CC BY
2Victor V. Rozen
Saratov State University,
Astrakhanskaya St. 83, Saratov, 410012, Russia E-mail: rozenvv@info.sgu.ru
Abstract We consider a problem of construction of preference relations for models of decision making with quality criteria. A quality criterion one means as a function from a set of alternatives in some chain (i.e. linearly ordered set). A system of axioms for rule of preferences is given. It is shown that any rule for preferences satisfying these axioms can be presented as a rule for preferences based on some pseudofilter of winning coalitions of criteria. The section 4 contains main results of the article. In particular, necessary and sufficient conditions for transitive and for linear preferences are found. An interpretation of Arrow paradox in terms of filters is given. Keywords: Rule for preference relations, Axiom for preferences, Pseudofilters and filters of winning coalitions.
1. Introduction
We study a general model of multi-criteria decision making with quality criteria in the form of a system
G = (A (qj)jej), (i)
where A is an arbitrary set with |A| > 2 (named a set of alternatives or outcomes) and (qj)jeJ are criteria for valuation of these alternatives. Formally every criterion qj ,j G J can be presented as a function from A in some scale, points of which are results of measurement for criterion qj. It is well known that any scale has some set of acceptable transformations and measurements produce up a some acceptable transformation.
A criterion qj is called a quality one if its scale is some linearly ordered set (Cj, <j), i.e. a chain. In this case acceptable transformations are all isotonic functions defined on Cj.
In this article, we consider some problems concerning of preference relations for model (i).
Definition 1. A pair (A,p), where A is an arbitrary set with |A| > 2 and p a reflexive binary relation on A is called a space of preferences.
For any a, a' G A put
a < a' (a, a') G p,
a < a' (a, a') G p, (a', a) G P, (2)
a ~ a' (a, a') G p, (a', a) G p.
In (2), the sign < means a preference, < strict preference and ~ indifference between elements a and a'. A preference relation < is well defined by the pair (<, ~), namely < is the union of relations < and ~.
Given a model G in the form (1), one can define a preference relation on the set of alternatives A in different manners. Let K be the class of models of the form (1). We say that a rule R for preferences in the class K is given if for each G G K some reflexive binary relation R(G) = p on the set of alternatives of model G is defined. Indicate some known rules for preferences.
1. The most important rule for preferences is Pareto-preference <Par which is given by the formula
ai <Par a2 ^ (Vj G J) qj (ai) <j qj (a2) . (3)
2. Strict Slater preference is defined by
ai <Sl a2 ^ (Vj G J) qj (ai) <j qj (a2) . (4)
In this case, indifference is the identity relation.
3. Rule of simple majority can be introduced in the following way. Assume in model G the set of criteria is finite and IJ| = n. For any alternatives a, a' G A we denote
n (a, a') = j G J: qj (a) >j qj (a')| , n* (a, a') = j G J: qj (a) >j qj (a')| .
One can define two rules of simple majority M1 and M2 by formulas
ai ^Ml a2 n (ai, a2) > n (a2, ai) , ai ^M2 a2 n (ai, a2) > n/2.
It is easy to show that Mi coincides with M2 for any elements a, a' G A in the case when all inequalities for n (a, a') and n (a', a) are strict. In general case these relations are different. Particularly the condition a >Ml a' holds if n* (a, a') > n* (a', a) and the condition a >M2 a' if n* (a, a') > n/2.
4. Rule of a-majority is defined as follows. Fix a real number a > 1/2. For any a, a' G A put a > a' n (a, a') > r where r = an, if an is integer and r = [an] + 1 otherwise. The indifference relation can be given here by two manners: a) a ~ a' if and only if neither a > a' nor a' > a; b) a ~ a' if and only if a = a'.
