Voting for a coalition government: A game-theoretic view

EuropeanJournalof POLITICAL ELSEVIER

European Journal of Political Economy Vol. 13 (1997) 537-555

ECONOMY

Voting for a coalition government: A game-theoretic view Stefano Vannucci

*

Dipartimento di Economia Politica, Universita' di Siena, Piazza S. Francesco 7, 53100 Siena, Italy

Received 1 September 1996; revised 1 January 1997; accepted 1 February 1997

Abstract I investigate direct electoral choice of the governing coalition in a multiparty parliamentary system and analyse some double-ballot electoral systems for inducing winning coalitional agreements among parties before the last round. The effectiveness of such electoral systems is shown to depend heavily on both the actual vote allocation and rules for the allotment of power among the parties of the governing coalition. Some alternative singleballot coalition-selecting procedures based upon proportional representation are also described and discussed. {3 1997 Elsevier Science B.V. JEL classification: C71; C72 Keywords: Coalitions; Strong equilibrium; Strategy-proofness; Proportional representation; Electoral

systems; NP-completeness

I. Introduction

Electoral systems affect the quality of public decision-making. However, there seems to be no consent concerning the criteria for assessing alternative electoral systems. Disagreement may but need not arise from conflicting basic value judgements. Indeed, it may reflect both the ambiguity inherent in criteria such as

* Fax: + 39-577-298661; e-mail: [email protected]. 0176-2680/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S0176-2680(97)0001 8-9

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'fair representation' or 'political stability' and the like and a concern for different drawbacks as typical of different electoral systems 1. In countries which have traditionally used the plurality method with single member constituencies or similar systems with a definite majoritarian bias, e.g. U.K. and other countries of Anglo-Saxon traditions, the flaws of such systems (i.e. the phenomena of 'minority rule', unfair representation and voters' alienation which are arguably most typical of them) are recognized and there is a growing interest in proportional systems 2. In some parliamentary democracies which have been using proportional representation for a relatively long time (e.g. Italy and Israel) there is also a growing dissatisfaction, due to some of the characteristic effects of proportional systems in a highly pluralistic and politically fragmented society. In particular, it is often argued that in a multiparty environment proportional systems leave the selection of the governing coalition to a bargaining process amongst the parties (i.e. their leading groups) so that the citizens are regrettably dispossessed of their right of choice. Conversely, so the argument goes, an electoral system which endows the citizens with the power to choose the, typically coalitional, cabinet is better suited to confer authority and stability to the governing coalition. This enhanced stability in turn would allegedly permit a more accurate assessment of the governing coalition's performance on the part of the electorate. To be sure, this argument for direct electoral choice is far from compelling if applied to environments which do not exhibit a reasonably fair allocation of resources and opportunities for agenda formation, access to information and deliberation (see e.g. Fishkin, 1991; Luciani, 1991). In this paper I take the foregoing criticism of current proportional systems at face value (without endorsement) and address the following issue: are there acceptable electoral systems which permit voters to choose the governing coalition directly, while avoiding the presidential shortcut? Or, to put it in the simplest terms, is direct choice of the governing coalition by the electorate a feasible option in a parliamentary system? To be sure, presidential systems, namely political systems relying on the (essentially) direct election of a president who shapes to a considerable extent the composition and activities of the executive but is neither subjected to a confidence vote nor endowed with a majority within the legislature, do guarantee direct electoral choice of a stable government. Unfortunately, this result is achieved by trading efficiency for direct choice and stability. In fact, presidential systems are arguably plagued by the endemic risk of startling inefficiencies due to the formation of weak 'divided governments' or, conversely, to lack of effective

1 See e.g. Bogdanor and Butler (1983) for a detailed review of electoral systems and Lijphart (1984) or Sartori (1994) for a classification of democracies. 2 See e.g. the persistent influence exerted both in U.K. and the U.S. by movements advocating electoral reforms.

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checks on presidential policies. Therefore, the answer to the foregoing question is indeed consequential. If direct choice of the governing coalition turns out to be consistent with a multiparty parliamentary system then it is confirmed that a better combination of efficiency and stability than that achieved by presidential systems is in principle available, provided that government-appointment and governmentremoval procedures are firmly divorced from each other 3 According to a widely held contention 4, the plurality system with single-member constituencies may help to achieve an essentially direct electoral choice of the government in a parliamentary system, but only (i) at a cost in terms of 'fairness' of representation and (ii) provided that the relevant polity is sufficiently homogeneous to allot a majority of seats to a single party. It is therefore worth exploring the feasibility of alternative electoral systems having the property of allowing a direct choice of the governing coalition by the electorate even in a multiparty environment and, possibly, at a less severe cost in terms of 'fairness'. In particular, is direct choice of the governing coalition consistent with proportional representation systems? Since the discussion which follows focusses on direct electoral choice, one should be very clear about the meaning of the latter notion. Indeed, in a multiparty environment, a distinction has to be made between a weaker and a stronger version of direct electoral choice of the governing coalition. Under the weaker version, coalition proposals are made by the parties, while the electoral body merely ratifies (or rejects) those proposals. Under the stronger version, the roles of the parties and the electoral body are reversed. Namely, the electoral body chooses, through the electoral process, the governing coalitions (or perhaps an ordered list of candidate coalitions to be considered for the governing role). The parties can either ratify the electoral choice or perhaps object to it (by refusing to form the prescribed coalition). However, they are in any case obliged to enter a coalition formation process, the guidelines of which are quite strictly prefixed by the electoral outcome (but see also Section 3 for further considerations on this point). Thus, direct electoral choice (of the governing coalition) in a weak sense can be conceivably achieved through an electoral system that provides the parties with suitably strong incentives to coalesce. On the other hand, direct electoral choice (of the governing coalition) in a strong sense obviously refers to explicit coalition-coting procedures. I shall consider electoral systems of both types. Concerning 'weakly' direct choice, double-ballot systems have been frequently

3 This is not to say that any parliamentary arrangement based upon direct choice of the government, or premier, will do: for instance, it may be noticed that the recent Israeli neo-parliamentary reform with direct election of the premier fails to ensure a parliamentary majority to the latter thereby being at risk of reintroducing the typical source of inefficiency of presidential systems (see e.g. Sartori (1994)). 4 Indeed, many political scientists and politicians tend to regard the 'Westminster model' as a paragon of effective democracy.

