on {1, . . . , n}. Let be the strict order induced by , and let nonsatGB(x) = {1, . . . , n} \ satGB(x). A common choice (see, e.g., [3,29]), called the discrimin ordering when is complete, Rdiscrimin, is the following: x discriminGB y if and only if ∀i ∈ nonsatGB(x) \ nonsatGB(y) ∃j ∈ nonsatGB(y) \ nonsatGB(x) such that j i. Note that Rdiscrimin generalizes R⊆ – namely, it coincided with R⊆ when is chosen to be the relation defined by i j for all i, j . An alternative common way of inducing preference on candidates from priorities is the representation Rleximin based on leximin ordering [3]. It generalizes Rcard (and assumes that the priority relation is complete). We omit its definition. It is not hard to show ... |

95 | Background to qualitative decision theory,
- Doyle, Thomason
- 1999
(Show Context)
Citation Context ...are really new (especially those related to ceteris paribus desires) and some others are byproducts of already existing results. There are several problems whose complexity was not entirely identified in this paper, such as finding the upper bound for CW existence problems for simple languages such as Rcard (proposition 5). Another issue which is lacking here is a study of the representational complexity [12] of these languages, which would assess precisely their concision power: see [18]. 5.2. Qualitative decision theory The survey I gave in section 2 differs from Doyle and Thomason’s review [22] on qualitative decision theory. Qualitative approaches to decision theory and compact, logical approaches for preference representation are two distinct issues, even if some papers are concerned with both of them (especially [5]): qualitative decision theory aims at studying criteria for decision making under uncertainty that refer as little as possible to numbers (in contrast with the standard expected utility criterion) and it is not surprising that several approaches are based on non-classical logics whose semantics is defined by means of orderings (as [5]), or on nonmonotonic logics [9,50... |

88 |
The Boolean hierarchy I: structural properties,
- Cai, Gundermann, et al.
- 1988
(Show Context)
Citation Context ...H2 = (2) is the class of all languages L such that L = L1 ∩ L2, where L1 is in NP and L2 in coNP. The canonical BH2-complete problem is SAT-UNSAT: 〈ϕ1, ϕ2〉 is a positive instance of SAT-UNSAT if and only if ϕ1 is satisfiable and ϕ2 is unsatisfiable. NP(3) is the class of all languages L such that L = L1 ∩ (L2 ∪ L3), where L1 and L2 are in NP and L3 in coNP. The canonical NP(3)-complete problem is SATSAT-UNSAT: 〈ϕ1, ϕ2, ϕ3〉 is a positive instance of SAT-SAT-UNSAT if and only if ϕ1 is satisfiable and (ϕ2 is unsatisfiable or ϕ3 is satisfiable). Theses classes are members of the Boolean hierarchy [13]. – p2 = PNP is the class of all languages that can be recognized in polynomial time by a Turing machine equipped with an NP oracle, where an NP oracle solves whatever instance of an NP problem in unit time. p2 = p2 [O(log n)] is the class of all languages that can be recognized in polynomial time by a Turing machine using a number of NP oracles bounded by a logarithmic function of the size of the input data. – p2 = NPNP is the class of all languages recognizable in polynomial time by a nondeterministic Turing machine equipped with an NP oracle telling in unit time whether a given proposit... |

86 | Vote elicitation: Complexity and strategy-proofness.
- Conitzer, Sandholm
- 2002
(Show Context)
Citation Context ...e. 5.3. Preference elicitation In this paper we only briefly mentioned the key issue of automated preference elicitation (i.e., how to interact with a voter so as to obtain her preference relation). This issue, which has received some attention in the last years, is a necessary upstream task for combinatorial vote and relationships between both problems should be studied further. As well, identifying preferential independences between some variables for a given voter (see [2]) is extremely relevant in this context. Preference elicitation traditionally focus on one agent; now, the recent paper [17] considers the elicitation issue is a vote context; given some partial data about the votes of a number of agents, it studies the complexity of determining which piece of preference and from which voter in order to be able to determine the winner of the election. We did not consider this issue in the context of combinatorial vote, and no doubt that it is extremely relevant to it. J. Lang / Logical preference representation and combinatorial vote 57 5.4. Manipulation It has been known for long in social choice theory that there is no vote rule being both non-dictatorial and strategyproof (this ... |

