aa r X i v : . [ m a t h . L O ] A p r Aggregating Relational Structures
Harshit Bisht − − − and Amit Kuber − − − Indian Institute of Technology, Kanpur, U.P. 208016, India
Abstract.
We generalize the Arrow’s impossibility theorem–a key resultin social choice theory–to the setting where the arity k of the relationunder consideration is greater than 2. Some special but natural propertiesof k -ary relations are considered, as well as an analogue for such k -aryrelations of Endriss and Grandi’s result on graph aggregation is proved. Keywords:
Social choice · Arrow’s impossibility theorem · Aggregationof structures.
With the seminal result in [1], Arrow introduced the problem of aggregatingvoters’ preferences into a collective preference ordering over a set of alternatives(candidates) and showed that such an aggregation must be dictatorial to satisfyseemingly reasonable constraints, laying the foundations for social choice theory.The result has been proved in multiple different settings ([2],[3]), and general-ized to partially ordered preferences and graphs in [3] and [4] respectively. Ascomputational tools make it easy to survey and assess data, summarisation andaggregation tools will be required to make inferences or decisions based on theobserved relationships such as planning city routes given access to graphs cor-responding to each pedestrian’s activity. However, not all relationships can beadequately represented in the form of binary relations such as partial orders andgraphs. Consider the relationship of “knowing” someone on a social network in-ferred from being tagged in the same picture, a group of three friends in thesame photo carries more information than all combinations of pairs and can bebest expressed by a 2-simplex. With the motivation of extending the problem toaccount for such complex relationships, we generalize the impossibility result toaggregating relational structures of any arity satisfying some special properties. k -ary Relations Fix a non-empty finite set A of candidates. Consider the predicate language L A := { R, { c a | a ∈ A }} (for simplicity, we have assumed that L A containsa single relation symbol R ), where R is a k -ary relation symbol with k ≥ c a is a constant symbol. Let T consist of a single L A -sentence whoseinterpretation guarantees that each L A -structure has domain (in bijection with) A . We will deal with models of some L A -theory T extending T ; the extension H. Bisht, A.S. Kuber T will be specified in the due course. When the theory T is fixed, in order toemphasize on the set A , we denote the collection of models of the theory T by M ( A ).A social choice situation over L A consists of the following. Let I denote aset of voters/individuals. Each voter chooses A i ∈ M ( A ). An aggregation rule(also known as a social welfare function) is a map σ : D ⊆ M ( A ) I → M ( A )that satisfies some desirable properties, where D is the set of allowed ballots orprofiles. By appropriately choosing T and the properties of the aggregation rule,we will prove that the only legitimate choice of the aggregated structure is eithera filtered product or an ultraproduct of {M i } i ∈I ; the latter case corresponds toa “dictatorship” when I is finite. k -ary relations Since Arrow’s original result speaks of preference relations, it seemed natural tofirst attempt to extend it over relations. Owing at the lack of standard definitionsof properties for k -ary relations, we attempt to provide our own and arrive atthe result. We begin by setting up some notations. – ( a , ..., a m ) + i ,..., + i n , − j ,..., − j w δ r refers to the set of subsequences of ( a , ..., a m )of length m − r , with elements a i , ..., a i n necessarily present and elements a j , ..., a j w necessarily absent. – ¯ a − j , b refers to the singleton element in ( a , ..., a j , b, a j +1 , ..., a k ) − jδ .Over a set A , a k -ary relation R is simply a subset of A × ... × A (k-times) or A k . Now we define some desirable properties of k -ary relations that will makeup the extended theory T . Definition 1. A k -ary relation R is called connected if, for each pairwise-distinct a , ..., a k ∈ A , there is a permutation τ of { , ..., k } such that ( a , ..., a k ) τ is in R . Definition 2. A k -ary relation R is called simplicial transitive if for each ( k + 1) -ary sequence of pairwise-distinct elements ( a , ..., a k , a k +1 ) , we have that ( a , ..., a k +1 ) + jδ ⊆ R implies ( a , ..., a k +1 ) − jδ ⊆ R for each j ∈ { , ..., k + 1 } . Definition 3. A k -ary relation R is called path transitive if for k -ary se-quences of pairwise-distinct elements ( a , ..., a k ) and ( b , ..., b k ) with a i = b j where i > j , we have that ( a , ..., a k ) ∈ R and ( b , ..., b k ) ∈ R implies that ( a , ..., a i − , b j +1 , ..., b k ) δ i − j − ⊆ R . Definition 4. A k -ary relation R is called exclusive if R [¯ a τ ] does not hold forall permutations τ of , ..., k together. Several natural relationships are not adequately captured by binary relations.The following two examples satisfy the above defined properties and presentnatural aggregation scenarios where our result asserts that a desirable collectiverelation is impossible to produce. ggregating Relational Structures 3
Moderate Voters
Consider a collection of voters and a group of electoralcandidates. Each voter interprets the political inclination of each candidate asleft or right leaning compared to the others resulting in a total order over theset of candidates. If each voter prefers the moderate candidate in a group of3 candidates, then this voting behaviour can be captured by a “betweenness”relation, with ( a, b, c ) ∈ R i ↔ ( c, b, a ) ∈ R i representing the i th voters preferencefor b over a and c . Clearly, relations of this nature are both connected andexclusive. This relation is also simplicial transitive since for a sequence a , ..., a of candidates, specifying the betweenness of 3 restricted triples is enough toguarantee a , ..., a as strictly monotonous. Seating along a circular table
Consider a party of dinner guests to be seatedon circular table in groups of 4 (any cyclic arrangement is fine) and preferencesover how every subsets of 4 people should be seated. This quartenary relation R on the party of dinner guests is such that if ( a, b, c, d ) ∈ R then all cyclicpermutations ( b, c, d, a ) , ( c, d, a, b ) , ( d, a, b, c ) ∈ R and no other permutation of { a, b, c, d } is in R . Again, this is both connected and exclusive. Now, consider thesame voter’s preferences over seating 5 people on the same table. If he agrees onthe cyclic permutation of every collection of 4 people but one, he will also agreeto the final restriction of the original 5-cycle. This is the definition of simplicialtransitivity considered above. Thus, preferences on the seating of people aroundcircular tables thus also satisfies the properties defined above.Our main result implies that both these situations cannot be aggregated withcertain desirable properties of the aggregation map. We considered some properties of k -ary relations above. Now we state the appro-priate k -ary generalizations of the properties of the aggregation map that makethe proof of the Arrow’s impossibility theorem work.( UD ) ∀ a , ..., a k +1 ∈ A. ∀ p ∈ M ( { a , ..., a k +1 } ) I . ∃ q ∈ D .q | a ,...,a k = p Normally one expects the aggregation map to be defined on all I -tuples of T -models, but the property UD generalizes the result by requiring the domain D to only be large enough for the proof to go through.( P ) ∀ ¯ a ∈ A k . ∀ p ∈ D . ( ∀ i ∈ I .p i | = R [¯ a ]) ⇒ σ ( p ) | = R [¯ a ] P requires that the result must satisfy the atomic formulas satisfied by eachindividual voter.( IIA ) ∀ ¯ a ∈ A k . ∀ p, q ∈ D . ( ∀ i ∈ I .p i | = R [¯ a ] ⇔ q i | = R [¯ a ]) ⇒ ( σ ( p ) | = R [¯ a ] ⇔ σ ( q ) | = R [¯ a ]) IIA requires each atomic formula’s satisfaction to be independent of otherformulas.
H. Bisht, A.S. Kuber ( D ) ∃ i ∈ I . ∀ ¯ a ∈ A k . ∀ p ∈ D . ( p i | = R [¯ a ] ⇔ σ ( p ) | = R [¯ a ]) D posits that the result satisfies the same atomic formulas as one specificvoter. k -ary relation Now we are ready to state and prove the generalization of Arrow’s theorem to thesituation for a single k -ary relation symbol. We deal with two sets of propertiesof k -ary relations, and prove the theorem in both cases parallelly. Theorem 1.