In this article, we study a construction of preference relation with help of some
family of criteria, indices of which form so-called pseudofilter. Remark that pseud-
ofilter is a certain generalization of well known conception of filter which is made use in algebra, mathematical logic and topology (see Birkhoff, 1967; Kelley, 1957; Kuratowsski and Mostowski, 1967). Using some properties of filters, we indicate an interpretation of Arrow paradox.
2. Axioms for rules of preference relations
We now state axioms for a rule R of preferences in the class K defined above.
(A1) Axiom of independence. Consider two models G = (A, (qj)jEJ) and G1 = (B, (qj)je^ of class K. Suppose for elements ai, a2 G A and bi, b2 G B the following equivalence
qj (ai) <j qj (a2) ^ qj (bi) <j qj (b2)
holds (j G J). Then the equivalence a1 <p a2 bi <p1 b2 is truth (we denote by
p = R (G) ,pi = R(G^).
Axiom of independence means that the preference between two alternatives in any model of class K is well defined by the set of criteria under which one of them is more preferential than another and does not depend on comparison of these alternatives with other alternatives of the model.
(A2) Axiom of monotony. Consider two models G = (A, (qj)jEJ) and G1 = {A, (q])jej) of class K. Fix two elements ai, a2 G A and assume for any j G J the following implication
qj (ai) <j qj (a2) ^ qj (ai) <j q1 (0,2)
holds. Then the condition a1 <p a2 implies the condition a1 <p a2.
Axiom of monotony states that the preference between two alternatives in models of class K is increasing under an enlargement the set of corresponding criteria.
(A3) Axiom of strict monotony. Consider two models G = (A, (qj )jEJ) and G1 = (A, (qi)jeJ) of class K. Fix two elements ai,a2 G A and suppose for any j G J the following implication
qj (ai) <j qj (a2) ^ qj (ai) <j q1 (a2) holds. Then the condition a1 <p a2 implies the condition a1 <p a2.
Remark 1. Formally, axioms (A2) and (A3) are independent one from another since (A3) has more strong assumption but more strong consequence also.
(A4) Axiom for absence of attachment. Let A be an arbitrary set. Fix two elements a1,a2 G A with a1 = a2. Then there exist two models G = (A, (qj )jEJ) and
G1 = (A, (qi )jeJ) of class K such that conditions
a1 <p a2 and — ^a1 <p a2^j
hold.
We now show that a rule R for preferences satisfying axioms (A1) - (A4) can be defined for models of the form
GQ = (A (aj )j€J) (5)
where aj is some linear quasi-order on A. Indeed, let G = (A, (qj )jEJ) be a model
of class K. Put
J(ai,a2) = {j G J: qj (ai) <j qj (a2)} .
It follows from axiom (A1) that for any fix elements a1 ,a2 G A, truth of assertions a1 <p a2 (where p = R (G)) is well defined by the subset J(ai,a2). Define a linear quasi-ordering aj on A by the formula
a <aj a' qj (a) <j qj (a') .
It is evident that subsets J(ai,a2) can be presented in the form
J(a1,a2) {j G J: a1 < j a2}
(6)
Thus any rule R for preferences in the class K can be given as a mapping which for each model Gq = (A, (aj)jEJ) some reflexive preference relation R (Gq) = p on the set A assigns. By this reason, sometimes we will consider the class K as a class of models of the form (5). Axioms (A1) - (A4) in this case can be written as follows.
(A1)* Consider two models Gq = (A, (aj)jEJ) and GQ = (B, (ai)jeJ) of class K. Denote by R (Gq) = p,R{GQ) = p1. Assume for elements a1,a2 G A and b1,b2 G B the following equivalence
ai <Jj a2 ^ bi <JJ b2
holds for each j G J. Then the equivalence a1 <p a2 ^ b1 <p b2 is truth also.
(A2)* Consider two models Gq = (A, (aj)jEJ) and GQ = (A, (ai)jeJ) of class K. Fix elements a1, a2 G A and assume for any j G J the following implication
ai <Jj a2 ^ ai <Jj a2
holds. Then the condition a1 <p a2 implies the condition a1 <p1 a2.
