equitable agendas: agendas ensuring identical sincere and sophisticated voting decisions

Equitable agendas: agendas ensuring identical sincere and sophisticated voting decisionsAuthor(s): K.B. ReidSource: Social Choice and Welfare, Vol. 14, No. 3 (1997), pp. 363-377Published by: SpringerStable URL: http://www.jstor.org/stable/41106218 .

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Soc Choice Welfare (1997) 14: 363-377 ~ « - « • Social ~ « - Choice « •

máWÍBtt © Springer-Verlag 1997

Equitable agendas: agendas ensuring identical sincere and sophisticated voting decisions K.B. Reid

Department of Mathematics, California State University, San Marcos, San Marcos, CA 92096-0001, USA

Received: 31 August 1993/Accepted: 28 August 1995

Abstract Sophisticated voting under amendment procedure using majority rule usually results in a decision that is distinct from the decision obtained through sincere voting. In this article it is shown that the underlying majority tournament (determined by the voters' preferences) admits an agenda so that the sincere and sophisticated decisions are identical if and only if the initial strong component of the tournament is not a 3-cycle. As a result, most tournaments, in an asymptotic sense, admit an agenda so that the sincere and sophisticated decisions are identical.

1. Introduction

Suppose that each voter in a finite, nonempty set of voters linearly orders, according to their preferences, a set of m > 1 alternatives, where m > 1. Majority rule is a common group aggregation rule in which one alternative is preferred by the population of voters over another alternative if and only if the former alternative is preferred to the latter alternative by a majority of the voters. Group preferences obtained by majority rule may exhibit some bothersome inconsisten- cies. For example, if the preference orders for three voters for three alternatives a, by c are (a, fc, c), {by c, a' and (c, a, b), then the group preference is a over b, b over c, and c over a (instead of a over c, as would be consistent with a over b and b over c), all by a two to one vote. Thus, there is often no single alternative which can be identified as "most preferred" by the group using majority rule alone. This "paradox of voting" was apparently first pointed out in 1785 by the Marquis de Condorcet, an eighteenth-century French philosopher, economist, social scientist, and mathematician (see Black [2, pp. 159-180]). Indeed, "cycles" of the alternatives in the group preference order are rather common, so additional means are needed to facilitate the decision problem of voters employing majority rule on a set of alternatives.

Research supported by the U.S. Office of Naval Research Grant No. N000 14-92- J- 1400.



364 K. B. Reid

One direction of study in the voting theory literature focuses upon the notion of an agenda, an ordering of the alternatives from which pairwise comparisons are made. For example, the use of an agenda is central in voting under amendment procedure. It is generally understood that the agenda setter has considerable influence over the voting outcome. Indeed, it is only the specific procedure em- ployed and the structure of the collection of outcomes of all possible pairwise comparisons of the alternatives by the voters that logically constrain the agenda setter. There may be other external constraints placed on the agenda setter by the rules of the electorate, but such considerations are not treated here. The investiga- tion of the form that this structure can take has been a major theme in the work on majority voting via an agenda (see the articles in the list of references). The purpose of this work is, in some sense, to mull the strategic potential of what is known as sophisticated voting under amendment procedure which, via most agendas, produces a decision distinct from the sincere voting decision under amendment procedure. The main result shows that usually (depending on the structure of the outcomes of all pairwise comparisons of the alternatives by the voters) there is an agenda so that the decision reached by sincere voting is identical to the sophisticated voting decision. Thus, if the agenda setter chooses such an agenda, then there is no incentive for strategic behavior of the voters since sincere voting leads to the same decision as sophisticated voting.

Throughout this paper it is assumed that there are no ties in any pairwise comparison of alternatives by the voters. For example, if there are an odd number of voters, then no ties are possible via majority rule.

Majority rule, or majority voting, with no ties can be modeled using tournaments, or oriented complete graphs. Each voter linearly orders the alternatives according to their preferences. The vertices of the tournament represent the alternatives, and there is an arc from one vertex to another if a majority of the voters prefer the alternative representing the former vertex over the alternative representing the latter vertex. The resulting tournament is called a majority tournament. However, every (finite) tournament arises as a majority tournament for some set of voters (see [4, 7, 11, 21]), so there is no loss of generality in the present discussion in dropping the adjective "majority". Throughout this paper a tournament will be thought to have arisen from majority rule by some set of voters on the set of alternatives given by the vertices of the tournament.

Sincere voting under amendment procedure consists of a sequence of votes between various pairs of alternatives as determined by an agenda. The first vote is taken between the first two agenda alternatives; the winner, determined by majority rule, is paired for a second vote with the third agenda alternative, and so forth. The winner of the vote involving the last agenda alternative is known as the sincere decision. A formal description of this process is given in Sect. 4. Each vote taken in sincere voting corresponds to an arc in the underlying tournament Indeed, the collection of such arcs forms a rooted spanning tree of the tournament, rooted at the vertex representing the sincere decision. The possible rooted spanning trees that can possibly arise via sincere voting relative to some agenda have been character- ized, and a formula is known for the number of different agendas that give rise to the same rooted spanning tree (see [16]).

A strategic approach to the amendment procedure, known as sophisticated voting under amendment procedure, involves systematic anticipation of decisions by utilizing pairs of subagendas that might possibly remain at each stage of the voting. The precise description, given in Sect. 4, involves a labeled, balanced, binary, rooted tree, called the division tree, whose vertices are labeled by certain



Equitable agendas 365

attainable subsequences of the agenda. In brief, decisions (which are alternatives) are anticipated at higher levels of the division tree in order to anticipate decisions at lower levels, until an anticipated decision at the root is made. That final decision is known as the sophisticated decision.

