Proposal Powers, Veto Powers, and the Design of Political Institutions

Transcription

1 Proposal Powers, Veto Powers, and the Design of Political Institutions Nolan McCarty Department of Political Science Columbia University New York, NY 007 Version.0 Paper prepared for the 998 Meetings of the American Political Science Association, Sept. 3-5, Boston, MA.

2 I. Introduction The major preoccupation of rational choice theorists is the study of the mapping between formal political powers and actual political outcomes. Using tools drawn from economics, social choice, and game theory, rational choice theorists have attempted to deduce propositions about the performance of many different types of political systems and institutions. A number of recurring themes have emerged from the rational choice enterprise. Two such themes are the roles of formal prerogative and sequence. An important part of any rational choice model is an assignment of prerogatives, formal powers, or more colloquially who gets to do what. In most rational choice models of policy process, these questions boil down to who may initiate policy change (proposal power) and who may block policy changes (veto power). In these models, the relative influence of different political actors often depends on whether the actor serves as an agenda setter or a veto player. Sequence also plays an important role in rational choice models by adding the question when to who gets to do what. When can a certain actor make a new proposals? At what stage can new legislation be blocked? What is the time horizon of the players? Using the tools of extensive form games and subgame perfection, the importance of anticipated reactions and credibility have been established. The downside is that many of these models are not robust when assumptions about sequence or time horizons are changed. Unfortunately, much of the theoretical work on political institutions has tended to focus only on subsets of these questions. For example Baron and Ferejohn (989b) have

3 analyzed the effects of proposal power in a dynamic model, but only in a majority rule setting with no veto players. Works such as Hammond and Miller (987) and Tsebellis (994) have focused on the role of veto players, but have placed less attention on issues of proposal power or sequence. Analyses based on the work of Romer and Rosenthal (978) have focused both on proposal power and veto power, but has dome so in typically static contexts with very stylized sequences of actions. In this paper, we attempt to synthesize these concerns into a single model that incorporates very general allocations of proposal and veto powers while also accounting while relaxing some of the assumptions about sequence and timing. The model we propose is a generalization of the sequential choice model developed Baron and Ferejohn (989a). In this model a legislative assembly must decide how to allocate a fixed set of resources across the legislative districts. Each member of the assembly has a set of prerogatives which may include proposal powers, veto powers, and voting rights. In each period, a member is chosen at random to make a proposal with a probability determined by her proposal power. Each member then decides whether to support the proposal. The veto rights of the opponents determine whether or not the proposal passed. If the proposal passes, the game ends and resources are allocated according to the latest proposal. In this model, expectations of the future will play an important role in the process. Each players support or opposition to the current proposal will be based on expectations about future proposals that are in turn are affected by the allocation of parliamentary rights. The paper proceeds as follows. In section II, we lay out the framework of the basic model and make our conceptualization of proposal and veto powers much more

4 precise. In section III and IV, we focus exclusively on veto power and draw a number of prediction about when more veto power will be valuable. In section V, we compare these results with the Shapley- Shubik procedure for measuring the distribution of power within a legislature (Shapley and Shubik (954)). In section VI, we analyze the effects of proposal power and compare these results to those on veto power in section VII. II. The Basic Model The model we employ is a generalization of the majority rule divide-the-dollar game pioneered by Baron and Ferejohn (989a). In each period a member of the legislature is chosen via a random recognition rule to make a proposal as to the division of the dollar. The proposal is then evaluated by the legislature who may reject it according to some veto rule. If a veto of the proposal is sustained, the game continues to the next period and a new proposal is made. Following any proposal that is not successfully vetoed, the game ends and the dollar is divided according to the last proposal. We assume that the legislative body contains N members. Let the proposed division by legislator at time t and a history of play h be denoted as where x ith is the share going to district i under proposal x th. x th = x th, x th,, x Nth Obviously, we require that N i= x ith. In equilibrium this constraint will always bind. Rather than arbitrarily limit the opportunities for legislative consideration of this project, following Baron and Ferejohn (989a), we assume that the legislature may As we will see, voting rights are a special case of veto powers. 3

