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Majority tournaments: sincere and sophisticated voting decisions under amendment procedure
Mathematical Social Sciences 21 (1991) l-19 Not-t h-Holland
K.B. REID California State University, Sun Marcos, CA 92069, USA and Louisiann Stute Cfr~i~yrsitj~, Peon Rouge, LA 70803, USA Communicated
by M.F. Janowitz
Received 14 June 1989 Revised 27 July 1990
In this work we give a careful mathematical treatment of two majority voting procedures in which a voting agenda is utilized: sincere voting and sophisticated voting. We elaborate and espand on work by Banks (1985), Milker (1977,1980), Miller, Grofman and Feld (1986), and Shepsle and Weingast (1984). W’e discuss algorithms to obtain these two decisions under such procedures, the location of these two decisions in the underlying majority tournament, the relative locations of these two decisions in the voting agenda, and the possibility of an alternative which is both of these decisions. Key words: Tournaments; division tree; sophisticated
voting agenda; amendment proced w-e; sincere voting: decision voting; Banks points; kings.
tree;
Suppose that each voter in a finite, nonempty set V of voters linearly orders, according to their preferences, a finite, nonempty set of alternatives 4. A common group aggregration rule is majority preference, in which an alternative is preferred over another alternative if and only if the former alternative is preferred to the latter alterantive by a majority of the voters. That is, given several transitive tournaments on the same vertex set ,4 (one for each voter), the majority preference rule gives rise t.o the majority digraph, an orieneed graph with vortex set A in which for all xfy in A, x dominates y if and only if x dominates y in a majority of the transitive tournaments. If no ties result (e.g. if IV1 is odd), then a tournament results (e.g. see Fig. I). McGarvey (1953) proved that every oriented graph with n vertices and q arcs is the majority digraph of n alternatives by at most 2y voters. NOX that Zq L FT(PI- I ). cGarvey’s theorem by showing that Stearns (1959) improved tively, pa+ 1) voters are needed if n is even (respectively, odd). proved that, for large n, the number of voters nee If m(n) denotes the least integer m such that at most m n vertices (n > I ), t era 01654896/91/$3.50
%I 1991 -Elsevkr
ScierLe Publishers
B.V. (North-Holland)
K.B. Rerd / Majority
tournaments
and votiq
a
Voters’ preferences Voter Voter Voter Voter
Fig. I. Majority
2
b,G,a,c,e
3
c,e,b,a,d
4 5
d,a,c,e,b e,c,d, t?,a
tournament.
there exist constants cl and c2 such that c&log
n < m(n) < c&log
n.
These results are discussed by Moon (1968). In the remainder of this paper, it is assumed that the majority digraph is a tournament . The issue of which alternatives should be considered most desirable by the group in majority voting procedures has been a major concern to voting theorists (e.g. see the references to the papers listed at the end of this paper). The majority tournament may contain (directed) cycles which may be seen as inconsistencies. Indeed, the majority tournament may even e strongly connected (e.g. see Fig. l), and hence every vertex is in cycles of every possible length (see Moon, 1968, p. 6). Work has focused on descriptions of sets of vertices, called solution sets, in which the ultimate decision(s) must lie when several important majority voting processes lead to a decision in the initial strong component of the majority tournament. Later Miller (1980) suggested a r- ‘inement, carAzdthe uncovered set, as the solution set for one voting process. Authors writing on tournaments have called elements of the uncovered set, kings. More recently, Banks (1985) and Miller, Crofman and Feld (1986) have suggested a further refinement which has been dubbed the Banks set. We treat these issues in this work. In Section 2 we discuss the combinatorics of a majority voting procedure, called sincere voting, under an amendment procedure. In Section 3 we discuss a strategic roach to majority voting under an amendment procedure which is called ting; we include a treatment of the algorithm due to Shepsle and
e sincere and sophisticated
decision.
K. B. Reid / Majoriry
totrmarnerits and voting
3
Before continuing we recall some of the basic terminology that will be employed subsequently. Let T denote a (finite) tournament. For vertices x and y in T, write xRy if and only if there are (directed) pat s in T from x to Y and from y to x. It is easy to see that R is an equivalence relation on V(T), the vertex set of T, and that if there is more than one equivalence class, then the equivalence classes Vt, . . . , Vk can be ordered so that each vertex in !$ dominates each vertex in 5 whenever 1 s i
2. Sincere voting Under amendment procedure, some order, say (al, a2, . . . , a,), of the m alter. . niklves 1s gtven. In the first vote alternatives ai and a2 are paired for a majority vote, the defeated alternative being eliminated from further consideration; at the ith vote, 25&m1, the alternative surviving the (i- 1)th vote is paired with a,+1 for a majority vote, the defeated alternative being eliminated from further consideration. The alternative surviving the (in - 1)th vote is called the sincere decision. That is, given the voting order (ai, a2, . . . , a,,,) and a majority tournament T with vertex set {ai, Q2,. . . , a,}, the sincere order (b,, b2, . . . , b,) is defined inductively by b,=at and, for 2(&m, bi
=
“’ bi-1,
bi_ 1 in T, if bi_ ,dominates ai in T.
if ai dominates
3.he alternative b, is the sincere decision. The m - 1 votes that occur in the amendment procedure correspond to m - 1 arcs in the majority tournament T. These m- l arcs induce a subdigraph of T in which exactly one vertex, the decision, has indegree 0 and all other vertices, which were eliminated in some vote, have indegree 0. As the voting order causes a se elimination of vertices, this digraph is acyclic. Thus, these yb/r - E arcs in oriented spanning subtree of T, rooted at the decision, called the decision tree of ore can be said about the structure of th ed) tree such that the re vertices results i
X is an orientation of the edges of X so that either the spine becomes a directed pat and all other arcs are directed away from the spine or a longest path in X (consisting of the spine and two ‘end’ ed:?es) becomes a directe directed away from that path. Fig. 2 illustrates these t caterpillar. Note that a special orientation of a cater rooted at the unique initial vertex of a longest directed path.
Let ‘9’be a toumament and let X be a spanning subdigraph of T. X is the decision tree of Treilative to some voting ordw if and only if X is a special orient&on of CIcaterpillar.
roof. Our proof is designed to yield considerably more information. Suppose that Qi,,ai.,...,U;,, 1 sil 2, then ~1~dominates exactly a?, . . . , q, _ I. Agld for 3 C;,CS, if ii = i, _ I+ 19 then ai, dominates ai, , and ai, , dominates only a,/ _,; and if .$> ii_ 1 + 1, inates exactly ai, ~ and ai, , + !, . . . , Oi, _ I. Moreover, if &< HI, then ajS dominates exactly Qi, , and a;, I,, . . . , a,,,, and if is= m, then a,,, dominates oniy ai Consequently, the decision tree X has the form of one of the rooted trees’ i Fig. 3(a)-(d). In any case X is a special orientation of a caterpillar. Conversely, suppose that X is a spanning special orientation of a caterpillar of T with vertex set A. First suppose that in X the root is not an endvcrtex. Then there is h of the form shown in Fig. 3(a) oc 3(b) to X, say SM. Consider the voting order given in Fig. 3(a), determine ex-
a
am
a $4
aij+l
(c 1 i,=2 and
"j
am 3
:
000
*l
I~=
m
ai. JK
000
(d) 192 and ts=m
Fig. 3.
g4 = &,, 1 S&S, where s is as in Fig. 3(a). First 8, = al, so take i, = 1. Assume that i &,+l,thensincea4 ,dominates8i,+I ,..., ii,,_I inX(andin T), 4 ,=b,, ,+]--: . . . = E4_ 1; and since @,dominates 5, , = b;, , = hb __1 in X (and in T), f,, = a4. In any case, we have now determined cl, . . . , l+, ,, . . . ,5;,. y induction, bl, . sa, bi, is is< tn (i.e. the root of X is not an endv tex), and 6i, = “i, domin X (and in T), SO hi,=Eiiy_,= . . . =6,?!. T determines (6,, e , 6,,J, sion tree of T wit respect to voting or Q, . = , &!s This sa fices if X is isun phic to the digraph in ’ . 3(b). A very simil or 3(d) (i.e. the root the same result if X is isomorphic to either endvertex), except that the last step above (i,< rn) is not nee is roof is now complete. El l
l
l
K. B. Reid / Majority tournaments
6
and voting
e proof enables us to enumerate the voting orders for which a fixed labelled X is the decision tree. If T has a transmitter x and X consists of x and all the HZ- 1 arcs fro ust be s to the other nr - 1 vertices (i.e. a special orientation of K,,,,l_ 1), t either first or second in the voting order if X is to be the decision tree, and the other ahernatives may occur in any order. So, there are 2[(m - I)!] voting orders yielding X as the decision tree. .
f X is as in Fig> 3(b)s then X is not only the decision tree relative to voting order (ai, a2, . . . q g&, but it is also the decision tree relative to each of the (i,6 - 2)! = (i2 - ir - l)! voting orders
where d,, d2, . . . 9di,c_ z is any (i2 - 2) permutation of a2, . . . , aj,_ 1. Moreover, for 2~j<~, if ij+i-$>I and ei ,..., ei ,_ i/ , is any of the (4, 1 - $ - l)! permutations of ail+], . ..) a&+,_ 1, then X is also the decision tree relative to the voting order
(q,Q2,~-.,Q~,el,...,e~+,_i-],Qj J
/+
,,...,
a,).