Let ( A, I , D , σ ) be a social choice situation over k -ary relation R language L with | A | ≥ k + 1 , satisfying UD , P , and IIA where R is (simplicialor path) transitive, exclusive, and connected. Then for finite I , it also satisfies D . We prove this theorem in a series of propositions along the lines of [5]. First weintroduce some notation. For any U ⊆ I , define U ¯ a := { p ∈ D| p U | = R [¯ a ] ∧ p U c = R [¯ a ] } . Define a new k -ary relation D U on A by D U [¯ a ] := distinct(¯ a ) ∧ ∀ p ∈ U ¯ a .σ ( p ) | = R [¯ a ], where distinct(¯ a ) is short for V ≤ i For any tuple of pairwise distinct elements ¯ a , D U [¯ a ] ⇒ D U [¯ a − j , b ] for each j ∈ { , ..., k } .Proof. If a j = b , there is nothing to prove and we are done. If R is simplicial transitive: For a j = b , we construct a profile p with ¯ a , (¯ a − j , b ), and ( a , ..., a j , b, ..., a k ) j,j +1 δ holding (or not) at U and U c according to the following table. The p as con-structed can be posited to exist due to UD .¯ a ¯ a − j , b ( a , ..., a j , b, ..., a k ) j,j +1 δ U ✓ ✓ ✓ U c ✗ ✗ ✓ Clearly, p ∈ U ¯ a ∩ U ¯ a − j ,b , along with p ∈ I ¯ c for each ¯ c ∈ ( a , ..., a j , b, ..., a k ) j,j +1 δ .Thus, σ ( p ) | = R [¯ a ] (definition of D U ) and σ ( p ) | = R [¯ c ] for each ¯ c ∈ ( a , ..., a j , b, ..., a k ) j,j +1 δ ( P ). By transitivity, we can now conclude σ ( p ) | = R [¯ a − j , b ].Now, for any other profile q ∈ U ¯ a − j ,b , we know that ∀ i ∈ I .p i | = R [¯ a − j , b ] ⇔ q i | = R [¯ a − j , b ], making us conclude σ ( q ) | = R [¯ a − j , b ] ( IIA ) and thus that D U [¯ a − j , b ] holds. If R is path transitive: For a j = b , we construct a profile p with various combinations of a , ..., a k , b hold-ing (or not) at U and U c according to the following table. The p as constructedcan be posited to exist due to UD .¯ a ¯ a − j , b ( a , ..., a j , b, ..., a k ) U ✓ ✓ ✓ U c ✗ ✗ ✓ ggregating Relational Structures 5 Clearly, p ∈ U ¯ a ∩ U ¯ a − j ,b , along with p ∈ I ¯ c for ¯ c = ( a , ..., a j , b, ..., a k ). Thus, σ ( p ) | = R [¯ a ] (definition of D U ) and σ ( p ) | = R [¯ c ] for ¯ c ( a , ..., a j , b, ..., a k ) ( P ).By transitivity, we can now conclude σ ( p ) | = ¯ a − j , b . Now, for any other profile q ∈ U ¯ a − j ,b , we know that ∀ i ∈ I .p i | = R [¯ a − j , b ] ⇔ q i | = R [¯ a − j , b ], making usconclude σ ( q ) | = R [¯ a − j , b ] ( IIA ) and thus that D U [¯ a − j , b ] holds.Note that this proof does not work for j = 1. For that, the same proof with( b, a , ..., a k − ) (reversing position of a j and b ) works. Proposition 2. For any k-tuples of pairwise distinct elements ¯ a and ¯ b , D U [¯ a ] ⇒ D U [¯ b ] .Proof. We will work through the proof in two cases. Case 1: Let ¯ b have the same elements as ¯ a , only in a different permutation.Since | A | ≥ k + 1, we can get hold of c ∈ A which is distinct from all elementsin ¯ a (or ¯ b ). Consider the following inductive procedure: Algorithm 1: Concluding D U [¯ b ] from D U [¯ a ] Beginning from D [¯ a ] Identify a m such that b = a m Conclude D U [¯ a − m , c ] (by proposition 2) Conclude D U [(¯ a − m , c ) − , b ] (by proposition 2) Conclude D U [((¯ a − m , c ) − , b ) − m , a ] (by proposition 2) You have now shown D U holds for a permutation agreeing with ¯ a on thefirst element Repeat similarly for b , b , ..., b k Case 2: Let ¯ b contain c , ..., c n and ¯ a contain d , ..., d n as the elements notshared between them. Conclude that D U holds at a sequence with each d i in¯ a replaced arbitrarily by c i ’s using proposition 2. What is obtained now is apermutation of ¯ b , allowing the conclusion to hold as shown in Case 1.The