If the tournament contains an alternative which is majority preferred by the voters over all other alternatives (in voting theory literature such an alternative is often called a Condorcet winner), then regardless of the agenda, the sincere decision and the sophisticated decision are identical (and equal to the Condorcet winner). However, usually the sincere decision and the sophisticated decision are distinct. For example, in the example above of the paradox of voting, none of the six possible different agendas yield the same alternative as both the sincere decision and the sophisticated decision.

Of course, a tournament and an agenda completely determine the sincere decision; this is also true for the sophisticated decision, which is the outcome of the sophisticated equilibrium in a game of successive elimination (e.g., see the discussion in Ordeshook and Palfrey [13]). So, the decision by the electorate to use amendment procedure implies that the tournament is the only logical constraint on a centralized agenda setter. Shepsle and Weingast [20] remarked that work by Miller [10] and McKelvey [8] suggested that voter sophistication limits the influence of a centralized agenda setter since the set of possible sophisticated voting decisions is a subset (proper in many cases) of the set of possible sincere voting decisions. There may be some concern about the outcome of the strategic behavior in sophisticated voting, particularly if the sophisticated decision is distinct from the sincere decision. However, if most tournaments allow for an agenda for which the sincere and sophisticated decisions are identical, then the decision to limit the centralized agenda setter to a choice from such agendas, if they exist, would help to dispel the impression of possible manipulation by strategic behavior. This would further limit the influence of the centralized agenda setter. Thus, a meaningful problem in voting theory, first posed by Reid [18], is to determine which tournaments admit an agenda so that the sincere decision and the sophisticated decisions are identical. This problem is answered herein. Indeed, a corollary of the main result is that most tournaments, in an asymptotic sense, admit such an agenda.

Section 2 contains some definitions and some notation regarding tournaments. Section 3 contains some results on tournaments that are basic to the paper. Section 4 contains formal descriptions of sincere and sophisticated voting under amendment procedure, some definitions and notation regarding voting, and several results on voting. The main result of the paper is posed in terms of tournaments and is established in Sect. 5. Section 6 contains some directions for further study and some open questions.

2. Definitions and notation

A tournament T is an orientation of a complete graph (i.e., T is a complete, irreflexive, asymmetric, binary relation on a set called its vertex set). The vertex (arc) set of a digraph D is denoted V(D) (A(D' respectively). For disjoint subsets R and S of V(T' R dominates S, denoted R =*> S, if for every vertex xe R and every vertex y e S, A(T) contains an arc from x to y. If R = {x} (X = {y}' then x => S (R => y) is used in place of {x} => S (R => {y}, respectively); if R = {x} and S = {y}, then x ->y is used in place of x => y. The out-set of a vertex x of T, denoted O(x) is the set of all vertices y of T such that x -> y. Vertex x is a transmitter



366 K. B. Reid

of T if O(x) = V{T) - {x}. If S e V(T' then T[S] denotes the subtournament of T induced by the vertices in S9 T - S denotes the subtournament T'V{T) - S]. In case S = {x}, T - x is used for T - {x}. In case there is no confusion, if there is an arc in A(T) from x to y, then x-*yis also used to denote that arc, and if W is a subtournament of T, then T - W is used for T - V{W) (i.e., T[K(T ) - V{Wy'). If FT is a proper subtournament of T and S is a set of vertices of T disjoint from V{W' then WkjS denotes the subtournament rCFXWJuS]; if S = {s}, then W'js is used in place of Wu{s}.

A rooted tree is a (directed) tree which contains a vertex, called the root, from which every other vertex is reachable via a (directed) path. A subdigraph D of a tournament T is a spanning subdigraph if K(D) = V(T). If P and ß are vertex disjoint paths in a tournament T such that the terminal vertex of P dominates the initial vertex of ß, then P -♦ ß denotes the path in T from the initial vertex of P to the terminal vertex of ß that uses all arcs of both P and ß. If P = {x} (ß = {y}), then x -♦ ß (P -> y) is used in place of {x} -► ß (P - {y}, respectively); if P = {x} and ß = {y}, then the use of x -♦ y is consistent with the meaning given earlier. If p = 0 (ß = 0), then P -> ß denotes ß (P, respectively). A tournament T is strongly connected (abbreviated strong) if for each pair of distinct vertices x, y of T there is a path in T from x to y (and hence, from y to x). A tournament T is transitive if for any three distinct vertices x, y, z of T, x -♦ y and y -♦ z imply x -► z. A transitive subtournament W of a tournament T is a maximal transitive subtournament if no transitive subtournament of T properly contains W.

The order of a (finite) digraph D is the integer | V(D) |. A tournament of order » is called an n-tournament. A (directed) cycle C of order k is called a k-cycle. The score of a vertex x in a tournament T is the integer |O(x)|. A transitive subtournament W of T is a maximum transitive subtournament if T contains no transitive subtournament whose order is larger than the order of W. Of course, a maximum transitive subtournament is also maximal, but the converse is generally not true.

All tournaments discussed in this paper have finite order. See Reid and Beineke [18] for a survey on tournaments.

