Formal preliminaries of the political game

148 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 ˜ some of the eventual policy outcomes, associated with the lottery g , are changed to the 1 advantage of the party principal. Note however, in the example, that the probability that VVD will be a member of a 6 24 ] ] majority coalition drops from to as the party moves to the more extreme position. 7 31 If coalition members obtain perquisites from being in government, then for sufficiently high value perquisites, the expected utility of the VVD principal at VVD0 would be lower than at VVD. Thus coalition perquisites may transform the centrifugal tendencies generated by policy negotiations to centripetal tendencies, similar to those generated by electoral concerns. The above discussion concentrated on the best response of a single party, namely the VVD, to the location of the other two parties. To deal formally with the strategic choices of parties in the bargaining game just described we now consider the nature of the Nash equilibria in such a game.

3. Formal preliminaries of the political game

In the previous section we supposed that the set of parties N 5 h1, . . . , i, . . . , nj was exogenously given, as was the set K 5 h1, . . . , j, . . . , kj of voters. In fact a deeper model would be to suppose that there is a subset K9 of K of elite actors who form coalitions called parties, thus generating N. Some of the results presented below suggest how groups of elite actors might cohere to form parties. The political game takes place in a policy space, Z. For ease of analysis we shall assume that Z is two-dimensional, but there is no impediment to assuming Z is of higher dimension. Each voter, v, has a quasi-concave utility function u :Z → R, which we can v without great loss of generality assume is Euclidean, and takes the form u y 5 2 v 2 a iy 2 x i , where x is the voter’s ideal point and a is a positive constant. Each party, i, v v n makes a declaration z , so z 5 z , . . . , z [ Z is the policy or ‘‘manifesto’’ profile. Let i 1 n D be the n 2 1 dimensional unit simplex. A vector v 5 v , . . . , v [ D represents N 1 n N the shares of the vote for the parties, and a vector ev 5 e , . . . , e [ D represents the 1 n N share of seats of each of the parties. The vector ev is, of course, determined by the nature of the electoral system in use. The two vectors v and ev represent the post-election realizations of the behavior of the electorate. Prior to the election, all agents hold common knowledge beliefs about the n stochastic relationship between the vector z [ Z of party declarations and the electoral response. In particular we shall assume that the response by voter v is described by a continuous probability function n x : Z → D . v N Thus x z 5 . . . , x z, . . . , where x z is the probability that voter v picks party v v i v i i, at the manifesto profile z. The MNP model Quinn et al., forthcoming; Schofield et al., 1998a, 1999 mentioned previously does not assume that voter behavior is pairwise independent; nor does it assume away the possibility of strategic voting in the electorate. n ˜ However MNP does permit an estimate of the stochastic vote function C :Z → D . Here N N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 149 Cz is a continuous random variable, whose components are the random variables characterizing the vote shares of the various parties. Obviously C is determined fundamentally by the probability functions x , and the nature of the covariance h j v structure on the error function. n In particular, there is an expectation operator E C :Z → D , where EC z 5 ECz N is the expectation of the vote shares at the profile z. In the same way e Cz is the random variable describing the seat shares of parties at z, and Ee Cz is the expectation of this vector. Given any realized vector, e, of seat shares, we can compute the class of winning coalitions, say, where coalition M is a member of iff N 1 ] o e . . Let 2 be the family of all subsets of N. Clearly a family of winning i [M i 2 N coalitions, say, is a subset of 2 . Thus the information encoded in e Cz can be N interpreted as a probability distribution over subsets of 2 . In other words the choice of the policy vector z determines a finite lottery p z, : t 