Similarly, any of the (m-Q! permutations of a.,s+1,. . . , a,,, may occupy the final m - i, positions in a voting order that has X as the decision tree. Consequently, X is the decision tree for 2i
(ij-ij_,
- I)! (m - is)! voting orders.
j=2
Since O! = 1, this formula holds also if X is any of the decision trees of the form given by Fig. 3(a), (c), or (d). If T is a tournament considered as a majority tournament on alternatives 442, =**,a*, define the (sincere) decision set D(T), as follows: D(T) = {a ) there is some voting order so that u is the sincere decision relative to T and that order}. Theorem 1 above provides a new proof of results by Miller (1977, Propositions
I
FQ~ any tournament T, D(T)=
1
V(P).
ED(T); say a is the sincere decision under voting order (al, a2, . . . , a,). g decision tree. By Theorem 1, X is a spanning directed at is, every vertex in hence in T, is reachable via
h’.B. Reid / Majority
tournaments and voting
d:;ected path in from a. Since no vertex in * is reachable from a vertex outside of T*, aE V(T*). On the other hand, let XE V(T*’ ). Then x is the origin of a spanning path in ?“. But a spanning path is a special orientation of a spanning caterpillar rooted at x. By Theorem 1, this spanning path is the decision tree of T relative to some voting order. And by a remark preceding Theorem I the root of this decision tree is the sincere decision. Thus, SE D(iT’). Hence V(T*) C_D(T). The result follows. c3 a
Note that any alternative x in the initial component of T may be the decision under many different voting orders. Any of the many spanning caterpillars with special orientations rooted at x gives rise to such a voting order according to Theorem 1. The spanning path given in the previous proof is just one such, and was the one used by Miller (1977, Proposition 2, p. 781). Of course, if a tournanznt contains a transmitter x, then x is the unique vertex in T*. Consequently, an immediate corollary follows. orollary 3. If x is a transmitter of a majority tournament T, then x is the sincere decision regardless of the voting order. With HI alternatives the sincere voting process under amendment procedure can also be described via a balanced, binary rooted tree on 2”‘- 1 vertices, cakd the division tree. Vertices in the division tree are labelled by subsequences of the voting order (aI, a2, . . . , a,,,) as follows: the root at level 0 is labelled (al, a2, . . . , a,,,); for 05 js m - 2, a vertex at level j which is labelled by a subsequence (of the voting order) of length m -j, say (6,, b2, ..-, b,,_J, dominates two vertices at level j + 1, one labelled (6,, bJ, . . . , btpl_J and one labelled (b2, b3, . . *, b,,, +). The division tree depends solely on the voting order; it is independent of any majority tournament. See Fig. 4(a) for an example with m = 4 (ignore the underlined alternatives). Note that several distinct vertices in the division tree may share a common label. Given a majority tournament T and a voting order (a,, a2, . . . , a,), the sincere amendment procedure yields a path in the division tree from the root to an endvertex labelled with the sincere decision. For example, if T is as in Fig. 4(b), then the (sincere) amendment procedure yields the following path in the division tree in Fig. 4(a): (a, b, G d), (a, c, d), (a, d), Cd)A census of the vertices in the division tree is described next. Let (al,a2, . . . . a,,,) be the voting order. For each k, 2 5 k< m, and for each j, 1(: j 5 k - 1, level k - 2 of the division tree contains
1,
ifj=l
2-i-2,
otherwise
K’.d. Reid / Majorit>* iournatttents and voring
Level
a
C
d
0
d
(b) Fig. 4.
vertices lube/led (a_,, a,, ak + I) . . . , u),~). Moreover, for each k, 2 I kr m, every vertex at !evel k - 2 is labelled (aj, a,+ a, + ,, . . . , a,,,) for some j, 15 j zzk - 1.
The final statement follows from the first claim, since 1 + ~~~~ 2j-’ =2”-‘. Induct on kz2. If k= 2, the root of the division tree, the only vertex at level 0, is labelled (a, a2, . . . , a,,,), as desired. Assume that 25 k
vertices labelled (q,, uk + ,, ak +2, . . . , a,,,). Suppose j< k. By the induction hypothesis, level k - 2 contains if j=l
1, 2j~ 2,
if 2sj
vertices labelled each of which dominates two vertices at level k - 1, one labelled (aj, ak + I, ali+z, ,.. ,a,,,), and one labelled (ali, ak + 1, . . . , a,,,). So there are ifj=l 1, (Uj,
ah,
2J-2 9
uk
+
,,
.
if2sjik-l
.
.
,
a,,,),
vertices at level R:- 1 labelled (a,, ak + ], ak +2, . . , a,,,). NOW suppose that j z=k. As remarked at the outset, every vertex at level k - 2 is labelled (a,, ak, ak +], . . . , a,,,) for some i, 1 Si5k-- i. And as just seen, each such vertex dominates exactly one vertex labelled (ak, ak + ], . . . , a,,,). As there are 2” ’ vertices at level k - 2, the result follows for j = k. By induction the result follows. II l
3.
The amendment procedure discussed above involves sincere voting decisions in contrast to so isticated voting decisions under the amendment procedure, to be discussed next. i II brief, decisions are anticipated at higher levels of the division tree in order to anticipate decisions at lower levels, until an anticipated decision at the first vote is made. A formal description follows. Assume that there are III alternatives. Recall that each vertex of the division tree at level rn - 2 is labelled with an ordered pair of alternatives; the anticipated decision at each vertex of level 177 - 2 is defined to be the majority choice between those two alternatives (as given by a fixed majority tournament T). Inductively, for Osj< m - 2 the anticipated decision at each vertex v of level j is defined to be the majority choice between the two alternatives which are the anticipated decisions at the two vertices at level j+ 1 which are dominated by v in the division tree. The wphisticated voting decision (under the amendment procedure) is the anticipated decision of the root at level 0. For exampie, given the majority tournament in Fig. 4(b), the anticipated decisions are underlined in the division tree in Fig. 4(a). So, in the sophisticated voting decision there is alternative 6. Note that the sincere voting decision in that example is alternative d. a vertex x of the division tree is labelled (aj,ak,ak+], . . ..a.,,), then the anticipated decision at x depends only on T[ai, ak, ak + l, . . . , a,,,]. Consequently, the anticipated decision at x is the sophisticated decision for T[ai, a,+,ak+ 1, . . . , a,,,] relative to the agenda (aj, a& ak+ 19. . . , a,,,). 3. If
mark
Before describing an easier procedure for finding the sophisticated decision we give a preliminary technical result. Let Dj denote all of the anticipated decisions at level j, 0 5 js 777- 1, in the division tree, relative to some fixed majority tournament T and some agenda (voting order). For example, D,,,_ I = T and Do contains only the sophisticated decision. ma 5. Let Z c T.
I”’
for some
0 5 j<
171-
I,
sophisticated decision is irl Z*.
Induct on j. If j = @,the result follows fr only the sophisticated decision. Let 0 < k ISr71- B. all values of j< k. Suppose that .
~j
c
Z and pi f7 2 *Z D8,then thP
h’.B. Reid 1 Ilia_ioi-i’tytsrmaments
10
and voting
the definition of an anticipated decision, so Dk__1 5 2. Let u E 62, f3 Z*, say a is the anticipated decision at vertex v at level k in the division tree. Vertex v is dominated by some vertex t at level k - 1 in the division tree; and t dominates exactly one other vertex in level k in the division tree, say u. Let b be the anticipated Note that a, b E Z and CIE Z*. If a = b or a dominates b in T, then a is the anticipated decision at t so a E Dk_ 1t3 Z*. If b dominates a in T, then b must lie in Z*, so b is the anticipated decision at t, i.e. Dk _ l n Z* # 0. In any case, Dk _ l C_Z and Dk _ 1n Z* # 0. By the induction hypothesis, the sophisticated decision is in Z*. By induction the result follows. 0 As an immediate corollary of Lemma 5, we can characterzie transmitters. oroh-y 6. Alternative a E T is a transmitter if and only if a is the sophisticated decision relative to every agenda.
