A model of three-party bargaining

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

The general model just described is intended to extend a two-party model presented earlier by Cox 1984. In the two party case described by Cox there are only three possible states of the post-election world: 5 h1j so 1 wins, 5 h2j so 2 wins, or 1 2 5 h1, 2j where 1 and 2 have exactly the same number of seats. In the case 1 wins, 1 then s 5 0 so 2 receives nothing. When neither wins, it is natural to assume that 2 12 12 1 1 ] s 5 s 5 s . 1 2 1 2 Thus the utility function for 1, say, is 1 U z , z 5 p zu z 1 s 1 p zu z 1 1 2 1 1 1 1 2 1 2 z 1 z 1 1 2 1 ]] ] S S D D 1 1 2 p z 2 p z u 1 s . 1 2 1 1 2 2 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 155 In the case 5 h1, 2j, party 1 and 2 compromise over policy and adopt the midpoint 1 ] z 5 z 1 z . Cox argued that natural assumptions on the nature of the electoral M 1 2 2 probabilities would lead to a PSNE, z. Clearly Cox’s model concentrated on ‘‘electoral risk’’, induced by the uncertainty associated with the electoral probabilities, p z and 1 p z. 2 In contrast, recent models Baron, 1989, 1991; Banks and Duggan, 1998 have concentrated on ‘‘coalitional risk’’ associated with the probabilities r , introduced in h j M Assumption 2. In general, the idea behind models of coalition bargaining is that the true ideal points o of the parties are declared, and the coalition probabilities and h j i compromise points a o are deduced from a bargaining game. Although existence of h j M these bargaining solutions is known, it is often unclear whether the solution is continuous in the ideal points o . A model currently being analyzed Winter and h j i Schofield, 1999 indicates that, in the three party case, the compromise point for coalition i, j will be linear in o , o and the coalition probabilities will be h j h j i j 2 approximately inversely proportional to io 2 o i . As indicated above, we suppose that i j the parties declare, not the vector of ideal points o , o , o but a vector z , z , z of 1 2 3 1 2 3 leader positions, chosen with the knowledge of the bargaining that will then occur. Since we are interested in the effect of coalitional incentives on electoral motivations, it seems appropriate to examine the three party case without electoral risk. This suggests that we impose a fixed coalition structure . For symmetry, we suppose 5 1, 2 , hh j 2, 3 , 1, 3 . We suppose further that the ‘‘costs of bargaining’’ c for coalition hi, jj h j h jj ij 2 are proportional to iz 2 z i and that the probability that coalition hi, jj forms is i j 21 proportional to c . When this coalition M 5 hi, jj forms it adopts the policy point ij hi, jj 9 a z 5 z 5 z 1 z 2, and assigns a perquisite s 5 s to party i. M M i j i ij Model 1 which we examine, satisfies Assumptions 1–4 above, but not Assumption 29, since the ‘‘core convergence’’ property is not assumed. That is, in Model 1 we do not ˜ suppose that when z 5 hz , z , z j are colinear, then g z is the singleton hz j. 1 2 3 2 3 3 Model 1. Suppose parties declare z [ Z and have ideal points given by o [ Z , 2 where Z is a disc D in R of sufficiently large radius. We assume: 1. p z 5 1 and p z 5 0 for all t ± 1, where t 5 1 corresponds to the state where each 1 t two-party coalition wins. 2. the Pareto set of the party leaders is the convex hull of hz , z , z j. 1 2 3 ˜ 3. the selection g 5 hr , a j is a lottery selected from the Pareto set. If M 5 hi, jj, M M 21 21 21 21 21 then r z 5 c c 1 c 1 c , and a z 5 z 1 z 2 5 z . The perquisite to M ij ij ik jk M i j ij i in coalition hijj is s [ R. ij 9 An important first step in formulating such a game can be found in Austen-Smith 1986 in the one- dimensional case. Just as in the model here, Austen-Smith assumes that each minimal winning coalition, M, forms with probability r , inversely proportional to its variances and adopts the mean policy point z . An M M obvious motivation in Austen-Smith’s paper was to model the creation of parties that is coalitions of heterogeneous candidates. The convergence phenomena found in the model presented here may provide a theoretical framework for the coalescence of elites into parties. 