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invoked as the most practical means to achieve a direct electoral choice of the governing coalition without giving up fair representation. It is sometimes argued 5 that the first ballot might be used to allot part of the seats: after that, the parties should declare the coalitions into which they would be willing to enter, determining a set of alternative coalitions. Then the second ballot would take place, allowing the voters to allot the remaining seats amongst alternative coalitions, to the effect of determining the winning coalition by a direct choice. Different combinations of proportionality and plurality coalition premia might be tried in order to achieve the desired balance between fairness of representation and encouragement to coalition formation. A few objections against double-ballot systems of this type can be immediately raised. It is in fact at least conceivable that neither the first ballot's plurality-winner list nor the coalition which is most successful at the second ballot (perhaps due to the relatively poor performance of its members at the first round) command a majority of seats. To put it in more general terms, can we expect that the incentives to coalesce embodied in such systems are sufficiently strong to generate an overall (possibly trivial i.e. single-party) winning coalition? A negative answer to this question would entail a quite serious objection to such two-ballot systems, since in this case the only way to form a coalition government, short of repeating the electoral process would be, again, a bargaining process amongst the parties: just the kind of procedure to be avoided! Section 2 is devoted to an (partial) answer to the foregoing question. It is shown that indeed the emergence of a winning coalition as a (strong) equilibrium of the coalition formation game induced by a double-ballot system cannot be guaranteed. The same result obviously applies to similar single-ballot systems with (reliable) polls replacing the first round of the double-ballot electoral process. The somewhat negative flavour of this result suggests that one should perhaps consider more explicit coalition-selecting algorithms, if a direct selection of the governing coalition by the electoral body is to be ensured. Section 3 describes some such coalition-selecting procedures that rely on single-ballot systems with single-transferable vote and cumulative transferable vote, respectively (as well as Brams and Fishburn's multiple approval coalition voting: see Brams and Fishburn, 1989). Since coalition-selecting electoral systems are considerably more complex than standard voting methods currently used, their computational viability is apparently a prominent issue. Moreover, given the emphasis of the paper on fair representation, it follows that non-manipulability of the proposed coalition-selecting procedures is also a most relevant requirement. We establish some nice properties of such electoral systems, including computational tractability and (computational) strategy-proofness. Admittedly, the coalition selecting procedures we consider are quite complex and even somehow concocted.

5 See for example the recent Italian reformproposals as reviewed in Luciani (1991).

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However, they largely suffice to establish that, in principle, direct choice of the governing majority coalition is by no means precluded in a parliamentary system and can be implemented through proportional voting procedures.

2. Relying on incentives: Encouragement of coalitions by means of a doubleballot system As mentioned in Section ! we shall focus on double-ballot systems in order to scrutinize the viability of 'weakly' direct electoral choice of the governing coalition in a parliamentary setting without giving up fair representation. The reasons fi~r that move should be clear enough: single-ballot plurality systems do give up the latter without being able to cope with a resilient multiparty environment, whereas double-ballot systems may be expected to favour, after the first round, a more realistic assessment of the relative strengths of the parties, thereby enhancing the probability of a successful coalitional agreement before the last round. Moreover, if the incentives to coalesce embodied in a double-ballot system are strong enough and reasonably accurate predictions are readily available before the last round, the electoral system may conceivably afford a more substantial role for proportional representation (e.g. a certain percentage of the available seats) without substantially affecting the scope for coalitional agreements. To be sure, double-ballot systems come in a wide variety of types, depending on the nature of the relevant districts (e.g. single member or multi-member), on the kind of majority or coalitional premia that are possibly provided, and on the criteria that are used to select the last-round contestants. However, I shall be able to avoid most of the details concerning the particular double-ballot system in use by assuming perfect foresight on the part of the players. Thus, for any particular partition of the contestant lists, the winning list will be assumed to be predictable by the relevant actors. As a result, the vagaries of expectation formation processes will be in a sense ruled out as a possible source of perverse behaviour on the part of the electoral system. Nevertheless, it will be formally shown that formation of a (stable) winning coalition before the last round is not to be expected, unless certain quite particular conditions obtain: however strong, the incentives to coalesce may not be strong enough in a genuine multiparty system. We consider the following scenario: N party lists have obtained votes at the first electoral ballot, but none of them commands a majority of seats. The outcome resulting at the second ballot from each possible partition of N into coalitions is reliably predictable and common knowledge among the players. Parties get their payoff in terms of portIblios (or, more generally, of jurisdiction over policy issues whose 'weights' are measured by the amount of budget units allocated to them). This amounts to assuming that the parties are interested in policy issues (perhaps due to reputation effects), that each minister is given jurisdiction over a certain class of issues (with the understanding that the same