. Suppose that a is a transmitter in T, so that T*= (a]. Regardless of the agenda, a is the label of some vertex at levei m - i in the resulting division tree, so a is the anticipated decision at some vertex at level m - 2 in the division tree. That is, if Z = T, D,, _ 2 c Z and D,,I_2 n Z*#0. By Lemma 5, the sophisticated decision is in Z*= T*= {a}. On the other hand, suppose that a is not a transmitter, so that I(a)#O. Hence (I(a))*#O, say b E (I(a))*. Consider any agenda and the resulting division tree in which CIis the last alternative. By Proposition 4, a vertex at level m - 2 of the division tree is labelled (x, a), for some XE:T. Of course, one of these vertices is labelled (b,a). Thus, the anticipated decision at each vertex at level m - 2 is in I(a) W {a}. Moreover, b is the anticipated decision at the vertex labelled (b, a). Let Z = I(a) U {a}, so that Z* = (I(a))*. Since Dnl_2 c Z and D,71_2 n Z*#0, Lemma 5 implies that the sophisticated decision is in Z* = (I(a))*. That is, a is not the sophisticated decision relative to every agenda. This completes the proof. 0 Perhaps the easiest way to determine the sophisticated decision is to use the algorithm given in the ‘Equivalence Theorem’ of Shepsls and Weingast (1984). Let T be a majority tournament with vertex set A and let (a,, a2, . . . , a,,,) be ir fixed agenda. Define the sophisticated agenda (zl, ~2, . . . , z,,,) as follows: zlll = a,, and for 1
r’ Qj,
Zj=
I
Zj+l9
tt1
if a, E n
QZ;),
i=j+l
otherwise. (Zj, z29 *-* 9 z,,,) is the sophisticated srgen-
K. B. Reid / Majority
da for a fixedtournament decision.
tournaments and voting
II
T and agenda (al, a2, . . , a,,,), then q is the sophisticated l
Induct on ypz.The case m = 1 is trivially true. Assume that in > 1 and that t result holds for all agendas of length less than m. Form the sophisticated agenda (Zr&, .**,z,). In the division tree, the root is labelled (aI, a2, . . . , a,,) 3 .rd dominates two vertices, one labelled (al, a3, . . . , a,) (to be considered as an agenda for T- a2) and one labelled (az, a3, . . . , a,,) (to be considered as an agenda for T--a{). The sophisticated agenda relative to T- al and agenda (a2, a3, . . . , a,) is (12~23,..* J,~). And, the sophisticated agenda relative to T- a2 and agenda (al,+, . ..) a,,) is (b,, ~3, z4, . . , z,), where .
l
n I(Zi), n1
b, =
al,
if al E
i=3 239
otherwise.
By the induction hypothesis, the anticipated decision at (al, a3, . . . , a,) is bl, and the anticipated decision at (al, a3, . . . , a,) is z2. In order to determine the sophisticated decision, bl is compared to z2. Case I: b1 = al. Note that z2 #al (more generally, by the definition of the sophisticated agenda, for 1 s i
z1 is the sophisticated
decision.
Case 2: bl = z3. If 22 = z3, then ~,2(=13) is the sophisticated decision. And, by definition of bl, al $ n;! 3 I(Zi) = n;12 I(Zi), so z1 = z2 by the definition of q. So, if z2 = z3, then q(=z2 =z3) is the sophisticated decision. If t2#z3, then z2 =a2 and a2 E nr3 I(Zi) by definition of z;. So, z2(= a2) dominates z3 and z2(= a,) is the sophisti-
cated decision. Now, by the definition of bl, al $ n:,l, 3 I(zi), so a1 $ n;=, I(q); this implies that q #aI by the definition of q, so z1= z2. That is, q is the sophisticated decision. In either case q is the sophisticated decision, so b;: induction the result follows. 0 le. Given the majority tournament in Fig. 4(b) and the agenda (a, 6, c, d), first entry of the sophisticated agenda (b, b, c, d) is b, the sop The proof of the
revious theorem,
like t
K.B. Reid / Majorirjf
12
forrrrrarnenCs and voting
valves only one level beyond the root in the division tree. The proof given by Shepsle and Weingast (1984) involved two levels beyond the root. The Shepsle-Weingast Theorem can be used to give short proofs of two observations made by Miller (1977, pp. 794-795). Miller’s explanations are incomplete, since they are based on an unsubstantiated algorithm (which we will treat below). We state these observations as a single proposition. a,,,) be a fixed agenda and !et T be a majority tournament on A. If ai is the sophisticated decision and aj is the sincere decision, then either i = j or i < j and ai dominates aj in T. roof. Assume that i# j. Let (q, . . . , z,J denote the sophisticated agenda. By the Shepsle-Weingast Theorem, z1= ai, 22= ai, . . . , zi = ai. If j= m, then i< j, z,,~= anI and q E I(z,?J by the definition of zi. So, assume that j< m. As aj is the sincere decision, aj dominates aj+ 1,. . . , a,,,. But then zj = aj, by the definition of +. Since each of q, zz, , z, is ai and ai f aj, i < j. SO zi E I(Zj), by the definition of zi In any case, i< j and ai dominates aj in T. q l
l
l
l
A more fundamental proof of Proposition 8 can be given by showing that all of the anticipated decisions at level j- 2 of the division tree are contained in {aI, 02, . . . , aj} fl (I(aj) U {aj}). But ai is the anticipated decision at the root, SO it must be the anticipated decision at least once at level j- 2. So, ai is in that intersection.
iller’s algorithm Miller (1977, pp. 789-791; and 1980, pp. 81-82) alluded to an algorithm for determining the sophisticated decision. That process was given only for the first two steps, no explicit proof was given for its termination but, nevertheless, the validity of the algorithm was assumed in his explanation of his Propositions 11 and 12 in his 1977 paper. We have proved those two results as Proposition 8 above. Our aim here is to describe the algorithm precisely for the first time and to prove explicity that the algorithm terminates v;iih the sophisticated decision. Let (a,,a2, . . . . a,,,) be a fixed agenda (voting order), and let T denote a fixed majority tournament on A. For Or is rn, define sets of vertices (alternatives) w as follows: IQ 3’ and b. is the last vertex of T* in the agenda; for ir0, having defined ,..., 4, if W,=0, then kI$+t = 0 and bi is not defined, and if I+$#0, then b, is the last vertex of IJ$* in the agenda and P$+ , = I(bi) n l4$*. The proof that this process terminates with the sophisticated decision is rather involved, so first we give a preliminary result which shows that the process does terinate.
K.B.
Reid / Majority
tournantents
and voting
(i) ;,cW* . and (ii) For O
13
and b, dot?linates
(i) Suppose that & # 0. Since bi E wi*, but bi $ I(bi), we have
(ii) For 0 QC
i, by (i) we have
bi E w* s q*+ 1 = I(bj) f7 q*, SObi dominates !Ij and bi E WjT But, bj is the last vertex of wj* in the agenda, precedes bj . (i)), a straightforward induction yields:
SO
b;
for 05&m. In particular, l&J 5 lP&ol-m=m-m=O, so W,,,=0. (iv) Since JVk#0, w,*#O. But IVk, 1 = I(b,& n WA?= 0, so bk is a transmitter IV”, i.e. Wc={bk}. q 0. If k-maxii!
u/;#5;,
in
A. I &L” soghisricated decision. bAcn _hl;gj
roof. If k = 0, then IV, = I(b,) n I#$,*=I(bo) fl T* = 0, so b. is a transmitter in T. By Corollary 6, b. is the sophisticated decision. So, in what follows assume that k>O. We recall the notation of Lemma 5 where Di denotes the set of all anticipated decisions at level i in the division tree, OS ic nz - 1. The alternatives bk, bk _ ], . . . , bo correspond to alternatives ajA,aj, ,, . . . , aj,,, respectively, in the agenda, where j, < j,_ 1< . . . 0, j, _ 12 2. We claim that (a) for OS&k-1, Dj,_zc Wi+lU{bi}, and bi+il=aj ,+,I is the anticipated decision at (b) for O&Sk-l and O
Wi*,,#0.
