Cooperative Games and Finite Geometries 57 Cooperative Games, Finite Geometries and Hyperstructures Antonio Maturo Università di Chieti - Pescara Abstract In this paper some relations between finite geometric spaces and cooperative games are considered. In particular by some recent results on blocking sets we have new results on blocking coalitions. Finally we introduce a new research field on the possible relations between quasihypergroups and cooperative games. Keywords Cooperative Games, Finite Geometries, Blocking sets, Quasihypergoups 1. Cooperative games Let M = {1, 2, …, n} be a finite non-empty set, called the set of players. A function v: ℘(M) → R such that: (C1) v(∅) = 0; (C2) (superadditivity) ∀A, B∈℘(M), (A∩B = ∅) ⇒ v(A∪B) ≥ v(A) + v(B); is called characteristic function on M. The pair (M, v) is called cooperative game with n players and the subsets of M are called coalitions. For every A∈℘(M) the number v(A) is the total gain that the players of A can have certainly forming a coalition, independently on the actions of the players not belonging to A. We assume the condition of “side payment”, that is in every coalition A any player can transfer an amount of his gain to another player belonging to A and so it is important only the total gain of the coalition. The condition (C2) is a consequence of the fact that the total gain obtained with an alliance between two disjoint coalitions is not inferior to the one without cooperation. We write v(i) to denote v({i}). By (C2) it follows that in a cooperative game (M, v) we have v(M) ≥ Σi∈M v(i). If v(M) > Σi∈M v(i) the game (M, v) is said to be essential, if the equality holds (M, v) is inessential. It is easy to prove that a cooperative game is inessential if and only if: (AD) (additivity) ∀A, B∈℘(M), (A∩B = ∅) ⇒ (v(A∪B) = v(A) + v(B)) and so there are no advantages by the cooperation. 58 Two cooperative games (M, v) and (M, v’), with the same set of n players, are called strategically equivalent, we write (M, v) ≈ (M, v’) if there exist n+1 real numbers k>0 and c1, c2,…, cn such that: (SE) ∀A∈℘(M), v’(A) = k v(A) + Σi∈A ci. We obtain the game (M, v’) by the game (M, v) with an initial payment cr to any player r and by multiplying the total gain of any coalition by the scale factor k. Then we can assume the same strategies to solve (M, v) or (M, v’). Proposition 1.1. Let (M, v) a cooperative game. The system with n+1 equations and n+1 unknowns k > 0 and c1, c2,…, cn: (ESI) k v(M) + Σi∈M ci = 1, k v(i) + ci = 0, i∈M has determinant v(M) - Σi∈M v(i) and so has solutions if and only if (M, v) is essential. In this case k = 1/(v(M) - Σi∈M v(i)) and so k > 0. The system: (NES) k v(M) + Σi∈M ci = 0, k v(i) + ci = 0, i∈M has not trivial solution if and only if v(M) - Σi∈M v(i) = 0. The relation ≈ is an equivalence relation among the cooperative games with the same set of players M. By proposition 1.1 we have that, for any equivalence class K with respect to ≈, we have a unique cooperative game (M, v)∈K, called normal element of K or normal form of the elements of K, such that v(i) = 0, ∀i∈M and v(M)∈{0, 1}. Precisely, v(M) = 1 if the game is essential and v(M) = 0 if it is inessential. The inessential games are in the same equivalence class and the normal form is such that v(A) = 0, ∀A∈℘(M). On the contrary, for n > 2, the essential games are in different classes. 2. Simple cooperative games and projective spaces Let (M, v) be an essential cooperative game in normal form. Then v(i) = 0, ∀i∈M, and v(M) = 1. We say that (M, v) is a simple game if, ∀A∈℘(M), v(A)∈{0, 1}. By (C2), for any coalition A, if Ac = M-A, we have three possibility: (a) v(A) = 1 and v(Ac) = 0; (b) v(A) = 0 and v(Ac) = 1; (c) v(A) = 0 and v(Ac) = 0. 