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may apply to internal subunits) and, somehow crudely, that the salience of each issue-portfolio unit to each party is approximately measured by the size of its budget. Between the first and the second ballot the players engage in a coalition formation game: we are interested in strong (i.e. coalitional Nash) equilibria of that game, with the understanding that strong equilibria qualify as 'stable' arrangements of parties into one winning coalition and one or more losing ones, while (possibly winning but) disequilibrium outcomes are bound to give rise to further unwelcome post-electoral bargaining and re-arrangements. The proposed model is the following. Let N = { 1. . . . . n} be the set of parties and I I ( N ) the set of all possible partitions of N. The winning opportunities open to the feasible coalitions are represented by a partitional simple game V = (N, g ) where ~ is a set of embedded winning coalitions (S, 7r) ~ 2 s × I I ( N ) ((we say that a coalition S ___N is embedded in partition ~-~ H ( N ) iff S ~ ~-). Obviously, V represents the relevant features of the (reliably) predictable distribution of electoral votes at the second round, under different possible scenarios. Thus, V is assumed to be such that the grand coalition is winning, i.e. (N, { N } ) ~ 6 After the first round and before the second one the parties, under common knowledge of V, play a coalition formation game in strategic form: each player i ~ N proposes a coalition S ~ 2 N, with i ~ S, that he is willing to enter and claims a certain fraction qi ~ [0, l] of the available cabinet portfolios. Thus, the strategy set of a player i ~ N is 2 g × [0, 1]. A coalition S is formed if each member-player i ~ S proposes S itself and the claims of the players in S are consistent, i.e. ~i e sqi ( (r ) <- 1. Otherwise the players in S stay alone. As a result a partition ~-= ~'(~r) of N obtains for any strategy profile ~r = ((r 1. . . . . (r). The partition may or may not include a winning coalitional agreement i.e. a coalition S such that (S, ~-) ~ 9 . If it does then the payoff ui((r) is qi for any i e S and 0 for the other players, otherwise the payoff is 0 for every player (the governing coalition will definitely be formed after further post-electoral negotiations). As mentioned above, we are interested in stable winning coalitional agreements, in order to discard those coalitions and cabinets that are undermined from the very start by disruptive forces. Therefore, we look for strong equilibria of the coalition formation game G ( V ) = (N,(2N × [0, 1])i~ N, (ui)i~ N) (a strategy profile ~ = ((r I . . . . . o-n) is a strong equilibrium of the coalition formation game if for any S ~ N and any T s ~ (2 u X [0, 1]) s if (ui(o'))i~ s ¢ (ui( Ts, o'U\S))i~ S then a player i ~ S exists such that ui(o-) > Ui(T s, o" N/S)).

6 The following requirements, though not necessary natural for V: (i) regularity (i.e. for no rr, S, T: superadditivity, namely for any pair of coalitions S, T then both (S U T, ~rs / T) ~ ~ and (S U T, ~r'r / S ) ~ ~ denotes the partition {U t_) T}U{S • ( N / T ) : S ~ ~-}).

for the results presented below, are also quite S ~ T and {(S, 7r), (T, ~-)}_c~) and (ii) s.t. S ¢q T =~i, if (S, ~-) ~ ~ and (T, ~ ' ) ~ V (here, for any (U, 7r) ~ ~ and T c N, 7r v / T

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The following result establishes that, whatever the details of the double-ballot electoral system, the existence of strong equilibria of G(V), hence of 'ex-ante' stable and winning coalitional agreements, is not generally warranted.

Proposition 1. Let G(V)=(N, ((2N×[o,1])i)i~N, (ui)i~ N) be the coalition formation game induced by V as defined above. Then G(V) has strong equilibria if and only if V is weak i.e. has some vetoer (a vetoer of V = (N, ~ ) is a player i ~ N such that (S, 7r) ~ 9 only if i ~ S. Proof.

Let V be non-weak and 00 = (o°1 . . . . . o-n) a strong equilibrium of G(V). Since (N, {N}) ~ 3 , it must be the case that ~ i ~ sUi(00) = 1 for some S ~ 7r(00). Now, by non-weakness of V, for any i ~ S a coalition T(i) exists such that i ~ T(i) and (T(i), rr') ~ 9 for some 7r' ~ II(N). W.l.o.g. take a i ~ S such that ui(00) > 0. Then, consider any strategy profile ~- s.t. for any j ~ T(i): 7j = (T(i), uj(00) -~- tli(00)" t-J)), where t = v~ T(i). By construction, Y~j~T(i)Uj(T) 1 and uj(-c) > uj(00) for any j ~ T(i): it follows that 00 is not a strong equilibrium, a contradiction. Conversely, let V be weak, i ~ N a veto player of V and S a coalition s.t. (S, ~-) ~ 9 for some 7r~ II(N). Then, take a strategy profile Cr with 00, = (S, 1) and 009= (S, 0) for any j ~ S/{i}. For any coalition T c N and any strategy profile r r for T, if (Uj("FT, O-N/T))j¢ T :'7t=(Uj(00))j¢ T then ui(r) < ui(o'), by construction. Thus 00 is indeed a strong equilibrium of G(V). [] =

Remark l.

The strategic coalition formation game presented above is very similar to the strategic claim game introduced in Borm and Tijs, 1992. The basic difference between the two of them boils down to the fact that the former construction builds upon partitional simple games, while the latter relies on general coalitional games with non-transferable utility. Concerning our model of coalition formation, its most restrictive assumptions consist perhaps in requiring the cabinet portfolio units to be perfectly fungible for each party, quite independently of the contents of their jurisdiction and the number of such units to be expandable at will in order to accommodate the parties' claims.