3. Tournament preliminaries

It is well known (e.g., see Harary, et al. [5]) that there is a unique partition of the vertices of a tournament T into non-empty blocks Kt, Vl9 ... , Vk> for some k > 1, so that T[K3 is strong, 1 ̂ i < k, and K =^ Vi% whenever 1 ̂ i <j < k. Of course, if k = 1, then T itself is strong. The strong subtournaments r[F,], 1 <* i < K are called the strong components of T. In particular, r[F|] is called the initial strong component of T and is denoted T*, and F(T*) is denoted V* (after Miller [9]). In some of the social choice literature T * is called the top cycle of T (e.g., see the discussions in Miller [9] and Moulin [12]) or the dominant cycle of T (see Ordeshook and Schwartz [14]), and V* is called the top cycle set or the dominant cycle set.

Two classical results that are basic to the study of tournaments and their applications are the following:

Proposition 1. Every tournament contains a spanning path.

Proposition 2. Every vertex of a strong n-tournament, n ̂ 3, is contained in a cycle of length kyfor each k = 3,4, ... ,n.




Proposition 1 is a special case of a deep result due to Rédei [15]: every tournament has an odd number of spanning paths. Proposition 2 is Moon's [11] generalization of a result of Camion [3] who established the spanning cycle case, i.e., k = n. Some authors refer to spanning paths and spanning cycles as hamiltonian paths and hamiltonian cycles, respectively.

Note that the initial vertex of a spanning path P of a tournament T is the root of a rooted spanning tree of T, namely P. This leads to the following result, the proof of which is left to the reader:

Proposition 3. Let v be a vertex in a tournament T. The following statements are equivalent, (i) tier (ii) v is the root of a spanning path of T. (iii) v is the root of a rooted spanning tree of T.

Two additional easily established results which are included for reference are the following:

Proposition 4. If a tournament T contains the vertices v and w, where w e V* and v -► w, then veV*.

Let T be a tournament, let B(T ) denote the set {x: x e V(T ), x is the transmitter of some transitive subtournament W of T, and no vertex of T dominates every vertex of W), and let M(T) denote the set {jr. ye V(T)y y is the transmitter of a maximal transitive subtournament of T}.

Proposition 5. (Reid [16]). For any tournament T, B(T) = M(T).

4. Sincere and sophisticated voting

As indicated in the Introduction, every m-tournament T can be thought of as the majority tournament by some set of voters on a set of m alternates given by V(T ). Let the sequence u = (aua2y ... ,am) be an agenda of the m alternatives in V{T). Sincere voting, as described informally in the Introduction, gives rise to a (usually new) sequence as follows. The sincere sequence b = {bl9b2, -,bm) is defined inductively by fri = ax and, for 2 ̂ i <, m,

¿ f 0< if o, -> 6,-1 in 7, 1 (Vi if frf-i-^o, in 7'

The alternative bm is the sincere voting decision under amendment procedure (abbreviated sincere decision). Note that b = a if and only if the reyersc order of a describes the order of the vertices encountered in a spanning path of T with initial vertex am and terminal vertex at.

Sincere voting can also be described using a labeled, balanced, binary, rooted tree on 2m - 1 vertices, called the division tree. All of the 2j vertices reachable from the root by paths, each with exactly/ arcs, make up level; of the tree, 0 < ; ^ m - 1. An agenda (aÌ9a29 ..,<O is fixed. Vertices in the division tree are labeled by subsequences of the voting agenda {aua2, ...,4m) as follows: the root at level 0 is labeled (aita2, ... ,aw); for 0 ̂ j f ̂ m - 2, a vertex at level ; which is labeled by a subsequence of the voting agenda of length m- ;, say (c!,c2,c3, ...9cm^j' dominates exactly two vertices at level ; + 1, one labeled {cx , c2, . . . , cm-j) and one



368 K. B. Reid

labeled {c2,c3, ... ,cm-¡). The division tree depends solely on the voting agenda; it is independent of any tournament. Note that several distinct vertices in the division tree may have a common label. A tournament T determines a unique path in the decision tree; the path starts at the root of the tree and terminates at an endvertex labeled with the sincere decision as follows: for 0 < ; < m - 2, if the path has been determined from the root to level; and if the vertex of the decision tree encountered at level; by the path is labeled {cuc2> ... ,cm_,), then the next arc of the path joins this vertex to the vertex labeled

(cucz, ...,cm-,), iic1^c2 in T or (c2,c3, ... ,cm-j) if c2 -+ct in T.

Thus, the last vertex on this path is labeled with the sincere decision. The sophisticated voting procedure is defined in terms of the division tree of

the agenda (aua2, ... ,am) and the m-tournament T with vertex set given by the m alternatives. Pick a vertex of the division tree at level m - 2. It is labled with an ordered pair of alternatives; if the last vote were to be taken between these two alternatives, then it is clear from the appropriate arc in T what the decision would be, so that decision is called the anticipated decision at that vertex. The anticipated decision at each vertex at level m - 2 of the division tree is determined this way. Inductively, for 0 <; < m - 2, the anticipated decision at each vertex v at level; is defined to be the majority choice between the two alternatives which are the anticipated decisions at the two vertices at level; + 1 which are dominated by v in the division tree (if these two anticipated decisions at level; 4- 1 happen to be equal, then the anticipated decision at v is also this common alternative). The sophisticated voting decision under amendment procedure (abbreviated sophisticated decision) is the anticipated decision of the root at level 0.

