In this section, we apply Theorem 5 to a social choice problem. We show (Corollary 17) that computability of a simple game entails a restriction on the number of alternatives that the set of players (with the coalition structure described by the simple game) can deal with rationally.
For that purpose, we define the notion of a simple game with (ordi-nal) preferences, a combination of a simple game and a set of alternatives and individual preferences. After defining the core for simple games with preferences, we extend (Theorem 16) Nakamura’s theorem [34] about the nonemptyness of the core: the core of a simple game with preferences is al-ways (i.e., for all profiles of preferences) nonempty if and only if the number of alternatives is finite and below a certain critical number, called the Naka-mura number of the simple game. We need to do this extension since what we call a “simple game” is not generally what is called a “simple game” in Nakamura [34].
We show (Corollary 15) that the Nakamura number of a nonweak simple game is finite if it is computable, though (Proposition 14) there is no upper bound for the set of the Nakamura numbers of such games. It follows from Theorem 16 that (Corollary 17) in order for a set of alternatives to always have a maximal element given a nonweak, computable game, the number of alternatives must be restricted. In contrast, some noncomputable (and nonweak) simple games do not have such a restriction (Proposition 18), and in fact have some nice properties. These results have implications for social choice theory; we suggest its connection with the study of Arrow’s Theorem [6].
5.1 Framework
Let N′ be an arbitrary nonempty set of players and B ⊆2N′ an arbitrary Boolean algebra of subsets (called “coalitions” in this section) ofN′. A
B-simple game ω is a subcollection of B such that ∅ ∈/ ω. The elements of ω are said to be winning, and the other elements inB are losing, as before.
Our “simple game” is a B-simple game with N = N and B = REC, if it does not contain ∅. Nakamura’s “simple game” [34] is one with B = 2N′. The properties (such as monotonicity and weakness, defined in Section 2.1) for simple games are redefined forB-simple games in an obvious way.
LetX be a (finite or infinite) set ofalternatives, with cardinal number
#X ≥ 2. Let A be the set of (strict) preferences, i.e., acyclic (for any finite set {x1, x2, . . . , xm} ⊆X, if x1 ≻x2, . . . ,xm−1 ≻xm, then xm 6≻x1; in particular,≻is asymmetric and irreflexive) binary relations ≻on X. (If
≻is acyclic, we can show that the relation º, defined by x ºy ⇔ y 6≻ x, is complete, i.e., reflexive and total.) A (B-measurable) profile is a list p = (≻pi)i∈N′ ∈ AN′ of individual preferences ≻pi such that {i ∈ N′ : x≻pi y} ∈ B for all x,y∈X. Denote by ANB′ the set of all profiles.
A B-simple game with (ordinal) preferences is a list (ω, X,p) of a B-simple gameω ⊆ B, a setX of alternatives, and a profile p= (≻pi)i∈N′ ∈ ANB′. Given the B-simple game with preferences, we define the dominance relation≻pω byx≻pωy if and only if there is a winning coalitionS ∈ω such that x ≻pi y for all i∈ S. (In this definition, {i∈ N′ :x≻pi y} need not be winning since we do not assume ω is monotonic. Andjiga and Mbih [3]
study Nakamura’s theorem, adopting the notion of dominance that requires the above coalition to be winning.) The core C(ω, X,p) of the B-simple game with preferences is the set of undominated alternatives:
C(ω, X,p) ={x∈X:6 ∃y∈X such thaty ≻pω x}.
A (preference) aggregation ruleis a map ≻:p7→ ≻p from profilesp of preferences to binary relations (social preferences) ≻p on the set of al-ternatives. For example, the mapping ≻ω from profiles p ∈ ANB′ of acyclic preferences to dominance relations≻pω is an aggregation rule. We typically restrict individual and social preferences to those binary relations ≻on X that are asymmetric (i.e., completeº) and either (i) acyclic or (ii) transitive (i.e., quasi-transitiveº) or (iii) negatively transitive (i.e., transitiveº). An aggregation rule is often referred to as asocial welfare function when indi-vidual preferences and social preferences are restricted to the asymmetric, negatively transitive relations.
5.2 Nakamura’s theorem and its consequences
Nakamura [34] gives a necessary condition for a 2N′-simple game with pref-erences to have a nonempty core for any profilep, which is also sufficient if the setX of alternatives is finite. To state Nakamura’s theorem, we define the Nakamura numberν(ω) of a B-simple game ω to be the size of the smallest collection of winning coalitions having empty intersection
ν(ω) = min{#ω′:ω′ ⊆ω and ∩
ω′ =∅}
if∩
ω =∅ (i.e., ω is nonweak); otherwise, set ν(ω) = #(2X) > #X. Note that the Nakamura number is independent ofX andp.
