In Paul E. Johnson (ed.), Mathematical and Computer Modelling (Formal Theories of Politics II: Mathematical Modelling in Political Science) 10 (August/Septmeber 1992): 15-26.

COALITION VOTINGl

Steven J. Brams and Peter C. Fishburn

l. Introduction

Coalition voting (CV) is a voting procedure for electing a parliament, and choosing a governing coalition, under a party-list system of proportional representation (PR). Under CV, voters cast two different kinds of votes:

l. A party vote, divided equally among all parties the voter approves

of, which determines the seat shares of the parties in parliament.

2. A coalition vote for all coalitions of parties that are acceptable

because they include all parties that the voter designates Y ("yes"),

no parties that the voter designates N ("no")--and possibly some

parties that are neither Y nor N (call them M for "maybe," but as a

residual class they do not have to be identified on the ballot).

A majority coalition (of all Y and possibly some M parties) that has no superfluous parties, and is acceptable to the most voters according to the coalition vote, becomes governing, provided that the parties it comprises agree. We shall say more about the process of forming governing coalitions later.

We assume there to be no necessary relation between party and coalition votes. Thus, a voter might give the green party his or her entire party vote--in order to maximize its representation in parliament--but not want the greens to be in the governing coalition. By designating the green party to be N, and one or more mainstream parties to be Y, this voter can say that the greens should have voice (i.e., seats in parliament), but other parties should be in the government. We think, however, that the parties most voters will support with their party votes and their Y coalition votes will substantially overlap if not coincide.

Of the two votes under CV, the party vote is more in keeping with standard practice. As in most party-list systems, parties win seats in proportion to the number of votes they receive. This is the same under CV, except that the usual restriction on voting for exactly one party is lifted so that voters can vote for as many parties as they like.

This feature of CV mimics approval voting (AV), whereby voters can vote for as many candidates as they like in multicandidate elections without PR (Brams and Fishburn, l978, l983). But unlike AV, if a voter votes for more than one party, each party does not receive one vote--it depends on how many parties the voter voted for under CV, as we illustrated earlier.

Since AV was first proposed more than ten years ago, it has generated much interest as well as a good deal of controversy (see, e.g., the recent exchange between Saari and Van Newenhizen, l988, and Brams, Fishburn, and Merrill, l988); we shall draw comparisons between it and CV later. Although we shall often use the language of "approval" in discussing CV, we emphasize here that AV and the party-vote aspect of CV resemble each other only in the physical act of voting: the voter indicates on a ballot all the alternatives (candidates under AV, parties under CV) of which he or she approves.

It is coalition votes that make CV radically different from AV. They allow voters to use two different sieves, Y and N votes, to indicate that, for a coalition to be acceptable, some parties must always be included (Y) and some always excluded (N). All coalitions that survive this straining process receive a coalition vote, but only coalitions whose parties have a majority of seats can become governing.

By giving voters an ability to construct such majority coalitions, CV also encourages parties to coalesce, even before an election. Indeed, because coalition votes place a premium on precisely the majority coalitions that are most acceptable--as defined by the voters--they make it advantageous for parties to urge their supporters not only to cast their party votes for them alone but also to vote for and against certain other parties with their Y and N coalition votes. In this manner, the bane of multiparty systems under PR--disincentives to coalition formation--is attenuated.

The plan of the paper is as follows. In section 2, we distinguish "majority" from "minimal majority" coalitions; "governing" coalitions, which are a subset of minimal majority coalitions, are also defined and illustrated. In section 3, the "congruence" and "sincerity" of different voting strategies are described.

In section 4 we derive some properties of CV and prove several theorems, focusing particularly on the rationality of voting insincerely--even for one's worst party. In section 5 we narrow the focus to three-party systems and give conditions for different kinds of sincere voting. We also show that single-peaked preferences give each voter a dominant strategy.

The apportionment of integer numbers of seats to parties is discussed in section 6, where a measure of the "bargaining strength" of parties is illustrated. In section 7 we consider possible uses of CV in party-list systems like those of Israel and Italy, both of which currently have more than a dozen parties represented in their parliaments and in which consensus on major policy issues has been difficult to achieve. In section 8 we conclude with an assessment of CV as a practicable reform, comparing it with AV, and also mention some possible modifications.

Because CV gives voters two different kinds of votes, it is certainly more complex than AV. But this increase in complexity is balanced by an increase in the opportunity that CV affords voters to accomplish two goals:

* select with their party vote one or more parties that can best

represent them in parliament;

* designate with their coaliton vote one or more majority coalitions to

form the government.

Whereas parties will be motivated to solicit party votes exclusively for themselves, they will at the same time be motivated to strike compromises and reach policy agreements with other parties to try to build majority coalitions that voters will support with their coalition votes.

Insofar as they are successful, CV should mitigate the voters' choice problem. Indeed, precisely because voters can indicate acceptable coalitions directly on their ballots--and these help to determine the election outcome--CV forces the parties to pay greater heed to voters' collective as well as individual preferences. In particular, parties will have a strong incentive to iron out differences with potential coalition partners before the campaign in order to pose reasonable coalition-vote strategies to the voters during the campaign.

Extant voting systems do not register electoral support for potential governing coalitions, which perhaps explains the paucity of models linking electoral competition and coalition formation in legislatures (Austen-Smith and Banks, l988; Laver, l989). By contrast, CV encourages voters to consider what coalitions might govern effectively, without denying them the opportunity to vote for a favorite party to maximize its seat share and to lend support to its ideology.

2. Majority, Minimal Majority, and Governing Coalitions

We begin by defining some concepts described in the introduction:

C: set of parties (j = l, . . .,m)

V: set of voters (i = l, . . .,v).

Party vote: Each i V votes for a subset Pi of C. With pi = , each party j Pi receives l/pi votes from i; the other parties receive no votes from i. Let v be the number of nonabstaining voters, and let vj be the number of votes for party j. Then .vj = v. The proportion of seats won by party j, sj, is vj/v. Thus, .sj = l. (Fractional seat assignments will be discussed later.)