Banks [1], Jung [6], Miller [9, 10], Moulin [12], Reid [16, 17], and Shepsle and Weingast [20], among others, have studied the location of the sincere and sophisticated decisions in the underlying tournament, the relative positions of the sincere and sophisticated decisions in the agenda, and the result on the outcomes by some rearrangements of the agenda, among other things. Specific examples of sincere and sophisticated voting can also be found in those papers. One important existing result, the so-called Equivalence Theorem, gives a fairly easy way to determine the sophisticated decision without use of the division tree. Let T be an m-tournament whose vertex set is given by the alternatives in the agenda a = (ahû2, ••• ,am)- Recursively define the sophisticated sequence (relative to the agenda a) as the sequence z = (zliz2, ... ,zm) as follows:

zm = am

and for 1 < ; < m

_ f a, if aj -> Zi for all i, ; + 1 < i < m, Zj _ ~~

' Zj+i otherwise.

Note that z = a if and only if T is transitive and the order of the alternatives in a gives the order of the vertices encountered in the unique spanning path in T. The sophisticated sequence gives a particularly transparent way of determining the sophisticated decision as is seen in the next result.

Theorem 6 (Shepsle and Weingast [20]). // z is the sophisticated sequence for tournament T and agenda a, then zx is the sophisticated decision.




Alternate proofs of the Equivalence Theorem can be found in Moulin [12] and in Reid [16, 17].

Another well-known fact is that the sincere and sophisticated decisions must lie in V* relative to any agenda. This can be made more precise. If a is an agenda of the alternatives in V(T) for some tournament T, let a* denote the subsequence of a involving only V * (i.e., a* is the restriction of a to F *). For example, if T is strong, then a* = a.

Proposition 7. Let a = (û1,a2> ... ,am)be an agenda of the alternatives in V(T)for some m-tournament T. Then the sincere (sophisticated) voting decision under amendment procedure relative to T and a is the sincere {sophisticated) voting decision under amendment procedure relative to T* and a*.

Proof. This is clear for m ̂ 3, so assume that m > 4. Let a* = (a*,d£, ... , a?), so that for some indices 1 < it < i2 < •• < ik < m, a* = ûî,, ö* = 0¿2> •• ,<** = <V Let b* = (b* , b* , ...,bt) denote the sincere sequence relative to T * and a*. Recall that fcf = a? .

Then the sincere sequence b = (bi,b2, • .,fcw) relative to T and a satisfies bij = bjy 1 <>j <>k. This can be seen by induction on;. First treat the case ; = 1. Vertex a¡t = a* belongs to V * and no alternative prior to aix in a, if any, is in V *, so bix = air This implies that bix = a*. As recalled above, a* = b*, so bit = b* as required. For the induction step, suppose that j > 1 and that the equality holds for values less than;. In particular, assume that bijv = b*-i. Now, each entry of a following aij_i = a*_ x and preceding atj = a*9 if any, is not in K*, but bijx = fc*_ x is in V*. So, each entry of b following fci^I and preceding bij9 if any, must be equal to bij- 1 = b*- 1 . So, by the definition of b, the vote to determine bi} is between btj_ t and aip i.e. between b*-i and a*. But, by the definition of b*, alternative b* is the notation for the majority choice between fc*_ ! and a*. Consequently bij = b*, and the induction step follows. By induction b¡. = b*, 1 <j < k.

In particular, bik = bjf. Since bik = b* is in V* and no entry of a following aik, if any, is in F*, all entries of b following bik, if any, are bik = bf. In particular, bm = bik = b* . That is, the sincere decision relative to T and a is the sincere decision relative to T * and a*.

To complete the proof, let z* = (z*, ... ,z?) denote the sophisticated sequence relative to T * and a*. The Equivalence Theorem (i.e., Theorem 6) is used repeat- edly. Recall that z? = a* . The sophisticated sequence z = (zx , z2 , . . • , zm) relative to T and a satisfies zfjk-> = z£_,, 0 <; < k - 1. This can be seen by induction on ;. Treat the case; = 0 first. Vertex aik = a* is in F*, but no vertex of a following alfc, if any, is in K*, so aik dominates any entry of a following aik, i.e. by the definition of z, zik = aik. This implies that zik = a*. But, as recalled above, a* = z*, so zifc = z?, as required. For the induction step, suppose that ; > 0 and that the equality holds for values less than ;. In particular, assume that zik_J+l = z*^J+i. Any entries of a following ai~j = a¡k^J and preceding ak-j+ x = a¿w+1 , if any, are not in V(T *), but zik.J+l = 2t-J+1 is in V*f so by the definition of z all entries of z following z//k . and preceding zÍJk _J4l, if any, are equal to zijk_J+l . Two cases are treated: either aijk_, 'does not dominate all of the vertices z¡k , 2¿á_>t2, ... ,zíjt, or else aífc_> dominates all of those vertices. In the former case, tne previous remark and the use of the definition of z lead to zik_j = zik_J+l. As at-j = aik_J and, by the induction hypothesis zi-j+i = Zit-j+t, z*-j+2 = z¡k_^2i ... ,zf = zIfc, this case also means that a*_, does not dominate all of z?_i+1,z*_7+ 2, ... , z?, so by the definition of z*,zi_j = zf_j+1. Combining these observations yields zik_J = ziàj+1 = zí-j^ i = z*_7, or zifc-J = z?_¿



370 K. B. Reid

in this case. To complete the induction step, suppose that aik¡ dominates all of z«à_J+I, zià-,+2> ••• >z¡k* Since a¡k_J = at-j is in V* and all entries, if any, of a following aikj which are distinct from <*¿a_>+I,flia_i+a, ••• ,ûjà are not in V*, aik_J dominates all such entries. So, in this case z,w = a^ , by definition of z. Also, since a*^j = aik_j9 and, by the induction hypothesis, zf_i+i = z¡ , z^^2 = zik_J+2, ... ,z? = zifc, this case also means that af-j dominates all of z?_i+1, zjLj+2, ••-,**• By the definition of z*, z?-, =» a?-,. Combining these observations yields zià> = aifc_y = ai-j = z£_;, or ziâ ̂ = z£_ ¿ in this case as well. This completes the induction step. By induction the equality holds for all ;, 0 < j < k - 1.