156 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 Thus for principal of party 1, the von Neumann-Morgenstern utility at z 5 z , z , z 1 2 3 is U z 5 r zu z 1 s 1 r zu z 1 s 1 r zu z . 1 12 1 12 12 13 1 13 13 23 1 23 Since the utility functions are smooth we show in Appendix A how to compute their critical points, and thus to determine the best response correspondences. The following four theorems whose proofs are presented in Appendix A 10 characterize Model 1. 2 Theorem 1. If Z is a compact, convex subset of R , then for each vector of ideal points 3 o 5 ho , o , o j [ Z , and scheme G 5 s : i, j [ 1, 2, 3 , i ± j of perquisites, there h j h j 1 2 3 ij 3 exists a mixed strategy Nash equilibrium. For each o [ Z , there exists s . 0, dependent on o, such that whenever s . s for each s in G, then there exists a stable ij ij PSNE. We show in Appendix A that the utility functions U need not be quasi-concave in the i strategy variable, z . Nonetheless the U are continuous indeed differentiable in the i i strategy variables and thus MSNE exist. We also show in Appendix A, when the 3 perquisites are sufficiently large, that, for each profile o [ Z , the joint best response correspondence, ho, is single-valued and continuous in the strategies of players other than i. This implies that ho has a fixed point, which corresponds to a PSNE. Moreover specific configurations of bliss points give rise to unique, stable PSNE. Two different cases giving unique PSNE are considered in detail. A Consider first the symmetric case in which io 2 o i 5 r constant for all pairs i j hi, jj. We suppose Z is a disc D centered at the barycenter of ho , o , o j with radius 1 2 3 4 r. If perquisites are zero, then the best response correspondence of each player is single-valued. The fixed point z of the joint best response function satisfies iz 2 z i 5 i j 2 io 2 o i for each pair i, j . Since iz 2 z i . io 2 o i for each pair, say divergence h j i j i j i j occurs. See Fig. 3. Note in particular, in this case, that no joint strategy, where all principals choose the same policy, can be stable. B In the second case, suppose the bliss points are colinear. Write the bliss points as o 5 0, y and suppose y . y . y . 0. Assuming zero perquisites, the best response i i 1 3 2 1 ] by 3 is to choose y closer to the mid-point y 1 y . On the other hand the best 3 1 2 2 response by both 1 and 2 is to move closer to y . In this second, degenerate, case we 3 find, as we might expect, that convergence occurs. By ‘‘convergence’’, we mean that iz 2 z i , io 2 o i for each pair i, j. i j i j In this colinear case, there is an attractor z, z, z of the best response function, where each player chooses the same policy. By definition this is a stable PSNE. Note that this is a form of Downsian convergence. To solve the general case, we consider the problem of the best response by party 3 to z 5 0, r 2, z 5 0, 2 r 2. The best response in x is essentially a function of 1 1 2 2 r 1 r , while the best response in y is a function of r 2 r . Either the response 1 2 1 2 equation in x or the equation in y will dominate, giving either divergence or convergence. Full mathematical analysis of the general equations has not been possible, 10 2 q Although the analysis is performed in R , it is evident that the analysis is valid if Z , R for any q 2. In this case all party behavior will lie in the two-dimensional affine subspace of Z generated by o , o , o . h j 1 2 3 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 157 Fig. 3. Divergence in Nash equilibrium declarations. but computer simulation indicates that for almost any assignment of bliss points, a stable pure strategy Nash equilibrium occurs in the interior of the space. Even in the symmetric case A, if there are positive perquisites from coalition membership, and if these perquisites are sufficiently large, then the Nash equilibrium z is convergent. The results obtained in the Appendix on the relationship between bliss points, perquisites and Nash equilibria are described in Theorems 2, 3, and 4. We first need generalizations of the notions ‘‘colinear’’ and ‘‘symmetric’’. Definition 4. Say three points hz , z , z j are e-bounded in linearity if i j k min iz 2 l z 2 l z i e. h j l , l [R i j j k k j k Say three points z , z , z are e-bounded in symmetry if h j i j k max uiz 2 z i 2 iz 2 z iu e, i, j,k i k j k where max means across all permutations of i, j, k. i, j,k In the case e 5 0, say simply that the points are symmetric. 