(A3)* Consider two models Gq = (A, (aj)jEJ) and GQ = l^A, (ai)jeJ) of class K. Assume for elements a1,a2 G A and any j G J the following implications
ai <Jj a2 ^ ai <Jj a2
hold. Then the condition a1 <p a2 implies the condition a1 <p a2.
(A4)* Let A be an arbitrary set. Fix two elements o1,o,2 G A with a1 = a2. Then there exist two models Gq = (A, (aj)jEJ) and GQ = l^A, (ai)jeJ) of class K such that conditions
a1 <p a2 and — ^a1 <p a2^j
hold.
We now indicate some consequences of axioms (A1)* - (A4)*.
Corollary 1. Consider two models Gq = (A, (aj)jeJ) and GQ = (B, (ai)j^^ of class K. Assume for elements a1,a2 G A and b1,b2 G B the following equivalences
ai <Jj 02 ^ bi <Ji b2
02 <Jj ai ^ b2 <JJ bi
hold for each j G J. Then the equivalences
ai ^p 02 ^ bi ^p b2
i (7)
ai <p 02 bi <p b2
are truth also.
For the proof put
J(ai,a2) = {j G J: 01 a2}
J(a2,a i) = {j G J : 02 °1} (8)
J(ai,a2) = {j G J : 01 ~Jj a2} .
Assumption of corollary 1 means J(ai,a2) = J(bi,b2) and J(a2,ai) = J(b2,bi). Hence
J(ai ,a2) = J(ai,a2) ^ J(a2,ai) = J(bi,b2) ^ J(b2,bi) = J(bitb2).
We obtain J0ai a2) = J0bi b2) that is the first equivalence in (7). It follows from the assumption of corollary 1 and the equality J°aua2) = J(bub2) that J(+ai,a2) = J(bub2) that is the second equivalence in (7).
Corollary 2 (Pareto optimality). For each model Gq = (A, (aj )jeJ) of class K following inclusions hold:
D aj C R (Gq) C J aj. (9)
jeJ jeJ
Proof (of corollary 2). Fix a pair (a1,a2) G p| aj. By axiom (A4)* there exists
jeJ
a family of linear quasi-orders (aj) ,eJ on A such that (a1,a2) G R (GQ) where GQ = (A, (aj)jeJ). For arbitrary j G J the following implication
0i <Jj 02 ^ 0i <Jj 02
holds (since the conclusion of this implication is truth). Put p = R (Gq) and p1 = R(GQ). According to axiom (A2)* the condition a1 <p a2 implies the condition a1 <p a2. Because the first condition is truth by assumption, we have that the second condition is truth also. Thus the first inclusion in (9) is proved. To prove the second inclusion, fix a pair (a3, a4) G U aj. By axiom (A4)* there exists a family of linear
jeJ
quasi-orders (aj) ,eJ on A such that (0,3,04) G R(Gq) where GQ = l^A, (aj )jeJ.
For arbitrary j J the following implication
2
03 <Jj 04 ^ 03 <Jj 04 (10)
holds (since the condition of this implication is false). Assume (a3,a4) G R (Gq). Then using (10) we receive by axiom (A2)* (a3, a4) G R (GQ) in contradiction with our assumption. Hence (a3,a4) G R (Gq) and the implication
(03,04) G[J a j ^ (03,04) GR (Gq) (11)
jeJ
is shown. It remains to note that (11) is equivalent to the second inclusion in (9).
3. Pseudofilters and filters
In this section we will study a notion of pseudofilter which can be used for construction of some rule of preferences in models of the form (1).
Definition 2. Let J be an arbitrary set. A family W of its subsets is called a pseudofilter over J if it satisfies the following conditions:
(PF1) Nonemptiness: W = 0;
(PF2) Majorant stability: S gW,T D S ^ T G W;
(PF3) Anticomplement: S G W ^ S' G W.