In particular, zi% = z?. Since z(j = z? is in K*, but no entry of a preceding aix = a? is in K*, z(l dominates all such entries, if any. By the definition of z, all entries of z that precede z,lf if any, are equal to z¡r In particular Zi = z,v Thus, z1 = zix = z? or Z! = zj. By Theorem 6, Zi is the sophisticated decision relative to T and a, and z* is the sophisticated decision relative to T * and a*. So, the result follows. D

One consequence of Proposition 7 is that the study of tournament structure involved with sincere and sophisticated voting under amendment procedure may be restricted to strong tournaments since the important "action" concerning these issues occurs in the initial strong components of tournaments. No alternative outside of the initial strong component can be either the sincere or sophisticated decision relative to any agenda of the alternatives.

As indicated in the Introduction, the object of this paper is to determine the class of tournaments for which there exists a very special type of agenda. Given a tournament T, an equitable agenda is one for which the sincere voting decision under amendment procedure and the sophisticated voting decision under amendment procedure are identical. Such an agenda insures against the strategic aspects of sophisticated voting, for regardless of the process, sincere or sophisticated, the ultimate decision is the same. T admits an equitable agenda if an equitable agenda exists for T. For example, if T contains a transmitter x, then every agenda is equitable (see [16]). However, not every tournament admits an equitable agenda as is seen in the next result.

Proposition 8. IfT is an m-tournament so that T * isa 3-cycle (i.e., | V*' = 3), then T admits no equitable agenda.

Proof. Suppose that T is an m-tournament so that T * is a 3-cycle. By Proposition 7, T admits an equitable agenda if and only if T * admits an equitable agenda. Since T ♦ is a 3-cycle, the sincere decision relative to any agenda for T * is always the last (i.e., third) entry of the agenda. On the other hand, the sophisticated decision relative to any agenda for T * is never the last (third entry). Indeed, if the order of the agenda for T * agrees with the order of the vertices of a spanning path in T ' then the sophisticated decision relative to that agenda is the second entry of the agenda; otherwise (i.e., the reverse of the agenda for T ♦ agrees with the order of the vertices of a spanning path in T *), the sophisticated decision is the first entry of the agenda. So, T* has no equitable agenda. Consequently, T has no equitable agenda. D

The search for an equitable agenda in a tournament T can be replaced by the search for a special structure (i.e., a special subdigraph) in T as described next.

Proposition 9. Let T be an m-tournament, m ̂ 2. T admits an equitable agenda if and only ifT contains a spanning subdigraph consisting of a transitive tournament




y with transmitter y9 a rooted spanning tree inT - Y with root v, and arc y -> v, such that no vertex ofT-Y dominates every vertex of Y.

Proof. Suppose that m-tournament, m > 2, T admits an equitable agenda, say agenda a = (ûi,û2, •• ,0») in which alternative ah, for some h, 1 < h < m, is both the sincere decision and the sophisticated decision. If T has a transmitter, then that transmitter is ah (by Proposition 7), so any transitive subtournament Y # T with transmitter ah and any spanning path in T - Y (by Proposition 1) gi'e rise to the subdigraph required. So, assume in the remainder of the proof of the necessity of the condition that T has no transmitter. Then the sincere sequence has the form b = (bi,b2, ... ,bh-uah,ahi ... ,ah' where h > 1 andòj = ax. This implies that (for h<m) ah => {ah+uah+2, •• ,««}, that ah-+bh-u and that (by [16, Theorem 1]) 7T{ui>u2> ••• >0/t-i}] contains a rooted spanning tree R, rooted at bk-x. As ak is also the sophisticated decision. Theorem 6 implies that the sophisticated sequence has the form z = (ahiah, ... ,ah,zh+x, ... ,zm) where h < m and zm = am. Let Y be the subtournament of T induced by ah and the distinct vertices among zh+1,zh+2» ••• , zm (such z,'s are contained in {af,+ i,ah+2, ... ,am}). Note that by the definition of z, none of al9a2, ... ,flj,-i dominates all of the vertices of Y, Y is transitive, and by the above, ah is the transmitter of Y. If V(Y) = K(T) - V(R) (i.e., all of zfc+1,Z/»+2> • •• >zm ai*e distinct), then Y, öh, /£, and bh-i are as required. Otherwise, by Proposition 1, the subtournament T[{ah+ i,a*+2> ••• ,ûm} - ^(^)] contains a spanning path P, say rooted at ak, for some fe, h < k <m. Now a/, => K(P), so no vertex of P dominates every vertex of Y. Let S denote the spanning rooted tree of T - Y which consists of R and P and the arc between bh- 1 and ak. S is rooted at bh- ! if bh- x -> a*, and S is rooted at ak'i ak-+bh-x. In either case, ûh dominates the root of S. Consequently Y, ah, S and the root of S (either ak or bh-l) yield an appropriate spanning subdigraph.