158 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 Note of course that if three points are e-bounded in linearity, for e . 0, then the degree of symmetry they exhibit will be low. These two definitions attempt to capture the difference between the extreme cases A and B. Theorem 2. Suppose that perquisites are zero. There exists e . 0 such that, if the ideal points are e-bounded in linearity, for any e , e, then there exists a unique, stable, pure strategy Nash equilibrium which is convergent. That is hz , z , z j all lie within 1 2 3 the convex hull of ho , o , o j. Moreover, the Nash equilibrium strategies are also 1 2 3 e-bounded in linearity. Theorem 3. Suppose that perquisites are zero and Z is the disc, D, as above. Then there exists e . 0 such that, if the ideal points are e 2 bounded in symmetry, for e , e, then there exists a unique stable pure strategy Nash equilibrium z in the interior of Z which satisfies iz 2 z i i j ]]] 5 b e, for each pair i, j . ij io 2 o i i j Here, b is a continuous real-valued function of e, with b e . 1 and b 0 5 2. ij ij ij Theorem 3 asserts that the Nash equilibrium z is divergent in the sense that iz 2 z i . io 2 o i for each pair. Moreover, if the ideal points are symmetrically i j i j located e 5 0, then z is symmetric, i.e., iz 2 z i 5 iz 2 z i 5 iz 2 z i. 1 2 2 3 1 3 Note that the Nash equilibrium positions do not lie in the Pareto set of the party principals, namely the convex hull of o , o , o . h j 1 2 3 The effect of perquisites is captured by the following result. Theorem 4. If perquisites are non-zero and constant s 5 s for all i, j, then for each ij ho , o , o j, which is e-bounded in symmetry, there exists a unique, stable, pure strategy 1 2 3 Nash equilibrium z which satisfies iz 2 z i 5 d e, sio 2 o i, for each pair i, j. i j ij i j Here d :R 3 R → R is a positive-valued continuous function, for each i, j, which ij satisfies the following properties: 1. d e, s decreases as s increases for each e; ij 2. if the ideal points are symmetric e 5 0, then d 0, s 5 d0, s for each pair, so that ij the Nash equilibrium will be symmetric; 2 4 ] 3. in this case, there is a bound s , io 2 o i such that d0, s , 1 for all s . s. i j 25 See Fig. 4 for an illustration of the convergence result. Theorem 4 shows that if the ideal points are symmetric, and the perquisites are N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 159 Fig. 4. Convergence in Nash equilibrium declarations. sufficiently high, then there exists a stable symmetric, convergent Nash equilibrium which is uniquely determined by the parameters o , o , o , G . Clearly z , z , z all h j h j 1 2 3 1 2 3 lie in the convex hull of the principals’ bliss points. Observe that, in the symmetric case, the expectation of the coalition outcomes is the center of the distribution of ideal points. Since the probability associated with each 1 2 ] ] coalition is , the expectation of private benefits or portfolios of each party is s. 3 3 These four theorems all use the properties of the reaction functions, together with Fort’s Theorem Fort, 1950, to assert that the Nash equilibrium mapping n n ˜ :Z → Z G defined by the scheme G of private benefits, with uNu 5 3, and which maps the ideal n points to the Nash equilibria, is a continuous function on specific open domains in Z . In order to understand the foundations of the convergence and divergence results, let us consider an example which is not covered by these theorems but which does illustrate the nature of the equilibria. Consider again Fig. 2, and let us relabel the CDA and PVDA as 1, 2 and the VVD as h j 160 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 ] ] ] Œ Œ Œ 3 . Suppose that 1 and 2 adopt positions 0, 3 and 0, 2 3 a distance r 5 2 3 h j apart. To generalize Fig. 2, suppose the ideal point of 3 is on the x-axis at L, 0, say, and consider the best response of 3. By symmetry, it must be on the x-axis at x, 0, say. We can normalize by letting x 5 ar and L 5 br for some parameters a, b . 