Coalition vote: Each i V votes for two disjoint subsets, Yi and Ni, of C. The parties in Yi are those that the voter indicates must be included in an acceptable governing coalition. The parties in Ni are those that the voter indicates must be excluded from an acceptable governing coalition. A coalition A C is acceptable for voter i if and only if Yi A and Ni A = . Coalition A receives l vote from voter i if A is acceptable for i; otherwise, A receives no vote from voter i. Let c(A) denote the number of coalition votes for coalition A.

A coalition A is said to be a majority coalition if it receives at least as many party votes as its complementary coalition = C\A, the set of all parties that are in C and not in A. Equivalently, A is a majority coalition if the number of party votes it receives is

v(A) = . vj [[threesuperior]] v/2.

Let M denote the set of all majority coalitions. Since there are m parties, there are 2m coalitions, including the empty set and the grand coaltion C. Moreover, because either a coalition or its complement will have a majority of party votes, the number of members of M will be at least 2m-l; it will be greater if and only if there is some A such that v(A) = v/2, in which case there is a tie and both A and are members of M.

Define A M to be "minimal" if no B M is a proper subset of A. Let M* denote the set of minimal majority coalitions. It will usually have far fewer members than M, but in all cases it has at least one member. Put another way, M* eliminates from M all elements that are proper supersets of other elements in M--that is, majority coalitions that have superfluous members.

Presumably, if an A M does not need the votes of one or more of its parties to maintain its winning edge against , the superfluous members will not be adequately compensated and are likely to defect. Extra votes, however, may not be superfluous if there is uncertainty about who exactly is a loyal coalition member, so coalitions may form that are not minimal winning (Riker, l962). But having a bare majority of seats is not equivalent to being a minimal majority coalition, as the "national unity" governments in Israel (to be discussed later) have illustrated.

A is a governing coalition if A M* and c(A) [[threesuperior]] c(B) for all B M*. Thus, a governing coalition is a minimal majority coalition that maximizes the coalition vote c over all minimal majority coalitions.

Let G denote the set of governing coalitions. Except for ties in c among members of M*, G will have exactly one member.

The sets of majority (M) and minimal majority (M*) coalitions are based on party votes, whereas governing coalitions are chosen from M* based on coalition votes. The congruence between these two different kinds of votes will be analyzed in section 3, but first we indicate the two-fold rationale for evenly dividing a voter's party vote among all parties of which he or she approves:

l. Equality of voters. Each voter counts equally in the apportionment of parliamentary seats. If there are l00,000 voters and l00 seats to be filled, for example, each voter accounts for l/l000 seats, whether or not voters choose to concentrate their representation on one party or spread it across several.

2. No-breakup incentive for parties. Suppose that a voter's party vote is evenly divided among the parties voted for, but that a voter who votes for more parties casts more votes in toto. For example, if a voter votes for two parties, suppose that each party receives l vote (as under AV) instead of half a vote. Then it might be profitable for a party to split, assuming its supporters continue to vote for both parts, to maximize its vote total and, therefore, its seats in parliament.

Allowing voters to distribute one vote unevenly across parties--or several votes, as under cumulative voting (presently used in a few local jurisdictions in the United States) and panachage (used in Luxembourg, Norway, Sweden, and Switzerland)--would obviously give voters more freedom to express themselves. But it would introduce complexities that would not only make voting impracticable in many systems but also alter the strategic incentives that we shall analyze in detail later.

3. Congruence between Party and Coalition Votes and Sincerity

We indicated in section l that we presumed no particular connection between voter i's party vote Pi and coalition vote (Yi, Ni). For most voters, however, it is reasonable to expect that there will be strong linkages between party votes and coalition votes. For one thing, the Pi and Yi sets both reflect the support of parties, so it is plausible that they will often overlap. By the same token, it is unlikely that there will be much overlap between Pi and Ni. After all, why would a voter insist that parties in Ni be excluded from a majority coalition and yet cast a party vote for one or more of these parties?

To be sure, we suggested earlier that there may be voters who would support, say, the green party with Pi and exclude it from an acceptable coalition by putting it in Ni. In fact, we show later that, for reasons quite different from those suggested by this example, voting for and against a party at the same time (with different votes, of course) is not a completely outlandish idea.

CV does not assume that voters have a best-to-worst ranking of parties but only that they can put them into the categories of Y, N, and M (residual). A ranking suggests that voters can rate parties on a single dimension, which seems particularly untenable in multiparty systems wherein parties express a variety of viewpoints that have little in common. Indeed, this is the raison d'tre for allowing voters to divide their party vote--they may approve of different parties for different reasons--and award coalition votes to more than one majority coalition.

While conceding that voters might use their party vote and coalition vote for different purposes, it is nevertheless plausible that they will be closely connected. We say that a voter's party vote P and coalition vote (Y, N) are

(l) congruent if P [[opthyphen]] and P N =

(2) strongly congruent if P [[opthyphen]] , P N = , and P Y whenever Y [[opthyphen]] .

Thus, congruence and strong congruence require that voters cast one or more party votes, and that these party votes not overlap the parties excluded by the N coalition votes. Strong congruence requires, in addition, that all parties approved with party votes be included in the Y set of coalition votes whenever Y is nonempty. Alternatively, we could reverse the inclusion relation and assume that P Y, or even P = Y.

We analyzed at some length the "sincerity" of AV (Brams and Fishburn, l978, l983), but the meaning of this concept in the case of CV is not so clear-cut. Unlike congruence, sincerity as we shall use the term does not depend on the correspondence--or lack thereof--between party and coalition votes but rather on the agreement between a voter's ranking of parties and his or her party-vote and coalition-vote strategy.

To analyze sincerity, we assume that voters have strict preference orders on the parties, say,

(l) >i (2) >i . . . >i (m),

where (l)(2) . . . (m) is a rearrangement of l2 . . . m. We then say that party vote Pi is sincere if (j) Pi whenever (k) Pi and j < k, so that a voter does not vote for lower-ranked parties without also voting for those ranked higher. And coalition vote (Yi, Ni) is sincere if Yi contains (j) whenever it contains (k) and j < k, and Ni contains (k) whenever it contains (j) and j < k. Hence, a sincere (Yi, Ni) would look like

Yi = {(l), (2), . . .} (all top parties in Yi)

{. . ., (j), (j+l), . . .} {Yi Ni} = (middle parties in neither Yi nor Ni)

Ni = {. . ., (m-l), (m)}. (all bottom parties in Ni)

Any of the three subsets Pi, Yi, or Ni may be empty, but to simplify the later analysis--and satisfy the definition of congruence--we shall assume that Pi is not empty.