5 consider the passage of the project for an infinite number of sessions. If the project does not pass in a given session, it may be taken up again in the future. Each legislator discounts future payoffs by δ so that legislator i s payoffs are δ t x where t is the time period in which an agreement passes, is the proposer at time t. 3 The allocation decisions of the proposer are going to be determined in large part by the allocation of parliamentary rights. In this paper, we focus the effects of the recognition and veto rules. Let proposal powers be represented by a vector p = p,, p N where p i is simply the probability that legislator i is chosen to make the ith proposal in any given period. Obviously, we require that N p i = i=. We assume that p remains the same over time so that the identity of the proposer is independently and identically distributed. Thus, we differ from Baron and Ferejohn (989b) who analyze proposal power under the assumption that a committee has the right to make a proposal in initial periods, but that the recognition probabilities are subsequently equal. The notion of veto rights is a bit more subtle. Let k = k,, k N be the vector of veto rights where k i is the number of votes required to overturn a veto by player i. Thus, the larger k i, the more veto power held by legislator i. This formulation is quite general and can be used to represent a number of legislative institutions. 4 For example, suppose that ki = N+ for all i. This is simple majority rule as a simple majority is sufficient to offset player i s opposition. If k i = N for some i and 0 for all other legislators, 3 The results will not be qualitatively affected by assuming that the process infinite. Deviating from the assumption that utilities are linear in the legislator s share precludes any closed form solutions and does not substantively alter the comparative analysis of institutions. 4

6 legislator i is a dictator. Finally, a qualified veto such as that of the U.S. president can be represented by k N = 3 for the president and k = N + for the legislators.5 Following Baron and Ferejohn (989a), we only consider stationary and symmetric strategies. Stationary strategies are those that are time and history independent. 6 In this context, stationarity implies that proposals will only be a function of the identity of the proposer as well as the basic parameters k, p, N, and δ, but not the history of play. We also impose symmetry so that proposers treat all legislators of the same type identically and all legislators of the same type play identical strategies. 7 Since the only ex ante differentiation of legislators is based on type, the stationarity and symmetry assumptions suggest that strategies will depend only on the legislator s type. Given our setup, we can define a legislator s type by the pair {p i, k i }. If {p i, k i } = {p j, k j }, we can say that legislator i and j are the same type. Because of symmetry, we are free to talk about the strategies of types rather than individuals. Therefore, let 4 This formulation is more general than other formulations that dichotomize the legislature into veto players and non-veto players (e.g. Winter (996)). 5 Of course, the override requirement is concurrent /3 majorities. Concurrent majorities require a slight modification to the model. 6 Baron and Ferejohn (989a) Baron and Ferejohn (989a) show that any division can be supported in non-stationary equilibria using infinitely nested punishment strategies. The intuition for this result is straightforward. Any proposer who deviates from the prescribed distribution of benefits will be zeroed out by the next proposer who would be punished if she failed to do so. Because of the infinite number of periods, proposers will always find it rational to carry out the punishment and so that the original proposer will not deviate. Such strategies can only be implement if proposers can recall the entire (infinite) history of play so as to ascertain which proposers are to be zeroed out. Baron and Kalai (993) argue that this is an attractive restriction of the strategy space due to the fact that it requires the fewest computations by the agents in the model. Further, with legislative turnover, infinite recall of past histories is not a reasonable assumption. 7 To be more precise, symmetry does not imply that the proposer proposes exactly the same share to each member of a type class, but only that the mixed strategy probability distribution be the same for all members of each class. 5

7 m T where m τ is the number of legislators of type τ. Let p T = p,, p T m = m,, be the recognition probabilities and k T = k,, k T be the veto rules for each type where superscripts denote types rather than individuals. For each legislator, let v i be expected utility of the playing the game. Since we will be interested in the ex ante payoffs to legislators of different legislators, v i will be the main object of our analysis. However, these values play an important analytical role as well. Because of the assumption of stationarity, the game begins anew each period. Therefore, v i is also the expected utility to member i of defeating the current proposals. For this reason we will refer to v i as the continuation value of legislator i. The continuation value plays the same role as reservation utilities in spatial models of legislative bargaining. However, the key difference is that continuation values are endogenously derived through expectations of future play rather than given exogenously. Given a continuation value, we can specify each legislator s optimal strategy given a proposal. If legislator i receives a share x i δv i, then she will not veto the proposal as the share exceeds the discounted utility of continuing the game another period. 8 Our assumption of symmetry implies that if members i and j are type τ, then v i = v j = v τ. The proposer s strategy is a bit more complicated. Proposals may be formulated to generate many different types of coalitions. To characterize these strategies, we need some additional notation that focuses exclusively on veto types. Let k ~ = ~ k,, ~ k L be 8 Implicit in this statement is that fact that we are assuming subgame perfection and that players do not use weakly dominated strategies. Subgame perfection rules out noncredible threats to veto proposed shares that exceed the discounted continuation value. Elimination of weakly dominated strategies rules out bizarre equilibria where all members always vote in favor of every proposal because they are never pivotal. 6