59 The set A is called winning coalition if (a) holds and losing coalition if (b) holds. It is evident that M is a winning coalition and the complement of a winning coalition is a losing coalition. So the number of winning coalitions is equal to the number of losing coalitions. The set A is said to be a blocking coalition if (c) holds. If A is a blocking coalition then also Ac is a blocking coalition. So, if there exist blocking coalitions, their number is even. We have the following: Proposition 2.1 Let W be a subset of a set ℘(M), with M set of players. Then W is the set of the winning coalitions of a simple cooperative game (M, v) if and only if satisfy the following properties, called the “axioms of Shapley” (see [32], [34]): (W1) M∈W; (W2) ∀A, B∈℘(M), (A∈W, A⊆B) ⇒ B∈W; (W3) ∀A∈℘(M), A∈W ⇒ Ac∉W. Proof. Let (M, v) be a simple cooperative game in normal form and let W be the set of winning coalitions. Then (W1) and (W2) are trivial and (W3) follows by (C2). On the converse, let W be a subset of ℘(M) satisfying the axioms of Shapley. We put, for any A∈℘(M), v(A) = 1 if A∈W and v(A) = 0 otherwise. The pair (M, v) is a simple cooperative game and W is the set of winning coalitions. By previous proposition, in the sequel we consider a simple cooperative game indifferently as the pair (M, v) or the pair (M, W). All the properties of the losing coalitions are obtained from the ones of the winning coalitions by replacing any coalition A with Ac and ⊆ with ⊇ and vice versa. We can prove the following: Proposition 2.2 Let (M, W) be a simple cooperative game. A family Θ of subsets of M is the set of blocking coalitions of (M, W) if and only if: (BC) ∀X∈Θ, ∀A∈W, X∩A≠∅, Xc∩A≠∅. Now we introduce some useful definitions. Definition 2.1 Let M be a non empty set and let ℑ be a family of subsets of M. We say that ℑ has the “intersection property” if we have: (IP) ∀A, B∈ℑ, A∩B ≠ ∅. We say that ℑ has the “non-inclusion property” if we have: (NI) ∀A, B∈ℑ, A∩Bc ≠ ∅ and Ac∩B ≠ ∅. 60 Definition 2.2 Let M be a non empty set and let Φ and ℑ be two families of subsets of M. We say that “ℑ is a generator of Φ” or “Φ is the closure of ℑ”, and we write Φ = K(ℑ), if (GK) Φ = {A∈℘(M): ∃B∈ℑ / B⊆A}. We say that “ℑ is a minimal generator of Φ” if ℑ has the non-inclusion property and is a generator of Φ. By (W3) it follows that, in a simple cooperative game (M, W), W satisfies the intersection property. The following proposition shows that any family ℑ of subsets of M that has the intersection property generates the winning coalitions of a simple cooperative game. Proposition 2. 3 Let M be a n-set, whose elements are called players, and let ℑ be a family of subsets non-void of M satisfying the intersection property. Then if W is the closure of ℑ, the pair (M, W) is a simple cooperative game with W set of winning coalitions, called “the game generated by (M, ℑ)”. Proof. Let W be the closure of ℑ. Then (W1) and (W2) are evident. If A∈W, then there exists B∈ℑ such that A⊇B and so Ac∩B = ∅. Then, ∀C∈ℑ, Ac don’t contains C. Otherwise Ac must contain C∩B ≠ ∅, a contradiction. It follows that Ac∉W. Now we examine some relations between the cooperative simple games and the geometric spaces. Definition 2. 3 A geometric space is a pair (M, ∆), with M a non-empty set, called the support and ∆ a non-empty family of subsets of M. The elements of M are called points and the ones of ∆ are called blocks. If any block has at least two points and any two blocks have at most one point in common (M, ∆) is called “space of lines” and the blocks are called also lines. (M, ∆) is non-degenerate if there are at least two blocks. Definition 2.4 A projective space is a geometric space (M, ∆) such that (see [7]): (PS1) ∀P, Q∈M, P ≠ Q, there is exactly one block containing {P, Q}, called the line PQ; (PS2) (Veblen-Young axiom) Let A, B, C, D four distinct points such that AB intersects CD. Then AC intersects BD. (PS3) Any line contains at least three points. A non-degenerate projective space is a