In words, a voter's party vote and coalition vote are sincere when they mirror the voter's preference order: Pi and Yi each includes all parties ranked above the lowest party approved by each kind of vote; Ni includes all parties ranked below the highest disapproved party.

It is worth noting that when P [[opthyphen]] , both P and (Y, N) can be sincere without implying congruence. For example, assume C = {l,2,3,4,5}, and the voter ranks parties in the order 12345. Then if P = {l,2,3,4}, and N = {4,5}, the voter is sincere, but these party and coalition votes are not congruent because P N [[opthyphen]] .

4. Properties of Coalition Voting

In this section we prove several theorems that highlight certain properties of CV, including the fact that sincere voting may not always be rational. To begin with, it is easy to show that if more than half the voters cast their party votes for a single party, it is the sole governing coalition. More precisely,

Theorem l. If v(j) > v/2, then G = {j}.

Proof. If v(j) > v/2, then i must be in every A M. Hence, M* = {j}, so G = {j}.

Observe in this case that coalition votes play no part in the determination of G, because party j is the unique minimal majority coalition and therefore must be the unique member of G.

If at least half the voters support party j alone, then this party obviously receives approval from at least half the voters. In fact, this result for majority party j can be extended to any majority coalition.

Theorem 2. For every A M, Pi A [[opthyphen]] for at least half the voters who voted for at least one party with their party vote.

Proof. If fewer than half of the voters vote for any party in A, then v(A) < v/2, in which case A M.

Whichever A becomes the governing coalition, then, at least half the voters must have cast party votes for one or more of its members.

Recall that the governing coalition is that coalition A M that maximizes the coalition vote among members of M*. Curiously, the A that minimizes the coalition vote is a member of M* when Ni = .

Theorem 3. Suppose Ni = for all i in V (i.e., only Y votes are used to determine coalition votes). Then some coalition that minimizes the coalition vote among all members of M is a member of M*.

Proof. Suppose A M and c(A) = min c(B). If A M*, then there is a

proper subset A' M such that A' A. Now c(A') receives a contribution of l from voter i only if Yi A', and in this case Yi A also, so (with all Ni = ) we have c(A') [[twosuperior]] c(A). The minimization condition for A then implies c(A') = c(A), and therefore A' M* and c(A') = min c(B).

To illustrate Theorem 3, consider

Example l. Assume m = v = 3, and the three voters vote as follows for C = {l,2,3}.

l. P = {l}; Y = {l}

2. P = {l,2}; Y = {l}

3. P = {2,3}; Y = {2,3}

Then M = {l,l2,l3,23,l23}, M* = {l,23}, and G = {l} because c(l) = 2 and c(23) = l. Among all members of M,

* coalition l23 receives 3 coalition votes

* coalitions l, l2, and l3 each receives 2 coalition votes

* coalition 23 receives l coalition vote.

Hence, the coalition that minimizes the coalition vote (i.e., 23) is also a member of M*. Note, however, that the coalition that maximizes the coalition vote (l23) is not a member of M* since it is a superset of l and 23; party l maximizes the coalition vote only among members of M*.

We shall next prove two theorems by examples. They illustrate possibilities that may arise under CV that, at least initially, seem surprising if not perverse. We shall comment later on the realism of these situations after proving additional theorems in this and the next section.

Theorem 4. Suppose a voter has a strict preference order and prefers coalitions that contain higher-ranked parties. Then it may be rational for this voter to cast a party vote only for his or her worst party.

Example 2 (proof of Theorem 4). Assume m = 4 and that a focal voter has preference order l234. Although the example does not depend on a "story," the scenario that follows may lend it plausibility.

Polls show that coalition l2 is a shoo-in for M* (but not G); also, the

party-vote election is very close between l3 and 24, and between 23 and l4,

for M* (fifteen votes required to be in M--see figures below). On the basis of coalition votes (not shown), assume

* if l3 M*, then G = l3

* if l3 M* and 23 M*, then G = 23

* if l3,23 M* and l2 M* (as presumed), then G = 12.

We also assume that the focal voter strongly prefers G = 12 to either of G = 13 or G = 23 and would gladly sacrifice a fractional seat for his or her favorite party (parties) if that would ensure G = 12.

Assume that the party votes from all other voters are:

l: l0 l/12 Party Votes of Two-party Coalitions

2: 9 ll/12 l3: l5 l/3 vs. 24: l4 2/3

3: 5 l/4 23: l5 l/6 vs. 14: l4 5/6

4: 4 3/4

If the focal voter chooses P = 4, then l3,23 M*, so G = 12. For every other P, G is l3 or 23. For example, if P = {l,2,4}, coalition l3 receives l5 2/3 votes and coalition 24 receives l5 l/3 votes, so G = l3. By way of explanation, the "game" is to keep both l3 and 23 out of M*, and this can only be done by P = 4. In this example, the outcome does not depend on whether coalition votes are of the (Y, N) type or just the Y type.

Example 2 did not depend on N votes, but this will not always be the case, as illustrated by

Example 3 (another proof of Theorem 4). Assume m = 3. The focal voter has preference order l23 and prefers that coalition 12 be in G to each of 2, 13, and 23. The party votes from all other voters are

l: l/2 2: T 3: T - l/2,

where T is assumed to be large. For all possible ways in which the focal voter can cast P, the resulting M* are:

P: l 12 123 2 3 13 23

M*: {2,13} {12,13,23} {2,l3} {l2,l3,23} {2} {3,12} {l2,l3,23} {2,l3}

Suppose c(23) > c(12) > c(3) > 0. [This is possible if, for example, v = 24, (Y, N) = (23, ) for l0 voters, (Y, N) = (12, ) for 8 voters, and (Y, N) = (3, 2) for 6 voters; it is not possible without some nonempty N coalition votes to ensure that party 3 has fewer coalition votes than coalition 23.] If P = 3, then G = 12. If P is anything else, G will be one of 2, 13, 23, and each of these is less preferred than G = 12. Hence, based on G, the focal voter is uniquely best off by voting only for his or her worst party (3).