On the other hand, suppose that tournament T contains a subdigraph composed of Y, y, and v as described. Let V(Y) = {>> = )>i,)>2> • • ».Vp}, where y¡->yj whenever 1 <i<j < p. By Proposition 3, T - Y contains a spanning path Q rooted at v, say given by v = Vi -+ v2, v2-+v3i ... , vq- x -► vqt where ^ = |K(T)| - p. Consider the agenda a = (vqyvq-ly ... ,v3iv2)v1 = vt y = .Vi »^2^3» ••• ,ypy Since Q is a path, the first q entries of the sincere sequence b relative to T and a are identical to the first q entries of a; also, the last p entries of b are all equal to y since y-+v and y => {y2,y3> ••• >.Kp}- Thus, 6i, the sincere decision, is >^. Since Y is transitive, the last p entries of the sophisticated sequence z relative to T and a are identical to the last p entries of a; also, the first q entries of z are all equal to y since no vertex of T - Y dominates every vertex of Y. Thus, zu the sophisticated decision, is y. Consequently, a is equitable.

This completes the proof. Note that, by default, a 1 -tournament admits an equitable agenda.

Of course, for any tournament T there is a rooted spanning tree in T - Y for any transitive subtournament Y of T; for example, use any spanning path inT - Y (Proposition 1). Moreover, if Y is chosen to be a maximal transitive subtournament of T, then no vertex of T - Y dominates every vertex of Y. Now, there is no assurance that the transmitter of Y dominates the root of the rooted spanning tree in T - Y (otherwise Proposition 9 implies that T admits an equitable agenda). Indeed, in any tournament T for which | V * | = 3, such dominance never occurs (by Propositions 8 and 9). The main result in the next section implies that a spanning subdigraph of the type described in Proposition 9 can always be found in any tournament T for which 'V*' ^ 3.



372 K. B. Reid

5. The main result

Proposition 9 shows that the problem of the existence of an equitable agenda for voting under amendment procedure is equivalent to the existence of a special spanning subdigraph in the underlying tournament. In order to determine a structural characterization of tournaments that admit equitable agendas, it is both necessary and sufficient to determine a structural characterization of tournaments that contain special spanning subdigraphs as specified in Proposition 9. The following result gives a particularly clean characterization of such tournaments.

Theorem 10. Let T be an m-tournament, m>2.T contains a spanning subdigraph consisting of a transitive tournament W with transmitter x, a rooted spanning tree Q in T - W with root v, and arc x-+v, such that no vertex of Q dominates every vertex of W if and only if' V*' ï 3.

Proof If 'V*' = 3, then by Proposition 8, T admits no equitable agenda. So, by Proposition 9, T contains no spanning subdigraph of the type required.

On the other hand, suppose that | V*' ̂ 3. If T contains a transmitter, then as was seen in the first part of the proof of Proposition 9, T contains a spanning subdigraph of the type required. So, in the remainder of the proof assume that T contains no transmitter. Hence, 'V(T)'>4 (in fact, |K*|>4). Let W be a maximum transitive subtournament of T with transmitter x, and let a be the root of a spanning path P in T - W (Proposition 1). Note that | V(T - W)' f 0, since T has no transmitter. Also, | V(W)' > 3 as | V(T)' > 4. Because W is maximum, no vertex of P dominates every vertex of W. If x -► a, then the result follows. So, in the remainder of the proof assume that a->x. If V(W) £ O(a' then a => W and transitive subtournament W'ja properly contains W, a contradiction. This means that there is a vertex y in V(W) - x so that y -» a. Also, note that a is in the initial strong component of T[_V(Py] (by Proposition 3).

For each maximum transitive subtournament W of T and each spanning path P in T - W, choose the vertex y in V(W) as above so that the number, denoted n(WyP' of arcs on that part of the unique spanning path of W from x, the transmitter of W, to y is as small as possible. By choice of y9 x -► y and y -► a, where a is the root of P. Such a vertex y is assumed to exist by the previous remarks. Now, among all possible maximum transitive subtournaments W of T and all possible spanning paths P in T - W, pick y so that n{WyP) is as small as possible. Note that n( W, P) > 1, with equality if and only if y is the transmitter of the transitive subtournament W - x.

Since W - y is transitive with transmitter x, y -> P is a spanning path in T - (W - y) with root y, and x -> y. Certainly y does not dominate every vertex of W - y. If no vertex in V(P) dominates every vertex of W - y, then the result follows. So, in the remainder of the proof assume that

some vertex b in V(P) dominates every vertex of W - y. (1)

This implies that y -► b, as otherwise b => W, contrary to the choice of W. Next it is shown that either y is the root of a spanning path in T - ((W - y)vb)

or the result follows. If b is the terminal vertex of P, then path Q given by 3>->(P-i>) is such a spanning path in T - ((W -y)vb). On the other hand, if some vertex c follows b on P (i.e., b -* c is an arc of P), consider the arc between c and y. If c -► y, then c is the root of a spanning tree R in T - ((w - y)ub). Note that (W - y) 'jb is a transitive subtournament of order




|K(W^)|; hence (W - y)ub is a maximum transitive subtournament (as W is maximum). Its transmitter is fe. Thus, no vertex of R dominates every vertex of (W - y)ub. Since b -> c, the transitive subtournament (W - y)ufe and the spanning tree R satisfy the requirements of the theorem, so the result follows. So, in what follows, if some vertex c follows b on P, then assume that

y^c. (2)

This implies that y is the root of a spanning tree in T - {{W - y)ufe). Hence, by Lemma 3, y is the root of a spanning path ß in T - ((W - y)vb). So, it is assumed in the remainder of the proof that y is the root of a spanning path Q in T-((W-y)ub).