projective plane (or projective space with dimension 2) if the axiom (PS2) is replaced by the stronger axiom: 61 (PS2S) Two lines have at least a point in common. If a non-degenerate projective space (M, ∆) is not a projective plane, for any A, B, C∈M, distinct and such that C not belongs to the line AB, we define “plane ABC”, or “2-dimensional subspace ABC” of M, the union of the lines CX, with X∈AB. We say that (M, ∆) has dimension 3 if: (PSD3) A line and a plane have at least a point in common. For recurrence we can consider projective spaces and subspaces with greater dimensions. If (M, ∆) is a projective plane we have that ∆ satisfies both the intersection property and the non-inclusion property. So, by proposition 2.3, we have the following: Proposition 2.4 Let (M, ∆) be a finite projective plane and let W be the closure of ∆. Then (M, W) is a simple cooperative game, with W set of winner coalitions, and ∆ is a minimal generator of W. If (M, ∆) is a projective space of dimension 3 or 4 the planes have both the intersection property and the non-inclusion property. Then we have: Proposition 2. 5 Let (M, ∆) be a finite projective n-dimensional space with n∈{3, 4} and let ∆* be the set of all the planes of M. If W is the closure of ∆* then (M, W) is a simple cooperative game, with W set of winner coalitions, and ∆* is a minimal generator of W. Definition 2.5 Let (M, ∆) be a geometric space and let ℑ be a family of subsets of M. A subset X of M is called a blocking set with respect to ℑ if. (BS) ∀A∈ℑ, X∩A ≠ ∅ and Xc∩A ≠ ∅. If C is a subset of M containing a A∈ℑ, by (BS) we have: X∩C ≠ ∅ and Xc∩C ≠ ∅. Then it follows the: Proposition 2. 6 Let (M, ∆) be a geometric space, ℑ be a family of subsets of M and Φ be the closure of ℑ. Then X is a blocking set with respect to ℑ if and only if it is a blocking set with respect to Φ. Some corollaries of the previous propositions are: 62 Proposition 2. 7 Let (M, ∆) be a geometric space such that ∆ has the intersection property. Then the blocking sets with respect to ∆ are the blocking coalitions of the simple cooperative game (M, W), with W closure of ∆. Proposition 2. 8 In a finite projective plane (M, ∆) the blocking sets with respect the lines are the blocking coalitions of the simple cooperative game (M, W), with W closure of ∆. Proposition 2. 9 In a finite 3-dimensional or 4-dimensional projective space (M, ∆) the blocking sets with respect the planes are the blocking coalitions of the simple cooperative game (M, W), with W closure of ∆*, set of the planes. The previous propositions show the importance of the research of blocking sets in a finite projective space. In particular we have the fundamental problems to find: (a) the minimal or maximal blocking sets; (b) the spectrum of the minimal blocking sets, that is the set of all the possible cardinalities of the minimal blocking coalitions; (c) the minimal winning coalitions; (d) the winning coalitions containing blocking coalitions. By (BS) it follows that the complement of a blocking set is also a blocking set, so to find the maximal blocking sets is equivalent to find the minimal ones. Now we show some results in the particular case of projective planes. It is well known that, in a non-degenerate finite projective plane, all the lines have the same number of points. If q+1 is such number, the projective plane is said to be of order q and is noted πq. Moreover, the lines through a fixed point P are also q+1 and the points of πq are q 2 + q + 1. By (PS3) we have q ≥ 2. It is well known (see [7], [17]) that there exists a Desarguesian projective plane if and only if q is a prime or a power of a prime and such plane is unique. The first value of q with non-Desarguesian planes is q = 9. For small values of q we have: • in π2 there are not blocking sets; • in π3 there are exactly two blocking sets; • the blocking