Plan for Today General requirements on preference representation - PowerPoint PPT Presentation

Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Plan for Today • General requirements on preference representation languages • Distinguish cardinal and ordinal preference structures Computational Social Choice: Spring 2008 • Different classes of utility functions (cardinal preferences): monotonic, dichotomous, modular, concave utilities . . . Ulle Endriss • Review of languages for representing utility functions: Institute for Logic, Language and Computation explicit form , k -additive form , weighted goals , . . . University of Amsterdam • Discussion of properties of different representation languages: expressive power , comparative succinctness , complexity • Review of languages for ordinal preference representation: prioritised goals and ceteris paribus preferences Ulle Endriss 1 Ulle Endriss 3 Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation Preference Representation Languages Collective decision making is driven by the interests of individuals, The following questions should be addressed when you investigate a who must be able to communicate preferences (directly through full preference representation language: revelation, or indirectly via “moves” in a game). • Cognitive relevance: How close is a given language to the way • So far, we have treated this topic only very abstractly , by in which humans would express their preferences? saying that agents “have” some preference structure. • Elicitation: How difficult is it to elicit the preferences of an • Preferences representation in combinatorial domains: agent so as to represent them in the chosen language? – electing a committee of size k from amongst n candidates • Expressive power: Can the chosen language encode all the � n � requires expressing preferences over possible committees; preference structures we are interested in? k – negotiation over n goods requires expressing preferences • Succinctness: Is the representation of (typical) structures over 2 n alternative bundles. succinct? Is one language more succinct than the other? • We shall review several preference representation languages . • Complexity: What is the computational complexity of related Some will be discussed in more detail later on in the course. decision problems, such as comparing two alternatives? We shall be interested in the properties of these languages, We are going to concentrate on expressive power and succinctness. such as expressive power and comparative succinctness . Ulle Endriss 2 Ulle Endriss 4

Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Preferences in Resource Allocation Scenarios Cardinal and Ordinal Preferences A representative example for a combinatorial domain: A preference structure represents an agent’s preferences over a set Let R be a finite set of indivisible resources (goods) with |R| = n . of alternatives X . There are different types of preference structures: Assume there are no externalities: agent preferences only depend • A cardinal preference structure is a ( utility or valuation ) on their assigned bundle (not on, say, the allocation as a whole) ❀ function u : X → Val , where Val is usually a set of numerical need to model preference structures over X = 2 R values such as N or R . Hence, the explicit representation has exponential space complexity. • An ordinal preference structure is a binary relation � over the set of alternatives (reflexive, transitive and connected). Possible ways out: • only consider restricted classes of preference structures, which Note that we shall assume that X is finite. may allow for a more concise representation; and/or Remark: What I refer to as connectedness is mostly called • consider (and compare) different representation languages . completeness in the literature. We start with the case of utility functions . . . Ulle Endriss 5 Ulle Endriss 7 Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Some Observations Classes of Utility Functions • Intrapersonal comparison: ordinal and cardinal preferences allow for Now a utility function is a mapping u : 2 R → R . comparing the satisfaction of an agent for different alternatives • u is normalised iff u ( { } ) = 0 • Interpersonal comparison: ordinal preferences don’t allow for interpersonal comparison (“Ann likes x more than Bob likes y ”) • u is non-negative iff u ( X ) ≥ 0 • Preference intensity: ordinal preferences cannot express preference • u is monotonic iff u ( X ) ≤ u ( Y ) whenever X ⊆ Y intensity; cardinal preferences can (subject to Val being numerical) • u is dichotomous iff u ( X ) = 0 or u ( X ) = 1 • Representability: a connected ordinal preference relation � is • u is modular iff u ( X ∪ Y ) = u ( X ) + u ( Y ) − u ( X ∩ Y ) representable by a utility function u : x � y iff u ( x ) ≤ u ( y ) � • u is additive iff u ( X ) = u ( { x } ) • Cognitive relevance: hard to make general statements, but at least x ∈ X ordinal preferences don’t require reasoning with numerical utilities Important: For the above definitions, the respective (in)equalities • Explicit representation: the explicit representation of cardinal and are understood to hold for all bundles X, Y ⊆ R . ordinal preferences have space complexity O ( |X | ) resp. O ( |X | 2 ) Ulle Endriss 6 Ulle Endriss 8

Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Modular and Additive Utilities Observations Modularity and additivity are really just two different names for The following relationships amongst some of these classes of utility the same thing (well, almost): functions are easily checked: Proposition 1 A utility function is additive iff it is both modular • submodular ∩ supermodular = modular and normalised. • u submodular iff − u supermodular Proof: “ ⇒ ”: obvious � “ ⇐ ”: Let X ⊆ R , x ∈ X . • u concave iff − u convex ¿From modularity, we get u ( X ) = u ( X \{ x } ) + u ( { x } ) − u ( { } ). • concave ⊆ submodular (Proof: set Z = X ∩ Y ) As u is normalised, we obtain u ( X ) = u ( X \{ x } ) + u ( { x } ). � • convex ⊆ supermodular If we iterate this step | X | times, we get u ( X ) = u ( { x } ). � x ∈ X Ulle Endriss 9 Ulle Endriss 11 Preference Representation in Combinatorial Domains COMSOC 2008 Preference Representation in Combinatorial Domains COMSOC 2008 Explicit Representation The explicit form of representing a utility function u consists of a table listing for every bundle X ⊆ R the utility u ( X ). More Classes of Utility Functions By convention, table entries with u ( X ) = 0 may be omitted. A few more commonly used classes of utility functions: • the explicit form is fully expressive: • u is submodular iff u ( X ∪ Y ) ≤ u ( X ) + u ( Y ) − u ( X ∩ Y ) any utility function u : 2 R → R may be so described • the explicit form is not concise: it may require up to 2 n entries • u is supermodular iff u ( X ∪ Y ) ≥ u ( X ) + u ( Y ) − u ( X ∩ Y ) • u is concave iff u ( X ∪ Y ) − u ( Y ) ≤ u ( X ∪ Z ) − u ( Z ) for Y ⊇ Z Even very simple utility functions may require exponential space: – Intuition: marginal utility (of obtaining X ) decreases as we e.g. the additive function mapping bundles to their cardinality. move to a better starting position (namely from Z to Y ) Remark: Of course, any additive utility function could be encoded • u is convex iff u ( X ∪ Y ) − u ( Y ) ≥ u ( X ∪ Z ) − u ( Z ) for Y ⊇ Z very concisely: just store the utilities for individual goods + the information that this is an additive function ❀ linear space But this is not a general language (not fully expressive). Ulle Endriss 10 Ulle Endriss 12