Remark 2. The role of veto players in Proposition 1 mimics the role they play in establishing non-emptiness of the core in coalitional simple games and parallels the use made of various sorts of 'focal' players in other coalition formation theories, in order to shrink the set of plausible winning coalitions (see e.g. the role of 'dominant' players, i.e. players enjoying a 'strict majority' within a winning coalition, in Peleg's approach to coalition formation in simple games (see Peleg, 1981) and of 'strong' (and 'very strong') parties in Laver-Shepsle's multidimensional spatial model of coalitional cabinet formation in parliamentary systems (a party is 'strong' in Laver-Shepsle's sense if it retains some portfolios in any cabinet that commands a majority against its monopoly cabinet and 'very strong' if its monopoly cabinet is majority-unbeatable: see Laver and Shepsle, 1996)).

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Clearly enough, existence of vetoers is a very restrictive condition. In order to probe the robustness of such a (partly) negative result concerning 'stable' winning agreements under double-ballot systems, we define yet another coalition formation game in strategic form, slightly different from the above. We start from a partitional simple game V = (N, 9 ) as defined previously. However, we also assume that, again perhaps by convention, the allotment of portfolios takes place according to a prefixed proportional rule p: QN ._..>QN based upon the (strictly positive) voting weight vector v = (v! . . . . . vn) as determined by the votes gained by the party lists at the first ballot. The ensuing coalition formation game is

G(V, p, v) = ( U, ((2 u × [0, l])i)i~ N, (ui( p, V))i~N), where for any strategy profile o- and any player i ~ N : [ui( p, v)](o-)= [ p(vi/v(S))] i if (S, cr(o-)) c5~ obtains for the coalition S that includes i (for any coalition S we denote by v(S) its voting weight i.e. v ( S ) = Ei~ sVi); otherwise [ui( p, v)](o-)= 0. In words, party i gains its proportional lot of portfolios, according to rule p, if it happens to be a member of the winning coalition and nothing otherwise. The following result obtains:

Proposition 2. Let G(V, p, v) be a coalition formation game as defined above. Also, suppose that V and v be such that the set of winning-and-blocking coalitions {S ~ N." (S, 7r) ~ g for some e t c H ( N ) and S f3 T 4=O for any T s.t. (T, 7r') ~ 9 for some cr' ~ H(N)} has a unique coalition S* with minimum voting weight. Then, a strategy profile o" is a strong equilibrium of G(V, p, v) if and only if o-i = (S*, [ p ( v i / v(S* ))]i) for any i ~ S*. Proof. Indeed, let T c N , r r be such that ([ui( p, v)](o-)i~r4=([ui( p, v)](r T, O-N/T))i~ r and [ui( p, v)](r r, O-N/r)> [ui(P, V)](O-) for any i ~ T. Hence, it must be the case that (T, ~r(r r, O-N\r)) ~ . But then, by definition of S* and o-, T N S* 4= O. Now, for any i ~ T N S*, [Ui( p, U)](T T, O-N\T) ~ [Ui( p, V)](O-). However, by definition of S*, v ( S * ) < v(T); hence, by definition of ui( p, v), [ui( p, v)](o-) > [ui( p, v)](r r, o-N/r) for any i ~ Tel S*, a contradiction since T fq S* 4=Q~. Conversely, let r be a strong equilibrium s.t. r i 4=(S*, [p(v)] i) for some i ~ S*. Then S* ~ ~-(r), by definition of G(V, p, v). Obviously, (S, ~-(r)) ~ @ for some S ~ 7r(r) (otherwise [ui(p, v ) ] ( r ) = 0 for any i ~ N and the strategy profile r / = ((N, [ p(v/v(N)]i))i~ U would clearly ensure a strictly higher payoff to each player i ~ N, a contradiction since r is by hypothesis a strong equilibrium). But then S fq S* ~ Q and v ( S * ) < v(S), by definition of S*. It follows that [ p(v/v(S* ))]i > [ p(v/v(S)]i for any i ~ S A S* : hence [ui( p, v)](o-) > [ui( p, v)](r) for any i ~ S*, is a contradiction. [] Thus, Proposition 2 provides another set of quite restrictive, if not inconceivable, sufficient conditions that ensure the existence of 'stable' winning coalitional

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agreements: namely, if portfolio units can be arbitrarily small and their allocation is constrained to be proportional and the distribution of votes at the first round is such that there is a unique coalition having the minimum sum of votes among the predictably winning-and-blocking ones, then such a coalition is 'stable' as well. It is to be stressed that the requirement of 'perfect' portfolio divisibility cannot be easily be dispensed with: if integer portfolio-payoffs are used then one has to face the well-known fact that quota-monotonic proportional allotment rules do not generally exist (see Balinski and Peyton Young, 1982, Theorem 4.1). Remark 3. It should also be emphasized that Propositions 1 and 2 can be immediately rephrased and applied to a single-ballot electoral system, provided that the voting weight vector v is reinterpreted as the result of a (fully reliable) public poll. Thus, the somehow negative result embodied in those propositions should not be ascribed to two-ballot systems as opposed to single-ballot systems. Rather, it has to do with the fact that, under such systems, coalitions are merely encouraged, i.e. the final decisions concerning coalition formation (either between the two electoral rounds or between the last relevant polls and the beginning of the electoral process) are not explicitly and formally constrained by the electoral outcomes through the electoral rules.

Overall, the message conveyed by Propositions 1 and 2 has a somewhat negative flavour. They confirm that double-ballot systems may implement (weakly) direct choice of a winning stable coalition, but not necessarily so. Clearly enough, and perhaps unsurprisingly, the details of the relevant party system, of the vote distribution and even of the portfolio-allotment conventions are of the essence when it comes to establishing whether the incentives to coalesce embodied in the electoral system are strong enough to produce 'stable' winning coalitional agreements.