sets on π4 and π5 are classified, respectively, in papers of Berardi - Eugeni ([2]) and Berardi - Innamorati ([5]); • the blocking sets on π7 are classified in papers of Innamorati and Maturo (see [23], [24], [25]). If k is the cardinality of a minimal blocking set on π7 we have 12≤k≤19. In particular there are, up to isomorphism, only two 63 minimal blocking sets of order 12 and there is only a minimal blocking set with 19 points. In the general case there are the following results (Innamorati – Maturo, [23], [25]): Proposition 2.10 Let S(q) the spectrum of the minimal blocking sets in πq. Then, if q≥4, S(q) ⊇ [2q-1, 3q-5]∪{3q-3} and, if πq is Desarguesian, S(q) ⊇ [2q-1, 3q-3]. Proposition 2.11 A sufficient condition for the existence of a minimal blocking set with 3q–4 points on a non-Desarguesian plane πq is that πq contains a proper subplane of order two. In [29] H. Newmann conjectured that any finite non-Desarguesian plane contains a proper subplane of order two. By previous proposition, if the conjecture is true, we have that also for the non-Desarguesian plane of order q there exists a blocking set with 3q – 4 points. 3. Cooperative games and finite geometric spaces We introduce the following: Definition 3.1 Let M be a non-void set and let Ψ and ℑ be two families of subsets of M. We say that “ℑ is a intersection-generator of Ψ” or “Ψ is the intersection- closure of ℑ”, and we write Ψ = IK(ℑ), if (IK) Ψ = {A∈K(ℑ): ∀B∈ℑ, A∩B ≠ ∅}. Let (M, ∆) be a geometric space. If ∆ has not the intersection property, and W* is the closure of ∆, the pair (M, W*) is not a simple cooperative game because (W3) is not valid. But we have the following proposition, that generalizes proposition 2.3: Proposition 3.1 Let (M, ∆) be a finite geometric space and let W be the intersection- closure of ∆. Then (M, W) is a simple cooperative game, called “the game generated by (M, ∆)”. Proof. (W1) is evident. If A∈W and A⊆B⊆M, then ∀C∈∆, C∩A≠∅ ⇒ C∩B≠∅ and (W2) holds. If A∈W then there exists C∈∆: C⊆A and so C∩Ac=∅ and Ac∉W. The game (M, W) generated by a geometric space (M, ∆) has two types of blocking coalitions: (T1) the blocking sets with respect ∆; 64 (T2) the subsets of M containing at least a block and with intersection void with at least a block. The losing coalitions are the subsets Y of M non containing blocks and having intersection void with at least a block. Example 3.1 Let M be a n-set, whose elements are called players, and let ℑ be a family of subsets non-void of M, called companies. By an economic point of view, we assume that a player belonging to a company has a power of veto and a coalition containing a company has the control of such company. Then a winner coalition of the game generated by the geometric space (M, ℑ) has the control of at least a company and a right of veto on all the companies, a losing coalition don’t have a power of veto on at least one company and don’t control any company. Finally a blocking coalition of type (T1) has veto for any company but don’t control anyone, and a blocking coalition of type (T2) control at least a company but has not veto for all the companies. We can construct a simple cooperative game by a finite geometric space (M, ∆) also with a “geometric” procedure different from the one of proposition 3.1, by assigning the set ∆* of minimal winner coalitions. Precisely, we consider a set ∆* with the following properties: (DS1) any A∈∆* is a union of elements of ∆; (DS2) any element of ∆ is contained in at least an element of ∆*; (DS3) ∆* has the intersection and non-inclusion properties; and we assume W equal to the closure of ∆*. We have the following: Proposition 3.2 Let (M, ∆) be a finite geometric space and let ∆* be a family of subsets of M satisfying (DS1), (DS2) and (DS3). If W is the closure of ∆* then: (DW1) (M, W) is a simple cooperative game; (DW2) the blocking sets of (M, ∆) are blocking coalitions of (M, W). Proof. Property (DW1) follows by (DS3). Let X be a blocking set of (M, ∆). Then X and Xc intersect any block and so, by (DS1), any element of ∆*. By proposition 2.7 it follows that X is a blocking coalition of (M, W). Proposition 2.5 is a particular case of the proposition 3.2. Another important particular case is concerning the affine planes. 