8 the set of distinct elements of k ordered from highest to lowest. 9 Now, we can specify all of the minimum winning coalition sizes. 0 Clearly, if at least ~ k receive their discounted continuation values, the proposal will pass as all possible vetoes will be overridden. The proposer can optimally create this coalition by choosing the ~ k - members with the smallest continuation values and voting for the proposal herself. A second winning coalition could be constructed with the votes of members with veto type ~ k and max, ~ k m s other members. Thus, all ~ k types accept and any veto by other ~ & 0 s k > k types will be overridden. If the proposer is type ~ k, she gives discounted continuation values to the other legislators where members of her veto type class and to the max, ~ k m s lowest remaining continuation values. If the proposer is not type ~ & 0 s k > k ~ k, she gives discounted continuation values all members of the ~ k type and to the max, ~ 0 k m s lowest remaining continuation values. Both of these strategies ~ & s k > k take into account that the proposer will support her own proposal. Continuing inductively, we can see that all feasible minimum winning coalitions can be constructed from the votes veto types greater than ~ k j and from max, ~ j s 0k m others. Each such coalition will be denoted as a ~ k j - coalition. To ~ & s j k > k form a minimum cost winning coalition, the proposer will choose those from the 9 The ~ symbol is to distinguish these veto types from general types which are a function of both proposal and veto power. 0 Note that at this point we are not saying anything about the costs of different coalitions. A minimum cost coalition must be minimum winning, but not vice versa. 7

9 remaining types with the lowest continuation values. In case of ties, we assume that the proposer randomizes. Since all of these strategies correspond to different elements of k ~, we will use k ~ to represent the strategies of the proposer. Each proposer i will choose a strategy ~ k j to maximize her remaining share that is and τ τ τ s s ~ j z = m δv δm v MIN k m i s k s ~ k j k s ~ τ; k j τ τ τ s s j s z = ~ δm v δm v MIN k m i s k s ~ k j k s ~ τ; k j s if k > ~ k j [] if otherwise [] where MIN i r is a shorthand for the sum of the r smallest discounted continuation values excluding i. These expressions account for the fact that the proposer will vote for her own proposal. We will also allow for mixed strategies where the proposer randomizes over various coalition sizes that we represent by σ Σ where Σ is the set of probability distributions over ~ k. To complete the model, we turn to the computation of the continuation values. Consider any proposal strategy profile σ = σ,, σ T. Given σ τ, we can calculate the probability that a type s is chosen by a type τ proposer. Let this be denoted as π τ s. We can again contrast our model with that of Winter (996). Since he is not concerned with overrides, his veto players are in all winning coalitions. Since there are multiple levels of override in our model, we find that members with more veto power are in more, but certainly not all, winning coalitions. 8

10 Further, we can compute the probability that a type τ is chosen as a coalition partner of T some other proposer as Π τ s s τ τ τ = mpπs pπτ. Let Π = Π,, Π T. s= Given the values of Π τ we can compute the continuation value of a legislator i as: τ τ τ τ τ v = p z +Π δv [3] Combining equations [] or [] with [3], we have a system of T equations with T unknowns. Thus, given (Π,, Π T ), v τ and z τ are the solutions to this system which must be full rank. A stationary, symmetric, subgame perfect Nash equilibrium requires the following two conditions i) v τ and z τ solve [] ([]) and [3] given Π * ii) Π τ* maximizes z τ given Π ~ * τ Before turning to the analysis of veto and proposal powers, we state and (prove in the appendix) some results that hold regardless of the distribution of legislative prerogatives. Lemma : In any stationary, symmetric, subgame perfect Nash equilibrium: ) The bargaining with last only one round N ) v i = i= This assumption can be rigorously justified as an essential part of a mixed strategy equilibrium. See Harrington (990). 9