3. Turning to direct methods: coalition-selecting procedures We have just seen that all double-ballot electoral systems, including of course those based upon plurality or proportional voting and (possibly) coalitional premia, may well fail to be able to select the coalition cabinet in a multiparty environment. This suggests that if 'stability' of the governing coalition is indeed the main concern, then one should perhaps consider electoral systems that map ballots into coalitions in a more straightforward way. I shall describe below a few coalitionselecting procedures. Before that, some preliminary general comments are in order here. First, in what follows I will focus on voters and coalition of voters, as opposed to parties or lists, as the relevant agents: parties themselves can also count as agents, but only insofar they are able to act as coalitions or 'blocs' of voters. Thus, I will make no attempt at distinguishing between 'rigid', or strictly

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deterministic, and 'flexible' procedures (i.e. coalition-selecting algorithms for an ordered list of governing coalitions). Indeed, the latter do leave (in principle) some room for postelectoral strategic manoeuvering within the legislature. Since this section is focussed on voters, I will deliberately ignore this manoeuvering opportunity and leave a full-fledged strategic analysis of such procedures as a topic for further work. In my view, this simplifying move is in fact a relatively innocent one, but in any case a simple alternative route is readily available to the reader: just ignore the 'flexible' variants in the list of procedures to follow and only retain the 'rigid' ones (allowing for suitable adjustments in the relevant definitions whenever necessary). Second, the present section is largely devoted to establishing that at least some of the coalition-selecting procedures described below are both (computationally) 'easy' to operate and (computationally) 'difficult' to manipulate. The underlying presumption is that both these properties are indeed highly desirable ones. One may well wonder why that presumption should be accepted. However, computational tractability is always a major requirement for any voting method. A fortiori, this must be the case with apparently more-complex-than-usual mechanisms as the coalition-selection procedures I am going to consider. Concerning the desirability of nonmanipulability, it should be recalled that I am trying to establish whether, or to what extent, fairness of representation and stability of a coalition government can be reconciled. Indeed, concern for fair representation is the only reason why I am going to insist on proportional methods or outcomes. Now, fair representation arguably alludes to 'truly' held opinions and preferences (over and above possible strategic motives), so that strategy-proofness requirements are clearly most relevant when it comes to fairness. It is well known that, unfortunately, strategy-proofness is typically not achievable by nice voting methods: but then, (at least worst-case) computational intractabili~ of manipulation tasks can be regarded as a tolerably effective, if not fully reliable, safeguard against attempted manipulations 7. Finally, it should be noticed that the procedures described below refer to a single district. This is done in order to avoid complications that are entirely irrelevant to the main argument of this paper. Let us then suppose that a polity N of n voters with a set M of m < n parties has to select k representatives out of h > k candidates from those parties and a governing coalition of parties having at least ( k / 2 ) + 1 seats. (The following discussion refers to a single distriet in order to avoid complications which are conceptually irrelevant to our general argumen0.

7 Whatever the force of this argument concerning the desirability of 'computational' strategy-proofness, it remains to be assessed the relevance of the NP-completeness criterion (a worst-case criterion) in that connection. This is a very difficult question, which I am not prepared to discuss here. Thus, my argument concerning 'computational' strategy-proofness of the proposed coalition-selecting procedures is in fact a conditional one which depends heavily on the aptness of NP-completeness as a criterion of computational intractability.

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A reasonably simple single-ballot coalition-selecting procedure has been recently proposed by Brams and Fishburn, 1989, under the name 'coalition voting'. It works as follows. (a) Coalition voting through multiple approval (with transfer of" surplus and elimination) (MTE-CV) (see Brams and Fishburn, 1989): Each voter i ~ N is asked to indicate the party lists she approves of. Her vote is counted according to the following rule: if she approves a set C c M of 1 parties, every approved party c~ receives vi(o~)= 1/l votes from i; moreover one coalition vote is allotted to every nonempty subset of C (including C itself). Then, seats are allotted to party-lists 'proportionately' (w.r.t. total party votes). This is accomplished by the following procedure. Let h be the number of available seats: then the quota is the number q = (nh + 1)/(h + 1). Each candidate ol who receives v ( a ) > q votes is elected, while the difference ( v ( c ~ ) - q ) is c~'s surplus and ( v ( c ~ ) q)(vi( ol)/t,(o~)) is voter i's share of the surplus. If the quota is achieved by h' < h candidates then for any elected candidate ce and any supporter i of o~, i's share of c~'s surplus is distributed equally among the non-elected candidates voted by i. If after this transfer of surplus the number of elected candidates is still h" < h then a similar transfer is operated from lists (candidates) with minimum party votes, that are simply eliminated from the contest (see e.g. Bolger, 1982 for a detailed description of this procedure). Of course, at any ballot at least one set of parties will collect at least one half of party votes. Moreover, at least one such set of parties will not admit a proper subset enjoying the same property: call any such set minimal majority coalition . The governing coalition is the minimal majority coalition with maximum coalition votes (a suitable tie-breaking rule can be used whenever a tie obtains). If the governing coalition has exactly one half of the available seats, then an extra seat (or vote) is introduced and allotted to it. Variants of the foregone procedure obtain by varying the apportionment method used in order to achieve a proportional allocation of seats among party lists (i.e. replacing multiple approval voting with single transferable voting or with cumulative voting). Let us consider the following variants. (b) Coalition voting through cumulative voting (with transfer of surplus and elimination) (CTE-CV): Under this method, the voters are asked to allocate h votes among the m party lists, if h is the number of available seats (fractional votes are allowed). The allocation submitted by any player counts as one coalition vote for the set C of lists with a positive vote and for all nonempty subsets of C. Then seats are allotted to lists (candidates) 'proportionately', possibly transferring ('proportionately' again) surpluses from both elected and low-ranking candidates (see e.g. Bolger (1982) and the description of MTE-CV given above for details). The minimal majority coalitions are then computed. The governing coalition is selected among the minimal majority coalitions with maximum coalition vote (as under MTE-CV). (c) Coalition voting through single transferable voting with partitioning (STVPCV): Each voter is asked to submit a linear ordering of the party lists and a