The following useful lemma [34, Lemma 2.1] states that the Nakamura number of aB-simple game cannot exceed the size of a winning coalition by more than one.
Lemma 13 Let ω be a nonweak B-simple game. Then ν(ω) ≤ min{#S : S∈ω}+ 1.
Proof. Choose a coalition S ∈ ω such that #S = min{#S : S ∈ ω}.
Since ∩
ω = ∅, for each i ∈ S, there is some Si ∈ ω with i /∈ Si. So, S∩(∩
i∈SSi) =∅. Therefore,ν(ω)≤#S+ 1.
It is easy to prove [34, Corollary 2.2] that the Nakamura number of a nonweakB-simple game is at most equal to the cardinal number #N of the set of players and that this maximum is attainable if B contains all finite coalitions. In fact, one can easily construct a computable, nonweak simple game with any given Nakamura number:
Proposition 14 For any integer k ≥ 2, there exists a δ-computable, non-weak simple game ω with Nakamura number ν(ω) =k.
Proof. Given an integerk≥2, let S={0,1, . . . , k−1}be a carrier and defineT ∈ω iff #(S∩T)≥k−1. Thenν(ω) =k.
Since computable, nonweak simple games have winning coalitions, it has finite winning coalitions by Proposition 9. An immediate corollary of Lemma 13 is the following:
Corollary 15 Let ω be a δ-computable, nonweak simple game. Then its Nakamura number ν(ω) is finite.
Nakamura [34] proves the following theorem forB= 2N′:
Theorem 16 Let B be a Boolean algebra of sets ofN′. Suppose that ∅∈/ ω and ω 6= ∅. Then the core C(ω, X,p) of a B-simple game (ω, X,p) with preferences is nonempty for all (measurable) profilesp∈ ANB′ if and only if X is finite and #X < ν(ω).
Remark 3. At first glance, Nakamura’s proof [34, Theorem 2.3] of the necessary condition #X < ν(ω), does not appear to generalize to an arbitrary Boolean algebraB: he constructs certain coalitions from winning coalitions by taking possibly infinite unions and intersections, as well as complements; a difficulty is that the resulting set of players may not belong to the Boolean algebra B. However, it turns out that once we make use of
the other necessary condition (disregarded by Nakamura) thatXis finite, we only need to considerfinite unions and intersections, and his proof actually works. Though accessible proofs are readily available in the literature (e.g., [8, Theorem 3.2]) for B = 2N′ and finite sets N′ of players, we choose to give a proof here since most available proofs pay little attention to the measurability condition (p∈ ANB′) for the profilesp that they construct. k
Proof. (⇐=). Suppose thatX is finite, #X < ν(ω), andC(ω, X,p) =∅ for some measurable profilep∈ ANB′. Then follow the proof of Theorem 2.5 in Nakamura [34] to find a cycle with respect to ≻pω consisting of at most
#X alternatives.
(=⇒). SupposeC(ω, X,p)6=∅ for all p∈ ANB′.
(i) To show that X is finite, suppose it is infinite. Then X contains a countable subset X′ ={x1, x2, x3, . . .} ⊆X. Letp∈ AN′ be a profile such that all playersi∈ N′ have an identical preference ≻pi (e.g., the transitive closure of itself) satisfyingxj+1 ≻pi xj for allj ∈ {1,2, . . .} and x1 ≻pi y for all y ∈ X\X′. The measurability condition p ∈ ANB′ is satisfied since for allx,y∈X, we have{i∈N′ :x≻pi y}=N′ or∅, both inB. Choose any winning coalitionS∈ω, which exists by assumption. Then all players inS have the same preference ≻pi, implyingxj+1≻pωxj for alljand x1 ≻pω y for all y∈X\X′. It follows that C(ω, X,p) =∅; a contradiction.
(ii) To show that #X < ν(ω), supposer := #X ≥ν(ω). This excludes the possibility thatωis weak orν(ω) is infinite. We will construct a profilep such that the dominance relation ≻pω has a cycle. By the definition of the Nakamura number, there is a collection ω′ = {L1, . . . , Lr} ⊆ ω such that
∩ω′=∩r
k=1Lk=∅. DefineL0 =N′ and for allk∈ {1, . . . , r}, Dk= (L0∩L1∩ · · · ∩Lk−1)\Lk.