5 1, . . . , n . Here , . . . , h j t t 1 are all the coalition structures that are believed possible, and p x is the probability n t associated with at z. A particular election result is a realization, say , from this set t t of possible coalition structures. As noted above, one property that the MNP voting model possesses in the context of a proportional electoral system is that these probabilities, p , are smooth functions of z. We emphasize this by declaring it as an t assumption. n Assumption 1. The electoral probability function p:Z → D is a smooth function from n Z to the simplex D of dimension n 2 1. n Note that in general n is of order 2 2 1. As we observed in the Introduction, in the case n 5 2 there are three possibilities, either party 1 wins, or party 2 wins or 1 and 2 gain exactly the same number of seats. The situation with n 5 3 is obviously more complicated, since there are three cases where a single party wins, and three cases where one party gains no seats and the other two have the same number of seats. Finally there is the non-degenerate case where each two-party coalition wins, so 5 hh1, 2j, h2, 3j, h1, 3jj. For convenience we may ignore the grand coalition h1, 2, 3j. Clearly the formulation presented here is a natural generalization of the two party case. It is important to note that the usual probabilistic models of electoral behavior Coughlin, 1992 may not satisfy Assumption 1. However Assumption 1 is satisfied by the electoral probability function induced by the MNP model, constructed in Schofield et al. 1998a and Quinn et al. forthcoming, for any ‘‘strongly proportional’’ electoral system. By this term we mean an electoral system where vote shares and seat shares are almost identical. Of course, the coalition structure is determined by numbers of seats, and this may introduce some small discontinuities in empirical analysis. For theoretical work, the covariance structure on the errors and the resulting variance in vote shares can be interpreted to mean that the electoral probability function is indeed smooth. More importantly, Assumption 1 emphasizes that elections necessarily involve risk. We may suppose that the electoral information encoded in p is consistent with an MNP model of the form so indicated, and accessible to the parties through electoral sampling. Suppose then that the policy vector z is declared. Prior to the election the beliefs of the parties are described by the lottery p z, in the manner described. After the h j t t 150 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 election a particular winning or decisive coalition structure, say , is realized. For t convenience we shall let z denote this realized coalition structure. For the moment, let us ignore the portfolio payoff to coalition members. Given this electoral realization, we shall assume that all government policy outcomes which result will belong to a subset of Z termed the ‘‘heart’’, z. The ‘‘heart’’ is a concept from spatial ‘‘committee’’ voting theory Austen-Smith, 1996; Schofield, 1995b, 1999, forthcoming obtained by localizing the ‘‘uncovered set’’ Banks et al., 1998; Cox, 1987; McKelvey, 1986. For a fixed coalition structure, , it is known Schofield, 1999, that the heart n is a lower hemi-continuous lhc correspondence :Z → Z, and also admits a n continuous selection say g :Z → Z. In the discussion of the previous section based on Fig. 1 the heart, under , can generally be identified with the convex hull of the 1 declared position of the three large parties, hPVDA, CDA, VVDj. As in the previous section, we suppose that in choosing the policy declaration z , party i i chooses at the same time a leader or representative of the party called s , say, who has i a preferred point at z . Without loss of generality, we may suppose that this player, s , i i s 2 1 ] has Euclidean preferences derived from the utility function u y 5 2 iy 2 z i . To i i 2 determine the heart at the decisive structure it is sufficient to compute the ‘‘median’’ hyperplanes through the set of points z , . . . , z . h j 1 n Thus for example, if N 5 1, 2, 3 and is the family, 1, 2 , 2, 3 , 1, 3 , then h j hh j h j h jj 1 z will be the convex hull of the three points z , z , z whenever these are not h j 1 2 3 colinear. For preference profiles