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partition of the lists into two equivalence classes i.e. 'approved lists' and 'not approved lists' (the partition must be consistent with the linear ordering, namely it must be a suitable 'truncation' of the latter). Seats are allotted according to the single transferable vote procedure as follows. Let h be the number of available seats. Then, again, the quota is q = (nh + 1 ) / ( h + 1). For any candidate a the number of voters who choose a as their first choice is denoted v ( a ) . If v ( a ) > q then c~ achieves the quota and is elected. If the number of elected candidates is h' < h then for any elected candidate a and any voter i having a as her first choice an amount of (v(ce) - q ) / v ( a ) votes is transferred to i's second choice (according to i's submitted linear ordering). If the number of elected candidates is still lower than h then the candidate (lists) with minimum score are eliminated and their votes are allotted to the remaining candidates according to the same procedure outlined above (see e.g. Bolger, 1982 for more details and Bartholdi and Orlin, 1991 for an explicit algorithmic presentation of the STV procedure). After allotting the seats, the governing coalition is selected among the minimal majority coalitions according to the same procedure as under MTE-CV and CTE-CV. (d) Coalition voting through single transferable voting with dissent minimization (STVD-CV): The parties are asked to list the coalitions they are not willing to enter (this may, but need not, amount to vetoing a certain number of parties as possible partners in a cabinet). As a result, a set /7 * of feasible coalitions is determined. Each voter i ~ N is asked to submit a linear ordering R i of the set M of parties (plus possibly the preferred candidate from the preferred party). The seats are allotted according to the single transferable vote (STV) method (see the description above). The governing coalition is computed as follows. For any linear ordering R on M and any j ~ M let us define the 'dissent score' of j at R the positive integer r - 1, where r is the rank of j according to R. Then at each ballot (Ri)i~ N and for any coalition C M , each party j ~ C receives a 'dissent score' which is determined by the sum of 'dissent scores' of j at the linear orderings of voters whose first choices are other members of C. The governing coalition is a feasible majority coalition with minimum 'dissent score' (hence, in particular, a minimal majority coalition). If there are many feasible coalitions with minimum 'dissent score', a suitable selection method is used. If there are no feasible winning (i.e. majority) coalitions, then the (unfeasible) coalition with minimum 'dissent score' is considered. If this coalition is vetoed again by some of its putative members, the voting parties and their votes are canceled, and the procedure restarts by ignoring them. (Alternatively, one might consider the case of no feasible winning coalitions as a failure calling for new elections; if so the 'reconsider or perish' subroutine mentioned above might be used as a default procedure to be activated only after a certain number of failures.) Remark 4. It should be stressed that multiple approval voting, cumulative voting and single transferable voting tend to result in proportional outcomes in the

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following sense: under those methods a coalition of c voters (out of a population of n participant voters) can guarantee the election of [hc/n] candidates (where h is the number of available seats and Ix] denotes the largest integer not larger than x; to be sure, this is true of single transferable voting provided that hq < n; on the other hand, proportional outcomes cannot be achieved by means of plurality voting: see Bolger, 1982). Therefore any coalition voting method which is based upon such seat allocation procedures is indeed fully consistent with proportional representation. A detailed analysis of the normative and strategic properties of the foregoing procedures is left as a topic for another paper. However, it should immediately be noticed that such procedures are clearly anonymous, efficient and (essentially) neutral. Moreover, they enjoy an interesting complexity-theoretic property (with possibly significant strategic implications). Unfortunately, making this last point clear requires a few preliminary notions. We recall that a computational decision problem is in P ( or is deterministic polynomial) if there is a deterministic algorithm (e.g. a Turing machine) M such that (i) M always stops after a finite number of steps (ii) M solves the problem (giving a definite yes-no answer) in 'polynomial time' i.e. after a number of steps which is bounded above by a polynomial function of the problem's input length (under a 'reasonable' encoding of the latter). If such an algorithm M solves the given problem after a number of steps which is bounded above by a polynomial function of both the problem's input length and the maximum absolute values of the numbers occurring as inputs then M is said to be a pseudo-polynomial time algorithm. P-problems are usually regarded as the main instance of computationally tractable problems (as opposed to, say, exponential problems, that are defined in a similar way) (see e.g. Garey and Johnson, 1979; Bartholdi et al., 1989). If, however, a priori bounds are given on the numerical inputs of a problem that is solvable in pseudo-polynomial time, then such a problem is also to be regarded as computationally tractable (see again Garey and Johnson, 1979). A class of problems whose computational tractability status is somehow uncertain is NP, the class of nondeterministic polynomial problems (a decision problem is in NP if it is solvable in 'polynomial time' by a nondeterministic Turing machine i.e. a Turing machine which algorithmically checks in 'polynomial time' a candidate solution which is randomly selected at a previous 'guessing' stage). It is easily shown that P c NP, while the validity of the converse inclusion is unknown. However, there is strong 'empirical' evidence that P ~ NP, hence that N P \ P ~ (~. If this is indeed the case, then some problems in NP, including in particular the 'hardest' NP-problems, are to be regarded as computationally intractable which means that in some instances they may require a huge amount of time in order to be solved (and are therefore practically unsolvable). Now, 'hardest' NP-problems can be precisely defined and have been shown to exist: they include the NP-complete decision problems, namely those NP-decision problems such that any other NP-problem can be