One consequence of the preceding discussion is that (W - y)vb is a maximum transitive subtournament of T with transmitter fe, Q is a spanning path of T - ((W - y)ub) with root y, y -> fr, and no vertex of ß dominates every vertex of (W - y)'jb (as (W - y)vb is maximum). Recall that x -> y. So, there is only one arc (namely, fe -* x) on that part of the unique spanning path of (W - y) ufe from fe, the transmitter of (W - y)vb, to x, a vertex of (W - y) ufe that dominates the root y of ß. That is, using the function n described above, n((W - )>)ufr, ß) = 1. Thus, n(W,P) = 1, and consequently.

y is the transmitter of the transitive subtournament W - x. (3)

In the event that P consists of a single vertex (so a = fe), x => V(T) - {x,y,a} (as x is the transmitter of W = T - a), y => V(T) - {x,y,a} (as by (3), y is the transmitter of W - x = (T - a) - x' and a => V(T) - {x,y,a} (as a = fe and in (1), W - y = (r - a) - y). Moreover, T[x,)/,a] is the 3-cycle given by arcs x-*y, y -► a, and a -> x. This implies that T * is T[x,y,û] and | V*' = 3, contrary to the original hypothesis. So, in the following it is assumed that

'V(P)>2. (4)

If vertex fe is the terminal vertex of P, then (P - fe)ux is a rooted spanning tree of T - ((W - x) ufe) with root a, (W - x)ufe is a transitive subtournament of T with root y, y-+a, and no vertex of (P - fe)ux dominates every vertex of (W - x)ufe (as | V((W - x)ufe)| = I V(W)' and W is maximum). That is, the result follows.

So, assume that vertex fe is not the terminal vertex of P, i.e. some vertex c follows b on P. Then (P - fe) u x is a spanning forest of T - ((W - x) u fe) which consists of exactly two rooted trees. One, denoted U and rooted at c, consists of the subpath of P from c to the terminal vertex of P. The other, denoted V, consists of either x alone, if a = b, (and hence is rooted at x) or if a # fe, V is the union of the arc a -♦ x and the subpath of P from a to the vertex of P which immediately precedes vertex fe (and hence V is rooted at a). In the case that a # fe, let /? denote the rooted spanning tree of T - {(W - x)ufe) consisting of U and V and the arc between a and c; R is rooted at a (if a->c) or c (if c->a). Recall that (W ~x)ufe is a maximum transitive subtournament with transmitter y, that >> -♦ a (by original choice of y), and that 3; -► c (by (2)). Thus, {W - x)ufe and /? satisfy the requirements in the statement of the theorem and the result follows.

Finally, suppose that a = fe, so that U = P - a and V = {x}. If some vertex u of U dominates x, then T - (W -x)uû) contains the spanning tree R, rooted at c, consisting of U and the arc u -► x. As y -+ c (by (2)), (W - x)ua and R satisfy the



374 K. B. Reid

requirements of the theorem and the result follows. Assume that u -► x for no u in K(U), i.e. x => V(V). Recall that x => V(W - x), so

x=>F(T)-{x,y,a}. (5)

In particular, x -+ c. If some vertex u of U dominates a, then T - W contains a spanning tree S, rooted at c, consisting of U and the arc u -♦ a. As x -+ c, W and S satisfy the requirements of the theorem and the result follows. So, assume that u -> a for no u in K(U), i.e. a => K(U). Recall that a => F(W - y) (by (1), since a = f>), so

a=*K(T)-{x,>;,a}. (6)

Now, a is the transmitter of the maximum transitive subtournament (W -y)'ja (by (1), since a = fc, or by (6) and the fact that a -> x), and y -► U is a spanning path of T - ((W - y)ua) (since ö = b, c is the root of U = P - a; and y -► c by (2)). If some vertex u of U dominates y, then T - {(W -y)ua) contains the spanning tree Q, rooted at c, consisting of U = P - a and the arc u^>y. As a -+ c,{W - y)Kja and g satisfy the requirements of the theorem and the result follows. So, assume that u -+ y for no u in K(U), i.e. y => K(U). Recall that y => V(W -x)-y) (by (3)), so

y=>V(T)-{x,y,a}. (7)

Now, recall that x -> y, y -+ û, and a -> x form the arcs of a 3-cycle C. So. (5)-(7) imply that T * is C, i.e. | F*| = 3, a contradiction to the hypothesis. That is, one of the earlier cases must arise, and the result follows. D

Theorem 11. Let T be an m-tournament9 m>2.T admits an equitable agenda if and onlyif'V*' #3.

Proof. Use Proposition 9 and Theorem 10. G

It should be pointed out that the transitive tournament W in the statement of Theorem 10 need not be maximal (and hence, not maximum). Indeed, there are strong tournaments T which contain no maximal transitive subtournament W and no rooted spanning tree Q in T - W satisfying the conditions of Theorem 10. For example, for k> 1 let V(T ) = Z2k+ x , the integers modulo 2/c 4- 1, and for i e V(T ), O(i) = {i + l,i + 2, ... ,i + k} (addition is modulo 2/c 4- 1). Such a tournament is known as a rotational tournament with symbol {1,2, ... ,fc} (see [15]). There are exactly 2k + 1 maximal transitive subtournaments of T, each of order k + 1; they are T[i,i + 1, ... J + fc], 0 < i < 2k. So, the transmitter of a maximal transitive subtournament W of T dominates no vertex outside of W. In particular, it dominates no root of any rooted spanning tree in T - W. However, substructures as described in Theorem 10 do exist in this tournament. That is, for 0 < i < 2k, the transitive subtournament W = T[U + 2, i + 3, ... ,i + k] of order k and the spanning path Q with the k arcs (i + 1) -► (i + k + 1), (i + k + 1) -> (i + k + 2), . . . , (i + 2/c - 1) -► (i + 2/c) satisfy the conditions of Theorem 10, so by Proposition 9, each vertex of this tournament T is both the sincere and the sophisticated decision relative to some equitable agenda.