Before verifying (a) and (b), we show how the proof is completed. Let Z= I4$ U {bk_ $ Since wk C_I(bk_ 1) (definition of IV,+),Z*= I+‘:. Sd, 5y (a) and (b) ut, then by Lemma 5, with i=k-1 and /i=l, Bji, ,_2~Z and the sophisticated decision is in Z*= IV’;. owever, by Lemma 9(iv), So, bk is the sophisticated decision. We now verify (a) and (b) by induction on i, 0 L is k - I. irst we 0-m the case i = 0. By Proposition 4, a typical vertex t at level j, - 2 in the division tree is label-
h’. B. Reid 1 Majority
1-l
townaments
and voting
1, and t dominates exact a,,,), for some p, idp5j0tices in level j,, - I, say II and v labelled (a,, ai0+1, . . , a,,,) and (q,,, ajo+i, . . . , a,,,), respectively. Since aj,, is the last vertex in the -agenda which is in IKf trans..ritter in T[aj,,,a,,,+1,..*, a,,,]. So by Remark 3 in Section 3 and C is the anticipated decision at v. If, in addition, aj,, dominates a, in T, t anticipated decision at t, regardless of the decision at u. If a, domi then a, E Wo*since aj(,E WO* = T*. Since none of aj,,+1,. . . , a,,, is in T* of a& a!, is a transmitter in T[a,, aj(,+1, . . . , a,,,]. SO, again by Rema ollary 6, a, is the anticipated decision at II, and hence at t. In this a,E I(aj,,) n Wo*= WI. So, recalling that aj,,= &, we see that all antici sions at level j,-- 2 are in W, U {b,}. That is, (a) follows for i= 0. F W,;*#0 by the definition of Wk) and bh = aj,, exists in WI:. If verte chosen so that aP=aj,, (there is such a t by Proposition 4), then since is the anticipated decision at U. But ap = aj,,= bll dominates aj,)= bo (by L in T, SO aj,, = ap is the anticipated decision at t. Thus, some anticipated decision at level j, - 2 is in WI,*.That is, (b) follows for i= 0. Assume that O
led tap,q,,,a,,,+1,
. . . y
l
l
l
l
Next ‘vi:’+ow that the anticipated decision at u is in H$U {bi_ l}. This is much easier than the case just considered at v. By part (a) of the induction hypothesis, the anticipated decisions at all the vertices at level j;_ 1- ? of the divison tree which can be reach from u (i.e. ‘descendents of u’) are in K$U Ibi_ 1). So, the anticipated at u, denoted a, u ibi- I>* Now, the anticipated decision at t depends on a, the antrcipated decision at M,
K. B. Reid 1 Mujority tournaments and voting
1s
an aj,, the anticipated decision at U. rite U {bi- r) as the disjoint union (K - H$“) U UI;”W {!I;_ ]}. The anticipated decision at t is a,,, if either a = &_ I = And, if ae &*, then the an% I or clrE I+$- wi* since ticqated decision is either a, if a d ;*, or a,,, if a,, dominates a ip *. So, if a is the anticipated decision at I, then a E I(aJ,)f3 4’” = !4;+1. Otherwise, the anticipated decision at t is aj, e Since t was arbitrary, part (a) is established for the value i. To complete the proof (i.e. the induction step for part (b)) fix h, O
E
Wi*
=
The Shepsle and Weingast algorithm and the Miller algorithm are intimately linked, as we see next.
be an agenda, and let T be a majority tournament on A. Let (zl, z2, . . . , 2,) be the sophisticated agenda obtained by the Shepsle and Weingast algorithm. Let (bk, bk _ !, . . . , bO) be the sequence of distinct alternatives obtained by the Miller algorithm. The bi 3 define a sequence of indices where bk=aj,, bk_1 =ajk ,, . . . . b,=aj,,. Then I sj,< . . .
). Since The proof of this result is not given here, but it may be found i the proof is independent of Theorem 7 and I’heorem 10, the validity of each of the algorithms described above can be used to establish the validity of the other.
h’.B. Reid / Majority
16
tournarnen ts and voting
The previous results can be employed to describe the set of alternatives in a tournament T which are possible sophisticated decisions. As in Banks (1985), let S(T) denote such a set, i.e. S(T) = {a~ T ( a is the sophisticated voting decision relative to some agenda}. A Banks point x (see Miller et al., 1986) in a tournament T is a vertex of T such that x is the transmitter of some transitive subtournament IV, and no vertex in T dominates every vertex in IV. We will use B(T) to denote the set of Banks points in T. The language used by Miller et al. (1986) to describe points in B(T) is slightly different but entirely equivalent to this definition. a 12. Let M(T) = (y E T 1y is the transmitter of a maximal transitive subtournament).
Then B(T) = M(T).
Clearly, M(T) c B(T). Let x E B(T), say x is the transmitter of transitive subtournament W, and no vertex in Tdominates every vertex in IV. If Wis maximal, then x E iti( Otherwise, there is a maximal transitive subtournament W’ properly containing IV. Now x must be the transmitter in IV’, by the choice of IV. So XE M(T). Thus, B(T) CM(T), and the result follows. 0 l
Since every tournament T contains a maximum transitive subtournament, which is, of course, maximal, B(T) #0. For example, if T itself has a transmitter x, then B(T) = (x}. In general, B(T) c T*. A bit more can be said. Recall that a king of T is a vertex x in T so that for each vertex y#x in T there is a (directed) path from x to y of length 1 or 2. And, the outset of x, denoted O(x), is the set {YE T 1x dominates y}. 3. Let K(T) = (YE T 1y is a king of T).
Then B(T) c K(T).
Let x&(T). Of course, x can reach each vertex in O(x) in one step. If there is a z in I(x) such that x cannot reach z in two steps, then z dominates every vertex in O(x) U (x}* So, xt$ M(T), contrary to Lemma 12. So x&C(T). Thus, (T) c_K(T). ‘J This observation was also made by Banks (1985, p. 300), without the ‘kings’ terminology. anks (1985, p. 300) characterized S(T) as follows: a&s, 1985). S(T) = B(T). is the transmitter of a maximal transitive sub. . . , wk, where dominates ere k+k=m, order of
K.B.
Reid / Mujority
fourt~attrett~s and voting
17
Under agenda (u,, . . , u,, wI, w2, . . , wk)I, the so isticated agenda (z,, . . . , z,~,) is given by z1= . . . =z/=z/+~ = wi, and Z, = wJ for / 5 jl tn. Thus, by Theorem 7, x = w1E S(T). Consequently, Let aES(T), say a is the s e&ion via agenda (a,, a?, . . . , a,,,). In the sophisticated agenda (zi, . . . ,z,,,), q = a by Theorem 7. If q = z,,), then a is a transmitter, and, clearly, a E B(T). So, assume that q # zm. Let q =Tz;,,z,,, . . . , Zikbe the distinct alternatives among q, z2, . . . , z,,~, where 1 c i2< ...< iks rn.-Note that z& dominates zi,, whenever 15 j< e’s k, SO{Zi,,Zi2,. . . , zi,> induces a transitive subtournament, denoted with transmitter q =a. Suppose that some vertex y in T dominates every vertex in W. Note that y is not among z,,, . . . , ziA, so y is nc:t in {z*,z2, **9z,~}. But by the definition of the sophisticated agenda, since y dominates any subset of the zi’s, y must arise as a zi, a contradiction. So, aEB(T). Consequently, S(T) c B(T), and the proof is complete. q l
l
l
The previous results can be utilized to give a new proof of another observation of Miller (1977, p. 793). First a preliminary result is established. emma 15. If an alternative aj is unanimously preferred to another alternative a,, then O(ai)F O(aj).
roof (Miller, 1977, p. 793). If XE O(ai), then a majority of the voters prefer ai over X. But all the voters prefer aj to ai, SO, by the transitivity of voters’ preferences, the same majority prefer aj over X, i.e. XE O(aj). Since ai E O(aj) - O(ai), the result follows. El carem 16. No altemative is unanimously preferred to the sophisticated decision. roof. By Theorem 14 and Lemma 12, the sophisticated decision, denoted ai, is the transmitter of a maximal transitive subtournament X of T. If aj is unanimously preferred to ai, then by Lemma 15 aj dominates every vertex in X, contrary to the maximality of X. The result follows. q For completeness we give an alternaLive proof that avoids the Shepsle and Weingast Theorem 7 (used in the proof of Theorem 14) and instead utilizes Theorem 10. This is the approach attempted by Miller (1980, p. 793). Suppose that aj is an alternative which is unanimously preferred . Let the index k be as in Theorem 10. We prove by induction that for OGr k, if a,E &*, then a, E *. If a,E Wz= T*, then aj dominates a,. Assume that, for 9 k, if a/c &“, then aj Then a,E &* by Lemma 9 y the induction hypothesis, ai E a/E lf-+$TI. WL =(I(aj,)f’l l-4$*)*(where aj, is as ?n the Y the Lemma abcve, aj, ~O(aj aj,E Otadl-
K. B. Reid / Majoriry tournamenrs and voting
I8
Since a,~ (I(u,,)~ cV,*)* and aj dominates a,, aj E (I(Uj,)fI K*)*== tion our claim follows. Thus, whenever a,~ I#$*, / &* 1~2. Since 1 q the sophisticated decision. The result follows.
1. By in&x= I, a, is not
We have seen that if a majority tourna both the sincere and sophisticated decisio check that if Tis the cyclic triple, then no an alternative which is both the sincere a a bit more effort, one can see that for e which T* is not the cyclic triple there is an agenda which contains an alternative which is both ihe sincere and sophisticated decision. For what tournaments T is there an agenda which contains an alternative which !s both the sincere and sophisticated decision? the cyclic triple, then it can be seen that there is no such agenda. .~ny sther instances there is such an agenda. For suppose that X is tournament of T which is as large as possible and that CIis the transmitter of X. Let the 2p uence ops op+ . . . . aI describe a Hamiltonian path in T- X. In the event ‘chat Ik’and a, can be found so that a dominates ap and if {x1,. . . ,xq} = X - (a), where .q dominates Xj if and only if 1 s i
The author acknowledges some partial research support from Contract 86LBR(39)-039-05 from the Louisiana Education Quality Support Fund.
ces J.S. Banks, Sophisticated voting outcomes and agenda control, Social Choice and Welfare i (1985) 295-306. P. Erdtis and L. Moser, On the representation of directed graphs as unions of orderings, Publ. Math. Inst. Hung. Acad. Sci. 9 (1964) 125-132. D.S. McGarvey, A theorem on the construction of voting paradoxes, Econometrica 21 (1953) 608-610. N.W. Miller, Graph-theoretical approac es to the theory of voting, Am. J. Polit. Sci. 21 (1977) 769-803. N.R. apilier, A new solution set for t urnaments snd majority voting: further graph-theoretical proaches to the theory of voting, Am. J. Polit. Sci. 24 (1980) 68-96. N. rofman and S. d, Cycle avoiding trajectories in majority voting tournaments, Paper e 1986 Annual eeting of the Public Choice Society, Baltimore.
