Endogenous Voting Agendas

John Duggan∗ Department of Political Science and Department of Economics University of Rochester Rochester, NY 14627

July 18, 2005

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John Duggan Wallis Institute of Political Economy University of Rochester Rochester, NY 14627

Running title: Endogenous Agendas

∗I thank David Austen-Smith and an anonymous referee for their feedback. Support from the National Science Foundation, grant number SES-0213738, is gratefully acknowledged. Abstract

Existence of a “simple” pure strategy subgame perfect equilibrium is established in a model of endogenous agenda formation and sophisticated voting; upper hemicontinuity of simple equilibrium outcomes is demonstrated; and connections to the set of undominated, or “core,” alternatives are examined. In one dimension with single-peaked preferences, the simple equilibrium outcome is essentially unique and lies in the core, providing a game-theoretic foundation for the median voter theorem in terms of endogenous agenda setting. Existence of equilibrium relies on a general characterization of sophisticated voting outcomes in the presence of “majority-ties,” rather than the standard tie-breaking convention in voting subgames in favor of the alternative proposed later. The model is illustrated in a three-agent distributive politics setting, and it is shown there that the standard tie-breaking convention leads to non-existence of equilibrium. 1 Introduction

There is currently no general equilibrium existence result for the canonical model of endogenous agenda formation. This paper provides such a result and examines several fundamental modelling issues, including continuity of the equilibrium correspondence and connections to the core. Indeed, one contribution of the paper is a game-theoretic foundation for the core in terms of agenda formation. The proof of existence requires attention to difficult technical issues, one having to do with the choices of agents with multiple optimal proposals, and another having to do with the treatment of ties in voting subgames. We show by example that the usual approaches to these problems lead to the absence of equilibria in some voting subgames and, therefore, in the game as a whole. With respect to the treatment of ties, our response is to endogenize the votes of indifferent voters, but this demands a deeper understanding of equilibrium voting behavior in amendment agendas. Along the way to existence, we accordingly give a characterization of all equilibrium outcomes of any given amendment agenda, without relying on a fixed tie-breaking assumption.

The agenda game proceeds as follows. In a fixed sequence, a number of agents propose alternatives to be considered by the group. Let x0 denote a status quo alternative, let x1 denote the alternative proposed first, followed by x2, . . . , and finally xk. Voting proceeds as in an amendment agenda: a vote is held between xk and xk−1, and then a vote is held between the winner of the first round and xk−2, and so on, with the final winner being the alternative chosen.1 The alternatives are assumed to lie in an arbitrary compact metric space, capturing both a finite set of alternatives and the spatial model of politics, where alternatives are assumed to lie in some convex subset of finite-dimensional Euclidean space. Preferences of the agents are only assumed to be continuous, which imposes no restriction in the case of a finite set of alternatives and generalizes the usual assumptions used in the spatial model. We establish the existence of a simple type of subgame perfect equilibrium of the agenda game, in which equilibrium voting strategies depend on proposals in a restricted way.2 Indeed, we show that every simple equilibrium outcome is supported by an equilibrium exhibiting Austen-Smith’s (1987) “sophisticated sincerity,” where behavior along the path of play is observationally equivalent to myopic, or “sincere,” voting.

We prove existence and Pareto optimality of simple equilibria, and we examine the rela- tionship between these outcomes and the core, the set of alternatives that are undominated with respect to the voting rule. If the core is “strong,” meaning some core point dom-

1See Ordeshook and Schwartz (1987) for a discussion of other common types of agendas. 2Existence of a pure-strategy subgame perfect equilibrium follows from Harris (1985), but his results do not deliver any speciﬁc properties of the equilibrium. Since he considers perfect information games, his result yields equilibria in the version of our model in which voting is sequential, but these equilibria could use schemes to “punish” individual voters, which we do not rely on.

1 inates all other alternatives, then the strong core point is the unique simple equilibrium outcome of the amendment agenda game. Such alternatives typically fail to exist when voting is by majority rule and the set of alternatives is multidimensional,3 but we prove a continuity result with the following implication: if preferences are close to admitting a strong core point, then the simple equilibrium outcomes must be close to that point. For the case of a convex one-dimensional set of alternatives and single-peaked preferences, the simple equilibrium outcome is unique (even if the core is not strong) and lies in the core, providing a game-theoretic foundation for Black’s (1958) median voter theorem in terms of endogenous agenda setting. Moreover, we show by example that the result is tight: if even one agent’s preferences violates single-peakedness, then there may be simple equilibrium outcomes outside the core.