Let us note some consequences of these axioms.
(C1) JgW.
(C2) 0 GW.
(C3) S,TgW ^ S n T = 0.
Indeed, by (PF1) there exists a subset S C J with S gW .By (PF2) we have JgW, i.e. (C1). Using (C1) and (PF3) we obtain (C2). Prove (C3). Suppose S n T = 0 then T C S' and by (PF2) we obtain S' G W. Because S G W that contradict (PF3).
Example 1. A game in the form of characteristic function can be given as a pair (J, v) where J is an arbitrary set (named a set of players) and v is a function which any subset S C J assigns a real number v (S). In the game theoretical terminology, any subset S C J is called a coalition. The characteristic function v is said to be superadditive if for any subsets S,T C J with S n T = 0 the inequality
v (S) + v (T) < v (S U T) (12)
holds. A game (J, v) is called prime one if values of the function v are 0 and 1 only. The following assertion is noted by Herve Moulin (Moulin, 1981).
Lemma 1. Let (J,v) be a prime game and W be a family of winning coalitions (i.e. coalitions S C J with v (S) = 1). The characteristic function v is superadditive if and only if W satisfies conditions (PF2) and (PF3).
Proof (of lemma 1). Let v be superadditive. Check (PF2). Suppose S G W and T D S. Put T1 = T n S'. Since S n T1 = 0 and S U T1 = T, by using (12) we have v (S) + v (T1) < v (T). Because S G W, we obtain v (S) = 1 and v (T) > 1 i.e. T g W. Check now (PF3). Suppose S gW and S' G W for some coalition S C J. Then by using (12) we have v (J) > v (S) + v (S') = 1 + 1=2, i.e. v (J) > 2, that is impossible. Necessity is proved.
To prove the sufficiency consider two coalitions S,T C J with S n T = 0. The case S,T GW is trivial. In the opposite case according the condition (C3) we can put S gW,T G W. Then by (PF2) we have S U T gW hence the left and the right parts of (12) are equal to 1 and (12) holds. □
A prime game (J, v) is said to be trivial, if v (S) = 0 for all coalitions S C J. Obviously, a prime game is non-trivial if and only if W = 0 i.e. when axiom (PF1) holds. Then using Lemma 1, we obtain
Lemma 2. Let (J, v) be a prime game and W be a family of its winning coalitions. A game G is non-trivial with superadditive characteristic function v if and only if W is pseudofilter.
We now consider some questions concerning a construction of pseudofilters. First of all note an important connection between the notion of pseudofilter and the notion of filter; the last is made use in various branches of algebra, mathematical logic and topology.
Definition 3. Let J be an arbitrary set. A nonempty family F of subsets J is called a filter over J if the following conditions hold:
(F1) S G F,T G F ^ S n T gF;
(F2) S gF,T D S ^ TgF ;
(F3) 0 G F.
Lemma 3.
1. Any filter is a pseudofilter.
2. A pseudofilter is a filter if and only if it is stable under intersection of its subsets.
Proof (of lemma 3). 1. Let F be a filter. Then axioms (PF1), (PF2) evidently hold. Check (PF3). Assume S,S' G F. Then by (F1) we have 0 = S n S' G F that contradicts (F3).
2. If a pseudofilter W is a filter the required condition holds (see (F1)). Conversely let W be a pseudofilter for which axiom (F1) holds. Axiom (F2) is equivalent to (PF2). Axiom (F3) is a consequence of (PF1) and (PF2) (see (C2)). □
We now consider some method for construction of pseudofilters. Let J be an arbitrary set and B a family of its subsets. We denote by M (B) the family of all oversets for sets belonging to B:
M (B) = {T C J: (3S gB) S C T} .
Definition 4. Let W be a pseudofilter over J and B a non empty family of some sets belonging to W i.e. B C W. We say that B forms a base of the pseudofilter W if M (B) = W.