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transformed to them in 'polynomial time' by a deterministic Turing machine. More generally, NP-hard computational problems can be defined in a similar way: they are those computational problems (not necessarily of the yes-no type or in NP) such that any NP-problem can be transformed to them in 'polynomial time' by a deterministic Turing machine (see Garey and Johnson, 1979). The foregoing discussion can therefore be summarized as follows: whenever a certain computational problem is proved to be NP-complete or, more generally, NP-hard, it can be concluded that there is strong evidence that at least in some instances such problem will prove to be intractable, or practically unsolvable. We are now ready to state the last result of this paper, concerning the computational complexity of (some of) the coalition voting procedures described above. It says that (some of) those coalition-voting procedures, while clearly more complex than any electoral system currently used, cannot be easily dismissed on computational grounds.

Proposition 3. (i) Computing the governing coalition under STVP-CV and STVD-CV is solvable in pseudo-polynomial time, while manipulating any such coalition voting procedure (i.e. finding an insincere or 'strategic' vote which ensures a better outcome than the sincere vote) is an NP-hard problem. (ii) Suppose that when a voter approves of only one party list, she lexicographically ranks parliamentary strength (i.e. number of seats) of that party above its participation to the governing coalition. Then, under MTE-CV, CTE-CV, STVP-CV and STVD-CV, even if a voter knows the (sincere) votes of all voters, deciding whether the situation is non-monotonic, namely whether there is scope for turning a loser into a winner by downgrading its positions on some individual ballots, is an NP-hard problem. Proof. (i) Indeed STVP-CV and STVD-CV can be decomposed into two subproblems. The first one is common to both procedures. It amounts to allotting seats to party lists according to the STV procedure: a P-problem (see Bartholdi and Orlin, 1991). The second subproblem of STVP-CV is at least as hard as the following decision problem: (P1) given are the set M = {1 . . . . . m} of parties (or party lists), a ballot response profile / / = (Tr I . . . . . ~'m) describing the distribution of votes among parties, and its corresponding seat allocation W = (w~ . . . . . Win); determine a majority coalition M ' ~ M , i.e. a coalition M' s.t. ~,i~M,wi > [(Y'./~ Mwi)/2], with maximum (possibly zero) coalition votes. Clearly enough, this problem is at least as hard as the corresponding decision problem, namely given an integer z and M, H , W as defined above determine whether there is an m ' c m s.t.F.i~ M,"['1"i~ 7. and Ei~ M'Wi ~-~ [(~iE M W i ) / 2 ] • Similarly, the second subproblem of STVD-CV is at least as hard as the following: ( P 2 ) given are the set of parties M = {1 . . . . . m}, a seat allocation (concerning parties) W = (w I. . . . . win), a dissent score profile (concerning parties

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and depending upon a certain ballot response profile H ) D ( I I ) = ( d l ( H ) . . . . . dm(11 )) and an integer z; determine whether a feasible majority coalition Mj _c_M exists such that £i~ v ' d i ( 1 1 ) <- z. Thus, it appears that both P1 and P2 are typical instances of the following decision problem: zero-one integer programming (with a single constraint and integral parameters): given are c, x ~ Z m, b, u ~ Z; determine whether there is y ~ {0, 1}" such that (x, y) _< b and (c, y) >_ u. Indeed, zero-one integer programming obtains from P 1 by putting x = - W , b = )Zi~ M wi/2, C = 7r, u = Z while it obtains from P2 by putting x = D ( H ) , C = W , u ~- ~ i ~ M w i / 2 ,

b = z.

Now, while general zero-one integer programming is known to be NP-complete (see Garey and Johnson, 1979, p. 245) it reduces to a pseudo-polynomial time solvable problem when the relevant parameters x, c, b, u are integral (see Papadimitriou, 1981, Schrijver, 1986). It follows that computing the governing coalition under STVP-CV and STVD-CV is pseudo-polynomially time-solvable. On the other hand, manipulating STVP-CV or STVD-CV involves manipulation of the STV procedure. But STV-manipulation is an NP-hard problem, because it typically requires exhaustive search in the set of preference orderings, which grows exponentially w.r.t, the number of alternatives (see Bartholdi and Orlin, 1991). (ii) Given our 'lexicographic' assumptions on the voters' preferences, the result of Bartholdi and Orlin, 1991, on NP-hardness of effective manipulation of STV extends to STVP-CV and STVD-CV. Moreover, the Bartholdi-Orlin proof can be easily adapted to MTE and CTE (in fact both MTE and CVE are non-monotonic if the number of available seats is at least 2 (see Fishburn, 1982; Bolger, 1983). Thus NP-hardness of non-monotonicity extends to MTE-CV and CTE-CV by virtue of our 'lexicographic' assumption concerning the voters' preferences. [] Corollary. The STVP-CV and STVD-CV coalition-selecting procedures are 'easy' to compute and 'difficult' to manipulate. Proqf~

The thesis follows from the proof of Proposition 3 (i) and from the fact

that s i z e ( / / ) : = m + Ei[log21~ril + 11 (size( D( 11 )): = m + Ei [log2 Idi( 11)l + 1], respectively) are clearly both bounded above by size ((nm)i= j ....... ), while Y2iwi = h (the fixed size of the legislature), so that in any case we are endowed with natural bounds for the relevant parameters of our problem. [] Proposition 3(i) and its corollary have a certain positive flavour, in that they show that there are direct coalition-selecting procedures which are simultaneously faithful to proportional representation, 'easy' to operate in actual Remark 5.