11 proof: See appendix Lemma is consistent with well-known efficiency results for bargaining models without transaction costs or asymmetric information. Part ) rules out any inefficiency due to delay while part ) suggests that aggregate welfare will be maximized at $. While these results are not terribly novel, they will often prove useful in the subsequent analysis. III. The Effects of Veto Power We will begin our analysis by looking exclusively at the effects of veto power while holding proposal power constant at p i = /N for all i. The primary focus will be on how the payoffs of different types of legislators are effected by veto power. Our first result is that a legislator s payoffs must be at least weakly increasing in her veto power. Proposition : Suppose that p i = /N for all i. Then, in any symmetric, stationary, subgame perfect Nash equilibrium, v τ v s if k τ < k s. Proof: See appendix The logic of proposition is quite straightforward. Suppose type τ did get lower utility than type s who has less veto power. Then each type τ would be an especially attractive coalition partner relative to type s. First, type τ is less expensive because of her lower continuation values. Secondly, it is more feasible to satisfy the override threshold of k s 0

12 since the votes of type s are expensive. Thus, it must be the case that the probability of being selected into a coalition is higher for τ implying that Π τ Π s. Since the probability that a type s is selected is weakly lower, z s must be sufficiently higher than z τ for v s > v τ. However, it can be shown that this is impossible or otherwise a proposer of τ could defect to the strategy used by type s proposers. It is possible however for v τ = v s even if k τ > k s. This can occur only when Π τ = Π s. Thus, if both types equally attractive coalition partners a difference in veto powers gives them no relative advantage over one another. As an example of this possibility, suppose that there are three types, and 3 where m = m = and m 3 = N-. Further, let k = N, k = N-, and k 3 =. 3 Then, clearly all types have a dominant strategy to build coalitions with type τ and s. The system of equations implied by equations [], [], and [3] is i) z = δv iv) Nv = z + N δv ii) z = δv v) Nv = z + N δv iii) z 3 = δv δv vi) Nv = z 3 3 Equations i) and ii) imply that z δv = z δ v. From iv) and v) this implies that N δn v = N δn v or v = v. Note that this calculation has only used up only two equations so we can still plug ii) into iv) to get v = v = N δ N. These results 3 Note that for type 3 to be distinct, we require N 4.

13 along with iii) and vi) imply v = 3 N δ N δ. To validate that the optimality of excluding type 3 in all of the coalitions is a best response, we only need to check to see that Nv 3 > v + v and (N-)v 3 + v > v + v 3 δ which holds when N max, δ δ As this example shows, there is no guarantee that more veto power leads to strictly higher payoffs. If N is sufficiently large, it is too costly to ignore either types or under any circumstances. The fact that type has slightly more veto power is irrelevant. As we will see in the more extensive analysis below, this irrelevance finding holds in a wide range of cases -- not just in the (admittedly) special one here.. IV. The Case of Two Veto Types We proceed now to a more elaborate example of the effects of veto powers. Let there be two types with veto powers k and k where k > k. Let the group sizes be m and m where m + m = N. Proposition presents the ratio of the continuation values of each type as a function of the veto powers, groups sizes and discount factors. Proposition : The following are the ratios v v for the symmetric, stationary Nash equilibrium in the two type game: If k m, v v =.

14 If k > m, k m, and k m, then v v If k > m, k > m, and k < m, then v v If k > m, k < m, and k m, then v v If k > m, k < m, and k m, then v v m δ k m N k = min, m δ N k m δ k m m + k k = min, m δ m N m = min, δ N k k = min, δ m.... The exact continuation values can be found by applying the result of Lemma that mv + mv =. Proof: See appendix In order to visualize the results of Proposition, Figure contains results from the case where m = m = 00 and δ =.9. Once again we find that under certain conditions higher veto powers may be irrelevant. In the case where k < k < m, both types have the same continuation values. In this case it will always be cheaper to satisfy the k override requirement than satisfy all m members of type. If v > v, the type s would have a much lower probability of selection than type s which would tend to lower their continuation values and raise those of type until v we find that as the gap between k and k narrows, the ratio v v = v. Not surprisingly, tends toward unity. 3

15 Insert Figure About Here In order to see the effects of discount factors on veto power, Figure plots the continuation value ratios for different values of δ and k where N = 400, m = m = 00, and k = 00. Clearly, patience benefits the members of type. Intuitively, veto power is power to continue the bargaining in hopes of a better outcome in the future. If the future is valueless, the veto is valueless and the ratio v v converges to. Insert Figure About Here V. An Aside: Veto Power and Power Indices One of the earliest methods for measuring the distribution of power within a legislative body was proposed by Shapley and Shubik (954). The Shapley-Shubik (SS) index relates the probability that an individual s support is pivotal to that member s power within the body. The SS index is the proportion of the permutations of the legislators for which a given legislator converts a losing coalition to a winning coalition. Formally, let S be the set of winning coalitions that are losing coalitions if member i is removed. Then SS i s N s =!! N! s S where s is the number of members in coalition s. 4