that are convex representable by quasi-concave utilities the heart lies within the Pareto set of the ‘‘committee’’ of party leaders. Moreover if z is a profile of positions such that the parliamentary game described by 6 has a legislative core , then z and the core coincide. In a situation characterized by , the policy z can be at a legislative core only if z lies within the arc [z , z ]. With 1 1 1 2 3 four parties, more complicated situations can arise. For example, under the coalition structure in the Dutch example, the PVDA position, z , will be a legislative core 2 1 point if it lies within the convex hull of the positions hCDA, VVD, D66j. The concept of the heart is utilized because of the theoretical and empirical inference that bargaining between parties must result in a ‘‘core’’ policy, when it happens that the legislative core is non-empty Banks and Duggan, 1998; Schofield, 1995a. We view as a natural ‘‘cooperative’’ theory that, in a sense, constrains the possibilities of coalition bargaining over policy in the post-election situation. Current work Winter and Schofield, 1999 suggests that the outcome of such bargaining will be ˜ a lottery g x, say, from the set of alternatives in z. n ˜ ˜ For a fixed coalition structure, , let :Z → Z be the correspondence that assigns n to each policy profile z [ Z the space of Borel probability measures on the set z. With respect to the topology of weak convergence Fudenberg and Tirole, 1991; ˜ Parthasathy, 1967 on Z the set of Borel probability measures in Z , this corre- spondence is also known to be lhc Schofield, forthcoming, and to admit a continuous 6 As indicated in the Dutch example in Section 2, if there is a legislative core at z, then there will be some party i whose position z cannot be beaten by a winning coalition M in , offering a policy x, say, in Z which is i preferred by all the leaders of the parties in M. N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 151 n ˜ ˜ ˜ selection Michael, 1956, namely a continuous function g :Z → Z such that g z [ n ˜ ˜ z, for all z [ Z . Assumption 29 requires that g z be a lottery chosen from z, n for each z [ Z, which is continuous on the space Z . The weaker Assumption 2 merely ˜ requires that g z be chosen from the Pareto sets associated with the coalitions in . ˜ ˜ Assumption 2. The policy outcome g z [ Z resulting from the realized fixed ˜ coalition structure at a policy profile z is a finite lottery g z 5 hr z,a z:M [ M M j, chosen in such a way that each a z is a ‘‘compromise point’’ for coalition M, in M the convex hull hz :i [ Mj, while r z is the probability that coalition M forms. i M n ˜ ˜ Moreover, the function g :Z → Z is continuous with respect to the specified topologies. n ˜ ˜ Assumption 29. In addition, the function g :Z → Z is a continuous selection of the n ˜ ˜ heart correspondence :Z → Z. ˜ ˜ It is important to note that we assume that the lottery g z [ Z does not depend directly on the seat or vote shares of the parties, but rather on the coalition structure . That is, we view the negotiation game between the parties to be based on coalition possibilities. If we assumed instead that a party could gain power by increasing its vote or seat share, even though remained unchanged, then parties would have a strong incentive to seek the electoral center. No centrifugal tendency would be possible. As we have emphasized, there may be a legislative core to the bargaining game given by the Euclidean preferences based on the vector of leader positions, z, and coalition structure, . In particular, if the legislative core is non-empty, then it will typically be at ˜ the declared point of one of the parties, z , say. Under Assumption 29, the lottery g z i 7 will be a singleton , with z occurring with probability 1. i To examine existence of equilibria in this model we must introduce the preferences of the parties, and the private perquisites that result from coalition membership. Assuming therefore that the coalition structure is realized, then coalition M [ forms with M probability r z, chooses a point a z [ Z, and allocates a vector s :i [ M of shares h j M M i of portfolios or perquisites to the members of the coalition M. In the most general case it would be appropriate to make hs j dependent on the realized seat shares Browne and i Franklin, 1973 within coalition M. In Model 1 below we shall just assume that the 8 portfolio shares are prespecified. Of course the behavior of the model will depend on the balance between policy preferences and the utility gained from government M perquisites. For convenience below, we sometimes specify s in terms of a scale h j i factor, which we shall denote by ‘‘r’’. 