J.W. Moon, Topics
orl
Tourn
1oice and ~~‘et~are 3 (IW6) 27 I-291. tween two algorirhms for decisions via soptljst~c~te~ rmjority an agenda, Discrete Agpl. Math. (199 K.A. Shepde and Weingast, Uncocered sets and sophirticatcd agenda instituti s. Am. J. Polit. Scl. 2X (1984) 49-7-I.
. Stearns,
The voting
probiev,
Am. hhth.
Iz%
aotiry
Lotinp
ktith
outComk3 iiith irnplic tlioik\ tar
66 (I9W) 761-?63.
K.B. REID California State University, Sun Marcos, CA 92069, USA and Louisiann Stute Cfr~i~yrsitj~, Peon Rouge, LA 70803, USA Communicated
by M.F. Janowitz
Received 14 June 1989 Revised 27 July 1990
In this work we give a careful mathematical treatment of two majority voting procedures in which a voting agenda is utilized: sincere voting and sophisticated voting. We elaborate and espand on work by Banks (1985), Milker (1977,1980), Miller, Grofman and Feld (1986), and Shepsle and Weingast (1984). W’e discuss algorithms to obtain these two decisions under such procedures, the location of these two decisions in the underlying majority tournament, the relative locations of these two decisions in the voting agenda, and the possibility of an alternative which is both of these decisions. Key words: Tournaments; division tree; sophisticated
voting agenda; amendment proced w-e; sincere voting: decision voting; Banks points; kings.
tree;
Suppose that each voter in a finite, nonempty set V of voters linearly orders, according to their preferences, a finite, nonempty set of alternatives 4. A common group aggregration rule is majority preference, in which an alternative is preferred over another alternative if and only if the former alternative is preferred to the latter alterantive by a majority of the voters. That is, given several transitive tournaments on the same vertex set ,4 (one for each voter), the majority preference rule gives rise t.o the majority digraph, an orieneed graph with vortex set A in which for all xfy in A, x dominates y if and only if x dominates y in a majority of the transitive tournaments. If no ties result (e.g. if IV1 is odd), then a tournament results (e.g. see Fig. I). McGarvey (1953) proved that every oriented graph with n vertices and q arcs is the majority digraph of n alternatives by at most 2y voters. NOX that Zq L FT(PI- I ). cGarvey’s theorem by showing that Stearns (1959) improved tively, pa+ 1) voters are needed if n is even (respectively, odd). proved that, for large n, the number of voters nee If m(n) denotes the least integer m such that at most m n vertices (n > I ), t era 01654896/91/$3.50
%I 1991 -Elsevkr
ScierLe Publishers
B.V. (North-Holland)
K.B. Rerd / Majority
tournaments
and votiq
a
Voters’ preferences Voter Voter Voter Voter
Fig. I. Majority
2
b,G,a,c,e
3
c,e,b,a,d
4 5
d,a,c,e,b e,c,d, t?,a
tournament.
there exist constants cl and c2 such that c&log
n < m(n) < c&log
n.
These results are discussed by Moon (1968). In the remainder of this paper, it is assumed that the majority digraph is a tournament . The issue of which alternatives should be considered most desirable by the group in majority voting procedures has been a major concern to voting theorists (e.g. see the references to the papers listed at the end of this paper). The majority tournament may contain (directed) cycles which may be seen as inconsistencies. Indeed, the majority tournament may even e strongly connected (e.g. see Fig. l), and hence every vertex is in cycles of every possible length (see Moon, 1968, p. 6). Work has focused on descriptions of sets of vertices, called solution sets, in which the ultimate decision(s) must lie when several important majority voting processes lead to a decision in the initial strong component of the majority tournament. Later Miller (1980) suggested a r- ‘inement, carAzdthe uncovered set, as the solution set for one voting process. Authors writing on tournaments have called elements of the uncovered set, kings. More recently, Banks (1985) and Miller, Crofman and Feld (1986) have suggested a further refinement which has been dubbed the Banks set. We treat these issues in this work. In Section 2 we discuss the combinatorics of a majority voting procedure, called sincere voting, under an amendment procedure. In Section 3 we discuss a strategic roach to majority voting under an amendment procedure which is called ting; we include a treatment of the algorithm due to Shepsle and
e sincere and sophisticated
decision.
K. B. Reid / Majoriry
totrmarnerits and voting
3
Before continuing we recall some of the basic terminology that will be employed subsequently. Let T denote a (finite) tournament. For vertices x and y in T, write xRy if and only if there are (directed) pat s in T from x to Y and from y to x. It is easy to see that R is an equivalence relation on V(T), the vertex set of T, and that if there is more than one equivalence class, then the equivalence classes Vt, . . . , Vk can be ordered so that each vertex in !$ dominates each vertex in 5 whenever 1 s i
2. Sincere voting Under amendment procedure, some order, say (al, a2, . . . , a,), of the m alter. . niklves 1s gtven. In the first vote alternatives ai and a2 are paired for a majority vote, the defeated alternative being eliminated from further consideration; at the ith vote, 25&m1, the alternative surviving the (i- 1)th vote is paired with a,+1 for a majority vote, the defeated alternative being eliminated from further consideration. The alternative surviving the (in - 1)th vote is called the sincere decision. That is, given the voting order (ai, a2, . . . , a,,,) and a majority tournament T with vertex set {ai, Q2,. . . , a,}, the sincere order (b,, b2, . . . , b,) is defined inductively by b,=at and, for 2(&m, bi
=
“’ bi-1,
bi_ 1 in T, if bi_ ,dominates ai in T.
if ai dominates
3.he alternative b, is the sincere decision. The m - 1 votes that occur in the amendment procedure correspond to m - 1 arcs in the majority tournament T. These m- l arcs induce a subdigraph of T in which exactly one vertex, the decision, has indegree 0 and all other vertices, which were eliminated in some vote, have indegree 0. As the voting order causes a se elimination of vertices, this digraph is acyclic. Thus, these yb/r - E arcs in oriented spanning subtree of T, rooted at the decision, called the decision tree of ore can be said about the structure of th ed) tree such that the re vertices results i
X is an orientation of the edges of X so that either the spine becomes a directed pat and all other arcs are directed away from the spine or a longest path in X (consisting of the spine and two ‘end’ ed:?es) becomes a directe directed away from that path. Fig. 2 illustrates these t caterpillar. Note that a special orientation of a cater rooted at the unique initial vertex of a longest directed path.
Let ‘9’be a toumament and let X be a spanning subdigraph of T. X is the decision tree of Treilative to some voting ordw if and only if X is a special orient&on of CIcaterpillar.
roof. Our proof is designed to yield considerably more information. Suppose that Qi,,ai.,...,U;,, 1 sil
a
am
a $4
aij+l
(c 1 i,=2 and
"j
am 3
:
000
*l
I~=
m
ai. JK
000
(d) 192 and ts=m
Fig. 3.
g4 = &,, 1 S&S, where s is as in Fig. 3(a). First 8, = al, so take i, = 1. Assume that i
l
l
K. B. Reid / Majority tournaments
6
and voting
e proof enables us to enumerate the voting orders for which a fixed labelled X is the decision tree. If T has a transmitter x and X consists of x and all the HZ- 1 arcs fro ust be s to the other nr - 1 vertices (i.e. a special orientation of K,,,,l_ 1), t either first or second in the voting order if X is to be the decision tree, and the other ahernatives may occur in any order. So, there are 2[(m - I)!] voting orders yielding X as the decision tree. .
f X is as in Fig> 3(b)s then X is not only the decision tree relative to voting order (ai, a2, . . . q g&, but it is also the decision tree relative to each of the (i,6 - 2)! = (i2 - ir - l)! voting orders
where d,, d2, . . . 9di,c_ z is any (i2 - 2) permutation of a2, . . . , aj,_ 1. Moreover, for 2~j<~, if ij+i-$>I and ei ,..., ei ,_ i/ , is any of the (4, 1 - $ - l)! permutations of ail+], . ..) a&+,_ 1, then X is also the decision tree relative to the voting order
(q,Q2,~-.,Q~,el,...,e~+,_i-],Qj J
/+
,,...,
a,).
Similarly, any of the (m-Q! permutations of a.,s+1,. . . , a,,, may occupy the final m - i, positions in a voting order that has X as the decision tree. Consequently, X is the decision tree for 2i
(ij-ij_,
- I)! (m - is)! voting orders.
j=2
Since O! = 1, this formula holds also if X is any of the decision trees of the form given by Fig. 3(a), (c), or (d). If T is a tournament considered as a majority tournament on alternatives 442, =**,a*, define the (sincere) decision set D(T), as follows: D(T) = {a ) there is some voting order so that u is the sincere decision relative to T and that order}. Theorem 1 above provides a new proof of results by Miller (1977, Propositions
I
FQ~ any tournament T, D(T)=
1
V(P).