16 To build some intuition about the relation between the SS index and our model, consider the following example. Suppose that there are three legislators,, 3 where k = k = and k 3 = 3. There are the following six permutations of these legislators: 3* 3* 3* 3* 3 * 3 * Note that in all of the cases denoted by * the indicated legislator is pivotal because the coalition of the members preceding it is a loser. Therefore, we can compute the SS indices are {/6, /6, 4/6} for,, and 3 respectively. While the SS index has received strong support and has well developed behavioral foundations (see Roth (977) ), our model suggests some of its liabilities. 4 To see these, we will compare the SS indices from our example with the equilibrium continuation values of our model. From Proposition, we get v = v = δ 6 5δ and δ v 3 = 6 5δ from our example. Thus, the continuation values only match the SS index at δ = 6/7. Such a result might be less troubling if the SS index represented a limiting case 4 These liabilities, as we shall see, are not peculiar to the SS index but are true other indices such as the Banzhaf. See Rapoport and Golan (985) for a good review of the literature on power indices and an application. 5

17 such δ 0 or δ, but δ = 6/7 is rather arbitrary. Not only does the SS index ignore the time horizons of legislators, its accuracy depends on them. A second problem with the SS index is that it assumes that coalitions are drawn randomly. This assumption is more problematic than the assumption of randomization over proposers. Because of random coalitions, the SS index for type τ will be strictly greater than that of type s if k τ > k s. Proposition 3 which is stated and proven in the appendix proves this claim for the case of two veto types. The logic of this result is that τ will be a part of more possible coalitions which raises her pivot probability. However, as we saw above, raising the override threshold may not increase the continuation values. For example, in Proposition, if m > k, then v = v. The reason more veto power is moot in this case is that the coalition with all members of type never forms in equilibrium. In general, SS indices may overstate the effect of veto powers. VI. Asymmetric Proposal Power Allocations of proposal power also may effect the distribution of power within a legislative body. The proposer is advantaged in two ways. First, the proposer can guarantee at least her discounted continuation value by making a proposal that would be defeated. Secondly and most importantly, the proposer is the residual claimant on any resources that remain once a majority coalition is formed. In fact, the proposer is the only person who receives more than her continuation value as all other legislators either receive 0 or δv i. So it would seem that the probability of proposing would have a 6

18 potentially large effect on a legislators payoffs. This logic turns out, however, to be only partially correct. Clearly, some proposal power is crucial. In our model, any legislator with a zero recognition probability must receive a zero benefit share in equilibrium. This result, summarized in Proposition 4, holds regardless of veto powers. Proposition 4: If p i = 0, v i = 0. The proof of this proposition is extremely straight forward. Equation [3] implies that when p τ = 0, v =Π δv. Since δ and Π τ are less that equal 0, the only solution is τ τ τ v τ = 0. Logically, if a legislator never receives a payoff higher than her average payoff, her average payoff must be zero. It is interesting that no similar result exists for veto powers. Even if k τ = 0, v τ may be positive since the votes of type τ may be used to override the vetoes of other types. For a more general examination of proposal powers, we focus on the effects of proposal power for the case of k i = k. Like veto power, we establish that enhanced proposal power increases continuation values, but only weakly. Proposition 5: Suppose that k i is constant across i. Then, in any symmetric, stationary Nash equilibrium, v τ v s if p τ > p s. proof: See appendix 7

19 The logic of Proposition 5 is clear. If p τ > p s and v τ < v s, then type s would have a lower probability of selection as both a proposer and as a coalition partner. However, this is inconsistent with v τ < v s. Nevertheless, just as in the case of veto powers, it is possible for p τ > p s and v τ = v s. If v τ > v s, the probability that τ is selected as a coalition partner may be substantially lower than that of s. If p τ is only slightly larger than p s, the negative effect will dominate driving v τ down to v s. The proposal power advantage must therefore be substantial in order to overcome the effect of a lowered probability of coalition participation. Proposition 6 presents the ratios of the continuation values necessary conditions for proposal power to be valuable for the two type case. Proposition 6: In a symmetric, stationary, subgame perfect Nash equilibrium for the model with two types, i) v v p N m k m = δ max, p N m δ p = max, p m m δk when m k when m < k ii) v > v if and only if p N m δk m > p N m δ when m < k p p > m m δk when m k 8