Examples 2 and 3 demonstrate that an exclusive party vote for a voter's worst choice can, paradoxically, help his or her best choice win. We next show how coalition votes can have equally bizarre effects through two more examples, which proves

Theorem 5. Suppose a voter has a strict preference order and prefers coalitions that contain higher-ranked parties. Then it may be rational to choose an insincere coalition-vote strategy over a sincere strategy.

Example 4 (proof of Theorem 5). Assume m = 4, the focal voter has preference order1234, and prefers 24 to l34 in G. Also, assume that polls show that M* = {24,134} and that G will be one or both of these coalitions--too close to call. The focal voter, therefore, wants to contribute to c(24) but not to c(134), which requires that {1,3} Y = and {2,4} N = . Two ways are insincere strategies (Y, N) = (2, l) and (Y, N) = (4, 3), whereas sincere strategy (Y, N) = (l, 4), for example, fails. For the two insincere strategies illustrated, insincerity arises not because a preferred party is left out of Y (see Example 5) but rather because N includes parties preferred to those in Y (l versus 2 in one case, 3 versus 4 in the other case).

We next give an example of insincerity due to the exclusion of some higher-ranked parties from Y:

Example 5. Assume m = 3 and that the focal voter has preference order l23. Also assume that polls show that M* = {l2,l3,23}, regardless of what the focal voter does, and that G will be either l3 or 23. Now if the focal voter chooses sincere coalition-vote strategy (Y, N) = (l, 3), this strategy has no effect on determining G because it singles out coalition 12 for a coalition vote. On the other hand, if this voter chooses as (Y, N) any of (l, ), (l, 2), (l3, ), and certain other coalition-vote strategies, then he or she increases c(13) by l and leaves c(23) unchanged, which is beneficial. But (Y, N) = (l3, ), in particular, is insincere because Y does not include 2.

Having illustrated some "worst case" scenarios of insincere voting under CV, we shall next prove some more "positive" results in three-party systems. In general, as we shall argue, sincere and congruent voting will be the norm rather than the exception under CV.

5. General Results for Three Parties

Although most multiparty parliamentary systems contain more than three parties, the parties often fall into ideological groupings perceived in terms of a left-center-right continuum. We shall assume such a continuum later when we postulate that voters have single-peaked preferences, but here we simply posit three parties. However, insofar as additional parties can be grouped with these parties, the analysis has relevance for larger party systems. We assume throughout this section that each of the parties has a nonzero party vote.

We shall prove two theorems on sincerity for three-party systems. Whereas Theorem 6 ("sincerity I") postulates, among other things, that the focal voter prefers a coalition of his or her best and worst parties to a middle party (or is indifferent between them), Theorem 7 ("sincerity II") does not postulate this relation. Although the proofs of each theorem turn on the focal voter's preference for parties in G, not on his or her preferences for the individual parties, the voter's most plausible underlying order for the parties--consistent with his or her G preferences--is l23.

Theorem 6 (Sincerity I). Assume m = 3, and the focal voter, for G, has the following preference-indifference relations: 1 [[threesuperior]] 23; l2 [[threesuperior]] each of 3, l3, and 23; and l3 [[threesuperior]] each of 2 and 23. Then this voter cannot improve his or her coalition by choosing a coalition-vote strategy other than sincere strategies (Y, N) = (l, 3) or (Y, N) = (l, ).

Proof. There are seven possibilities for M*: l, 2, 3, {l,23}, {2,13}, {3,12}, {12,13,23}. (Note that there are no coalitions of the type {12,13}, because if 12 and 13 are in M*, so must 23 be: its two minority parties must have a majority since party l is a minority.) The focal voter's coalition vote has no effect on the first three cases. Now consider the remaining four cases:

M* = {l,23}. Both (Y, N) = (l, 3) and (Y, N) = (l, ) increase c(l) by l and

do not change c(23).

M* = {2,13}: (Y, N) = (l, ) increases c(13) by l and does not change

c(2).

M* = {3,12}. Both (Y, N) = (l, ) and (Y, N) = (l, 3) increase c(12) by l

and do not change c(3).

M* = {l2,l3,23}. (Y, N) = (l, 3) increases c(12) by l and does not change

the others; (Y, N) = (l, ) increases c(12) and c(13) and does not

change c(23).

Thus, (Y, N) coalition-vote strategies of (l, 3) and (l, ) cannot be improved upon by the focal voter, but which of these two strategies is better depends on who the members of M* are and on specific c counts. Because there is no single strategy that is at least as good and sometimes better than all others, (l, 3) and (l, ) are undominated but not dominant strategies.

Theorem 7 (Sincerity II). Assume m = 3, and the focal voter, for G, has the following preference-indifference relations: l [[threesuperior]] 23,12 [[threesuperior]] each of 3, 13, and 23; and l3 [[threesuperior]] 23. Then this voter cannot improve his or her coalition than to have: 3 Y, l N, and 2 Y N. In particular, sincere strategies (Y, N) = (l, ) and (Y, N) = (, 3) are undominated.

Proof. We follow the lines of the proof of Theorem 6 for the relevant four cases:

M* = {l,23}. (Y, N) = (l, ) increases c(l) by l and does not change c(23).

M* = (2,l3}. If 2 [[threesuperior]] l3, (Y, N) = (, 3) increases c(2) by l and does not

change c(13); if l3 [[threesuperior]] 2, (Y, N) = (l, ) increases c(l3) by l and does

not change c(2).

M* = {3,12}. (Y, N) = (l, ) increases c(12) by l and does not change

c(3).

M* = {l2,l3,23}. Changes in the coalition votes are as follows:

(Y, N) c(12) c(13) c(23)

(l, ) +l +l 0

(,3) +l 0 0

In the last case, because 12 [[threesuperior]] l3 [[threesuperior]] 23, (Y, N) = (l, ) and (Y, N) = (, 3) cannot be improved upon by the focal voter. Combining this case with the three previous cases verifies the theorem.