0. Eq. 4 of the Appendix shows, for zero perquisites s 5 0, that a and b are related by the 2 a 9 ] ] quadratic expression a 1 2 5 0 with solution b 4 ]] 1 1 1 ] ] ] a 5 2 6 9 1 . 1 S D 2 2 b b œ This is obtained by setting dU 50. In fact the term involving the negative sign 3 corresponds to a minimum, as we need only consider the positive term. An easy ] 1 Œ ] calculation shows that a 5 b iff b 5 5. This defines a fixed point of the best response 2 ] ] r r Œ Œ ] ] function, h . In particular if L [ 0, 5, then a . b but x [ 0, 5. In other words 3 2 2 ] r Œ ] h3j diverges from z , z but z is still bounded by the limit 5. On the other hand if L 1 2 3 2 ] ] r r 3r 1 Œ Œ ] ] ] ] [ 5, ` then a , b and x [ 5, . Of particular interest is the solution b 5 ] 2 2 2 Œ ] 3 Œ 3 ] and a 5 . As Fig. 3 illustrates, if the bliss points are symmetrically distributed, so 2 io 2 o i 5 r for each pair i, j , and we choose coordinates so that for player 3, o 5 h j i j 3 ] Œ 2r 3 ] ] L, 0 5 , 0 then 3’s best response to z , z is at z, 0 5 2r , 0. By symmetry, ] 1 2 2 Œ 3 the three best responses in fact are symmetrically located, so iz 2 z i 5 2io 2 o i for i j i j each pair. This gives a symmetric, unique and divergent PSNE, from which Theorem 3 can be derived. To see how the Nash equilibrium is changed as L is decreased, let us modify Fig. 3 by r r ] ] supposing o 5 0, , o 5 0, 2 and o 5 L, 0. If we choose o to be a distance 1 2 3 3 2 2 ] Œ r 3 ] from the y axis, then the equilibrium is z , z , z with z approximately 1 2 3 3 4 ] Œ 3 ] 0.93r , 0 while z is close to the y-axis but with its x-coordinate at approximately 1 2 r ] 0.64 . In other words, 3 still ‘‘diverges’’, but 1 and 2 converge. On the other hand if 2 ] Œ r 3 ] L 5 , then 3 initially diverges, but as 1 and 3 continuously converge in best response, 8 the scale parameter, iz 2 z i drops until 3 goes through its ‘‘best response fixed point’’. 1 2 After this stage, 3 also converges towards the positions of 1 and 2. Computation of the round of best responses shows all three players converging towards the bliss point of party 3. If we regard r 5 io 2 o i as an indication of the initial scale parameter, then by 1 2 round 33 of best responses, all three players are within 0.001 of this point of convergence. Thus the degree of colinearity required for Theorem 2 is not high. To determine the effect of perquisites in determining best response, we may suppose that the perquisite s 5 tr, where t is a coefficient and r is the scale parameter. Instead of 1 above we now obtain ]]]]] 2 1 1 1 2t 1 1 2t ] ]] ]] a 5 2 6 b 1 9 . 2 S D S D S D 2 b b œ ] ] Œ 3 Œ ] For example, if t 5 1 and b 5 3, then a 5 . This immediately implies that the 2 5 ] parties converge. Indeed it is easy to see that if t . , then there is no solution to the 8 fixed point requirement, and so b . a. In the symmetric case, presented in Fig. 4, if N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 161 2 2 4 ] s 5 r 5 io 2 o i , then there is a convergent Nash equilibrium with iz 2 z i 5 i j i j 25 2 ] io 2 o i. Theorem 4 follows by continuity. i j 5 On the other hand, in the skew-symmetric case if io 2 o i 5 io 2 o i . io 2 o i, 3 1 3 2 2 1 then in the Nash equilibrium 3 diverges from 1 and 2, but 1 and 2 may converge to one another. Sufficiently high perquisites will force convergence of all parties. Computation of this three party game always resulted in stable PSNE. 3 In computing Nash equilibria the procedure we adopted was, for each profile o [ Z of bliss points, to start the process of best response from random initial points and to terminate once the process of