The literature on voting in agendas focuses mainly on the case in which there are no majority-ties among the alternatives on the agenda. When we endogenize the agenda and the alternatives are picked from a convex subset of Euclidean space, however, we can no longer assume the absence of ties — indeed, there will always be subgames in which ties occur, and we must address the existence of equilibria in those subgames as well as others. Banks and Bordes (1988) consider a fixed agenda and allow for ties, in the sense of indif- ferences in a social preference relation, and they consider several conventions for resolving ties in the voting game defined by the agenda: one used in other work on agenda setting is to always resolve ties in favor of the alternative earlier in the agenda (proposed later, in our framework).4 An implication of our equilibrium analysis of the endogenous agenda game is that the distinction — not drawn in other work — between two kinds of ties becomes critical: there are “procedural” ties, where two alternatives receive the same number of votes, and there are “preferential” ties, where, possibly because some agents are indifferent, the number of agents with a strict preference for either alternative is not sufficient to decide the vote. For procedural ties, the usual convention of resolving ties in favor of the alternative proposed later is problematic, because it introduces a type of discontinuity into the problem of a proposer, and we in fact adopt the opposite convention. In any case, this is probably more natural: since the initial alternative is a status quo, it might well serve as a default; and a bill to change the status quo would likely have greater priority than an amendment to the bill or amendments to the amendment introduced later.

In a preferential tie, the votes of the indifferent agents may be decisive, and since these agents are indifferent, voting for either alternative is a best response. In such cases, the usual convention of resolving in favor of the alternative proposed later is not a structural assumption, but rather it is a significant behavioral restriction. The construction used

3See Plott (1967), Schoﬁeld (1983), Cox (1984), Le Breton (1987), Banks (1995), and Saari (1997). 4This convention is actually deﬁned earlier by Shepsle and Weingast (1984), who allow ties but do not delve into the details.

2 to prove existence of a simple equilibrium makes use of the flexibility of specifying votes arbitrarily for indifferent agents, and we demonstrate that this flexibility is critical for the solution of the existence problem: in a distributive politics example, where three agents must decide on a division of a dollar, we show that the usual convention leads to the absence of a best response for the second agent in some subgames and, therefore, to the non-existence of a simple equilibrium. Thus, for a general existence result, indifferent voters must be employed in a more subtle way. In pursuing this approach, we extend the results of Shepsle and Weingast (1984) and Banks and Bordes (1988) to characterize the outcomes of sophisticated voting when agents may be indifferent and the votes of such agents are not a priori fixed in favor of one alternative.

In earlier work on endogenous amendment agendas, Banks and Gasmi (1987) characterize the unique equilibrium outcome of a three-agent model, assuming majority rule, Euclidean preferences, and a two-dimensional space of alternatives. They consider several protocols, including one where the agents sequentially propose alternatives to form the agenda, as in this paper, but they resolve all preferential ties in favor of the alternative proposed later.5 It is straightforward to extend the distributive politics example to their Euclidean setting, with the conclusion that, for some configuations of ideal points, there is in fact no subgame perfect equilibrium under their convention. Using our methods, however, their conjectured equilibrium can be corrected with minor modifications: at points where the convention leads to the absence of a best reponse for one agent, behavior in the voting subgame can be specified to repair the discontinuity. Austen-Smith (1987), in his analysis of sophisticated sincerity, takes up endogenous amendment agendas in the context of the spatial model with an arbitrary odd number of agents, allowing a randomly chosen agent to propose one alternative, then randomly choosing another to propose a second alternative, and so on. He claims that there is a unique subgame perfect equilibrium of the endogenous agenda game, but his proof omits consideration of the difficult issues surrounding existence of an equilibrium. McKelvey (1986) considers a model in which proposers are “pessimistic,” rather than strategic, about the outcome of agenda-setting. Dutta, Jackson, and Le Breton (2002) consider endogenous agendas in a less structured setting, imposing a consistency condition on proposals, rather than using a game-theoretic equilibrium analysis.

For the case of just two proposers, the endogenous agenda model has an interesting interpretation in terms of an election with policy-motivated candidates: here, we view the proposers as candidates who sequentially commit to policy platforms before an election. This extends the models of Wittman (1983), Calvert (1985), and Duggan and Fey (2005), who consider the case of simultaneous platform choice. In contrast to the results of Duggan

5Procedural ties are not an issue under majority rule with three agents. Austen-Smith (1987) also assumes majority rule with an odd number of agents, ruling out procedural ties.

3 and Fey, which give extremely restrictive conditions for the existence of a Nash equilibrium in multidimensional policy spaces, we establish existence of a simple equilibrium in the model with sequential platform choice. And in environments with a one-dimensional policy space and single-peaked preferences, where simultaneous platform choice leads the candidates to locate at the median, our results on the core generate the same prediction in the model with sequential platform choice.