ED(T); say a is the sincere decision under voting order (al, a2, . . . , a,). g decision tree. By Theorem 1, X is a spanning directed at is, every vertex in hence in T, is reachable via
h’.B. Reid / Majority
tournaments and voting
d:;ected path in from a. Since no vertex in * is reachable from a vertex outside of T*, aE V(T*). On the other hand, let XE V(T*’ ). Then x is the origin of a spanning path in ?“. But a spanning path is a special orientation of a spanning caterpillar rooted at x. By Theorem 1, this spanning path is the decision tree of T relative to some voting order. And by a remark preceding Theorem I the root of this decision tree is the sincere decision. Thus, SE D(iT’). Hence V(T*) C_D(T). The result follows. c3 a
Note that any alternative x in the initial component of T may be the decision under many different voting orders. Any of the many spanning caterpillars with special orientations rooted at x gives rise to such a voting order according to Theorem 1. The spanning path given in the previous proof is just one such, and was the one used by Miller (1977, Proposition 2, p. 781). Of course, if a tournanznt contains a transmitter x, then x is the unique vertex in T*. Consequently, an immediate corollary follows. orollary 3. If x is a transmitter of a majority tournament T, then x is the sincere decision regardless of the voting order. With HI alternatives the sincere voting process under amendment procedure can also be described via a balanced, binary rooted tree on 2”‘- 1 vertices, cakd the division tree. Vertices in the division tree are labelled by subsequences of the voting order (aI, a2, . . . , a,,,) as follows: the root at level 0 is labelled (al, a2, . . . , a,,,); for 05 js m - 2, a vertex at level j which is labelled by a subsequence (of the voting order) of length m -j, say (6,, b2, ..-, b,,_J, dominates two vertices at level j + 1, one labelled (6,, bJ, . . . , btpl_J and one labelled (b2, b3, . . *, b,,, +). The division tree depends solely on the voting order; it is independent of any majority tournament. See Fig. 4(a) for an example with m = 4 (ignore the underlined alternatives). Note that several distinct vertices in the division tree may share a common label. Given a majority tournament T and a voting order (a,, a2, . . . , a,), the sincere amendment procedure yields a path in the division tree from the root to an endvertex labelled with the sincere decision. For example, if T is as in Fig. 4(b), then the (sincere) amendment procedure yields the following path in the division tree in Fig. 4(a): (a, b, G d), (a, c, d), (a, d), Cd)A census of the vertices in the division tree is described next. Let (al,a2, . . . . a,,,) be the voting order. For each k, 2 5 k< m, and for each j, 1(: j 5 k - 1, level k - 2 of the division tree contains
1,
ifj=l
2-i-2,
otherwise
K’.d. Reid / Majorit>* iournatttents and voring
Level
a
C
d
0
d
(b) Fig. 4.
vertices lube/led (a_,, a,, ak + I) . . . , u),~). Moreover, for each k, 2 I kr m, every vertex at !evel k - 2 is labelled (aj, a,+ a, + ,, . . . , a,,,) for some j, 15 j zzk - 1.
The final statement follows from the first claim, since 1 + ~~~~ 2j-’ =2”-‘. Induct on kz2. If k= 2, the root of the division tree, the only vertex at level 0, is labelled (a, a2, . . . , a,,,), as desired. Assume that 25 k
vertices labelled (q,, uk + ,, ak +2, . . . , a,,,). Suppose j< k. By the induction hypothesis, level k - 2 contains if j=l
1, 2j~ 2,
if 2sj
vertices labelled each of which dominates two vertices at level k - 1, one labelled (aj, ak + I, ali+z, ,.. ,a,,,), and one labelled (ali, ak + 1, . . . , a,,,). So there are ifj=l 1, (Uj,
ah,
2J-2 9
uk
+
,,
.
if2sjik-l
.
.
,
a,,,),
vertices at level R:- 1 labelled (a,, ak + ], ak +2, . . , a,,,). NOW suppose that j z=k. As remarked at the outset, every vertex at level k - 2 is labelled (a,, ak, ak +], . . . , a,,,) for some i, 1 Si5k-- i. And as just seen, each such vertex dominates exactly one vertex labelled (ak, ak + ], . . . , a,,,). As there are 2” ’ vertices at level k - 2, the result follows for j = k. By induction the result follows. II l
3.
The amendment procedure discussed above involves sincere voting decisions in contrast to so isticated voting decisions under the amendment procedure, to be discussed next. i II brief, decisions are anticipated at higher levels of the division tree in order to anticipate decisions at lower levels, until an anticipated decision at the first vote is made. A formal description follows. Assume that there are III alternatives. Recall that each vertex of the division tree at level rn - 2 is labelled with an ordered pair of alternatives; the anticipated decision at each vertex of level 177 - 2 is defined to be the majority choice between those two alternatives (as given by a fixed majority tournament T). Inductively, for Osj< m - 2 the anticipated decision at each vertex v of level j is defined to be the majority choice between the two alternatives which are the anticipated decisions at the two vertices at level j+ 1 which are dominated by v in the division tree. The wphisticated voting decision (under the amendment procedure) is the anticipated decision of the root at level 0. For exampie, given the majority tournament in Fig. 4(b), the anticipated decisions are underlined in the division tree in Fig. 4(a). So, in the sophisticated voting decision there is alternative 6. Note that the sincere voting decision in that example is alternative d. a vertex x of the division tree is labelled (aj,ak,ak+], . . ..a.,,), then the anticipated decision at x depends only on T[ai, ak, ak + l, . . . , a,,,]. Consequently, the anticipated decision at x is the sophisticated decision for T[ai, a,+,ak+ 1, . . . , a,,,] relative to the agenda (aj, a& ak+ 19. . . , a,,,). 3. If
mark
Before describing an easier procedure for finding the sophisticated decision we give a preliminary technical result. Let Dj denote all of the anticipated decisions at level j, 0 5 js 777- 1, in the division tree, relative to some fixed majority tournament T and some agenda (voting order). For example, D,,,_ I = T and Do contains only the sophisticated decision. ma 5. Let Z c T.
I”’
for some
0 5 j<
171-
I,
sophisticated decision is irl Z*.
Induct on j. If j = @,the result follows fr only the sophisticated decision. Let 0 < k ISr71- B. all values of j< k. Suppose that .
~j
c
Z and pi f7 2 *Z D8,then thP
h’.B. Reid 1 Ilia_ioi-i’tytsrmaments
10
and voting
the definition of an anticipated decision, so Dk__1 5 2. Let u E 62, f3 Z*, say a is the anticipated decision at vertex v at level k in the division tree. Vertex v is dominated by some vertex t at level k - 1 in the division tree; and t dominates exactly one other vertex in level k in the division tree, say u. Let b be the anticipated Note that a, b E Z and CIE Z*. If a = b or a dominates b in T, then a is the anticipated decision at t so a E Dk_ 1t3 Z*. If b dominates a in T, then b must lie in Z*, so b is the anticipated decision at t, i.e. Dk _ l n Z* # 0. In any case, Dk _ l C_Z and Dk _ 1n Z* # 0. By the induction hypothesis, the sophisticated decision is in Z*. By induction the result follows. 0 As an immediate corollary of Lemma 5, we can characterzie transmitters. oroh-y 6. Alternative a E T is a transmitter if and only if a is the sophisticated decision relative to every agenda.
. Suppose that a is a transmitter in T, so that T*= (a]. Regardless of the agenda, a is the label of some vertex at levei m - i in the resulting division tree, so a is the anticipated decision at some vertex at level m - 2 in the division tree. That is, if Z = T, D,, _ 2 c Z and D,,I_2 n Z*#0. By Lemma 5, the sophisticated decision is in Z*= T*= {a}. On the other hand, suppose that a is not a transmitter, so that I(a)#O. Hence (I(a))*#O, say b E (I(a))*. Consider any agenda and the resulting division tree in which CIis the last alternative. By Proposition 4, a vertex at level m - 2 of the division tree is labelled (x, a), for some XE:T. Of course, one of these vertices is labelled (b,a). Thus, the anticipated decision at each vertex at level m - 2 is in I(a) W {a}. Moreover, b is the anticipated decision at the vertex labelled (b, a). Let Z = I(a) U {a}, so that Z* = (I(a))*. Since Dnl_2 c Z and D,71_2 n Z*#0, Lemma 5 implies that the sophisticated decision is in Z* = (I(a))*. That is, a is not the sophisticated decision relative to every agenda. This completes the proof. 0 Perhaps the easiest way to determine the sophisticated decision is to use the algorithm given in the ‘Equivalence Theorem’ of Shepsls and Weingast (1984). Let T be a majority tournament with vertex set A and let (a,, a2, . . . , a,,,) be ir fixed agenda. Define the sophisticated agenda (zl, ~2, . . . , z,,,) as follows: zlll = a,, and for 1
r’ Qj,
Zj=
I
Zj+l9
tt1
if a, E n
QZ;),
i=j+l
otherwise. (Zj, z29 *-* 9 z,,,) is the sophisticated srgen-
K. B. Reid / Majority
da for a fixedtournament decision.
tournaments and voting
II
T and agenda (al, a2, . . , a,,,), then q is the sophisticated l
Induct on ypz.The case m = 1 is trivially true. Assume that in > 1 and that t result holds for all agendas of length less than m. Form the sophisticated agenda (Zr&, .**,z,). In the division tree, the root is labelled (aI, a2, . . . , a,,) 3 .rd dominates two vertices, one labelled (al, a3, . . . , a,) (to be considered as an agenda for T- a2) and one labelled (az, a3, . . . , a,,) (to be considered as an agenda for T--a{). The sophisticated agenda relative to T- al and agenda (a2, a3, . . . , a,) is (12~23,..* J,~). And, the sophisticated agenda relative to T- a2 and agenda (al,+, . ..) a,,) is (b,, ~3, z4, . . , z,), where .
l
n I(Zi), n1
b, =
al,
if al E
i=3 239
otherwise.