The only difference between sincerity I and sincerity II is the assumption of sincerity I that l3 [[threesuperior]] 2. When this assumption is violated in Theorem 6, congruence as well as sincerity may break down, as shown by

Example 6 (Insincerity and Incongruence). Assume m = 3, the focal voter has preference order l23, and the party votes from the other voters are vl =l0, v2 = l9, and v3 = l0. Assume the polls indicate that if M* = {2,13}, G is equally likely to include either member. However, if coalition 23 is in M*, then G = 23. For the focal voter, 2 > l3; furthermore, this voter very much prefers both 2 and l3 to 23.

If P = 2, the focal voter can ensure that M* = {2,13} and then, with his or her coalition vote, increase 2's chances by selecting a coalition-vote strategy that increases c(2) by l and does not increase c(13). This requires l Y as a necessary condition.

Now if P is anything other than 2, then M* = {l2,l3,23}, and G = 23. Hence, the focal voter's best strategy is to vote insincerely by choosing P = 2 and l Y.

Consistent with the latter strategy, if 2 Y or 3 Y, or 1 N or 2 N but 3 N, (Y, N) will be insincere. Finally, note that when P = 2 and 2 N, party and coalition votes will not be congruent.

In summary, sincerity and congruence may be violated when 2 > 13 for the focal voter. The fact that one's middle party is better than a coalition of one's best and worst parties makes it rational to vote for it without also voting for one's best party, or to vote for it with one's party vote and against it with one's N coalition vote.

Now assume that the three parties can be arrayed on a left-right continuum such that party l is on the left, party 2 is in the center, and party 3 is on the right. Furthermore, assume that voters have single-peaked preferences: there are no voters who rank the parties l32 or 3l2, such that nonadjacent parties l and 3 are ranked first and second. Rather, voters have either left (123), right (321), or center (213 or 231) preferences.

Theorem 8 (Single-peakedness). Assume m = 3 and voter preferences are single-peaked over order l23. Assume also that coalition l3, consisting of the extremes, is not a viable contender for G, and neither is any single party. Suppose that the left focal voter has the following preference-indifference relations: l [[threesuperior]] each of12 and 23; and l2 [[threesuperior]] each of 2, 3, l3, and 23. Then this voter's dominant strategy is to choose sincere and strongly congruent strategies P = l and (Y, N) = (l, 3). Analogous results apply to right focal voter 321 and center focal voters 213 and 23l: their optimal strategies are to cast P and Y votes for their first choices, N votes for their worst choices.

Proof. Call the dominant strategy given by Theorem 8 S, and consider the four relevant cases in the proofs of Theorems 6 and 7:

M* = {l,23}. S increases c(l) by l and does not change c(23).

M* = {2,l3}. By assumption, G = 2 in this case.

M* = {3,12}. S increases c(12) by l and does not change c(3).

M* = {l2,l3,23}. S increases c(12) by l and does not change the others.

Since l3 is out of the running, this is best for the focal voter.

Because (Y, N) = (l, 3) is unconditionally best, it is a dominant coalition-vote strategy. Moreover, P = l is always better than P = 12 because the latter strategy awards l/2 a party vote each to parties l and 2, which is worse for the left focal voter with preference order l23 than giving party l alone l party vote. The derivation of dominant strategies for the right focal voter and two center focal voters are based on similar arguments.

Theorems 8 establishes that all voters--left, center, and right--have dominant strategies of casting both a party and a Y coalition vote exclusively for their best parties and an N coalition vote exclusively for their worst parties. This strategy is sincere and strongly congruent.

If no single party wins a majority of seats, Theorem 8 has an immediate implication:

Corollary (Possible Discrepancy between Party and Coalition Votes). In a three-party system in which no single party has a majority of party votes, in which voter preferences are single-peaked, and in which voters choose their dominant party and coalition-vote strategies as in Theorem 8, either the left-center (12) or right-center (23) coalition will be in G, depending on which coalition receives more coalition votes. The other coalition, however, may receive more party votes.

Proof. By assumption, the left-right coalition l3 receives too few coalition votes to be in contention, so only the left-center (l2) and right-center (23) coalitions can be in G. The contest between l2 and 23 will hinge upon whether coalition 12 receives more coalition votes from l23 and 2l3 voters than coalition 23 receives from 32l and 23l voters. Because coalitions l2 and 23 receive the same numbers of party votes from center voters (213 and 23l), the party-vote contest between coalitions l2 and 23 will depend on whether there are more left (l23) or right (321) voters. This party-vote result will not necessarily agree with the coalition-vote result that determines which coalition is in G.

This corollary establishes that even when voters have single-peaked preferences in a three-party system, there may be a discrepancy between party and coalition votes: one may favor the left-center coalition, the other the right-center coalition, given that no single party wins a majority of seats. But the fewer the number of center voters, the less likely there will be a discrepancy because left and right voters (for whom there is no discrepancy--they help either l2 or 23 with both their P and Y votes) will be more determinative.

If there were not single-peakedness, and coalition l3 were therefore a possibility for inclusion in G, then the optimal party and coalition-vote strategies given by Theorem 8 would not be dominant. To be sure, a voter with preference order l23 would always choose P = l and Y = l, but N = 3 would not always be optimal. The reason is that if polls indicated that coalition 12 was out of the running and that the real contest for membership in G was between l3 and 23, then voter l23 would presumably prefer coalition l3 to 23. If so, the left focal voter's optimal coalition-vote strategy would be (Y, N) = (l, ) rather than (Y, N) = (l, 3) in order to boost c(13) by one vote over c(23). But because single-peakedness rules out coalition l3 as a viable contender, the focal voter does not have two undominated coalition-vote strategies but a (single) dominant strategy: (Y, N) = (l, 3).

We shall not attempt here to extend the formal results in this section to systems with more than three parties. One reason is that conditions that ensure sincerity, congruence, and dominance quickly become more complicated as the number of parties increases beyond three. Indeed, even for three parties, as Examples 3 and 4 demonstrated, casting party or coalition votes for one's worst party may, under certain circumstances, be optimal.

Generally speaking, we think these circumstances will rarely arise in practice. First, they depend on much specific information, which most voters will not possess. Second, the scenarios we developed to demonstrate these paradoxical results are quite contrived; they were designed to illustrate "worst case" instances of insincere or incongruent strategies--not to describe a typical situation. More typical, we believe, are the preferences assumed in Theorems 6-8, under which voters will choose sincere, congruent, or dominant strategies that closely mirror their preferences.