best response had ‘‘stabilized’’. This was interpreted to mean that no movement was observed greater than 0.001 times the scale parameter, r. In all cases we found that the computed PSNE was unique, and a global attractor. That is to say the process of best response led into a neighborhood of the PSNE, z, from all initial points. Our analysis suggests that the best response correspondence is a contraction mapping. Computation Fact 1. The Nash equilibrium map of Model 1 with uNu 5 3, given by : G 3 3 3 3 ˜ Z → Z is a function, with image in Z . Moreover, for each o [ Z , the PSNE o is G a global attractor of the joint best response function. Computation Fact 2. Suppose all perquisites are zero in Model 1. Consider the skew symmetric case with io 2 o i 5 io 2 o i 5 eio 2 o i. 3 1 3 2 2 1 If e . 0.616, then all parties diverge in Nash equilibrium, so iz 2 z i . io 2 o i, i j i j ; i, j. If 0.54 , e , 0.616, then 1 and 2 ‘‘weakly’’ converge, but 3 diverges, so iz 2 z i , 1 2 io 2 o i but iz 2 z i . io 2 o i for i 5 1, 2. 1 2 3 i 3 i If e , 0.54 then all three parties converge to a neighborhood of o . 3 While the specific details on the equilibria in Model 1 depend on our assumptions on ˜ r and g, we regard the above results as being qualitatively valid for any similar model. That is, the qualitative aspects of convergence or divergence in Nash equilibrium appear ˜ to be unchanged for certain small perturbations of r and g, with uNu 5 3. Model 2. To illustrate this observation, consider the following perturbation of the lottery 3 ˜g given in Model 1. For any z [ Z with z , z , z colinear and z the median, let h j 1 2 3 i 3 9 g z 5 z be a singleton. For any e . 0, let V be an open set in Z such that, for any h j i e z 5 z , z , z with z , z , z colinear, there is an open e-ball B containing z, and h j h j 1 2 3 1 2 3 e e 3 3 ˜ ˜ ˜ ˜ contained in V . Then we can construct a smooth selection g :Z → Z of :Z → Z, e e e ˜ ˜ ˜ ˜ 9 such that g z 5 g z whenever z , z , z are colinear, and g z ; g z for all h j 1 2 3 3 e ˜ ˜ z [ Z \V . Here g is the lottery outcome function of Model 1. Call g an e- e ˜ perturbation of g . The proof procedure of Schofield and Sened 1998 can then be e ˜ ˜ adapted to show that the Nash equilibrium mappings for g and g will be ‘‘close’’. ˜ Thus the qualitative results obtained for g in Model 1 will also hold for some smooth 162 N . Schofield, R. Parks Mathematical Social Sciences 39 2000 133 –174 e e ˜ ˜ perturbation g for e sufficiently small, where g satisfies the core convergence property of Assumption 29 e ˜ ˜ Theorem 5. For sufficiently small e . 0, there exists an e-perturbation g of g in e ˜ Model 1 such that g satisfies Assumption 29. Moreover, Theorems 1–4 are valid for e ˜ the political game induced by g . ˜ Note however that not every e-perturbation of g will necessarily satisfy the convergence result of Theorem 2. Whether convergence in the degenerate colinear case e ˜ does occur will depend on the precise details of the assumptions made on g in the neighborhood V . The following section mentions an example from Germany to illustrate e this point. Model 3. In Models 1 and 2, it has been assumed that the vector of perquisites obtained by the members of coalition M if it forms are prespecified in some fashion. Assumption 5 is designed to deal with situations where bargaining between party leaders generates the payoffs of perquisites see also Austen-Smith and Banks, 1988. If we use G to denote the total value of perquisites available to coalition members, and G is suitably constrained, the Nash equilibria policy outcomes under this more general game n n ˜ ˜ ˜ ˜ 9 form g : Z → W will be close to those generated by the game form g : Z → Z of Model 2. We therefore expect that when the coalition structure is held constant, as in n n ˜ ˜ ˜ ˜ 9 Models 1 and 2, and g :Z → W is a continuous selection of :Z → W, then the Nash equilibria of the induced political game will have qualitatively similar properties to those presented in Theorems 1, 3 and 4.

5. Extension of the model to include electoral risk