way we defined D U [¯ a ] talks only about preference profiles where only theindividuals in U support ¯ a . Intuitively, it is clear that profiles where even moreindividuals support ¯ a should also guarantee it to hold in the result. This iscaptured by the relation E U [¯ a ] := distinct(¯ a ) ∧ ∀ p ∈ D . ( p U | = R [¯ a ] ⇒ σ ( p ) | = R [¯ a ]). Proposition 3. For any tuple of pairwise distinct elements ¯ a , D U [¯ a ] ⇔ E U [¯ a ] . Proof. E U [¯ a ] ⇒ D U [¯ a ] is clear from the definitions. We will thus focus on theproof of D U [¯ a ] ⇒ E U [¯ a ]. If R is simplicial transitive: Consider a profile p , not necessarily in U ¯ a such that p U | = R [¯ a ]. We construct aprofile q with ¯ a , ( b, a , a , ..., a k ), and ( a , b, a , ..., a k ) , δ holding (or not) at U and U c according to the following table. The q as constructed can be posited toexist due to UD . H. Bisht, A.S. Kuber ¯ a b, a , ..., a k ( a , b, a , ..., a k ) , δ U ✓ ✓ ✓ U c mimics p ✓ ✗ Clearly, q ∈ U ¯ c for each ¯ c ∈ ( a , b, a , ..., a k ) , δ , along with q ∈ I b,a ,...,a k . Thus, σ ( q ) | = R [ b, a , ..., a k ] ( P ) and σ ( q ) | = R [¯ c ] for each ¯ c ∈ ( a , b, a , ..., a k ) , δ ( D U holds for each such ¯ c ). By transitivity, we can now conclude σ ( q ) | = R [¯ a ]. Itis now easy to see that ∀ i ∈ I .p i | = R [¯ a ] ⇔ q i | = R [¯ a ], making us conclude σ ( p ) | = R [¯ a ] ( IIA ). Since the initial choice of p was arbitrary, we can conclude E U [¯ a ] holds. If R is path transitive: Consider a profile p , not necessarily in U ¯ a such that p U | = R [¯ a ]. We construct aprofile q with ¯ a , ¯ a − ( k − , b , and ¯ a − k , b holding (or not) at U and U c accordingto the following table. The q as constructed can be posited to exist due to UD .¯ a ( a , ..., a k − , b, a k ) ( a , a , ..., a k − , b ) U ✓ ✓ ✓ U c mimics p ✗ ✓ Again, q ∈ U ¯ c for ¯ c = ( a , ..., a k − , b, a k ), along with q ∈ I a ,...,a k − ,b . Thus, σ ( q ) | = R [ a , ..., a k − , b ] ( P ) and σ ( q ) | = R [¯ c ] for ¯ c = ( a , ..., a k − , b, a k ) ( D U ).By transitivity, we can now conclude σ ( q ) | = R [¯ a ]. It is now easy to see that ∀ i ∈ I .p i | = R [¯ a ] ⇔ q i | = R [¯ a ], making us conclude σ ( p ) | = R [¯ a ] ( IIA ). Since theinitial choice of p was arbitrary, we can conclude E U [¯ a ] holds.Now, consider the following collection of voter coalitions U = { U ⊆ I | ∃ ¯ a ∈ A.D U [¯ a ] } . We wish to claim that this collection is an ultrafilter, which will allowus to find a ”dictator” in the social choice situation, as described above. Claim. U as defined above is an Ultrafilter. Proof. ( F1 : I ∈ U ) Because of P , it is easy to note that I ∈ U .( F2 : U is an upper set) Consider U ∈ U and U ⊆ V . Clearly, that gives us E U ⊆ E V , which allows us to conclude V ∈ U using Proposition 3.( F3 : Any two elements of U intersect) Consider U, V ∈ U such that U ∩ V = φ .Construct a preference profile p with all possible permutations of ¯ a holding (ornot) according to the following table. The p as constructed can be posited toexist due to UD ¯ a ¯ a τ ( = id ) U ✓ ✗ V ✗ ✓ U c ∩ V c ✗ ✗ This would imply that σ ( p ) | = R [¯ a τ ] for each permutation τ (Since U, V ∈ U ),contradicting the exclusivity of R . Thus, such U, V cannot belong in U together. ggregating Relational Structures 7 ( F4 : U is a prime filter) Consider U ∈ U such that U = W ⊔ V . We mustshow one of W or V also belong in U . If R is simplicial transitive: Construct a preference profile p with various permutations of ¯ a and a , ..., a k , b holding according to the following table:¯ a ¯ a τ = id ( b, a , a , ..., a k ) , δ ( b, a , a , ..., a k ) δ ( a , a , ..., a k , b ) δ V ✗ ✓ ✓ ✗ ✗ W ✓ ✗ ✗ ✓ ✗ U c ✓ ✗ ✗ ✗ ✓ Since R is connected, some permutation of ¯ a must hold at σ ( p ). We evaluate twopossible cases separately. Case 1: Let σ ( p ) | = R [¯ a τ ] for some τ = id . Clearly here, since p ∈ V ¯ a τ , we canconclude that V ∈ U . Case 2: σ ( p ) | = R [¯ a ]. Clearly, p ∈ U ¯ c for each ¯ c ∈ ( b, a , ..., a k ) , δ with U ∈ U .Therefore, σ ( p ) | = R [¯ c ] for each ¯ c ∈ ( b, a , ..., a k ) , δ . By transitivity, we can con-clude σ ( p ) | = R [ b, a , ..., a k ]. Since p also belongs to W b,a ,...,a k , we can conclude W ∈ U . If R is path transitive: Construct a preference profile p with various combinations of a , ..., a k , b holdingaccording to the following table:¯ a ¯ a τ = id ( a , ..., a k , b ) ( b, a , ..., a k ) ( a , ..., a k − , b ) V ✗ ✓ ✓ ✗ ✗ W ✓ ✗ ✓ ✗ ✓ U c ✓ ✗ ✗ ✓ ✗ Since R is connected, some permutation of ¯ a must hold at σ ( p ). We evaluate twopossible cases separately. Case 1: Let σ ( p ) | = R [¯ a τ ] for some τ = id . Clearly here, since p ∈ V ¯ a τ , we canconclude that V ∈ U . Case 2: σ ( p ) | = R [¯ a ]. Clearly, p ∈ U ¯ c for ¯ c = ( a , ..., a k , b ) with U ∈ U . Therefore, σ ( p ) | = R [ a , ..., a k , b ]. By transitivity, we can conclude σ ( p ) | = R [ a , ..., a k − , b ].Since p also belongs to W a ,...,a k − ,b , we can conclude W ∈ U .This leads us to conclude that while aggregating L -structures with single k -ary relation symbol R , UD , P , IIA , along with connectedness, exclusivity, andtransitivity (any definition) of R are sufficient conditions to conclude U is anultrafilter. This is equivalent to the existence of a dictator while aggregating k -ary relations. k -ary relations The above results have been established only for the properties we defined,restricting their usefulness. Borrowing the ideas put forward in [4], we define H. Bisht, A.S. Kuber metaproperties that collect a class of properties and show that an impossibilityresult follows for each of them. On a fixed set V , we will talk about k -ary relations R ⊆ V k . Since such relationscan be aptly described as uniform directed k -hypergraphs, we interchangeablycall them U k -graphs, for short. Any property P of k -ary relations can be iden-tified with the collection of all relations satisfying the property, i.e., a subset P of P ( V k ).The social choice situation has N as the set of individuals, with each i ∈ N contributing a relation R i ⊆ V k over a fixed set V . The collection of each voter’spreference (a preference profile) ( R i ) i ∈N will be denoted by R . An aggregationrule F : P ( V k ) N → P ( V k ) takes a preference profile R as input and outputs acollective relation F ( R ). In profile R , we will use N R ¯ a = { i ∈ N | ¯ a ∈ R i } todenote the collection of voters in N supporting tuple ¯ a in R . Definition 5. The dictatorship of an individual i ∗ is the aggregation rule F i ∗ such that for each profile R , F i ∗ ( R ) = R i ∗ . Definition 6. The oligarchy of a nonempty coalition C ∗ is the aggregationrule F C ∗ such that for each profile R , F i ∗ ( R ) = T i ∈ C ∗ R i . We now restate the properties of aggregation rules with slight modifications tomake it easier to work with them. Unanimity means that the aggregated U k -graph will contain all tuples included in every voter’s U k -graph. Groundedness can be seen as unanimity with respect to abstinence/silence, stating that the re-sult must contain a tuple only if at least one voter proposes it. Analogous to IIA, independence of irrelevant edges (IIE) requires that F pays no attention tothe other tuples of the U k -graphs while making a decision about the inclusionabout a particular