As the number of parties increases, one must balance casting fewer Y and N votes--thereby giving one's support to more majority coalitions--to casting more such votes and thereby narrowing the acceptable list to one or a very few favorite coalitions. While the latter strategy will most help these favorite coalitions, it could be an exercise in futility if none of these is likely to be in G. On the other hand, if the focal voter is less discriminating, he or she could hurt a favorite coalition by helping a nonfavorite (though acceptable) choice.

6. Apportioning Seats and Measuring Bargaining Strength

We turn in this section to two issues related to the use of CV. One issue of relatively minor importance concerns how one should assign integer numbers of parliamentary seats to parties, based on party votes. In section 2, we assumed an allocation rule that would assign fractional seats, giving each party exactly the proportion to which it was entitled on the basis of its party vote.

Methods for assigning integer numbers of seats to parties, and apportioning legislative seats to districts in district-based legislatures, have been analyzed by Balinski and Young (l978, l982). They recommend the so-called Jefferson method (also called d'Hondt) for party-list systems, because it encourages smaller parties to coalesce into larger parties--by combining their votes, smaller parties will collectively obtain at least as many seats as they would obtain separately. Because CV already incorporates a strong incentive for ideologically proximate parties to coordinate their policies--if not merge (more on this question later)--we think the incentive of the Jefferson method may be unnecessary. A more "neutral" method, such as the Webster method (also called Saint-Lague), which Balinski and Young recommend for apportioning district-based legislatures, may work as well.

Although CV is meant to give a boost to the most approved majority coalitions, it is the parties themselves--or, more properly, their leaders--that must decide whether a governing coalition will actually form. For example, assume M* = {12,l3,23}, as would be the case in a three-party system if no single party has a majority of seats. Whichever of these coalitions receives the most coalition votes and is therefore in G (say, l2), there is no guarantee that parties l and 2 will be able to reach an agreement to form a new government.

If they do not, assume that the next-most acceptable coalition is then given the opportunity to form a government. If voter preferences are single-peaked over order l23, the next-most acceptable coalition will be 23. One might think that party 2, which is a member of both coalitions l2 and 23, could make extravagant demands on party l (e.g., for ministerial posts), knowing that it would still have an opportunity with party 3 to form a government if coalition l2 failed to reach an agreement.

However, coalition l3, comprising the left and right parties and the coalition least acceptable to the voters, would still be a possibility--even if it received no coalition votes--should coalition 23 fail to reach an agreement. Moreover, since party 3 is the common member of coalitions 23 and 13, it replaces party 2 as the party in the advantageous position if coalition l2 fails to form initially.

Because of single-peakedness, coalition l3 will enjoy little if any support from the voters. But it is still a theoretical possibility if coalitions12 and 23 are not able to form a government. In fact, a temporary coalition of the conservative New Democracy party and two Communists parties formed in Greece in l989 in order to bring charges against members of the center (Socialists), so this phenomenon is not unprecedented. Its possibility should not be ruled out because, once formed, such a coalition may generate its own support and become quite popular, as have grand coalitions of the left and right in such countries as Austria and the Federal Republic of Germany.

How might one measure the bargaining strength of parties in a hierarchical ordering of coalitions, with the governing coalition at the top (12 in our example, followed by 23 and then l3). Such a measure could be a useful indicator of the payoff a party might expect if it became a member of the governing coalition.

First observe that there will always be at least one common member of every pair of coalitions in M* if the members do not tie in party votes.

Thus in our earlier example, party 2 is the common member of 12 and 23, which are the first and second coalitions in M* that would be asked to form a government, based on their coalition votes.

For illustrative purposes, assume that if two parties have the opportunity to form a governing coalition, there is a 50 percent chance that they will actually agree to do so. In our example, therefore, party 2 has a probability of being in a government of

.5 + (.5)(.5) = .75,

where .5 is the probability of l2's succeeding initially, and (.5)(.5) is the probability of 12's failing initially and 23's succeeding next. By similar reasoning, party l has a probability of being in a government of

.5 + (.5)(.5)(.5) = .625,

where the second term reflects 12's failing, 23's failing, and then 13's succeeding. Party 3 has a probability of being in a government of

(.5)(.5) + (.5)(.5)(.5) = .375,

where the first term reflects l2's failing and 23's succeeding, and the second term reflects l2's failing, 23's failing, and then l3's succeeding.

Of course, there may not be a government. The probability of this outcome, and the three other outcomes in which one of the two-party coalitions forms, are

l2 forms: .5

23 forms: (.5)(.5) = .25

13 forms: (.5)(.5)(.5) = .l25

No government forms: (.5)(.5)(.5) = .l25,

which necessarily sum to l since they are mutually exclusive and exhaustive events.

Consider again the probabilities that each party is in the government. Normalizing these probabilities of .75, .625, and .375 for parties 2, l, and 3, respectively, we can define the bargaining strength (BS) of each party to be

BS(2) = .43 BS(1) = .36 BS(3) = .21.

Plainly, party 2's common presence in the first two coalitions to be asked to form a government helps it the most, with party l's common presence in the first and third coalitions hurting it somewhat. But party 3 is hurt most, primarily because it suffers from the 50-percent chance that coalition l2 will form a government at the start and consequently exclude it.

In fact, the 50-50 chance that party 3 will be left out may be far too conservative. The fact that coalition l2 is in G may make it an overwhelming favorite to form (say, above 90 percent). The "50 percent assumption" is arbitrary and was made simply to illustrate one way of calculating the differential advantage that parties higher in the coalition-vote hierarchy have in negotiating for ministerial posts, concessions on policy, and so on.

BS might also be used to approximate a division of spoils between coalition partners. If coalition l2 forms, for example, the division between parties l and 2 would be 5 to 6 according to BS, though the party votes of l and 2 (l.5 and 2, respectively, in the example) would surely also have to be taken into account.

In turn, voters might well take account of these anticipated results in selecting a coalition (Austen-Smith and Banks, l988), but we shall not elaborate such a calculation here. Suffice it to say that a voter who prefers, say, l2 to 23 to l3 would presumably reconsider voting sincerely for l2 if the polls indicated the likely coalition-vote ordering to be c(13) > c(23) > c(12). More generally, a model of strategic voting under CV needs to be developed when the conditions of Theorems 6-8 are not satisfied.