Remark that any pseudofilter W has a base (for example B = W) and psedofilter is well defined by any its base. A base B0 is called the smallest base of pseudofilter W, if Bo C B for any base B. In the case the set J is finite, each pseudofilter W has the smallest base consisting of all minimal (under inclusion) subsets of W.
Lemma 4. Let J be an arbitrary set and B some family of its subsets. Then
1. B forms a base of some pseudofilter over J if and only if the following condition holds
SgB,TgB ^ S n T = 0; (13)
2. B forms the smallest base of some pseudofilter over J if and only if the condition (13) and the following condition
SgB,TgB,S C T ^ S = T (14)
holds.
Proof (of lemma 4). 1. Let B be a base of some pseudofilter. Because (13) holds in each pseudofilter (see (C3)) it holds for any its subset. Conversely, let B be some family of subsets of J for which (13) holds. Put W = M (B) and show that W is a pseudofilter. Axioms (PF1) and (PF2) are evident. Check (PF3). Fix T g W, i.e. T D S where S G B. Suppose T' g W i.e. T' D S1 for some S1 G B. Then S n S1 C T n T' = 0 hence S n S1 = 0 that is contradiction with (13). Thus W is pseudofilter and B is its base.
2.The necessity of condition (13) have shown above. To prove (14) remark that the smallest base of pseudofilter W consists of all minimal subsets of W hence the condition (14) for smallest base holds. Let us prove sufficiency. Put W = M (B). It is shown that W is pseudofilter and B is its base. We need to prove that B is the set of all minimal subsets of W. Indeed, fix S0 G B. Assume for T g W the
inclusion T C S0 holds. We need to check the equality T = S0. By definition of mapping M (B) we have T D S for some S G B. Then S C T C S0 hence S C S0 and by (14) S = S0. Thus T D S0 and because the inclusion T C S0 also holds we obtain the equality T = S0.
It remains to prove that each minimal subset of W belongs to B. Indeed let subset T1 G W be a minimal in W. We have T1 D S1 where S1 G B. The strict inclusion T1 D S1 is impossible and we obtain T1 = S1 G B. □
4. Rules for preferences based on pseudofilters of winning coalitions
Consider the class K of models G = (A, (qj)jeJ) of the form (1). Associate with each model G G K a model Gq = (A, (aj)jeJ) where aj is a linear quasi-order on A defined by
a <Jj a' ^ qj (a) <j qj (a') . (15)
It is shown above that we can consider K as a class consisting of models of the form Gq . The aim of this section is to introduce a fairly general rule for preferences in class K satisfying to some natural axioms. We solve this problem in the following manner.
Definition 5. Let W be a pseudofilter over J. Subsets belonging to W are called winning coalitions of criteria (briefly, winning coalitions).We now define a rule RW for preferences in the class K which any model G K assigns a binary preference relation RW (G) = RW (Gq) = pW on A given by the formula:
a <PW a' ^ {j G J: a <J a'} G W. (16)
The rule given by definition 5 is called a rule defined by pseudofilter W.
Example 2. Put W = {J} (obviously, W is a pseudofilter). Then preference relation pW coincides with Pareto-preference.
Example 3. Fix a real number a > 1/2. Let r = an if an is integer and r = [an] + 1 otherwise (where n = \ J|). Now put W = {S C J: |S|> r} (it is easy to show that W is a pseudofilter). Then preference relation pW coincides with rule of a-majority, see section 1.
Remark 2. Because J G W for any pseudofilter (see (C1), section 3), a preference relation pW is reflexive always. But axiom of transitivity for pW need not be holds. For example, preference relation for rule of a-majority is not transitive in general case.
It follows from the definition 5
Corollary 3. Fix a models Gq of class K and let for some a, a' G A the condition {j G J: a <Jj a'} G W holds. Then a <PW a' holds.