20 iii) The exact continuation values can be obtained from i) and the result from lemma that m v + mv =. proof: See appendix Figure 3 presents these ratios for 400 legislators and k = 00 as a function of m. Notice that the ratio of type s proposals power to type s must be substantially greater than one for it to have any effect. Unlike veto power that tends to increase the number of coalitions in which a member participates, proposal power tends to greatly reduce the number. Only if the increased probability of proposing compensates is proposal power valuable. Proposition 6 also indicates, not unsurprisingly, that proposal power is more important the more impatient the players are. Note that in the limit as δ 0, the critical value of p p goes to. In fact, in the limit continuation values are proportional to the recognition probabilities. VII. A Comparison of Veto and Proposal Powers A comparison of the effects of veto power and proposal power is at best a slippery exercise. After all, it s hard to say where or not a.0 increase in the probability of recognition is comparable to a 0 seat increase in an override threshold. Nevertheless, we will attempt to undertake such a comparison. We will compare a legislature with asymmetric proposal power with one with a asymmetric veto power. In each case, we 9

21 will limit ourselves to the two-type case. The primary question of interest is given an asymmetric in veto power, how asymmetric must proposal power be to generate the same legislative outcomes. 5 This can be accomplished by choosing p p to equate the ratio v v from Propositions and 6 for given values of k and k. Since the relevant algebraic solutions are messy and less than informative, Figure 4 presents these solutions for the case of N = 400, k = 00, and δ =.9 as a function of k for different numbers of type s (and therefore type s). Note that as k increases, p p must get substantially larger to compensate. Also the fewer members of type, the more asymmetric proposal power must be to compensate for a higher k. Insert Figure 4 Here Acknowledging the difficulty of making precise comparisons, our results make a much stronger case for veto powers over proposal powers than one of the most widely applied analyses. The agenda setting model of Romer and Rosenthal (978) has been used to study policymaking in a large number of contexts. In the Romer-Rosenthal model, a proposer chooses a policy alternative to an exogenous status quo. A veto player chooses whether to accept the new policy or to maintain the status quo. The game ends either with the proposal or the status quo as the outcome. The well-known result is that the proposer does better than the vetoer. The only situation in which the vetoer can 5 An alternative approach to this question would be compare two groups within the same legislature where one group is veto advantaged and the other is proposal advantaged. We leave this extension to future work. 0

22 increase her utility beyond that of the status quo is when the proposer gets her most preferred policy. It is the dynamism of the sequential choice model that undermines the predictions of the static Romer-Rosenthal model. As we pointed out before, veto powers are little more than the option to continue the game in the chance that a member will become the next proposer. In a static context, these powers are not as important. Proposal powers, on the other hand, are less potent for two reasons. First, they make the proposer more vulnerable to being omitted from future coalitions. Secondly, the option to continue the game prevents the proposer from making credible ultimatums. The effects of the future considerations can be seen by repeating the analysis of Figure 4 for different levels of the discount factor. Insert Figure 5 Here Clearly, any advantage that veto power has over proposal power dissipates as the future becomes more irrelevant. In the limit as δ 0, proposal power clearly dominates as v v p = for asymmetric proposal power while v p v = for any distribution of veto power. VIII. Conclusions To argue that institutions may effect outcomes is no longer very controversial in political science. The question now is to be able to make more precise and general

23 predictions about the relationship between certain procedures, rules, and outcomes. In this paper, we have attempted to formulate a relatively general model to capture two important institutional features: veto power and proposal power. Of the predictions that our model, perhaps the most important are those about when institutional prerogatives do not effect outcomes. For example, we find that veto powers are ineffective when they are shared by many members and when members are very impatient. In the same vein, we find that proposal powers are not important unless they are especially asymmetrically distributed or when legislators are quite impatient. Perhaps our model will provide a roadmap to explore which powers are likely to be determinative in various institutional settings. Another lesson of the model pertains to the question of how to design a legislative institution. Rather than provide a theory that predicts whether designers will focus on veto or proposal powers, our results suggest that any such theory is likely to be underdetermined. Either veto power or proposal power can in principal be used to engineer any set of (expected) outcomes in this model. The choice of institutional arrangements is likely to be determined by considerations outside the purely distributive setup we have employed.