Then{D1, . . . , Dr}is a family of (possibly empty) pairwise disjoint coalitions inB such that Lk ⊆Dkc :=N′ \Dk for allk and ∪r
k=1Dk =N′ (i∈ N′ is in the firstDk such that i /∈Lk).
Write X={x1, . . . , xr} and x0 =xr. Fix the cycle
≻={(xk, xk−1) :k∈ {1, . . . , r}}.
Define p ∈ AN′ as follows: for each k, all players i in Dk have the same (acyclic) preference ≻pi=≻ \{(xk, xk−1)}. Then for all (x, y) ∈ ≻, we have/ {i∈N′ :x≻pi y}=∅ ∈ B. On the other hand, for all (x, y) = (xk, xk−1)∈
≻, we have {i∈N′ :x≻pi y}=Dck∈ B and Lk⊆Dkc. Therefore, p∈ ANB′ and ≻pω=≻, a cycle. It follows that C(ω, X,p) =∅; a contradiction.
It follows from Theorem 16 that if a B-simple game ω is weak (and satisfies ∅ ∈/ ω and ω 6= ∅), then the core C(ω, X,p) is nonempty for all profilesp∈ ANB′ if and only ifXis finite. The more interesting case is where
ω is nonweak. Combined with Corollary 15, Theorem 16 has a consequence for nonweak, computable simple games:
Corollary 17 Let ω be a δ-computable, nonweak simple game satisfying
∅ ∈/ ω and ω 6= ∅. Then there exists a finite number ν (the Nakamura number ν(ω)) such that the core C(ω, X,p) is nonempty for all profilesp∈ ANREC if and only if #X < ν.
If we drop the computability condition, the above conclusion no longer holds. An example of ω that has no such restriction on the size of the set X of alternatives is a nonweak prefilter (e.g., the q-complement rule of Example 1, forq >1), which has an infinite Nakamura number.
In fact, we can say more, if we shift our attention from the core—the set of undominated alternatives with respect to the dominance relation≻pω—to the dominance relation itself.
Proposition 18 Let ω be a nonweak simple game satisfying∅∈/ ω andω6=
∅. (i) ω cannot be a δ-computable prefilter. (ii) If ω is δ-computable; then ν(ω)is finite, and ≻pω is acyclic for allp∈ ANREC if and only if #X < ν(ω).
(iii)If ω is a prefilter, then≻pω is acyclic for allp∈ ANREC, regardless of the cardinal number#X of X.
Proof. (i) If ω is a nonweak prefilter, then it has an infinite Nakamura number. But nonweak computable games have finite Nakamura number by Corollary 15.
(ii) and (iii) are obvious from the following corollary [34, Theorem 3.1]
of Theorem 16: ≻pω is acyclic for all p∈ ANB′ if and only if #X′< ν(ω) for all finiteX′ ⊆X. (This corollary can also be obtained from the well-known fact that≻pωis acyclic if and only if the setC(ω, X′,p) of maximal elements with respect to≻pω is nonempty for all finite subsetsX′ ofX.)
We can strengthen the acyclicity of the dominance relation ≻pω in state-ment (iii) of Proposition 18 by replacing the statestate-ment with one of the following: (iv) if ω is a filter, then ≻pω is transitive for all p such that all individuals have transitive preferences≻pi; (v) ifω is anultrafilter, then≻pω
is asymmetric and negatively transitive for all p such that all individuals have asymmetric, negatively transitive preferences≻pi. In fact, statements (iii), (iv), and (v) each gives an aggregation rule ≻ω:p7→ ≻p that satisfies Arrow’s conditions of “Unanimity” and “Independence of irrelevant alter-natives.” These results are immediate from the relevant definitions (Arm-strong [4, Proposition 3.2] gives a proof). According to Arrow’s Theorem [6], however, if the setN of players were replaced by afinite set, then social welfare functions given by statement (v) would be dictatorial (andω would be weak).
In an attempt to escape from Arrow’s impossibility, many authors have investigated the consequences of relaxing the rationality requirement (neg-ative transitivity of≻pω) for social preferences. In view of the close connec-tion between the raconnec-tionality properties of an aggregaconnec-tion rule and preflters [8, Theorems 2.6 and 2.7] (also [23, 4, 5]), Proposition 18 has a significant implication for this investigation.