7 Notice that each a z will lie in the convex hull of z :i [ M . If z → z9 for some profile z9 with a legislative h j M i 9 9 core z , then Assumption 29 requires that a z → z for any M containing i, and r z → 0 for any M that i M i M ˜ does not include i. We shall call this the ‘‘core convergence’’ property for g . 8 However, the expected portfolio share of a party, in the coalition situation , will depend on . In early empirical work attempting to explain the distribution of portfolios, it was found that a theory based on a transferable value bargaining set predicted portfolio payoffs Schofield, 1982; Laver and Schofield, 1990. This predictor was based on , and not on actual seat shares. 152 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 n Assumption 3. At the policy profile z [ Z the ‘‘pre-election’’ von Neumann-Morgenst- ern utility for the principal of party i is M U z 5 O p z O r z [u a z 1 s ] . i t M i M i S D t M [ t n Moreover each U :Z → R is a continuous function in the argument, and there exists a i n n n ˜ ˜ ˜ continuous extension U :Z → R to the space Z of Borel probability measures on Z . i Obviously, to extend U in this fashion, it is necessary to extend the electoral i n ˜ ˜ probability function p to p :Z → D. To complete the model we need to be precise about the nature of the ‘‘policy utility’’, u , of party i. Consistent with the earlier discussion we assume that each elite member, i k , of party i has Euclidean policy preferences generated by a ‘‘sincere’’ policy utility of i the form 1 2 ] u y 5 2 iy 2 o i k k i i 2 where o is k ’s ideal point. As above, the ideal point o for party i is chosen to be that k i i i point which is at the multidimensional median of ho : k belongs to party ij. This point k i i is a proxy for the sincere ‘‘internal’’ voting ‘‘equilibrium’’ within the party, using n majority rule. Let o 5 o , . . . , o [ Z be the vector of party ideal points. 1 n Assumption 4. For each party i [ N, the ‘‘policy utility’’, u , is given by i 1 2 ] u y 5 2 iy 2 o i . i i 2 Assumptions 29 and 3 characterize a model in which coalition bargaining gives outcomes within the ‘‘policy heart’’ z. However this model can in principle be extended to a situation where bargaining is over both policy and perquisites Schofield and Sened, 1998. To see this, let W 5 Z 3 D be the product space of policy and distributions of perquisites D is the n 2 1-dimensional simplex. In the obvious 9 fashion the policy utility u :Z → R of i can be extended to u :W → R by defining i i 9 u y, s 5 u y 1 p s i i i th ˜ where p : D → R is the projection onto the i component. Let W be the space of all i probability measures on W, endowed with the weak topology Parthasathy, 1967. We ˜ 9 assume u : W → R is measurable with respect to the Borel sigma-algebra on W. i n If we assume the vector z [ Z encodes information about the chosen party leader’s policy preferences and preferences for perquisites, then for each coalition structure , n the more general heart correspondence :Z → W can be constructed Schofield, 1999. z , W is conceived of as the general constraint on bargaining over policy and perquisites, in the context of the coalition structure and leader characteristics, z. n ˜ ˜ As in the simpler case, z:Z → W is a lhc correspondence Schofield, forthcoming n ˜ ˜ and admits a continuous selection g :Z → W. Combining hg :t [ Tj with the risk t function of Assumption 1, gives the game form N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 153 n ˜ ˜ ˜ g 5 P p , g :Z → W. t t t n ˜ ˜ ˜ Composition of u with g, and extension to Z gives the utility function i n ˜ ˜ U :Z → R. i We present this framework as our most general Assumption 5. n n ˜ ˜ Assumption 5. The political game U : Z → R is induced from a continuous game form n n ˜ ˜ ˜ ˜ ˜ g 5 P p , g : Z → W where each g : Z → W is a continuous selection of the general t t t t n ˜ ˜ heart correspondence : Z → W induced by the post-election coalition structure . t t This framework allows us to introduce the definition of Nash equilibrium. In the analyses that follow, the Nash equilibrium notion can be interpreted in the appropriate fashion for the simpler games that are studied. n n n ˜ ˜ ˜ ˜ Definition 1. Consider the game U, Z where U : Z → R is the utility ‘‘profile’’ for the committee of party principals. The induced preference correspondence for i, n ˜ ˜ f : Z → Z is given by i ˜ f z , . . . , z , . . . , z 5 z [ Z :U z , . . . , z , . . . , z h i 1 i n i i 1 i n ˜ . U z , . . . , z , . . . , z ;z [ Z , j i 1 i n i n ˜ ˜ and the best response correspondence h : Z → Z for i is given by h z , . . . , z , . . . , i i 1 i ˜ z 5 hz [ Z : f z , . . . , z , . . . , z 5 fj. Since h z , . . . , z , . . . , z 5 h z , . . . , n i i 1 i n i 1 i n i 1 9 9 z , . . . , z for any z , z , we can combine hh j to form the joint best response i n i i i n n ˜ ˜ correspondence h:Z → Z in the obvious way. n ˜ A mixed strategy Nash equilibrium MSNE z [ Z is a fixed point of h, such that n z [ hz. Since f , h , h can all be restricted to Z , we may define a pure strategy Nash i i n equilibrium PSNE as a fixed point, z [ Z , of h if one exists when h is restricted to n have domain Z . The models described by these Assumptions give a fully specified continuous game n ˜ ˜ U, Z . Using standard arguments Bergstrom, 1975, 1992; Glicksburg, 1952 there ˜ will, in general, exist a MSNE Fudenberg and Tirole, 1991 when Z is compact, n ˜ convex, and h is continuous on Z . However PSNE may fail to exist when h is not n continuous on Z . As we now observe, there may exist equilibria that are ‘‘unstable’’ in a certain sense, and we wish to exclude these. n n ˜ ˜ ˜ Definition 2. A MSNE z for U, Z is stable iff for any neighborhood V of z in Z , there is a proper subneighborhood V 9 of z in V such that for any z [ V, then hz [ n V 9. Similarly, a PSNE z is stable if hz [ V 9, where h is restricted to the domain Z . Clearly if z in V is not stable i.e., unstable then ‘‘best response’’ from a perturbation z in V of z may lead to a point outside V. In the model just described we have left unspecified the electoral probability functions ˜ ˜ ˜ ˜ h p j and the coalition selections g , where g 5 g . We shall denote these by p and g h j t t t t 154 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 respectively, and leave the assumptions on these terms until the next section. For the M model based on Assumption 2, let G 5 s denote the scheme of private benefits or h j i perquisites that describe the private benefit to player i in coalition M. The remaining parameters describe the distribution of ideal points of the elite members of each of the parties. By Assumption 4, we may restrict attention to the vector, o, of party ideal points. n ˜ Definition 3. The Nash correspondence, p, g , maps the vector o [ Z of party G n n ˜ ˜ principals’ ideal points to the stable MSNE, z. Thus p, g :Z → Z . G ˜ It is our contention that, under certain constraints on p, g, the Nash correspondence n n ˜ gives PSNE. In this case, and when p, g are specified we shall write :Z → Z or G n n simply :Z → Z . ˜ Assumption 6. In a game described by the parameters p, g, G , if there exists a unique n PSNE z 5 o at o [ Z , then each party, i, declares the policy z and chooses as a i leader of the party that member of the party elite whose ideal point coincides with z . i We have assumed that the principal of each party determines the pure strategy Nash equilibrium policy to declare, and chooses as a leader that party colleague whose ‘‘sincere’’ or ideal policy coincides with the party’s Nash equilibrium choice. The case with a mixed strategy Nash equilibrium can be less easily interpreted. However it is our contention that unique PSNE are obtained in simple versions of this model. In the general model, a result on generic existence of local PSNE can be obtained Schofield and Sened, 1998. Of course there may well be multiple equilibria of this kind, so the selection problem must be solved. However, the results of the simple model are offered in the following section, because they suggest that non-convergent Nash equilibria are possible.

4. A model of three-party bargaining