tuple. Finally, an aggregation rule being collectively ratio-nal with respect to a property P implies that its output satisfies P whenevereach relation in the profile does. This is especially useful when the result of theelection must be in the same form as the votes (when aggregating total orders,for example). The preservation of properties preserve the relational structure ofthe inputs.We wish to define suitable metaproperties that will both allow the proofs togo through and be general enough to be satisfied by a large class of relations.We wish for the properties to spread from a tuple in the U k -graph to its “neigh-bouring” tuples (contagious), force the inclusion of certain tuples when someother tuples are present (implicative), and force atleast some tuples to exist sowe forbid the empty relation on any induced U k -subgraph (disjunctive). To makestating the results of this section easy, we will denote by P [ S + , S − ] the collectionof all relations satisfying property P , containing all tuples in S + and none in S − for some disjoint S + , S − ⊆ V k . Definition 7. Let ¯ a, ¯ b ∈ V k . A U k -graph property P is called ¯ a/ ¯ b contagious ifthere exist two disjoint sets S + , S − ⊆ V k such that: ggregating Relational Structures 9 1. For every R ∈ P [ S + , S − ] , ¯ a ∈ R implies ¯ b ∈ R .2. There exist R , R ∈ P [ S + , S − ] with ¯ a ∈ R and ¯ b / ∈ R . Definition 8. A property P is called contagious if it satisfies either of theconditions below:1. For some j, P is ¯ a/ ¯ c contagious for all distinct elements a , ..., a k , b ∈ V forall ¯ c ∈ ( a , ..., a j − , b, ..., a k ) + jδ .2. P is ¯ a/ ¯ c contagious for all distinct elements a , ..., a k , b ∈ V for all j where ¯ c = ¯ a − j , b . Definition 9. A property P is called implicative if there exist two disjoint sets S + , S − ⊆ V k and three pairwise distinct tuples ¯ a , ¯ a , ¯ a ∈ V k \ ( S + ∪ S − ) suchthat:1. For every relation R ∈ P [ S + , S − ] , ¯ a , ¯ a ∈ R implies ¯ a ∈ R .2. There exist U k -graphs R , R , R , R , R ∈ P [ S + , S − ] which satisfy R ∩ { ¯ a , ¯ a , ¯ a } = φ , R ∩ { ¯ a , ¯ a , ¯ a } = { ¯ a } , R ∩ { ¯ a , ¯ a , ¯ a } = { ¯ a } , R ∩ { ¯ a , ¯ a , ¯ a } = { ¯ a , ¯ a } , and R ∩ { ¯ a , ¯ a , ¯ a } = { ¯ a , ¯ a , ¯ a } . Definition 10. A U k -graph property P is called disjunctive if there exist twodisjoint sets S + , S − ⊆ V k and two pairwise distinct tuples ¯ a , ¯ a ∈ V k \ ( S + ∪ S − ) such that:1. For every relation R ∈ P [ S + , S − ] , ¯ a ∈ R or ¯ a ∈ R .2. There exist relations R , R ∈ P [ S + , S − ] with R ∩ { ¯ a , ¯ a } = { ¯ a } and R ∩ { ¯ a , ¯ a } = { ¯ a } . Before using the metaproperties defined to prove the required results, itis important to justify the choice of definition by verifying if they are sat-isfied by the properties used to complete the proof earlier. For example, tosee that simplicial transitivity is contagious and implicative, we can choose S + = ( a , ..., a j , b, ..., a k ) + j, +( j +1) δ , S − = φ , and S + = ( a , ..., a j , b, ..., a k ) + j, +( j +1) δ \{ ( a , ..., a j , b, ..., a k ) } , S − = φ respectively.In a similar spirit, to see that connectedness is disjunctive, choose S + = { ¯ a τ | τ = τ , τ } for some distinct permutations τ , τ and S − = φ . For a tuple ¯ a , consider the set W ¯ a such that ¯ a ∈ F ( R ) ↔ N R ¯ a ∈ W ¯ a . Thus, W ¯ a is the collection of winning coalitions for the tuple ¯ a . An important propertyof elections is symmetry with respect to candidates. This would require thata coalition that could ensure the inclusion of one tuple is also able to ensureinclusion of all the other tuples or W ¯ a = W for all tuples ¯ a and some