79 |
Essai sur l’application de l’analyse a la probabilite des decisions rendues a la pluralite des voix.
- Condorcet
- 1785
(Show Context)
Citation Context ...eference representation and combinatorial vote Table 3 sB1 s B 2 s B 3 s B sp sv (a, b, c) 8 7 2 17 1 2 (a, b, ¬c) 8 3 6 17 1 3 (a, ¬b, c) 8 3 6 17 1 3 (a, ¬b, ¬c) 8 1 8 17 2 2 (¬a, b, c) 4 8 6 18 1 3 (¬a, b, ¬c) 4 7 2 13 0 2 (¬a, ¬b, c) 2 7 8 17 1 2 (¬a, ¬b, ¬c) 2 7 6 15 0 2 4.3. Condorcet-consistent rules 4.3.1. Condorcet winner A Condorcet winner (CW) for a profile P is a candidate x such that for any candidate y = x, there are strictly more agents preferring x to y than agents preferring y to x, i.e., ∀y ∈ Mod(K), |{i |x >i y} |> |{j |y >i x}|.12 This notion goes back to 1785 [19] and it is known since then that there are profiles for which there is no CW. Importantly, when there exists a CW, it is unique. A first problem is that the usual definition of a CW is not well-suited to combinatorial vote. This can be seen easily in the case where R = Rbasic (and holds as well for more sophisticated representation languages). We easily get that x is the Condorcet winner for B = 〈G1, . . . ,Gn〉 if and only if the number of i such that x |= Gi is maximal and no other interpretation than x maximizes this number of voters whose goal is satisfied. This has the consequence that ... |

78 | Complexity of manipulating elections with few candidates, in:
- Conitzer, Sandholm
- 2002
(Show Context)
Citation Context ...ble application of combinatorial vote is electronic vote concerning decisions about several dependent variables, that have to be taken in small organizations (small companies, laboratories, recruiting committees etc.). When the set of candidates has little or no combinatorial structure (e.g., choosing a person to be recruited), or when the variables are independent (or almost) from each other, preferences can be aggregated manually, but this is not the case as soon as the former has a strong combinatorial structure.18 17 Manipulation is already hard when the number of candidates is small, see [15,16]. 18 This can be seen for instance on the following real-world problem, concerning a decision to be taken by a recruiting committee: when not a single, but k individuals (out of n) can be recruited, the space of possible decisions has a combinatorial structure: a “candidate” is no longer an individual but a set of k individuals. The problem can be solved manually only if the dependencies between individuals are ignored, which means that the voters cannot express correlations between individuals, as for instance: “In the first place I prefer recruiting A, then B, then C, but since A and B work ... |

61 | Utility Independence in Qualitative Decision Theory.
- Bacchus, Grove
- 1996
(Show Context)
Citation Context ...pure preference relation on states, and most of these approaches degenerate when the assumption of perfect knowledge is made. 5.3. Preference elicitation In this paper we only briefly mentioned the key issue of automated preference elicitation (i.e., how to interact with a voter so as to obtain her preference relation). This issue, which has received some attention in the last years, is a necessary upstream task for combinatorial vote and relationships between both problems should be studied further. As well, identifying preferential independences between some variables for a given voter (see [2]) is extremely relevant in this context. Preference elicitation traditionally focus on one agent; now, the recent paper [17] considers the elicitation issue is a vote context; given some partial data about the votes of a number of agents, it studies the complexity of determining which piece of preference and from which voter in order to be able to determine the winner of the election. We did not consider this issue in the context of combinatorial vote, and no doubt that it is extremely relevant to it. J. Lang / Logical preference representation and combinatorial vote 57 5.4. Manipulation It ha... |

56 | Desires and defaults: A framework for planning with inferred goals. In
- Thomason
- 2000
(Show Context)
Citation Context ...al logic benefits from many well-worked algorithms (especially for satisfiability). 2. The set of possible decisions (candidates) X is identical to the set of physically realizable worlds (by the agents), Mod(K), where K is a propositional formula restricting the set of (physically) feasible worlds. This strong assumption implies that agents have a full and common knowledge of this set of feasible alternatives.4 It is important to keep in mind the strong distinction between K, which represents knowledge, and the formulas that represent goals (confusing both notions leads to “wishful thinking” [50]). Now we briefly survey different logical preference representation languages. For each of these languages we discuss its computational complexity. Since the problem at hand is not to reason about knowledge but to decide from preferences, the important problems are not quite the same as for logical languages for knowledge representation. In particular, inference is not as important, whereas the problems that are particularly important are the following: Definition 1 (COMPARISON). Given a logical specification GB (goal base) of the preferences of an agent, and two candidates x and y, the COM... |