65 Definition 3.2 A geometric space (M, ∆) is an affine plane if: (AP1) Through any two distinct points there is exactly one line; (AP2) (Parallel axiom) If g is a line and P is a point outside g then there is exactly one line through P that has no points in common with g; (AP3) There exist three points that are not on a common line. Let (M, ∆) be a finite affine plane. It is well known that all the lines have the same number q ≥ 2 of points. The plane is said to be of order q and is noted αq. The number of elements of αq is q 2 and the lines through a fixed point are q+1. Let ∆* be a set whose elements are union of two non parallel lines and such that any line of ∆ is contained in at least one element of ∆*. We say that ∆* is “a set of paired lines”. The set ∆* has the intersection and non-inclusion properties and so, by proposition 3.2, we have the following: Proposition 3.3 Let (M, ∆) be a finite affine plane and let ∆* be a set of paired lines. If W is the closure of ∆* then (M, W) is a simple cooperative game. Moreover, the blocking sets of (M, ∆) are blocking coalitions of (M, W). In general we can obtain simple cooperative games from block designs, in particular from Steiner systems. Definition 3.3 Let t, k, v be natural numbers such that 2≤k≤v. A finite geometric space (M, ∆) is a Steiner system with parameter t, k, v, noted S(t, k, v), if: (SS1) Through any t distinct points there is exactly one block; (SS2) Any block has exactly k points; (SS3) M has v points. It is well known that necessary conditions for the existence of a S(t, k, v) is the existence of natural positive numbers br, r∈{0, 1, …, t-1} such that: br       − − rt rk =       − − rt rv , r = 0, 1, …, t-1. (3.1) For any r∈{0, 1, …, t-1}, br is the number of blocks through r fixed points. In particular bo is the number of all the blocks. For t = k the blocks are the subsets of M with k elements and for k = v there is only a block. We say that S(t, k, v) is non- degenerate if t1. Let I[x] be the minimum integer not inferior to x. By previous proposition, we can find a set ρrs union of at most I[(d 2-d)(k-1)/(2k)] = I[α/2] lines, such that any line of S(2, k, v) intersects r∪s∪ρrs. Then we have the following: Proposition 3.5 Let (M, ∆) be a S(2, k, v) with d>1. For any pair (r, s) of incident lines let ρrs be the union of a minimal set L of lines such that L intersects all the lines not incident to r∪s. Let ∆* be the family of the sets r∪s∪ρrs, with r, s∈∆. Then ∆* satisfies (DS1), (DS2) and (DS3) and so generates a set W such that (M, W) is a simple cooperative game. Therefore every element of ∆* is the union of at most I[α/2] + 2 lines. Example 3.2 For k = 2, a S(2, k, v) is the trivial case of a graph complete with v elements and d = v-3. Any element of ∆* is the union of exactly I[(d2-d)/4] + 2 lines. For v = k a S(2, k, v) has only a line and d = 0. Now we consider the non-degenerate Steiner systems with small values of d>1. For d = 2, by proposition 3.4, is k =2 and so we don’t have non-degenerate Steiner systems. For d = 3, k is a divisor of 6 different from 2. If k = 3 we have a S(2, 3, 13). It is proved (see [17]) that there exists two non isomorphic S(2, 3, 13). In this case we have α = 4 and the elements of ∆* are the union of at most 4 lines. If k = 6 we have a S(2, 6, 46) and α = 5. Then there are at most 5 lines in any element of ∆*. 