By the induction hypothesis, the anticipated decision at (al, a3, . . . , a,) is bl, and the anticipated decision at (al, a3, . . . , a,) is z2. In order to determine the sophisticated decision, bl is compared to z2. Case I: b1 = al. Note that z2 #al (more generally, by the definition of the sophisticated agenda, for 1 s i
z1 is the sophisticated
decision.
Case 2: bl = z3. If 22 = z3, then ~,2(=13) is the sophisticated decision. And, by definition of bl, al $ n;! 3 I(Zi) = n;12 I(Zi), so z1 = z2 by the definition of q. So, if z2 = z3, then q(=z2 =z3) is the sophisticated decision. If t2#z3, then z2 =a2 and a2 E nr3 I(Zi) by definition of z;. So, z2(= a2) dominates z3 and z2(= a,) is the sophisti-
cated decision. Now, by the definition of bl, al $ n:,l, 3 I(zi), so a1 $ n;=, I(q); this implies that q #aI by the definition of q, so z1= z2. That is, q is the sophisticated decision. In either case q is the sophisticated decision, so b;: induction the result follows. 0 le. Given the majority tournament in Fig. 4(b) and the agenda (a, 6, c, d), first entry of the sophisticated agenda (b, b, c, d) is b, the sop The proof of the
revious theorem,
like t
K.B. Reid / Majorirjf
12
forrrrrarnenCs and voting
valves only one level beyond the root in the division tree. The proof given by Shepsle and Weingast (1984) involved two levels beyond the root. The Shepsle-Weingast Theorem can be used to give short proofs of two observations made by Miller (1977, pp. 794-795). Miller’s explanations are incomplete, since they are based on an unsubstantiated algorithm (which we will treat below). We state these observations as a single proposition. a,,,) be a fixed agenda and !et T be a majority tournament on A. If ai is the sophisticated decision and aj is the sincere decision, then either i = j or i < j and ai dominates aj in T. roof. Assume that i# j. Let (q, . . . , z,J denote the sophisticated agenda. By the Shepsle-Weingast Theorem, z1= ai, 22= ai, . . . , zi = ai. If j= m, then i< j, z,,~= anI and q E I(z,?J by the definition of zi. So, assume that j< m. As aj is the sincere decision, aj dominates aj+ 1,. . . , a,,,. But then zj = aj, by the definition of +. Since each of q, zz, , z, is ai and ai f aj, i < j. SO zi E I(Zj), by the definition of zi In any case, i< j and ai dominates aj in T. q l
l
l
l
A more fundamental proof of Proposition 8 can be given by showing that all of the anticipated decisions at level j- 2 of the division tree are contained in {aI, 02, . . . , aj} fl (I(aj) U {aj}). But ai is the anticipated decision at the root, SO it must be the anticipated decision at least once at level j- 2. So, ai is in that intersection.
iller’s algorithm Miller (1977, pp. 789-791; and 1980, pp. 81-82) alluded to an algorithm for determining the sophisticated decision. That process was given only for the first two steps, no explicit proof was given for its termination but, nevertheless, the validity of the algorithm was assumed in his explanation of his Propositions 11 and 12 in his 1977 paper. We have proved those two results as Proposition 8 above. Our aim here is to describe the algorithm precisely for the first time and to prove explicity that the algorithm terminates v;iih the sophisticated decision. Let (a,,a2, . . . . a,,,) be a fixed agenda (voting order), and let T denote a fixed majority tournament on A. For Or is rn, define sets of vertices (alternatives) w as follows: IQ 3’ and b. is the last vertex of T* in the agenda; for ir0, having defined ,..., 4, if W,=0, then kI$+t = 0 and bi is not defined, and if I+$#0, then b, is the last vertex of IJ$* in the agenda and P$+ , = I(bi) n l4$*. The proof that this process terminates with the sophisticated decision is rather involved, so first we give a preliminary result which shows that the process does terinate.
K.B.
Reid / Majority
tournantents
and voting
(i) ;,cW* . and (ii) For O
13
and b, dot?linates
(i) Suppose that & # 0. Since bi E wi*, but bi $ I(bi), we have
(ii) For 0 QC
i, by (i) we have
bi E w* s q*+ 1 = I(bj) f7 q*, SObi dominates !Ij and bi E WjT But, bj is the last vertex of wj* in the agenda, precedes bj . (i)), a straightforward induction yields:
SO
b;
for 05&m. In particular, l&J 5 lP&ol-m=m-m=O, so W,,,=0. (iv) Since JVk#0, w,*#O. But IVk, 1 = I(b,& n WA?= 0, so bk is a transmitter IV”, i.e. Wc={bk}. q 0. If k-maxii!
u/;#5;,
in
A. I &L” soghisricated decision. bAcn _hl;gj
roof. If k = 0, then IV, = I(b,) n I#$,*=I(bo) fl T* = 0, so b. is a transmitter in T. By Corollary 6, b. is the sophisticated decision. So, in what follows assume that k>O. We recall the notation of Lemma 5 where Di denotes the set of all anticipated decisions at level i in the division tree, OS ic nz - 1. The alternatives bk, bk _ ], . . . , bo correspond to alternatives ajA,aj, ,, . . . , aj,,, respectively, in the agenda, where j, < j,_ 1< . . .
Wi*,,#0.
Before verifying (a) and (b), we show how the proof is completed. Let Z= I4$ U {bk_ $ Since wk C_I(bk_ 1) (definition of IV,+),Z*= I+‘:. Sd, 5y (a) and (b) ut, then by Lemma 5, with i=k-1 and /i=l, Bji, ,_2~Z and the sophisticated decision is in Z*= IV’;. owever, by Lemma 9(iv), So, bk is the sophisticated decision. We now verify (a) and (b) by induction on i, 0 L is k - I. irst we 0-m the case i = 0. By Proposition 4, a typical vertex t at level j, - 2 in the division tree is label-
h’. B. Reid 1 Majority
1-l
townaments
and voting
1, and t dominates exact a,,,), for some p, idp5j0tices in level j,, - I, say II and v labelled (a,, ai0+1, . . , a,,,) and (q,,, ajo+i, . . . , a,,,), respectively. Since aj,, is the last vertex in the -agenda which is in IKf trans..ritter in T[aj,,,a,,,+1,..*, a,,,]. So by Remark 3 in Section 3 and C is the anticipated decision at v. If, in addition, aj,, dominates a, in T, t anticipated decision at t, regardless of the decision at u. If a, domi then a, E Wo*since aj(,E WO* = T*. Since none of aj,,+1,. . . , a,,, is in T* of a& a!, is a transmitter in T[a,, aj(,+1, . . . , a,,,]. SO, again by Rema ollary 6, a, is the anticipated decision at II, and hence at t. In this a,E I(aj,,) n Wo*= WI. So, recalling that aj,,= &, we see that all antici sions at level j,-- 2 are in W, U {b,}. That is, (a) follows for i= 0. F W,;*#0 by the definition of Wk) and bh = aj,, exists in WI:. If verte chosen so that aP=aj,, (there is such a t by Proposition 4), then since is the anticipated decision at U. But ap = aj,,= bll dominates aj,)= bo (by L in T, SO aj,, = ap is the anticipated decision at t. Thus, some anticipated decision at level j, - 2 is in WI,*.That is, (b) follows for i= 0. Assume that O
led tap,q,,,a,,,+1,
. . . y
l
l
l
l
Next ‘vi:’+ow that the anticipated decision at u is in H$U {bi_ l}. This is much easier than the case just considered at v. By part (a) of the induction hypothesis, the anticipated decisions at all the vertices at level j;_ 1- ? of the divison tree which can be reach from u (i.e. ‘descendents of u’) are in K$U Ibi_ 1). So, the anticipated at u, denoted a, u ibi- I>* Now, the anticipated decision at t depends on a, the antrcipated decision at M,
K. B. Reid 1 Mujority tournaments and voting
1s
an aj,, the anticipated decision at U. rite U {bi- r) as the disjoint union (K - H$“) U UI;”W {!I;_ ]}. The anticipated decision at t is a,,, if either a = &_ I = And, if ae &*, then the an% I or clrE I+$- wi* since ticqated decision is either a, if a d ;*, or a,,, if a,, dominates a ip *. So, if a is the anticipated decision at I, then a E I(aJ,)f3 4’” = !4;+1. Otherwise, the anticipated decision at t is aj, e Since t was arbitrary, part (a) is established for the value i. To complete the proof (i.e. the induction step for part (b)) fix h, O
E
Wi*
=
The Shepsle and Weingast algorithm and the Miller algorithm are intimately linked, as we see next.
be an agenda, and let T be a majority tournament on A. Let (zl, z2, . . . , 2,) be the sophisticated agenda obtained by the Shepsle and Weingast algorithm. Let (bk, bk _ !, . . . , bO) be the sequence of distinct alternatives obtained by the Miller algorithm. The bi 3 define a sequence of indices where bk=aj,, bk_1 =ajk ,, . . . . b,=aj,,. Then I sj,< . . .