54 | CP-nets: Reasoning and consistency testing. In
- Domshlak, Brafman
(Show Context)
Citation Context ...reported in table 1. The surprising point is that, for the general case, the comparison problem is much more difficult than the other ones (in contrast to other representation languages). This is due to possible exponentially long “preference paths” (see the proof in appendix). As we can see, imposing that goals are literals does not make the problems easier.8 Studying further the complexity of the comparison and the non-dominance problems for ceteris paribus desires under various assumptions is a promising topic. Some significant results have been obtained in the resticted case of CP-nets in [20,21], especially tractable cases (however, the general comparison problem for the restriction of ceteris paribus desires corresponding to CP-nets is still open). 3.3.6. Distances Let d be a (pseudo-)distance on X , i.e., a function from Mod(K) × Mod(K) to N such that (i) d(x, y) = 0 if and only if x = y and (ii) d(x, y) = d(y, x). Distancebased logical representations of preference [31,34,35], denoted by Rd , are based on the 8 This would however be the case with variants of the framework, that we do not consider here for the sake of brevity. J. Lang / Logical preference representation and... |

47 | A Logic of Relative Desire (Preliminary report).
- Doyle, Shoham, et al.
- 1991
(Show Context)
Citation Context ...nditional desire induces an explicit utility loss [36]. This principle is further generalized by introducing numerical strengths and polarities in [38,51]. We did not investigate complexity issues for these approaches. 3.3.5. Ceteris paribus preferences C, G and G′ being three propositional formulas and V being a subset of VAR such that Var(G) ∪ Var(G′) ⊆ V , the ceteris paribus desire C : G > G′[V ] is interpreted by: “all irrelevant things being equal, I prefer G ∧ ¬G′ to G′ ∧ ¬G”, where the “irrelevant things” are the variables that are not in V . The definitions proposed in various places [7,23,24,49] differ somehow. We take as a basis the definition of [23], slightly generalized, in the spirit of [49] but with less complications, by introducing the explicit set of variables V which expresses, in an explicit way, which variables are referred to when saying “all other things being equal” (namely, those not in V ). For natural reasons, and to remain consistent with the origiginal definitions, we impose that Var(G)∪Var(G′) ⊆ V . This modification is simple, it does not affect significantly the computational aspects of the framework and answers (to a certain extent) a criticism addressed to ce... |

43 |
Bipolar representation and fusion of preference in the possibilistic logic framework, in:
- Benferhat, Dubois, et al.
- 2002
(Show Context)
Citation Context ...i j for all i, j . An alternative common way of inducing preference on candidates from priorities is the representation Rleximin based on leximin ordering [3]. It generalizes Rcard (and assumes that the priority relation is complete). We omit its definition. It is not hard to show that as soon as x GB y can be decided in polynomial time from , which is the case for R⊆, Rdiscrimin and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 prefer... |

40 |
Logical representation of preferences for group decision making, in:
- Lafage, Lang
- 2000
(Show Context)
Citation Context ... is no candidate y ∈ Mod(K) satisfying the goals satisfied by x and at least another one. J. Lang / Logical preference representation and combinatorial vote 43 3.3.2. Weighted goals The refinement Rcard of Rbasic considers all-or-nothing but independent goals, and enables compensations. This representation can be generalized to Rwg by weighting goals with numerical valuations. The utility of a candidate is computed by first gathering the valuations of the goals it satisfies, the valuations of the goals it violated, and then by aggregating these valuations in a suitable way (see for instance [34]): GB = {〈α1,G1〉, . . . , 〈αn,Gn〉} and u F1,F2,F3 GB (x) = F1(F2({αi |x |= Gi}, F3({αj |x |= ¬Gj}))). When utility can be considered as a relative notion rather than an absolute one (which means that only differences of utilities between candidates are relevant), it can be assumed that only the violated goals count (see for instance [34,51]), which leads to uFGB(x) = F({αi |x |= ¬Gi}); F has of course to satisfy a number of desirable properties (see [34]).5 Usual choices for F are, for instance, sum (weights are then usually called penalties) or maximum (which corresponds to possibilistic... |