A factor complicating the determination of a party's bargaining strength, either as a member of G or as an heir apparent if G fails to form a government, is the size of the majority coalition. Insofar as the size principle is applicable (Riker, l962), having an overwhelming majority might be a coalition's Achilles heel. If, for example, 12 is such a coalition but 23 is minimal winning (making13 somewhere in between), party 2 might have good reason to sabotage a coalition with party l in order to consummate an agreement with party 3 at the next stage. Presumably, party 3 would be less demanding of party 2 because not only is 23 not in G but it also has fewer supporters to pay off.

It is certainly possible for coalitions other than those that are governing to strike a deal in the end, despite the wishes of the voters. With a majority of seats, these coalitions could presumably assume the reins of power, even if most voters did not approve of them.

There are at least two ways of countering deal-making that blatantly defies voter approval. One is to permit only governing coalitions to try to form a government; if they fail, new elections would be held.

We prefer the less drastic procedure, suggested by our example, of giving governing coalitions priority in trying to form a government. Only if they fail would other coalition possibilities be entertained, in descending order of the size of the coalition vote of the different minimal majorities.

In the latter case, the parties to the negotiations would know what alternatives, if any, there were to fall back on if a deal fell through. Because the numbers approving of the various minimal majority coalitions would also be known, the repercussions from flouting voter desires would be starkly evident.

The severe lack of approval of coalition l3, given that voters' preferences are single-peaked and they choose their dominant strategies, would surely render it a dubious choice. But perhaps the fact that it has a majority of seats and theoretically could form a government, if coalitions l2 and 23 do not form, would prevent party 2 from exploiting its common presence in coalitions 12 and 23 to force parties l and 3 to "give away the store" in order to be selected as a coalition partner. If worst came to worst, parties l and 3 would always have each other.

7. Possible Uses

CV seems to us an attractive procedure for coordinating if not matching voters' and parties' interests in a party-list system. Because the most compatible (i.e., acceptable) majority coalitions get priority in the formation of a government, both voters and parties are encouraged to weigh coalition possibilities. Even before the election, it seems, parties might negotiate alliances and offer joint policy positions, better enabling voters to know what a coalition stands for if it should form the next government. "Citizen electoral control," as Powell (l989) put it, varies considerably in democracies today and almost certainly would be enhanced.

Under AV, by contrast, candidates do not form alliances but instead enjoy a premium in votes by taking positions that are acceptable to as many voters as possible--without, at the same time, appearing bland or pusillanimous, which might cost them votes. The fractional approval votes of CV give it some flavor of AV, but, just as significant, the coalition votes help to ensure that both voters and parties think beyond single best choices to coherent coalitions that can govern. This is not a factor in single-winner elections, to which AV is best suited.

But, as noted earlier, AV and CV are equivalent in allowing the voter to make multiple choices. To be sure, CV imposes greater burdens on voters than AV by asking them to cast two different kinds of votes that fulfill different purposes. Party-vote strategies that determine seat assignments would seem unproblematic--most voters would probably vote for a single favorite party to maximize its seat representation. On the other hand, the choice of a coalition-vote strategy is not so straightforward, especially if voter preferences are not single-peaked and there are more than three parties.

As a case in point, consider the Israeli parliamentary election in November l988. Neither Labor on the left nor Likud on the right won a majority of seats; an agreement with different combinations of the religious parties could have given either major party the necessary seats to form a government. Consequently, both parties flirted with the religious parties, which won collectively 15 percent of the seats in the Knesset, over a period of several weeks in an effort to forge an alliance, but both failed in the end to reach an agreement. Another "national unity" government comprising Labor and Likud, which had formed in l984, resumed its shaky relationship, but it collapsed in March l990, this time to be replaced by a narrower coalition of Likud and some of the religious parties.

Under CV, Israeli voters, anticipating in the l988 elections that no single party would win a majority of seats, would have been able directly to express themselves on what coalitions of parties they found they could best live with, if not like. With 27 parties running, and l5 that actually gained one or more seats in the Knesset, the opportunities for building coalitions would have been manifold.

It seems likely, however, that broad coalitions on the left and right would have arisen, reflecting the main secular dimensions of Israeli politics (Bara, l987). Small and relatively extremist parties, while keeping their most fervent supporters and their seats proportionate to their numbers, would have hurt their chances of being in a governing coalition unless they broadened their appeal and allied themselves with other parties that could boost their chances.

Israel may be a special case--one observer has characterized her coalitions as "fragile" (Seliktar, l982)--but she is not alone in experiencing difficulties in establishing durable coalitions under a party-list system. Another example is Italy, which has had 49 governments since World War II, an average of more than one a year (Riding l989).

Lijphart (l984) classifies Italy as a "consensus" democracy precisely because cleavages in the society have necessitated restraints on majority rule. To be sure, the frequent shifts in Italian governments probably have not had a great impact on the country's economy, national defense, or foreign policy, in part because the same people--mostly Christian Democrats--have governed for over 40 years, shuffling ministry portfolios among one another.

In fact, there have been only l8 persons since World War II to serve as prime minister. But, as Haberman reports, with l3 parties in parliament and a style of government that is more patronage-driven than policy-driven,

many Italian political leaders sense that the national mood is

changing, that people have grown tired of the old politicking

and want governments that last longer than the average life of

a flashlight battery. There is a desire for predictable, enforceable

and sustainable policies, they say, and it has grown keener with

the approaching economic integration of Europe in l992 (Haberman,

l989).

Indeed, CV probably would help to promote what Haberman reported is "a widely shared desire: a restructuring of political forces to give Italian

voters a choice between a few dominant parties or coalitions, offering clear

programs and capable of taking turns at the helm" (Haberman l989).

In the concluding section we shall briefly recapitulate our findings and make our normative stance explicit. We shall also indicate some problems and possible alternative approaches.

8. Conclusions

CV diminishes the problem of politicians' ignoring voters' interests. Indeed, it induces party leaders to try to reconcile their differences, insofar as possible, before an election and might even encourage mergers of parties. Because voters benefit from forcing coalitions to surface before, not after, an election, their task of choosing a coalition, as allowed under CV, is better informed if such coalitions do indeed form.