24 Bibliography Baron, David P., and John A. Ferejohn. 989a. Bargaining in Legislatures. American Political Science Review 89: Baron, David P., and John A. Ferejohn. 989b. The Power to Propose. In Models of Strategic Choice in Politics, edited by Peter C. Ordeshook. Ann Arbor: University of Michigan Press. Baron, David P., and Ehud Kalai The Simplest Equilibrium of a Majority Rule Division Game. Journal of Economic Theory 6: Hammond, Thomas H., and Gary J. Miller The Core of The Constitution. American Political Science Review 8: Harrington, Joseph E The Power of a Proposal Maker in a Model of Endogenous Agenda Formation. Public Choice 64(): -0. Rapoport, Amnon, and Esther Golan Assessment of Political Power in the Israeli Knesset. American Political Science Review 79: Romer, Thomas, and Howard Rosenthal Political Resource Allocation, Controlled Agendas, and the Status Quo. Public Choice 33(): Roth, Alvin E The Shapley Value as a von Neumann-Morgenstern Utility. Econometrica 45: Shapley, Lloyd S., and Martin Shubik A Method for Evaluating the Distribution of Power in a Committee System. The American Political Science Review 48(3): Tsebellis, George Decision-Making in Political Systems: Veto Players in Presidentialism, Parliamentarism, Multicameralism, and Multipartism. British Journal of Political Science 5: Winter, Eyal Voting and Vetoing. American Political Science Review 90(4):

25 Appendix Lemma : In any stationary, symmetric, subgame perfect Nash equilibrium: ) The bargaining with last only one round N ) v i =. i= proof: First we wish to show that N i= v i. Because of discounting, the maximum sum of continuation values must come when agreement is reached immediately. When there is immediate agreement, v i = E(x i ) where E is the expectations operator over the probability distribution of proposals. Feasibility requires and optimality requires N x i = i= and the laws of the expectations operator therefore imply Ex i =. Thus, if we have immediate agreement, v i = otherwise v i <. z i N i= N i= N i= To show that we in fact do have immediate agreement, we need to show that δ v. The most undesirable case for any proposer is k i = N for all i so if we can i establish the claim for this case we are done. In this case, z j = δ v which may be i j i re-written as z v = v N δ δ j j i i= which is positive since N i= v i and δ <. Therefore, given immediate agreement v i =. N i=

26 The following lemma is useful in several of the propositions proven below. Lemma : In any stationary, symmetric Nash equilibrium, i) If π τ s τ τ s s > 0, then z δv z δv. ii) iii) iv) If π τ = 0 and π s τ s s s > 0, then z z δv If π τ = 0 and π s s s = 0, then z τ z s. If π τ = 0 and π τ s τ = 0, z s z τ. s proof: i) τ s τ Suppose that z + δv δv s < z. Then the type τ could raise her payoff choosing the same coalition as a type s proposer. Since π τ s > 0, one of type s s best responses must include a coalition with at least one type τ. Therefore, τ may mimic s with a coalition which requires one additional type s and includes one τ s τ fewer type τ. This defect will necessarily pay since z + δv δv s < z. τ τ Therefore, we must have z δv s s z δv. ii) τ s s Suppose that z + δv < z. Then τ can mimic s at the cost of an additional type s. iii) In this case, τ can exactly mimic s. iv) In this case, s can exactly mimic τ with perhaps one fewer type s.

27 Proposition : Suppose that p i = /N for all i. Then, in any symmetric, stationary, subgame perfect Nash equilibrium, v τ v s if k τ < k s. proof: Suppose not i.e. v τ < v s and k τ > k s. In this case, there is not optimal to include a type s in a coalition at the expense of a type τ. Type τ is cheaper and is required for more coalitions than s. Therefore, π τ r s π in for all types so that Π τ Π s as well. r τ τ τ τ s s s s Since p i = /N for all i, recall that Nv = z + δnπ v and Nv = z +δnπ v. Consider two cases. First, suppose π τ s s = π =0. From Lemma (iii), this implies that s z τ z s τ τ s s so that N δnπ v N δπ v which is inconsistent with v τ < v s and Π τ Π s. Now suppose that π τ s > 0. Then using the results of Lemma (i) and doing τ τ some algebra, we find that N N v N N s s δ Π δ δ Π δ v which is inconsistent with v τ < v s and Π τ Π s. Proposition : The following are the ratios v v for the symmetric, stationary, subgame perfect Nash equilibrium in the two type game: If k m, v v =. If k > m, k m, and k m, then v v If k > m, k > m, and k < m, then v v m δ k m N k = min, m δ N k m δ k m m + k k = min, m δ m..