36 | On the complexity of conditional logics, in:
- Friedman, Halpern
- 1994
(Show Context)
Citation Context ...x{i |d ∈ Di and x violates d} otherwise. Note that more preferred J. Lang / Logical preference representation and combinatorial vote 45 candidates have lower ranks, henceforth, cond,ZGB is defined by x cond,ZGB y if and only if κGB(x) κGB(y). Intuitively speaking, cond,ZGB is the preference relation, among those satisfying GB, maximizing preference world by world [5, p. 79]. The obtained relation cond,ZGB is much more discriminant (hence much better) than cond,SGB . Complexity results for Rcond,S and Rcond,Z can be derived as byproducts of complexity results for conditional logics [28] and for conditional knowledge bases [27]: Proposition 2. 1. For Rcond,S: COMPARISON is coNP-complete.6 2. For Rcond,Z: COMPARISONcond,Z , NON-DOMINANCEcond,Z and CAND-OPT-SATcond,Z are p2 -complete. One drawback of Rcond,Z is that, as for Rcond,S , a so-called “drowning effect” occurs (some goals are ignored while they should not); this can be remedied for instance by adding extra constraints expressing that violating a conditional desire induces an explicit utility loss [36]. This principle is further generalized by introducing numerical strengths and polarities in [38,51]. We did not inves... |

33 | Qualitative choice logic, in:
- Brewka, Benferhat, et al.
- 2002
(Show Context)
Citation Context ...in and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 preferred to ¬ϕ1 ∧ϕ2 preferred to ¬ϕ1 ∧ ¬ϕ2 ∧ ϕ3, . . . . Formulas expressed in this language can be translated into a set of prioritized goals, the preference relation then being based on the leximin ordering (however the transformation is not polynomial). Therefore the problems associated with RQCL are at least as hard as with Rleximin and possibly above (we did not investigate them). 3.3.4... |

29 |
Expected utility networks, in:
- Mura, Shoham
- 1999
(Show Context)
Citation Context ...ase I would prefer smoked salmon as first course”, “I prefer white wine if one of the courses is fish and none is meat, red wine if one of the courses is meat and none is fish, and in the remaining cases I would like equally red or white wine”, etc. Since the preference structure of each voter cannot be expressed explicitly by listing all candidates, what is needed is a compact preference representation language. Such preference representation languages have been developed within the AI community; they are often build upon propositional logic, but not always (see for instance utility networks [1,33] or valued constraint satisfaction [47] – however in this paper we restrict the study to logical approaches); they enable a much more concise representation of the preference structure, while preserving a good readability (and hence a proximity with the way agents express their preferences in natural language). Therefore, the first parameter to be fixed, for a combinatorial vote problem, is the language for representing the preferences of the voters. In section 3, we recall different logical representation languages proposed in the literature, we discuss their relevance to combinatorial vote, ... |

29 | BReLS: a system for the integration of knowledge bases, in:
- Liberatore, Schaerf
- 2002
(Show Context)
Citation Context ... relative notion rather than an absolute one (which means that only differences of utilities between candidates are relevant), it can be assumed that only the violated goals count (see for instance [34,51]), which leads to uFGB(x) = F({αi |x |= ¬Gi}); F has of course to satisfy a number of desirable properties (see [34]).5 Usual choices for F are, for instance, sum (weights are then usually called penalties) or maximum (which corresponds to possibilistic logic). The complexity of decision problems for weighted logics can easily be derived from the complexity of distance-based belief merging [30,39]. Assuming that the aggregation functions F1, F2 and F3 can be computed in polynomial time, we get the following results as byproducts from known results, especially the complexity of distance-based belief merging ([39], and especially [30]): COMPARISON is polynomial; NON-DOMINANCE is coNPcomplete; CAND-OPT-SAT is in p2 , and in p 2 when the set {uGB(x) |x ∈ Mod(K)} can be computed in polynomial time (and therefore has a polynomial size); CAND-OPT-SAT is p2 -complete even for the simple representation language Rcard. 3.3.3. Prioritized goals Instead of weighting formulas by numerical wei... |

29 | Parameters for utilitarian desires in a qualitative decision theory,
- Torre, Weydert
- 2001
(Show Context)
Citation Context ...ghting goals with numerical valuations. The utility of a candidate is computed by first gathering the valuations of the goals it satisfies, the valuations of the goals it violated, and then by aggregating these valuations in a suitable way (see for instance [34]): GB = {〈α1,G1〉, . . . , 〈αn,Gn〉} and u F1,F2,F3 GB (x) = F1(F2({αi |x |= Gi}, F3({αj |x |= ¬Gj}))). When utility can be considered as a relative notion rather than an absolute one (which means that only differences of utilities between candidates are relevant), it can be assumed that only the violated goals count (see for instance [34,51]), which leads to uFGB(x) = F({αi |x |= ¬Gi}); F has of course to satisfy a number of desirable properties (see [34]).5 Usual choices for F are, for instance, sum (weights are then usually called penalties) or maximum (which corresponds to possibilistic logic). The complexity of decision problems for weighted logics can easily be derived from the complexity of distance-based belief merging [30,39]. Assuming that the aggregation functions F1, F2 and F3 can be computed in polynomial time, we get the following results as byproducts from known results, especially the complexity of distance-based... |