Thus, CV seems well equipped to produce a convergence of voter and party/coalition interests, which Downs (l957) suggested, to the contrary, would be opposed in multiparty systems. In fact, parties often find it in their interest to be as ambiguous as possible about compromises they would make to enter a coalition, thwarting voters who seek clarity in order to make more informed choices in the election. If vagueness is dispelled at all, it is only after the election, when forced upon the parties by the necessity of forming a government.

The coalition-inducing and voter-responsive properties of CV should commend it to politicians in a party system like Israel's, which has been repeatedly torn by corrosive conflict that reflects in part the divided nature of that society, or Italy's, whose ephemeral governments seem ill-suited to making the tough choices required for the future. Similarly, in other countries (e.g., Belgium) that may be divided along ethnic, linguistic, racial, religious, or other lines--or simply have a tradition of factional conflict--CV would seem to offer hope for promoting greater reconciliation and compromise, not because it is in the public interest (however defined) but rather because it is in the parties' self-interest if they desire to participate in a governing coalition.

These advantages of CV, in our opinion, justify a somewhat more complex seat-determination and coalition-selection process. The intricacies of the Hare system of single transferable vote (STV), which has been widely used in both public and private elections, are no less easy to understand--even by mathematicians (Brams, l982)--but these do not seem to burden the average voter.

In fact, because the voters under CV have only to indicate approval (with P and Y votes) and disapproval (with N votes) of parties--but do not have to rank them, as under STV--CV facilitates the task of voting, particularly if there is a large number of parties. In addition, the parties, fueled by their desire to be in a governing coalition, will, in all likelihood, present the voters with reasonable coalition strategies, which should further ease the voters' burden in deciding upon acceptable majority coalitions. All in all, we see no insurmountable obstacles in implementing CV in party-list systems.

At the same time, our ideas about the best way to select coalition governments are not frozen in concrete. To encourage further thinking about the problem, we conclude by mentioning three possible modifications in CV that seem worthy of further analysis:

l. Dropping the restriction that governing coalitions be minimal majority. Today larger-than-minimal coalitions often form as a cushion against the possible defecton of some members or even of a party. Sometimes the opposite happens: minority governments, without a majority of seats, take office. Thus, stipulating that only minimal majority coalitions can be governing may be too restrictive. One consequence of dropping the minimal-majority restriction would be that voters who cast Y votes for more than a minimal majority, or who cast N votes against a majority (minimal or not), would not necessarily be disenfranchised. But for which coalitions their coalition votes would count would have to be spelled out.

2. Requiring that voters rank Y and N parties in the order that they desire them to be included or excluded from a governing coalition. This additional information would indicate what coalitions of parties, of varying size, the voters would choose as one descends their ranking until, say, the inclusion of a party would give the coalition a majority of seats or the exclusion of a party would give it less than a majority. Thereby voters' "favorite" majorities could be determined from their ballots without their knowing which parties, on the basis of the party vote, constitute a majority in the election.

3. Permitting the party-vote election to precede the coalition-vote election. The choice of favorite majorities would not require rankings of the Y and N parties if the electorate cast their coalition votes after the party vote and, therefore, knew what coalitions had a majority of seats. But a second coalition-vote election after the first party-vote election, in addition to being most costly, would give the parties little incentive to make compromises--and indicate to the voters reasonable coalition possibilities to consider--in the first election. On the other hand, if the second election were just among members of the elected parliament, it might be a more democratic way of selecting a governing coalition than having a president make this choice.

These and other possible modifications in CV will be desirable under different circumstances, including other kinds of social-choice situations. For example, allowing voters to use both Y and N votes might be a sensible way to elect a committee or a governing board that is intended to be both coherent and representative.2 NOTES

l. We thank the following for valuable comments on an earlier draft of this paper: Gideon Doron, Dan S. Felsenthal, Danny Kleinman, D. Marc Kilgour, Vasiliki Koubi, Arend Lijphart, Zeev Maoz, Samuel Merrill, III, Alex Mintz, Ben Moore, Jack H. Nagel, Hannu Nurmi, Richard F. Potthoff, Amnon Rapoport, Itai Sened, and Shlomo Weber. Steven J. Brams gratefully acknowledges the support of the National Science Foundation under grant SES-87l537.

2. "Constrained approval voting" (CAV), which is a very different procedure from CV or the modifications mentioned, has been proposed for the election of governing boards in Brams (l990); Potthoff (l990) gives an algorithm for implementing CAV.

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COALITION VOTING

Steven J. Brams

Department of Politics

New York University

New York, NY l0003

Peter C. Fishburn

AT&T Bell Laboratories

Murray Hill, NJ 07974

Prepared for delivery at the l990 Annual Meeting of the American Political Science Association, The San Francisco Hilton, August 30 through September 2, l990. Copyright by the American Political Science Association.

ABSTRACT

Coalition voting (CV) is a voting procedure for electing a parliament, and choosing a governing coalition, under a party-list system of proportional representation. Voters have two different kinds of votes. The first kind is a "party vote," which, as under approval voting (AV), voters can cast for as many alternatives as they like. Unlike AV, however, each party approved of does not receive one vote; instead, a party vote is divided evenly among all parties of which the voter approves. These fractional approval votes determine the seat shares of parties in the parliament.

The second kind of vote is a "coalition vote," which counts for all majority coalitions that are acceptable because they include all parties that the voter designates Y ("yes") and no parties that the voter designates N ("no"). The majority coalition acceptable to the most voters becomes governing, provided that the parties it comprises agree.

By placing a premium on precisely the most approved majority coalitions, CV encourages parties, before an election, to reconcile their differences and form coalitions that are likely to have broad appeal. Such coalitions, insofar as they formulate coherent policies, facilitate voter choices, producing a convergence of voter and party/coalition interests.

Theoretical properties of CV are analyzed, and optimal strategies of voters and parties are investigated; a measure of a party's "bargaining strength" is illustrated. CV's most likely empirical effects in faction-ridden multiparty systems, like those of Israel and Italy, are discussed. Finally, the practicality of CV as an election reform is considered, and some possible modifications in CV are mentioned.