Proof (of corollary 3). We have T = {j G J: a <Jj a'} D {j G J: a <Jj a'} = S. Since S G W, by axiom (PF2) we obtain T G W hence a <PW a' holds. On the other hand {j G J: a >Jj a'} = {j G J: a <Jj a'}' = S' /W hence by definition 5 the condition a' <pw a does not hold. Thus we obtain a <PW a'. □
We now state the following important result.
Theorem 1. Any rule for preferences in class K defined by a pseudofilter W satisfies axioms (A1)*-(A4)*.
Proof (of theorem 1). We need to check these axioms for rule (16).
(A1)* Consider two models Gq = (A, (aj)jeJ) and GQ = (B, (ai)jeJ) of class K. Denote by RW (Gq) = pW,RW (Gq) = pW. Suppose for elements a1,a2 G A and b1,b2 G B the following equivalences
Oi <Jj 02 ^ bi <Ji b2
hold for each j G J. Then {j G J: a1 <Jj a2} = |j G J: b1 <Jj b2^ hence conditions f ^
{j G J: ai <Jj 02} G W and |j G J: bi <jj b^ G W
are equivalent and by (16) conditions a1 <PW a2 and b1 <pW b2 are equivalent also.
(A2)* Consider two models Gq = (A, (aj)jeJ) and GQ = l^A, (ai)jeJ) of class K .Fix elements a1,a2 G A and let for every j G J the following implication
Oi <Jj 02 ^ Oi <Jj 02
holds. Then we have
S = {j G J: ai <Jj a2}C |j G J: ai <Jj 021 = T.
By (16) the condition a1 <PW a2 means S G W; using the inclusion S C T and axiom (PF2) we obtain T gW, that is a1 <pW a2.
(A3)* Consider two models Gq = (A, (aj)jeJ) and GQ = l^A, (ai)jeJ) of class K. Assume for elements a1,a2 G A and all j G J following implications
ai <Jj 02 ^ ai <Jj 02 (17)
hold. Then as above we obtain that a1 <PW a2 implies a1 <Pw a2. On the other hand, the condition a1 <PW a2 means that {j G J: a1 <Jj a2} G W then by axiom (PF3) U = {j G J: a1 <J a2}' / W.
It follows from (17) that |j G J: a1 <Jj a2 j C{j / J: a1 <Jj a2}' = U / W. Then we have
V = |j G J: 02 <Jj ai| = |j G J: ai <Jj a^ C U / W
and by axiom (PF2) V / W, i.e. the condition a2 <pW a1 does not hold. Thus the assumption a1 <PW a2 implies a1 <Pw a2 which was to be proved.
(A4)* Let A be an arbitrary set. Fix two elements a1,a2 G A with a1 = a2. Consider two families of linear quasi-orders (aj)jeJ and (aj)jeJ on A such that for
any j G J conditions a1 <Jj a2 and a2 <Jj a1 hold. Then we have
|j G J: a1 <aj 02^ = J G W and |j G J: a1 <aj a^ = 0 / W.
Put p'W = Rw ((A, (aj)jeJ)) and p'W = Rw {(A, (aj')je^). According with (16) we obtain that the condition a1 <Pw a2 holds and the condition a1 <Pj a2
does not hold which completes the proof of Theorem 1. □
We now state the converse of Theorem 1.
Theorem 2. Fix a family of scales (Cj, (<j)jeJ) for measurement of quality criteria. Let R be a rule for preferences which every models Gq = {A, (aj )jeJ) of class K assigns some reflexive preference relation R (Gq) = p on A and for R axioms (A1 )* - (A4)* hold. Then there exists a pseudofilter W over J such that R = RW.
Proof (of theorem 2). Let us define a family W of winning coalitions of criteria in the following manner. For any subset S Ç J, the condition S G W means that there exists a model Gq = (A, (&j)jej) of class K and elements ôi,ô2 G A such that
«1 and {j G J : Si <aj «2} = S (18)
(we denote by p = R (Gq)).