27 | Distance-based merging: a general framework and some complexity results, in:
- Konieczny, Lang, et al.
- 2002
(Show Context)
Citation Context ... relative notion rather than an absolute one (which means that only differences of utilities between candidates are relevant), it can be assumed that only the violated goals count (see for instance [34,51]), which leads to uFGB(x) = F({αi |x |= ¬Gi}); F has of course to satisfy a number of desirable properties (see [34]).5 Usual choices for F are, for instance, sum (weights are then usually called penalties) or maximum (which corresponds to possibilistic logic). The complexity of decision problems for weighted logics can easily be derived from the complexity of distance-based belief merging [30,39]. Assuming that the aggregation functions F1, F2 and F3 can be computed in polynomial time, we get the following results as byproducts from known results, especially the complexity of distance-based belief merging ([39], and especially [30]): COMPARISON is polynomial; NON-DOMINANCE is coNPcomplete; CAND-OPT-SAT is in p2 , and in p 2 when the set {uGB(x) |x ∈ Mod(K)} can be computed in polynomial time (and therefore has a polynomial size); CAND-OPT-SAT is p2 -complete even for the simple representation language Rcard. 3.3.3. Prioritized goals Instead of weighting formulas by numerical wei... |

17 |
Conditional desires and utilities – an alternative logical approach to qualitative decision theory, in:
- Lang
- 1996
(Show Context)
Citation Context ... Complexity results for Rcond,S and Rcond,Z can be derived as byproducts of complexity results for conditional logics [28] and for conditional knowledge bases [27]: Proposition 2. 1. For Rcond,S: COMPARISON is coNP-complete.6 2. For Rcond,Z: COMPARISONcond,Z , NON-DOMINANCEcond,Z and CAND-OPT-SATcond,Z are p2 -complete. One drawback of Rcond,Z is that, as for Rcond,S , a so-called “drowning effect” occurs (some goals are ignored while they should not); this can be remedied for instance by adding extra constraints expressing that violating a conditional desire induces an explicit utility loss [36]. This principle is further generalized by introducing numerical strengths and polarities in [38,51]. We did not investigate complexity issues for these approaches. 3.3.5. Ceteris paribus preferences C, G and G′ being three propositional formulas and V being a subset of VAR such that Var(G) ∪ Var(G′) ⊆ V , the ceteris paribus desire C : G > G′[V ] is interpreted by: “all irrelevant things being equal, I prefer G ∧ ¬G′ to G′ ∧ ¬G”, where the “irrelevant things” are the variables that are not in V . The definitions proposed in various places [7,23,24,49] differ somehow. We take as a basis the de... |

17 |
Complexity results for independence and definability in propositional logic, in:
- Lang, Marquis
- 1998
(Show Context)
Citation Context ... giving a little more expressivity to the framework, for instance by allowing ramifications of the goals to be taken into account (by default, V is considered to be the set of variables mentioned in G and G′7). 6 As to NON-DOMINANCE and CAND-OPT-SAT, I could not manage to identify exactly their complexity; the best I can say is that they are BH2-hard and (obviously) in p 2 , but since this is not very significant (the gap between BH2 and p 2 being large), I omit the technical details. 7 This could be refined further by considering the variables on which G and G′ do not semantically depend [37,49]. This is not be considered further in this paper. 46 J. Lang / Logical preference representation and combinatorial vote Table 1 COMPARISON NON-DOMINANCE CAND-OPT-SAT General case PSPACE-comp. coNP-comp. p2 -comp. Simple desires PSPACE-comp. coNP-comp. p2 -comp. Definition 4 (RCP). Let GB = 〈K,D〉 with D = {C1 : G1 > G′1[V1], . . . , Cm : Gm > G′m[Vm]} such that for all i, Ci , Gi and G′i are propositional formulas and Var(Gi) ∪ Var(G′i) ⊆ Vi ⊆ VAR. For two candidates x, y ∈ C, x is said to dominate y with respect to the desire Di = (Ci : Gi > G′i[Vi]), denoted by x >Di y, if and only i... |