collectionof coalitions W . We will only consider neutrality over the tuples with all distinctelements.Now we are ready to prove k -ary analogues of some of the important resultsof [4]. Lemma 1. (Neutrality lemma)(cf. [4, Lemma 12]) For | V | ≥ k + 1 , any unani-mous, grounded, and IIE aggregation rule that is collectively rational with respectto a contagious property must be neutral as defined above.Proof. Consider first a property P that is ¯ a/ ¯ b contagious for ¯ a, ¯ b ∈ V k . Consideran aggregation rule F that is unanimous, grounded, IIE, and collectively rationalwith respect to P . We will show that W ¯ a ⊆ W ¯ b . Let C be a coalition in W ¯ a and S + , S − ⊆ V k , R , R ∈ P [ S + , S − ] be appropriate constructions from the ¯ a/ ¯ b contagiousness of P . Consider a profile R where individuals in C propose R andothers propose R . Since C was a winning coalition for tuple ¯ a , ¯ a ∈ F ( R ). Also, S + ⊆ F ( R ) and S − ∩ F ( R ) = φ by unanimity and groundedness respectively. Bycollective rationality, F ( R ) ∈ P [ S + , S − ] and thus ¯ b ∈ F ( R ). Since only votersin C supported ¯ b , C ∈ W ¯ b .We know that all W ¯ a ’s are nonempty (since N ∈ W ¯ a for each ¯ a ). Considerarbitrary pairwise distinct ¯ a, ¯ b ∈ V k . We will prove that W ¯ a ⊆ W ¯ b , which issufficient to prove the claim. Notice the similarity to Proposition 2. Consider C ∈ W ¯ a . – If P is contagious by condition 1, we can use¯ a/ [ a , ..., a j , b , ..., a k ]-contagiousness to get C ∈ W [ a ,...,a j ,b ,...,a k ] , followedby [ a , ..., a j , b , ..., a k ] / [ a , ..., a j , b , b , ..., a k ]-contagiousnessfor C ∈ W [ a ,...,a j ,b ,b ,...,a k ] , and so on till you get C ∈ W [ b ,...,b j ,a j +1 ,...,a k ] .Following that, apply contagiousness in a similar fashion but choose compo-nents of ¯ b in reverse ( b k followed by b k − and so on) while replacing. After k steps, we can conclude C ∈ W ¯ b . – If P is contagious by condition 2, we can use ¯ a/ [ b , a , ..., a k ]-contagiousnessto get C ∈ W ¯ a − ,b . Similarly, use the definition of contagiousness for valuesof j increasing by 1 till you conclude C ∈ W ¯ b .This completes the proof.Below is the filter-version of the main result. Theorem 2. (cf. [4, Theorem 15]) (Oligarchy Theorem) For | V | ≥ k + 1 , anyunanimous, grounded, and IIE aggregation rule for k -ary relations that is col-lectively rational with respect to a contagious and implicative property must beoligarchic on pairwise distinct tuples.Proof. Take any property P which is contagious and implicative, along with anaggregation rule F that is unanimous, grounded, IIE, and collectively rationalwith respect to P . As shown in the lemma above, we can now talk about a com-mon collection of winning coalitions W such that ¯ a ∈ F ( R ) ↔ N R ¯ a ∈ W for anypairwise distinct tuple ¯ a . We will show that W is a filter, equivalent to claimingthat F is an oligarchic rule, with the oligarchy serving as the least element inthe filter.Clearly, by unanimity N ∈ W . ggregating Relational Structures 11 To show W is closed under intersections, consider arbitrary C , C ∈ W . Con-sider a profile R where individuals in C ∩ C propose R , those in C \ C pro-pose R , those in C \ C propose R , and others propose R where the U k -graphsare the ones from the definition of an implicative property. Since C , C are win-ning coalitions, ¯ a , ¯ a ∈ F ( R ). Also, S + ⊆ F ( R ) and S − ∩ F ( R ) = φ by unanim-ity and groundedness respectively. By collective rationality, F ( R ) ∈ P [ S + , S − ]and thus ¯ a ∈ F ( R ). Since only voters in C ∩ C supported ¯ a , C ∩ C ∈ W .To show W is closed under upper bounds, consider arbitrary C ∈ W and C ⊆ C . Consider a profile R where individuals in C propose R , thosein C \ C propose R , and others propose R where the U k -graphs are the onesfrom the definition of an implicative property. Since C is a winning coalitions,¯ a ∈ F ( R ). Also, ¯ a ∈ F ( R ), S + ⊆ F ( R ), and S − ∩ F ( R ) = φ by unanimityand groundedness respectively. By collective rationality, F ( R ) ∈ P [ S + , S − ] andthus ¯ a ∈ F ( R ). Since only voters in C supported ¯ a , C ∈ W .Thus, we have successfully shown that W is a filter under the given assump-tions. Theorem 3. (cf. [4, Theorem 16]) (Dictatorship Theorem) For | V | ≥ k + 1 ,any unanimous, grounded, and IIE aggregation rule for k -ary relations that iscollectively rational with respect to a contagious, implicative, and disjunctiveproperty must be dictatorial on pairwise distinct tuples.Proof. Take any property P which is contagious, implicative, and disjunctive,along with an aggregation rule F that is unanimous, grounded, IIE, and collec-tively rational with respect to P . As shown in the above theorem, we can nowtalk about a common collection of winning coalitions W such that ¯ a ∈ F ( R ) ↔ N R ¯ a ∈ W for any pairwise distinct ¯ a which is a filter. We will now show thatit is an ultrafilter, equivalent to claiming that F is a dictatorial rule, with thedictator serving as the least element in the ultrafilter.Consider arbitrary coalition C . Consider a profile R where individuals in C pro-pose R , and others propose R where the U k -graphs are the ones from the def-inition of a disjunctive property. It follows that S + ⊆ F ( R ) and S − ∩ F ( R ) = φ by unanimity and groundedness respectively. By collective rationality, F ( R ) ∈ P [ S + , S − ] and thus ¯ a ∈ F ( R ) or ¯ a ∈ F ( R ). Since only voters in C supported¯ a and only those in N \ C supported ¯ a , C ∈ W or N \ C ∈ W . Thus, W is anultrafilter.Thus, we have successfully extended the idea to k -ary relations. We have already seen that for social choice situations where our language carriesa single relation symbol, the collection of winning coalitions forms an ultrafilter. Now, consider a language with many relation symbols { R i } i ∈ I . For each R i , anargument similar to above would state that its interpretation is decided by anultrafilter. Thus, the coalitions that decide all relations of the aggregate form anintersection of ultrafilters which is a filter. In model theory, one often thinks of a filteredproduct of a collection of first-order structures, for a fixed language, as theiraverage/aggregate. So far we have only dealt with a special class of relationalstructures. It will be very interesting to see whether it is possible to find somesufficient conditions that the interpretations of function symbols satisfy in orderto force the aggregation to be a filtered product of those structures. Simplicial complexes A lot of applications model relationships as simpli-cial complexes with bounded dimension. Apart from aggregating social relation-ships, simplicial complexes are also useful in diverse areas including rendering3 D -graphics. Such aggregation problems could arise naturally in decentralizedcomputing setups when each unit produces a simplicial complex built on a pre-determined grid of points. References 1. Arrow, K.J.: Social choice and individual values. 2nd edn. Yale University Press(1963)2. Fishburn, P.C., Rubinstein, A: Aggregation of equivalence relations. Journal of Clas-sification (1), 61–65 (1986)3. Pini, M.S., Rossi, F., Venable, K.B. and Walsh, T.: Aggregating partially orderedpreferences (3), 475–502 (2008)4. Endriss, U., Grandi, U.: Graph aggregation. Artificial Intelligence245