28 If k > m, k < m, and k m, then v v If k > m, k < m, and k m, then v v N m = min, δ N k k = min, δ m.. proof: k < m : There are two cases m k and m < k. Consider the case of m k. Given any values of v and v, it is cheaper to build a k coalition. Suppose that v > v, then type s would not be included in any coalitions since there are at least k members of type. Therefore, Lemma iv) implies that z z. The continuation values for each type solve v z = and v N z k = + δ v. Together these imply that N N Nv N δk δ v which cannot hold if v < v. The only remaining possibility is v = v. The case of m < k is similar. k > m, k > m, and k > m : There are two possible types of equilibria in this case. There is a pure strategy equilibrium where both types form k coalitions and a mixed strategy equilibrium where type proposers mix between k and k coalitions. In the pure strategy equilibrium, both types make offers to both types since m < k. Lemma.i suggests that z δv = z δ v which implies that N δnπ δ v = N δnπ δ v. Since all proposers give to all type s NΠ = N-. Since type s are used to complete k coalitions, type proposers will select them with probability k m m and type s select

29 them with probability k m. Thus, N m m m k m k m = + Π. Substituting. these above and doing a little algebra yields v m δ = v m δ k m In the mixed strategy equilibrium, a type proposer must be indifferent between a k coalition which costs δk m v + δm v and a k coalition which costs. Indifference requires that N kv = N k v. If N k > δ N k N k < the mixed strategy does not exist. Therefore, δ N k m δ k m N k. m δ N k δmv + δ k m v m δ k m m m δ k m m v v = min,, then the pure strategy equilibrium does not exist and if The other cases can be proven in a similar fashion. Proposition 3: In the two type model if k > k, SS > SS. proof: Let there be two types where k > k. There are two distinct ways that a type can be pivotal:

30 . Appear in the k -th position following all m members of type. In this case, the vetoes of other type s will be overridden. The number of permutations of this form can be given as k N m!!. k m!. Appear in the k -th position following j < m members of type. In this case all vetoes will be overridden. The number of permutations of this form is N m m k! N k!. k j j Therefore, the Shapley-Shubik Index for Type is k N m m k N k N m m + k m N j 0 N k j!!!! j!! =! For a type to be pivotal, the possibilities are. Appear as the m -th member of type one in any position greater than or equal k and less than k. The number of permutations satisfying this requirement is k i N m!!. i m! i= k. Appear in the k -th position following j < m members of type. As above this is N m m given by k! N k!. k j j

31 Therefore, SS = i= k i N m m!! k N k N m m + i m N j 0 N k j!! j.!! =! k To prove the claim, note that SS SS = i= k i! N m! k N m i m N!!!! k m! N! k which must be positive for any k > k since the first term of the summand is greater than k! N m!. k m! Proof of Proposition 5: Suppose that k i is constant across i. Then, in any symmetric, stationary, subgame perfect Nash equilibrium, v τ v s if p τ > p s. proof: The proof is nearly identical to that of Proposition except that /N is replaced with p τ and p s. Proposition 6: In a symmetric, stationary, subgame perfect Nash equilibrium for the model with two types, i) v v p N m k m = δ max, p N m δ p = max, p m m δk when m k when m < k

32 ii) v > v if and only if p N m δk m > p N m δ when m < k p p > m m δk when m k Proof: We will consider only the case of m < k. The other case is very similar. From Proposition 5, we know that if p > p, v v. Suppose that v > v. Then all proposers will choose type s first. Since m < k, both proposers will have to choose some of both types. Therefore, from Lemma.i, z δv = z δ v which implies v δπ δ = δπ δ p v = p. p. Since all type s are always chosen, Π Type s are chosen with probabilities k m m and k m by types and m respectively. Therefore, Π = p k m + p m k m m algebra, we v v p N m δ k m = p N m δ. Plugging these in and doing which must be greater than if v > v. Otherwise, we must have v = v. The rest of the proposition follows trivially.

33 Figure Continuation Value Ratios 7 9 Ratio Veto Power of Group Veto Power of Group

34 Figure Effect of Discounting on Veto Power Override Threshold for Type Delta =.9 Delta =.7 Delta =.5 Delta = Ratio v/v

35 Figure 3 Critical Proposal Power Ratios Number of Type Members delta =.9 delta =.7 delta = Ratio p/p

36 Figure 4 Proposal Power Needed to Balance Veto Power Override Threshold for Type m = 00 m = 00 m= Ratio p/p

37 Figure 5 Proposal Power Needed to Balance Veto Power Override Threshold for Type Delta =.7 Delta =.5 Delta = Ratio p/p