16 | Nonmonotonic reasoning: from complexity to algorithms,
- Cayrol, Lagasquie-Schiex, et al.
- 1998
(Show Context)
Citation Context ...can be decided in polynomial time from , which is the case for R⊆, Rdiscrimin and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 preferred to ¬ϕ1 ∧ϕ2 preferred to ¬ϕ1 ∧ ¬ϕ2 ∧ ϕ3, . . . . Formulas expressed in this language can be translated into a set of prioritized goals, the preference relation then being based on the leximin ordering (however the transformation is not polynomial). Therefore the problems associated with RQCL are at least as hard... |

15 | Comparing space efficiency of propositional knowledge representation formalisms. In: KR,
- Cadoli, Liberatore, et al.
- 1996
(Show Context)
Citation Context ...would not be so far from those obtained, but again, this is left for further research. – Logical languages for qualitative decision theory (see next paragraph). As to complexity results, some are really new (especially those related to ceteris paribus desires) and some others are byproducts of already existing results. There are several problems whose complexity was not entirely identified in this paper, such as finding the upper bound for CW existence problems for simple languages such as Rcard (proposition 5). Another issue which is lacking here is a study of the representational complexity [12] of these languages, which would assess precisely their concision power: see [18]. 5.2. Qualitative decision theory The survey I gave in section 2 differs from Doyle and Thomason’s review [22] on qualitative decision theory. Qualitative approaches to decision theory and compact, logical approaches for preference representation are two distinct issues, even if some papers are concerned with both of them (especially [5]): qualitative decision theory aims at studying criteria for decision making under uncertainty that refer as little as possible to numbers (in contrast with the standard expected ... |

13 |
Propositional distances and preference representation, in: Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU
- Lafage, Lang
- 2001
(Show Context)
Citation Context ...f the comparison and the non-dominance problems for ceteris paribus desires under various assumptions is a promising topic. Some significant results have been obtained in the resticted case of CP-nets in [20,21], especially tractable cases (however, the general comparison problem for the restriction of ceteris paribus desires corresponding to CP-nets is still open). 3.3.6. Distances Let d be a (pseudo-)distance on X , i.e., a function from Mod(K) × Mod(K) to N such that (i) d(x, y) = 0 if and only if x = y and (ii) d(x, y) = d(y, x). Distancebased logical representations of preference [31,34,35], denoted by Rd , are based on the 8 This would however be the case with variants of the framework, that we do not consider here for the sake of brevity. J. Lang / Logical preference representation and combinatorial vote 47 intuitive idea that, when expressing a goal G, then ideally, x must satisfy G, and when it is no longer the case, then, the “further” x is from G, the less preferred x. Formally, a pair 〈{G}, d〉, where G is a propositional formula and d a pseudodistance, induces the utility function uGB (x) = −d(x,G) = − min y|=G d (x, y). When d is computable in polynomial time, on... |

12 | Specification and evaluation of preferences for planning under uncertainty, in:
- Tan, Pearl
- 1994
(Show Context)
Citation Context ...nditional desire induces an explicit utility loss [36]. This principle is further generalized by introducing numerical strengths and polarities in [38,51]. We did not investigate complexity issues for these approaches. 3.3.5. Ceteris paribus preferences C, G and G′ being three propositional formulas and V being a subset of VAR such that Var(G) ∪ Var(G′) ⊆ V , the ceteris paribus desire C : G > G′[V ] is interpreted by: “all irrelevant things being equal, I prefer G ∧ ¬G′ to G′ ∧ ¬G”, where the “irrelevant things” are the variables that are not in V . The definitions proposed in various places [7,23,24,49] differ somehow. We take as a basis the definition of [23], slightly generalized, in the spirit of [49] but with less complications, by introducing the explicit set of variables V which expresses, in an explicit way, which variables are referred to when saying “all other things being equal” (namely, those not in V ). For natural reasons, and to remain consistent with the origiginal definitions, we impose that Var(G)∪Var(G′) ⊆ V . This modification is simple, it does not affect significantly the computational aspects of the framework and answers (to a certain extent) a criticism addressed to ce... |

10 |
Reasoning with conditional ceteris paribus statements, in:
- Boutilier, Brafman, et al.
- 1999
(Show Context)
Citation Context |