Further we define a rule RW for preferences in class K and write RW (G) = RW (Gq) = pW by setting for any Gq = {A, (aj)jeJ) of class K and every a1,a2 G A
a1 <PW a2 ^{j G J : a1 <Œj a2} G W.
As the first step, we show the equality RW = R. It suffices to prove that for each model Gq = {A, (aj)jEJ) of class K the equivalence
a1 <p a2 ^{j G J : a^ <aj a2} G W. (19)
holds. In fact, the implication ^ in (19) is truth by definition of family W. Conversely, suppose the right part of (19) holds. Then there exists a model Gq = (.A, (&j)jej) of class K and elements âi,â2 G A such that
«1 «2 and {j G J: ôi <aj â2} = {j G J: a\ <aj a2} .
Then conditions a\ <ai a2 and â\ <‘Jj a2 are equivalent for any j G J; by axiom (Al)* the propositions ai <p a2 and â\ <p a2 are equivalent also and because ôi <p a2 is truth we obtain that ai <p a2 is truth.
It remains to be shown that W is a pseudofilter. Check axioms (PF1)-(PF3). (PF1) Let A be an arbitrary set with |A| > 2. Fix two elements a\,a2 G A. For any j G J let aj be a linear quasi-order on A with a\ <aj a2. Then condition (a\,a2) G p| aj holds and using Corollary 2 we obtain ai <p a2; since jeJ
{j G J : a\ <aj a2} = J then by (18) J G W, i.e. W = 0.
(PF2) Suppose S G W and T D S. We need to prove T gW .In fact, by (18) there exists a model Gq = {A, (aj)jeJ) of class K and elements a\,a2 G A such that al <p a2 and {j G J : al <aj a2} = S. Consider a family of linear quasi-orders (aj)jeJ on A defined as follows. For j G T n S', the quasi-order aj the condition
a,i <aj a2 satisfies and aj = aj for other j G J. Then
[jGJ: a1 <aj a2} = (T n S') U S = T. (20)
Let us show the following implication
ai <<Jj a2 ^ ai a2.
(21)
Indeed, for j GT n S' the implication (21) holds since its consequence is truth; in other cases the condition and the consequence of (21) are equivalent. Denote by GQ = (A, (aj)jeJ), R (Gq) = p,R(GQ) = p1. Since a1 <p a2 then by axiom (A2)*
and (21) we have a1 <p a2; using (20) and (18) we obtain T G W.
(PF3) Suppose S G W i.e. there exists a model Gq = {A, (aj )jeJ) of class K and elements a1,a2 G A such that a1 <p a2 and {j G J : a1 <Jj a2} = S. Assume S' G W. Consider a family (aj)jeJ of linear quasi-orders on A satisfying
| j G J : a1 <Ji a,2^ = S. Then for any j G J the implication
a1 <Jj a,2 ^ a1 <Jj a,2. (22)
is truth. Put GQ = (A, (a1)jeJ), R(Gq) = p, R (GQ) = p1. Using (22) and the
condition a1 <p a2 we obtain by axiom (A3)* the condition a1 <p a2. On the other hand, since
|j G J : a2 <jî a^ = | j G J : a,1 <Jj a^ = S ' G W
we have a2 <pW a1; because RW = R we obtain a2 <pl a1 in contradiction with
condition a1 <p a2 proved above. □
To conclude this section we consider a construction of rules for preferences based on filters of winning coalition.
Theorem 3. Let RW be a rule for preferences in class K which based on pseudofilter W. Then for every model Gq = {A, (aj )jeJ) of class K the preference relation pW = RW (Gq) is transitive if and only if the pseudofilter W is a filter.
Proof (of theorem 3). Necessity. Suppose W is not a filter then by Lemma 3 there exist subsets S,T G W such that S n T G W. Put A = {a1, a2, a3} and for every j G J let us define a linear order relation aj as follows:
a3 <Jj a1 <Jj a2 for all j G S n T';
a1 <Jj a2 <Jj a3 for all j G S n T;
a2 <Jj a3 <Jj a1 for all j GT n S ';
a3 <Jj a1 for all j G J n (S U T)'.