6 Examples
Propositions 7, 9, and 11 show that the class of computable games (i) in-cludes the class of games that have finite carriers and (ii) is included in the class of games that have both finite winning coalitions and cofinite los-ing coalitions. In this section, we construct examples showlos-ing that these inclusions are strict.
We can find such examples without sacrificing the desirable properties of simple games. We pursue this task thoroughly in a companion paper [26].
Thenoncomputable simple game example in Section 6.1 that has both finite winning coalitions and cofinite losing coalitions is a sample of that work. It is monotonic, proper, strong, and nonweak. An example of a computable simple game that is monotonic, proper, strong, nonweak, and has no finite carrier is given in that work [26].
6.1 A noncomputable game with finite winning coalitions We exhibit here a noncomputable simple game that is monotonic, proper, strong, nonweak, and have both finite winning coalitions and cofinite los-ing coalitions. It shows in particular that the class of computable games is strictly smaller than the class of games that have both finite winning coalitions and cofinite losing coalitions. In this respect, the game is unlike nonweak prefilters (such as the q-complement rules in Example 1); those examples do not have any finite winning coalitions. Furthermore, unlike nonprincipal ultrafilters—which are also monotonic, proper, strong, and nonweak noncomputable simple games—the game is nonweak in a stronger sense: it violates the finite intersection property.
LetA=N\ {0}={1,2,3, . . .}. We define the simple gameωas follows:
Any coalition except Ac ={0} extending the string 1 of length 1 (i.e., any coalition containing 0) is winning; any coalition except A extending the string 0 is losing; A is winning and Ac is losing. In other words, for all S∈REC,
S ∈ω ⇐⇒ [S=A or (0∈S &S 6=Ac)].
Remark 4. The reader familiar with the notion of repeated games (or binary rooted trees) may find the following visualization helpful. Think
of the extensive form of an infinitely repeated game played by you, with the stage game consisting of two moves 0 and 1. If you choose 1 in the first stage, you will win unless you keep choosing 0 indefinitely thereafter;
if you choose 0 in the first stage, you will lose unless you keep choosing 1 indefinitely thereafter. Now, you “represent” a certain coalition and play 1 in stage i ifi is in the coalition; you play 0 in that stage otherwise. Then the coalition that you represent is winning if you win; it is losing if you lose.
k
Lemma 19 ω is notδ-computable.
The following proof demonstrates the power of Theorem 5, although its full force is not used (Proposition 4 suffices). Proposition 8, which appeared earlier in Mihara [33], does not have this power.
Proof. If ω is δ-computable, then by Theorem 5 (or by Proposition 4), A has an initial segment A∩k that is a winning determining string. But A∩k itself is not winning (though it extends the string trivially).
Lemma 20 ω has both finite winning coalitions and cofinite losing coali-tions.
Proof. For instance,{0,1}is finite and winning. N\{0,1}={2,3,4, . . .}
is cofinite and losing.
Lemma 21 ω is monotonic.
Proof. Suppose S∈ω andS (T. There are two possibilities. IfS =A, then T = N, and we have N ∈ ω by the definition of ω. Otherwise, S contains 0 and some other numberi. The same is true ofT, implying that T ∈ω.
Lemma 22 ω is proper and strong.
Proof. It suffices to show that Sc ∈ω ⇐⇒ S /∈ω. From the definition ofω, we have
S /∈ω ⇐⇒ S6=A& (0∈/ S orS=Ac)
⇐⇒ Sc 6=Ac & (0∈Sc orSc =A)
⇐⇒ (0∈Sc &Sc 6=Ac) or Sc =A
⇐⇒ Sc∈ω.
Lemma 23 ω is not a prefilter. In particular, it is not weak.
Proof. We show that the intersection of some finite family of winning coalitions is empty. The coalitions{0,1},{0,2}, and A form such a family.
(Incidentally, this shows that the Nakamura number of ω is three, since ω is proper.)
6.2 A computable game without a finite carrier
We exhibit here a computable simple game that does not have a finite carrier.
It shows that the class of computable games is strictly larger than the class of games that have finite carriers.
Our approach is to construct r.e. (in fact, recursive) sets T0 and T1 of determining strings (of 0’s and 1’s) satisfying the conditions of Theorem 5 (the full force of the theorem is not needed; the easier direction suffices).