). Since The proof of this result is not given here, but it may be found i the proof is independent of Theorem 7 and I’heorem 10, the validity of each of the algorithms described above can be used to establish the validity of the other.
h’.B. Reid / Majority
16
tournarnen ts and voting
The previous results can be employed to describe the set of alternatives in a tournament T which are possible sophisticated decisions. As in Banks (1985), let S(T) denote such a set, i.e. S(T) = {a~ T ( a is the sophisticated voting decision relative to some agenda}. A Banks point x (see Miller et al., 1986) in a tournament T is a vertex of T such that x is the transmitter of some transitive subtournament IV, and no vertex in T dominates every vertex in IV. We will use B(T) to denote the set of Banks points in T. The language used by Miller et al. (1986) to describe points in B(T) is slightly different but entirely equivalent to this definition. a 12. Let M(T) = (y E T 1y is the transmitter of a maximal transitive subtournament).
Then B(T) = M(T).
Clearly, M(T) c B(T). Let x E B(T), say x is the transmitter of transitive subtournament W, and no vertex in Tdominates every vertex in IV. If Wis maximal, then x E iti( Otherwise, there is a maximal transitive subtournament W’ properly containing IV. Now x must be the transmitter in IV’, by the choice of IV. So XE M(T). Thus, B(T) CM(T), and the result follows. 0 l
Since every tournament T contains a maximum transitive subtournament, which is, of course, maximal, B(T) #0. For example, if T itself has a transmitter x, then B(T) = (x}. In general, B(T) c T*. A bit more can be said. Recall that a king of T is a vertex x in T so that for each vertex y#x in T there is a (directed) path from x to y of length 1 or 2. And, the outset of x, denoted O(x), is the set {YE T 1x dominates y}. 3. Let K(T) = (YE T 1y is a king of T).
Then B(T) c K(T).
Let x&(T). Of course, x can reach each vertex in O(x) in one step. If there is a z in I(x) such that x cannot reach z in two steps, then z dominates every vertex in O(x) U (x}* So, xt$ M(T), contrary to Lemma 12. So x&C(T). Thus, (T) c_K(T). ‘J This observation was also made by Banks (1985, p. 300), without the ‘kings’ terminology. anks (1985, p. 300) characterized S(T) as follows: a&s, 1985). S(T) = B(T). is the transmitter of a maximal transitive sub. . . , wk, where dominates ere k+k=m, order of
K.B.
Reid / Mujority
fourt~attrett~s and voting
17
Under agenda (u,, . . , u,, wI, w2, . . , wk)I, the so isticated agenda (z,, . . . , z,~,) is given by z1= . . . =z/=z/+~ = wi, and Z, = wJ for / 5 jl tn. Thus, by Theorem 7, x = w1E S(T). Consequently, Let aES(T), say a is the s e&ion via agenda (a,, a?, . . . , a,,,). In the sophisticated agenda (zi, . . . ,z,,,), q = a by Theorem 7. If q = z,,), then a is a transmitter, and, clearly, a E B(T). So, assume that q # zm. Let q =Tz;,,z,,, . . . , Zikbe the distinct alternatives among q, z2, . . . , z,,~, where 1 c i2< ...< iks rn.-Note that z& dominates zi,, whenever 15 j< e’s k, SO{Zi,,Zi2,. . . , zi,> induces a transitive subtournament, denoted with transmitter q =a. Suppose that some vertex y in T dominates every vertex in W. Note that y is not among z,,, . . . , ziA, so y is nc:t in {z*,z2, **9z,~}. But by the definition of the sophisticated agenda, since y dominates any subset of the zi’s, y must arise as a zi, a contradiction. So, aEB(T). Consequently, S(T) c B(T), and the proof is complete. q l
l
l
The previous results can be utilized to give a new proof of another observation of Miller (1977, p. 793). First a preliminary result is established. emma 15. If an alternative aj is unanimously preferred to another alternative a,, then O(ai)F O(aj).
roof (Miller, 1977, p. 793). If XE O(ai), then a majority of the voters prefer ai over X. But all the voters prefer aj to ai, SO, by the transitivity of voters’ preferences, the same majority prefer aj over X, i.e. XE O(aj). Since ai E O(aj) - O(ai), the result follows. El carem 16. No altemative is unanimously preferred to the sophisticated decision. roof. By Theorem 14 and Lemma 12, the sophisticated decision, denoted ai, is the transmitter of a maximal transitive subtournament X of T. If aj is unanimously preferred to ai, then by Lemma 15 aj dominates every vertex in X, contrary to the maximality of X. The result follows. q For completeness we give an alternaLive proof that avoids the Shepsle and Weingast Theorem 7 (used in the proof of Theorem 14) and instead utilizes Theorem 10. This is the approach attempted by Miller (1980, p. 793). Suppose that aj is an alternative which is unanimously preferred . Let the index k be as in Theorem 10. We prove by induction that for OGr k, if a,E &*, then a, E *. If a,E Wz= T*, then aj dominates a,. Assume that, for 9 k, if a/c &“, then aj Then a,E &* by Lemma 9 y the induction hypothesis, ai E a/E lf-+$TI. WL =(I(aj,)f’l l-4$*)*(where aj, is as ?n the Y the Lemma abcve, aj, ~O(aj aj,E Otadl-
K. B. Reid / Majoriry tournamenrs and voting
I8
Since a,~ (I(u,,)~ cV,*)* and aj dominates a,, aj E (I(Uj,)fI K*)*== tion our claim follows. Thus, whenever a,~ I#$*, / &* 1~2. Since 1 q the sophisticated decision. The result follows.
1. By in&x= I, a, is not
We have seen that if a majority tourna both the sincere and sophisticated decisio check that if Tis the cyclic triple, then no an alternative which is both the sincere a a bit more effort, one can see that for e which T* is not the cyclic triple there is an agenda which contains an alternative which is both ihe sincere and sophisticated decision. For what tournaments T is there an agenda which contains an alternative which !s both the sincere and sophisticated decision? the cyclic triple, then it can be seen that there is no such agenda. .~ny sther instances there is such an agenda. For suppose that X is tournament of T which is as large as possible and that CIis the transmitter of X. Let the 2p uence ops op+ . . . . aI describe a Hamiltonian path in T- X. In the event ‘chat Ik’and a, can be found so that a dominates ap and if {x1,. . . ,xq} = X - (a), where .q dominates Xj if and only if 1 s i
The author acknowledges some partial research support from Contract 86LBR(39)-039-05 from the Louisiana Education Quality Support Fund.
ces J.S. Banks, Sophisticated voting outcomes and agenda control, Social Choice and Welfare i (1985) 295-306. P. Erdtis and L. Moser, On the representation of directed graphs as unions of orderings, Publ. Math. Inst. Hung. Acad. Sci. 9 (1964) 125-132. D.S. McGarvey, A theorem on the construction of voting paradoxes, Econometrica 21 (1953) 608-610. N.W. Miller, Graph-theoretical approac es to the theory of voting, Am. J. Polit. Sci. 21 (1977) 769-803. N.R. apilier, A new solution set for t urnaments snd majority voting: further graph-theoretical proaches to the theory of voting, Am. J. Polit. Sci. 24 (1980) 68-96. N. rofman and S. d, Cycle avoiding trajectories in majority voting tournaments, Paper e 1986 Annual eeting of the Public Choice Society, Baltimore.
J.W. Moon, Topics
orl
Tourn
1oice and ~~‘et~are 3 (IW6) 27 I-291. tween two algorirhms for decisions via soptljst~c~te~ rmjority an agenda, Discrete Agpl. Math. (199 K.A. Shepde and Weingast, Uncocered sets and sophirticatcd agenda instituti s. Am. J. Polit. Scl. 2X (1984) 49-7-I.
. Stearns,
The voting
probiev,
Am. hhth.
Iz%
aotiry
Lotinp
ktith
outComk3 iiith irnplic tlioik\ tar
66 (I9W) 761-?63.