Then we have
{j G J : a1 <Jj a2} D (S n T') U (S n T ) = S GW ;
{j G J : a2 <Jj a3} D (S n T ) U (T n S') = T GW ; (23)
{j G J : a1 <Jj a3} = S n T /W.
According with Definition 5 and using (23) and axiom (PF2) we obtain a1 <PW a2, a2 <PW a3 but the condition a1 <PW a3 does not hold.
Sufficiency. Let Gq = {A, (aj)jeJ) be any model of class K. Put RW (Gw) = pW. Suppose a1 <PW a2, a2 <PW a3. Then by definition 5 we have {j G J : a1 <Jj a2} = S G W, {j G J : a2 <Jj a3} = T G W hence by axiom (F2) S n T G W. Obviously,
S n T Ç{j G J : a1 <Jj a3} and by axiom (F2) we obtain {j G J : a1 <Jj a3} G W,
i.e. a1 <PW a3 which was to be proved. □
We now consider the condition of linearity of preference relations. It connects with condition of maximality for filters. Recall that a filter W over J is a maximal one (or ultrafilter) if it satisfies the condition
either S G W or S' G W for every S C J. (24)
Theorem 4. Let RW be a rule for preferences in class K which based on pseudofil-
ter W. Then for every model Gq = (A, (aj )jEJ) of class K the preference relation pw = RW (Gq) is linear if and only if the pseudofilter W the condition (24) satisfies.
Proof (of theorem 4). Necessity. Assume (24) does not hold for pseudofilter W then there exists a subset S C J such that S / W and S' / W. Consider a model Gq = (A, (aj)jEJ) of class K where A = {a1,a2} and linear quasi-orders (aj)jEJ the following conditions satisfy:
a1 <Jj a2 for each j G S;
a2 <Jj a1 for each j G S'.
Then {j G J: a1 <Jj a2} = S / W and {j G J: a2 <Jj ai} = S' / W. Hence by definition5 both conditions a1 <PW a2 and a2 <PW a1 are false, i.e. the preference relation pW is not linear.
Sufficiency. Let Gq = (A, (aj)jEJ) be an arbitrary model of class K. Put RW (Gq) = pW. Fix two elements a1, a2 G A and suppose the condition a1 <PW a2 does not hold, i.e. {j G J: a1 <Jj a2} / W. Then by assumption of Theorem 4 {j G J: a1 <Jj a2}' G W, i.e. {j G J: a2 <Jj a]} G W; since {j G J: a2 <Jj a]} C {j G J: a2 <Jj a1} by axiom (PF2) we obtain {j G J: a2 <Jj a]} G W, that is a2 <PW a1. Thus the relation pW is linear. □
It follows from Theorem 3 and Theorem 4
Corollary 4. Let RW be a rule for preferences in class K which based on pseudofilter W. Then for every model Gq = (A, (aj )jEJ) of class K the preference relation pw = RW (Gq) is a linear quasi-order if and only if the pseudofilter W is an ultrafilter.
It follows from results of this section an interpretation of Arrow paradox in terms of filters. In fact, any rule for preferences in class models K which leads to linear quasi-order can be given by some ultrafilter. Since the set of criteria J is finite, every filter W over J is a principal one and a principal ultrafilter consists of all subsets which contain some fix element j* G J; namely this element j* is called a dictator in terms of Arroy.
References
Birkhoff, G. (1967). Lattice theory. Amer. Math. Soc., Coll. Publ., Vol. 25.
Kelley, J. (1957). General topology. D. Van Nostrand Comp., Princeton, New Jersey. Kuratowsski, K. and A. Mostowski (1967). Set theory. North-Holland Publ. Comp. Amsterdam.
Moulin, H. (1981). Theory of games for economics and politics. Hermann, Paris.