We first give a condition that any string in T0∪T1 must satisfy. We then specify each ofT0 andT1, and construct the simple game by means of these sets. We conclude that the game is computable by checking (Lemmas 24, 25, and 27) that T0 and T1 satisfy the conditions of the theorem. Finally, we show (Lemma 28) that the game does not have a finite carrier.
Let {ks}∞s=0 be an effective listing (recursive enumeration) of the mem-bers of the r.e. set{k:ϕk(k)∈ {0,1}}, whereϕk(·) is thekth p.r. function of one variable. We can assume that all elementsks are distinct. (Such a list-ing{ks}∞s=0 exists by the Listing Theorem [41, Theorem II.1.8 and Exercise II.1.20].) Thus,
CRec⊂ {k:ϕk(k)∈ {0,1}}={k0, k1, k2, . . .}, where CRec is the set of characteristic indices for recursive sets.
Let l0 = k0+ 1, and for s > 0, let ls = max{ls−1, ks+ 1}. We have ls ≥ls−1(that is,{ls}is an nondecreasing sequence of numbers) andls> ks
for eachs. Note also that ls≥ls−1 > ks−1, andls ≥ls−2 > ks−2, etc. imply thatls> ks,ks−1,ks−2, . . . .
For eachs, letFs be the set of stringsα =α(0)α(1)· · ·α(ls−1) (the *’s denoting the concatenation are omitted) of lengthls such that
α(ks) =ϕks(ks) and for eachs′ < s,α(ks′) = 1−ϕks′(ks′). (4) Note that (4) imposes no constraints onα(ks′) fors′> sand no constraints on α(k) for k /∈ {k0, k1, k2, . . .}, while it imposes real constraints for s′ ≤s, since|α|=ls> ks′ for such s′. We observe that ifα∈Fs∩Fs′, thens=s′.
Let F = ∪
sFs. (F will be the union of T0 and T1 defined below.) We claim that for any two distinct elementsαandβinF we have neitherα⊆β
(α is an initial segment ofβ) nor β⊆α (i.e., there is k <min{|α|,|β|}such that α(k) 6=β(k)). To see this, let |α| ≤ |β|, without loss of generality. If α and β have the same length, then the conclusion follows since otherwise they become identical strings. If ls = |α| < |β| = ls′, then s < s′ and by (4),α(ks) =ϕks(ks) on the one hand, butβ(ks) = 1−ϕks(ks) on the other hand. Soα(ks)6=β(ks).
The gameωwill be constructed from the setsT0 andT1of strings defined as follows:
α ∈T0 ⇐⇒ ∃s[α∈Fs and α(ks) =ϕks(ks) = 0]
α∈T1 ⇐⇒ ∃s[α∈Fs andα(ks) =ϕks(ks) = 1].
We observe thatT0∪T1 =F and T0∩T1 =∅.
Define ω by S ∈ω if and only if S has an initial segment in T1. Lem-mas 24, 25 and 27 establish computability ofω by way of Theorem 5.
Lemma 24 T0 and T1 are recursive.
Proof. We prove thatT0 is recursive; the proof forT1 is similar. We give an algorithm that can decide for each given string whether it is inT0 or not.
To decide whether a string σ is in T0, generate k0,k1,k2, . . . , compute l0, l1, l2, . . . , and determine F0, F1, F2, . . . until we find the least s such thatls≥ |σ|. If ls>|σ|, thenσ /∈Fs. Sincels is nondecreasing ins andFs
consists of strings of lengthls, it follows thatσ /∈F, implyingσ /∈T0. Ifls=|σ|, then check whetherσ ∈Fs; this can be done since the values of ϕks′(ks′) for s′ ≤s in (4) are available and Fs determined by time s. If σ /∈ Fs and ls+1 > ls, then σ /∈ T0 as before. Otherwise check whether σ ∈ Fs+1. If σ /∈ Fs+1 and ls+2 > ls+1 = ls, then σ /∈ T0 as before.
Repeating this process, we either get σ ∈Fs′ for some s′ or σ /∈Fs′ for all s′ ∈ {s′ :ls′ =ls}. In the latter case, we have σ /∈T0. In the former case, if σ(ks′) =ϕks′(ks′) = 1, thenσ ∈T1 by the definition ofT1; hence it is not in T0. Otherwiseσ(ks′) =ϕks′(ks′) = 0, and we haveσ ∈T0.
Lemma 25 T1 consists only of winning determining strings forω; T0 con-sists only of losing determining strings forω.
Proof. Let α ∈ T1. If a coalition S extends α, then by the definition ofω,S is winning. This proves that αis a winning determining string.