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practice and endowed with a obstruction to strategic manipulation. The relevance of this computational obstruction should be carefully assessed since NP-hardness results refer to a worst-case analysis. In general, NP-hardness does not rule out the possibility that, on average, the computational problem at hand be quite tractable. In any case, knowing that manipulation under those procedures may prove to be an exceedingly difficult task can be regarded in itself as a somewhat positive result, especially if one is prepared to concede that constraints on available computational resources may be more stringent for individual voters than for 'society' as a whole. Moreover, Proposition 3(ii) is definitely good news, in that it shows that it may be very difficult for a voter to decide whether perverse non-monotonic behaviour on the part of the electoral system does indeed take place. Since the coalition voting procedures considered above are non-monotonic and monotonicity failures are usually regarded as a most disturbing drawback for a voting procedure, Proposition 3(ii) does in fact assert that those possible failures may well be hard to detect. This, in a sense, partly disqualifies the non-monotonicity argument against the foregoing coalition voting procedures (see Bartholdi and Orlin, 1991 for a more detailed discussion of this point with reference to STV). Finally, it should be stressed again that, as mentioned above, the coalition voting procedures we are considering are on other respects definitely nice, enjoying suitable anonymity, neutrality and efficiency properties. All in all, Proposition 3 may be taken to confirm that there are nice procedures for direct coalition voting in a parliamentary system which are both consistent with proportional representation and reasonably resistant to strategic manipulation. Conjecture. It is an open problem whether the results of Proposition 3(i) can be extended to MTE-CV, CTE-CV and possibly to similar procedures based upon different apportionment methods such as so called 'quota methods' (where the quota qi of party list i is the size of the legislature times its fraction of total party votes (see Balinski and Peyton Young, 1982). Since all those methods are non-monotonic (see Balinski and Peyton Young, 1982; Bolger, 1983; Bartholdi et al., 1989) the author is led to conjecture that such an extension should be possible. 4. Related literature It is well-known that until recently the analysis of coalition formation (stricto sensu) has been largely neglected within the game-theoretic literature (however, see Greenberg, 1994 for an excellent survey focussed on 'stable' coalition structures.) Therefore, it is not surprising that the amount of literature addressing the problem considered in the second part of the present paper is not particularly rich. To the best of my knowledge, the most closely related attempts to formalize coalition formation processes are the strategic claim games introduced in Bonn and Tijs, 1992 and the model presented in Hart and Kurz, 1983, 1984 and Kurz, 1988, which combines a version of the Shapley value with use of the strong

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equilibrium concept (see also Aumann and Myerson, 1988 for a somewhat similar model in a more noncooperative vein, and Peleg, 1980, 1981 for a quite different approach that relies on various criteria, including the possible existence of 'dominant' players, for determining a small subset of plausible winning coalitions in simple games). Other contributions that provide analyses of coalition-cabinet formation processes in a strategic setting include Austen-Smith and Banks, 1988, 1990, Laver and Shepsle, 1990, Baron, 1991, Schofield, 1992. Mueller, 1993 offers some interesting ideas concerning the design of new institutional frameworks conducive to a reasonable compromise between fairness of representation and executive stability in a multiparty environment. Laver and Shepsle, 1996 offers a very detailed study, with applications, of a model of coalitional cabinet formation in a multidimensional spatial setting. The literature on coalition voting procedures is still in its infancy. Brams and Fishburn, 1989 is a pioneering paper on coalition voting. Brams and Fishburn, 1993 presents another version of their MTE-CV procedure illustrated above, called "yes-no voting', as a method for choosing a coalition-cabinet in a parliament.

5. Concluding comments The results presented in part 2 of this paper suggest that two-ballot electoral systems, including those based on coalition premia, may well be conducive to the formation of a ' winning' coalitional agreement before the last round under certain circumstances (i.e. vote allocations), but it is unlikely that this will be the case under any vote allocation. Similar results obtain, afortiori, for single ballot systems of a similar type (i.e. without explicit rules for coalition voting on the part of the voters). Hence, if one insists that, under any circumstances, the governing coalition is to be chosen by a direct electoral choice as opposed to a bargaining process among the parties, then it might turn out to be better to opt for a suitable single-ballot electoral system endowed with a coalition-selecting algorithm. The coalition-voting procedures for choosing the governing coalition by means of general elections which I consider in Section 3 are consistent with proportional representation in a parliamentary system and exhibit desirable symmetry and efficiency properties, in addition to a certain 'computational' resistance to manipulation. Such procedures may not be compelling, but they suggest that an imaginative approach to institutional design problems, as opposed to exclusive reliance on dubious empirical regularities and idiosyncratically selected paragons, should be regarded as most welcome in this politically sensitive field.

Acknowledgements This paper is a substantially revised and enlarged version of Vannucci (1993). Thanks are due to the participants at the 1996 Tel Aviv-Tiberias Meeting of the

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E u r o p e a n P u b l i c C h o i c e S o c i e t y for u s e f u l d i s c u s s i o n s a n d criticisms. C o s i m o S p e r a p o i n t e d o u t a fault in a p r e v i o u s draft. T h e d e t a i l e d a n d c o n s t r u c t i v e c r i t i c i s m s o f a n a n o n y m o u s r e f e r e e h a v e h e l p e d m e to i m p r o v e s i g n i f i c a n t l y the p r e s e n t a t i o n . O f course,

the usual d i s c l a i m e r applies. F i n a n c i a l s u p p o r t f r o m

M U R S T 6 0 % is g r a t e f u l l y a c k n o w l e d g e d .

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