10 | Complexity results for default reasoning from conditional knowledge bases, in:
- Eiter, Lukasiewicz
- 2000
(Show Context)
Citation Context ...can be decided in polynomial time from , which is the case for R⊆, Rdiscrimin and Rleximin, then COMPARISON is polyno5 However, this assumption is sometimes not made; in this case, positive preference items (goals) have to be formally distinguished from negative ones (constraints); see [4]. 44 J. Lang / Logical preference representation and combinatorial vote mial, therefore NON-DOMINANCE is in coNP and CAND-OPT-SAT in p2 . Moreover, NON-DOMINANCE is coNP-complete for R⊆. Lastly, from results in [26,41], it can be easily derived that CAND-OPT-SAT is p2 -complete for R⊆, and from results in [14,27], that CAND-OPT-SAT is p2 -complete for Rleximin. A recent approach [10] proposes a preference representation language QCL based on prioritized goals, where priority is expressed by means of a new connector ×: ϕ1 ×ϕ2 ×· · ·×ϕn has to be understood as the priority relation: ϕ1 preferred to ¬ϕ1 ∧ϕ2 preferred to ¬ϕ1 ∧ ¬ϕ2 ∧ ϕ3, . . . . Formulas expressed in this language can be translated into a set of prioritized goals, the preference relation then being based on the leximin ordering (however the transformation is not polynomial). Therefore the problems associated with RQCL are at least as hard... |

9 | On decision-theoretic foundations for defaults, in:
- Brafman, Friedman
- 1995
(Show Context)
Citation Context ... [22] on qualitative decision theory. Qualitative approaches to decision theory and compact, logical approaches for preference representation are two distinct issues, even if some papers are concerned with both of them (especially [5]): qualitative decision theory aims at studying criteria for decision making under uncertainty that refer as little as possible to numbers (in contrast with the standard expected utility criterion) and it is not surprising that several approaches are based on non-classical logics whose semantics is defined by means of orderings (as [5]), or on nonmonotonic logics [9,50]; however the goal in these approaches is not to describe a pure preference relation on states, and most of these approaches degenerate when the assumption of perfect knowledge is made. 5.3. Preference elicitation In this paper we only briefly mentioned the key issue of automated preference elicitation (i.e., how to interact with a voter so as to obtain her preference relation). This issue, which has received some attention in the last years, is a necessary upstream task for combinatorial vote and relationships between both problems should be studied further. As well, identifying preferential ... |

8 |
How many candidates are required to make an election hard to manipulate?, in:
- Conitzer, Lang, et al.
- 2003
(Show Context)
Citation Context ...ble application of combinatorial vote is electronic vote concerning decisions about several dependent variables, that have to be taken in small organizations (small companies, laboratories, recruiting committees etc.). When the set of candidates has little or no combinatorial structure (e.g., choosing a person to be recruited), or when the variables are independent (or almost) from each other, preferences can be aggregated manually, but this is not the case as soon as the former has a strong combinatorial structure.18 17 Manipulation is already hard when the number of candidates is small, see [15,16]. 18 This can be seen for instance on the following real-world problem, concerning a decision to be taken by a recruiting committee: when not a single, but k individuals (out of n) can be recruited, the space of possible decisions has a combinatorial structure: a “candidate” is no longer an individual but a set of k individuals. The problem can be solved manually only if the dependencies between individuals are ignored, which means that the voters cannot express correlations between individuals, as for instance: “In the first place I prefer recruiting A, then B, then C, but since A and B work ... |

8 | Modelling and reasoning about preferences with CP-nets,
- Domshlak
- 2002
(Show Context)
Citation Context ...reported in table 1. The surprising point is that, for the general case, the comparison problem is much more difficult than the other ones (in contrast to other representation languages). This is due to possible exponentially long “preference paths” (see the proof in appendix). As we can see, imposing that goals are literals does not make the problems easier.8 Studying further the complexity of the comparison and the non-dominance problems for ceteris paribus desires under various assumptions is a promising topic. Some significant results have been obtained in the resticted case of CP-nets in [20,21], especially tractable cases (however, the general comparison problem for the restriction of ceteris paribus desires corresponding to CP-nets is still open). 3.3.6. Distances Let d be a (pseudo-)distance on X , i.e., a function from Mod(K) × Mod(K) to N such that (i) d(x, y) = 0 if and only if x = y and (ii) d(x, y) = d(y, x). Distancebased logical representations of preference [31,34,35], denoted by Rd , are based on the 8 This would however be the case with variants of the framework, that we do not consider here for the sake of brevity. J. Lang / Logical preference representation and... |