Let α ∈ T0. Suppose a coalition S extends α ∈ T0 ⊂ T0∪T1 = F. If β ∈ F and β 6= α, we have, as shown before, α 6⊆ β and β 6⊆ α, which implies thatS does not extend β. So, in particular,S does not extend any string inT1. It follows from the definition ofω thatS is losing. This proves that α is a losing determining string.
Lemma 26 For each s, any string α of length ls such that α(ks) =ϕks(ks) extends a string in ∪
t≤sFt.
Proof. We proceed by induction on s. Let α be a string of length ls
such that α(ks) = ϕks(ks). If s = 0, we have α ∈ F0; hence the lemma holds for s = 0. Suppose the lemma holds for s′ < s. If for some s′ < s, α(ks′) =ϕks′(ks′), then by the induction hypothesis, the ls′-initial segment α∩ls′ of α extends a string in ∪
t≤s′Ft. So α extends a string in ∪
t≤sFt. Otherwise, we have for each s′ < s, α(ks′) = 1−ϕks′(ks′). Then by (4), α∈Fs⊂∪
t≤sFt.
Lemma 27 Any coalition S∈REC has an initial segment in T0 or T1. Proof. Suppose ϕk is the characteristic function for a recursive coali-tion S. Then k ∈ {k0, k1, k2, . . .} since this set contains the set CRec of characteristic indices. So k = ks for some s. Consider the initial segment S∩ls. It extends a string in ∪
t≤sFt by Lemma 26. The conclusion follows since∪
t≤sFt⊂F =T0∪T1.
Lemma 28 ω does not have a finite carrier.
Proof. We will construct a set A such that for infinitely many l, the l-initial segment A∩l has an extension (as a string) that is winning and for infinitely many l′, A∩l′ has an extension that is losing. This implies that A∩l is not a carrier of ω for any such l. So no subset of A∩l is a carrier. Since there are arbitrarily large such l, this proves that ω has no finite carrier.
LetAbe a set such that for eachkt,A(kt) = 1−ϕkt(kt). For any s′ >0 andi∈ {0,1}, there is ans > s′ such thatks> ls′ and ϕks(ks) =i.
For a temporarily chosens′, fixiand fix suchs. Then choose the greatest s′ satisfying these conditions. Since ls > ks > ls′, there is a string α of length ls extending (as a string) A∩ls′ such that α ∈ Fs. Since α(ks) = ϕks(ks) =i, we have α∈Ti.
There are infinitely many suchs, so there are infinitely many such s′. It follows that for infinitely manyls′, the initial segment A∩ls′ is a substring of some string α in T1 (by Lemma 25, α is winning in this case), and for infinitely manyls′,A∩ls′ is a substring of some (losing) stringα inT0.
References
[1] L. Anderlini and L. Felli. Incomplete written contracts: Undescribable states of nature. Quarterly Journal of Economics, 109:1085–1124, 1994.
[2] L. Anderlini and H. Sabourian. Cooperation and effective computabil-ity. Econometrica, 63:1337–1369, 1995.
[3] N. G. Andjiga and B. Mbih. A note on the core of voting games.Journal of Mathematical Economics, 33:367–372, 2000.
[4] T. E. Armstrong. Arrow’s Theorem with restricted coalition algebras.
Journal of Mathematical Economics, 7:55–75, 1980.
[5] T. E. Armstrong. Precisely dictatorial social welfare functions: Erratum and addendum to ‘Arrow’s Theorem with restricted coalition algebras’.
Journal of Mathematical Economics, 14:57–59, 1985.
[6] K. J. Arrow. Social Choice and Individual Values. Yale University Press, New Haven, 2nd edition, 1963.
[7] K. J. Arrow. Rationality of self and others in an economic system.
Journal of Business, 59:S385–S399, 1986.
[8] D. Austen-Smith and J. S. Banks.Positive Political Theory I: Collective Preference. University of Michigan Press, Ann Arbor, 1999.
[9] J. S. Banks. Acyclic social choice from finite sets. Social Choice and Welfare, 12:293–310, 1995.
[10] J. S. Banks, J. Duggan, and M. L. Breton. Social choice and electoral competition in the general spatial model. Journal of Economic Theory, 126:194–234, 2006.
[11] J. Bartholdi III, C. A. Tovey, and M. A. Trick. Voting schemes for
[11] J. Bartholdi III, C. A. Tovey, and M. A. Trick. Voting schemes for