Let t(m) denote the number of non-isomorphic m-tournaments (i(0) = 1). Then the number of non-isomorphic m-tournaments, m > 3, with initial strong component a 3-cycle is given by t(m - 3). Theorem 10 implies that the proportion




of m-tournaments that do not admit an equitable agenda is t(m - 3)/r(m). Since (see [11])

2d/2)(m2-m) 2^f2^m2~5m + S> i(m) ~ +

3(m-3)! (1+0(1))

the ratio t(m - 3)/t(m) rapidly approaches zero as m increases. Thus, the proportion of m-tournaments that do admit an equitable agenda rapidly approaches 1. For example, i(3) = 2, i(6) = 56, i(9) = 191, 536, and i(12) = 154, 108, 311, 168, so t(m - 3)/i(m) is approximately 0.5, 3.6 x 10~2, 2.9 x 10"4, and 1.2 x 10~6 for m = 3, 6, 9, 12, respectively. Corollary 12. Most tournaments admit an equitable agenda,

6. Open questions

Generalizing results attributed to Black and Farquharson, Jung [6] proved that if alternative x, in the jth position in the agenda, is the sincere (sophisticated) decision, then x is also the sincere (sophisticated) decision with respect to any agenda that is the same as the initial agenda in positions from 1 to j - 1 (from positions ; + 1 to the last, respectively). This gives rise to two problems: 1. For a fixed tournament, characterize agendas that yield the same sincere decision. 2. For a fixed tournament, characterize agendas that yield the same sophisticated decision.

Two more general problems along these lines are: 3. For a fixed tournament, characterize agendas that yield the same sincere sequence (rather than decision, as in Problem 1). 4. For a fixed tournament, characterize agendas that yield the same sophisticated sequence (rather than decision, as in Problem 2).

The main result in Sect. 5 yields existence of equitable agendas in tournaments T for which | V*' ̂ 3. The prevalence of such agendas would be of interest. 5. How many of the m! agendas of an m-tournament T such that 'V*' ^ 3 are actually equitable agendas? Of course, if p is an automorphism of T, then (aua2, ... ,am) is an equitable agenda if and only if (p{ax' p(a2), ... >p(am)) is an equitable agenda. So there are at least as many equitable agendas as there are automorphisms of T.

Let E(T) denote the set of alternatives each of which is both the sincere and the sophisticated decision relative to an appropriate equitable agenda. Banks [1] proved that for a tournament 7, the set of possible sophisticated decisions (using suitable agendas), denoted S(T), is equal to the set B(T) (which is M(T) by Proposition 5). Note that there is no claim in Proposition 9 and in Theorem 10 that the transitive subtournament Y be maximal, and hence there is no claim that every vertex in M(T), and hence in S(T), is in E(T). This suggests several questions. 6. If T is a tournament for which | V(T *)^3, which alternatives in B(T) must be in E(T)1 More generally, characterize the alternatives in E(T). 7. How does E(T) intersect other "solution concepts", such as the tournament equilibrium set of Schwartz [19]? 8. What properties are exhibited by the correspondence T ^>E(T)1 For examples if T' is a subtournament of T, when is E(T') a subset of E(T) (of course, assume that 'V*' ¿ 3 in order to ensure that E(T) is nonempty)?



376 K. B. Reid

In work to date on sincere and sophisticated voting under amendment procedure, the majority tournament is fixed throughout the voting. Of course, this does allow for some small changes in voters' preferences so long as the majority preferences for pairs of alternatives do not change. Suppose that an agenda is fixed and that voters' preferences are a bit more unstable as the voting proceeds, i.e. suppose that after each vote the voters are allowed to alter their preference orders subject to some well defined rule(s). For example, exactly one remaining alternative (e.g., the winner of the previous vote, or the next alternative in the agenda, or the last alternative in the agenda) might be allowed to be moved in each voter's (transitive) preference order. Or, interchanges of pairs of alternatives in voters' preference orders might be allowed, either the same pair for all voters or perhaps different pairs for different voters. A voter may choose to exercise the rule(s) or leave their preference order unchanged after each vote. Several questions arise. 9. What limitations need to be placed on the rules for modifying voters' preference orders so that the sets of potential sincere decisions and of potential sophisticated decisions do not become "too large"? Progress on such a question most likely will proceed by investigating specific rules. 10. Under a specific rule (e.g., interchanging pairs of alternatives in voters' preference orders after each vote) what can be said about the set of potential sincere decisions and the set of potential sophisticate decisions?

Note that since it is possible that at each stage of voting no voter exercises the rule (and hence the tournament remains unchanged throughout all voting stages), these new sets must contain the potential "ordinary" sincere and sophisticate decisions, respectively.

Acknowledgment. The author would like to thank the referees for some useful suggestions.

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equitable agendas: agendas ensuring identical sincere and sophisticated voting decisions

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