67 6. Cooperative games and hyperstructures In this paragraph we introduce some ideas on the possible relations between cooperative games and some particular commutative weak associative quasi- hypergroups, called “geometric hypergroupoids”. We think that it is a very interesting argument of research. Definition 4.1 Let M be a non-empty set and let ℘*(M) be the family of non-empty subsets of M. A hyperoperation on M is a function σ: M×M → ℘*(M), such that to every ordered pair (x, y) of elements of M associates a non-empty subset of M, noted xσy. The pair (M, σ) is called hypergroupoid with support M and hyperoperation σ. If A and B are non-empty subsets of M, we put AσB = ∪{aσb: a∈A, b∈B}. Moreover, ∀a, b∈M, we put, aσB = {a}σB and Aσb = Aσ{b}. Definition 4.2 A hypergroupoid (M, σ) is said to be: (SI) a semihypergroup, if ∀x, y, z∈M, xσ(yσz) = (xσy)σz (associativity); (QI) a quasihypergroup, if ∀x∈M, xσM = M = Mσx (riproducibility); (HY) a hypergroup if it is both a semihypergroup and a quasihypergroup; (CO) commutative, if ∀x, y∈M, xσy = yσx; (WA) weak associative, if ∀x, y, z∈M, xσ(yσz) ∩ (xσy)σz ≠ ∅; (CL) closed, if ∀x, y∈M, {x, y}⊆xσy; (IP) idempotent, if ∀x∈M, xσx={x}. Definition 4.3 We say that a hypergroupoid (M, σ) is geometric if it is commutative, closed and idempotent. A geometric hypergroupoid (M, σ) is said to be a join space if the following incidence axiom holds: (IA) ∀a, b, c, d∈M, (∃x∈M: a∈bσx, c∈dσx) ⇒ (∃y∈M: y∈aσd∩bσc). Definition 4.4 Let (M, σ) be a geometric hypergroupoid. A geometric space (M, ∆) is said to be “associated to (M, σ)” if ∆ is the set of the hyperproducts aσb with a≠b. Proposition 4.1 Let (M, ∆) be a space of lines. Then there exists only a geometric hypergroupoid (M, σ) with (M, ∆) associated geometric space. Precisely we have: (GHA) ∀x∈M, xσx = {x}, ∀x, y∈M, x≠y, xσy is the line containing {x, y}. Example 4.1 Let M be the support of a projective space and, for any x, y∈M, with x ≠ y, put xσx = {x} and xσy equal to the line xy. The hypergroupoid (M, σ) is 68 geometric and is a hypergroup. It is also a join space. The incidence axiom is the Veblen-Young axiom. Example 4.2 Let M be the support of an Euclidean space and, for any x, y∈M, with x ≠ y, put xσx = {x} and xσy equal to the segment xy. The hypergroupoid (M, σ) is geometric. It is a hypergroup and a join space, but not a space of lines, and the incidence axiom is the Pasch axiom. Example 4.3 Let M be the support of an affine space and, for any x, y∈M, with x ≠ y, put xσx = {x} and xσy equal to the line xy. (M, σ) is a geometric hypergroupoid and a space of lines but not a hypergroup. The incidence property is not valid. In the sequel we don’t distinguish from a geometric hypergroupoid and the geometric space associated. By previous considerations we have that the concept of geometric hypergroupoid generalizes the one of space of lines and so projective spaces, affine spaces and Steiner systems S(2, k, v) are particular cases. The space of lines that are hypergroups or join spaces, e. g. projective spaces, have very interesting properties. Also join spaces that are not spaces of lines such as the one of example 4.2, have important properties. Let (M, σ) be a geometric hypergroupoid. We call blocks of order 1 the singletons {a}, a∈M, blocks of order 2 the hyperproducts aσb with a≠b and, for n≥3, we call blocks of order n the hyperproducts HσK, with H block of order h1, we can obtain a cooperative game with particular properties dependent on the algebraic structure of (M, σ). A possible economic interpretation of a block B of order n is as the set of the players disposed to form a coalition because influenced by the set of players {a1, a2, …, an} that generates the block. If (M, σ) is not a hypergroup such coalition depend on the process of aggregation of the n players. Another possible interpretation of the block B is as a company controlled by {a1, a2, …, an}. 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