5 |
Handbook of Defeasible Reasoning and Uncertainty Management Systems (Kluwer Academic,
- Nebel
- 1998
(Show Context)
Citation Context ...ariants of the framework, that we do not consider here for the sake of brevity. J. Lang / Logical preference representation and combinatorial vote 47 intuitive idea that, when expressing a goal G, then ideally, x must satisfy G, and when it is no longer the case, then, the “further” x is from G, the less preferred x. Formally, a pair 〈{G}, d〉, where G is a propositional formula and d a pseudodistance, induces the utility function uGB (x) = −d(x,G) = − min y|=G d (x, y). When d is computable in polynomial time, one easily derives from the literature on the complexity of belief revision [26,42] and of distance-based belief merging [30] that COMPARISON (which amounts to deciding whether d(x,G) d(y,G)), NONDOMINANCE and CAND-OPT-SAT are in p2 (and, in particular, p 2 -complete when d is the Hamming distance). This principle can be generalized by considering a set of goals, each goal being associated with a pseudo-distance: GB = {〈G1, d1〉, . . . , 〈Gn, dn〉} and uGB(x) = F (d1(x,G1), . . . , dn(x,Gn)), where F is an aggregation function. This has no strong impact on complexity. 3.3.7. Discussion These results have a value of their own, since they enable a first comparison of p... |

2 | The computational complexity of propositional STRIPSplanning. ArtificialIntelligence 69:165–204 - Bylander - 1994 |

2 |
On the limits of ordinality in decision making, in:
- Dubois, Fargier, et al.
- 2002
(Show Context)
Citation Context ...inatorial blow up. Such languages are said to be factorized, or succinct, because they enable a much more concise representation of preference structures than explicit representations. In the rest of the paper, for simplicity reasons we make the following two important hypotheses: 1. The representation languages considered are logical (and propositional), i.e., each vi is a binary variable: D1 = · · · = Dn = {0, 1}. Languages based on propositional 1 More generally, this relation may be fuzzy so as to enable representing intensities of preference. We omit this eventuality in this paper. 2 See [25] for an extensive discussion on the limits of ordinality in decision making under uncertainty, multicriteria decision making and social choice. 3 Or possibly a subset of R from which R is drawn by transitive closure – this is only a detail since this does not enable escaping the combinatorial blow up. J. Lang / Logical preference representation and combinatorial vote 41 logic are not only compact, but also particularly expressive, thanks to the expressive power of logic, and therefore they are close to intuition (ideally, a preference representation language should be easily obtained from its ... |

1 | Berre: 2002, ‘Qualitative choice logic - Brewka, Benferhat, et al. |

1 |
Expressivity and succinctness power of preference representation language,
- Coste-Marquis, Lang, et al.
- 2003
(Show Context)
Citation Context ...arch. – Logical languages for qualitative decision theory (see next paragraph). As to complexity results, some are really new (especially those related to ceteris paribus desires) and some others are byproducts of already existing results. There are several problems whose complexity was not entirely identified in this paper, such as finding the upper bound for CW existence problems for simple languages such as Rcard (proposition 5). Another issue which is lacking here is a study of the representational complexity [12] of these languages, which would assess precisely their concision power: see [18]. 5.2. Qualitative decision theory The survey I gave in section 2 differs from Doyle and Thomason’s review [22] on qualitative decision theory. Qualitative approaches to decision theory and compact, logical approaches for preference representation are two distinct issues, even if some papers are concerned with both of them (especially [5]): qualitative decision theory aims at studying criteria for decision making under uncertainty that refer as little as possible to numbers (in contrast with the standard expected utility criterion) and it is not surprising that several approaches are based on ... |

1 |
Languages for negociation, in:
- Wooldridge, Parsons
- 2000
(Show Context)
Citation Context ...cerned with interaction, cooperation or negotiation within agent societies. For these problems, it often occurs that the set of all feasible states has a very large size, due to its combinatorial nature. For this reason, research has been done so as to develop representation languages aiming at enabling a succinct representation of the description of the problem, without having to enumerate a prohibitive number of states. Languages based on propositional logic have been proposed recently for some multi-agent problems, for instance for combinatorial auctions [8,43,46] and automated negotiation [52]. In this paper we focus on combinatorial vote. Combinatorial vote is located within the larger class of group decision making problems. Each one of a set of agents (called voters) initially expresses her preferences on a set of alternatives (called candidates); these preferences are then aggregated so as to identify (or elect) an acceptable common alternative in an automated way (without negotiation). Formulated as such, this can be identified as a vote problem. Vote problems have been investigated by researchers in social choice theory (see for instance [40] for an overview) who have studied... |