In the rest of the paper, Section 2 develops the agenda formation model and describe a simple algorithm for computing equilibrium outcomes of any given amendment agenda. Section 3 contains a discussion the technical diﬃculties in proving existence of an equilibrium. Section 4 contains the existence proof, a characterization of simple equilibria, and a result on the upper hemi-continuity of the simple equilibrium correspondence. Section 5 considers the Pareto optimality of simple equilibrium outcomes and connections to the core. Section 6 solves for simple equilibria in the context of distributive politics and illustrates the failure of the usual conventions adopted in the analysis of endogenous agendas. Section 7 extends the basic model to allow for random proposer selection. Section 8 concludes.

2 The Model

Let N denote a finite set of n agents, denoted 1, 2, . . . , n. Let X denote a compact metric space of alternatives, which may be finite or, as in the spatial model of politics, may be any compact subset of finite-dimensional Euclidean space. Suppose each agent i has a preference relation on X represented by a continuous utility function ui: X → R, with no further properties required. Thus, we capture continuous and strictly quasi-concave utilities, the usual assumption in the spatial model, and much more. Let x0 ∈ X denote an exogenous “status quo” alternative.6 Now consider any finite sequence of agents (possibly with repetitions), and without loss of generality let the sequence be 1, 2, . . . , k. The agents will propose alternatives (possibly the status quo) in that order, call them x1, x2, . . . , xk, and those proposals will then be voted on in an amendment agenda: xk vs. xk−1, winner against xk−2, and so on, with the last remaining proposal matched against the status quo, x0. The winner of that last vote is then the final outcome of the game.

In each pairwise vote, say xh vs. xh−1, each agent may vote for xh or xh−1, with no abstention. The outcome of each vote is decided by a collection D ⊆ 2N of decisive coalitions, where we assume only that D is non-empty, proper (C ∈ D implies N \ C/∈ D), and monotonic (C ∈ D and C ⊆ C0 implies C0 ∈ D). Thus, we capture majority rule (D = {C ⊆ N | #C > n/2}), any “quota rule” (where a ﬁxed number, greater than n/2,

6 The setup is general enough to capture situations without a status quo: simply add a point x0 to the set of alternatives and specify utilities so that every alternative is preferred by every agent to x0.

4 of voters is needed to be decisive), and rules that give some agents veto power as special cases. We say a coalition C is blocking if its complement is not decisive, and we denote the collection of blocking coalitions by

B = {C ⊆ N | N \ C/∈ D}.

In the case of majority rule, for example, the coalitions with at least n/2 members are blocking.

In our analysis of voting, we must distinguish between two types of ties. The ﬁrst is

“procedural,” when neither the set of voters voting for xh is decisive nor is the set of voters voting for xh−1, i.e.,

0 C = {i ∈ N | i votes xh} ∈ B and C = {i ∈ N | i votes xh−1} ∈ B.

In the case of majority rule, this occurs when n is even and the voters are evenly split between the two alternatives. In the event of procedural ties, we resolve the tie in favor of the lower indexed alternative (the alternative proposed earlier and appearing later in the agenda), which is xh−1 here. If D is strong (C/∈ D implies N \ C ∈ D), then the decisive and blocking coalitions are the same, and procedural ties cannot occur. Then a proposal passes if and only if the set of voters voting for it is decisive. This will be the case for majority rule when n is odd, and more generally D will be strong if the vote of a designated agent, a “president,” breaks any ties. The other type of tie is “preferential,” where neither the set of voters who strictly prefer xh is decisive nor is the set of voters who strictly prefer xh−1, i.e.,

0 C = {i ∈ N | ui(xh−1) ≥ ui(xh)} ∈ B and C = {i ∈ N | ui(xh) ≥ ui(xh−1)} ∈ B.

In this event, which can occur even if D is strong, any convention about which proposal wins implicitly restricts the behavior of indifferent voters. We will not assume any such convention, and, indeed, we will see that some flexibility in the behavior of indifferent voters is needed for the existence of equilibria.

In each pairwise vote, we assume for simplicity that voters cast their ballots simultane- ously.7 We assume that information is complete and that all proposals and voting winners are observed by all agents later in the game, and we analyze the agenda formation process as an extensive form game: each partial list (x1, . . . , xh) deﬁnes a subgame in which agent h + 1 proposes next; each complete agenda (x1, . . . , xk) deﬁnes a voting subgame; and an agenda and a partial list (w1, . . . , wh) of winners in the voting subgame (where w1 is the

7We could assume sequential voting with perfect information without aﬀecting the analysis. In that case, we would apply subgame perfection to voting subgames, rather than iterative elimination of weakly dominated strategies.

5 winner of the xk vs. xk−1 vote, etc.) define a smaller voting subgame. A pure strategy for agent h specifies a proposal for every (x1, . . . , xh−1) and specifies a vote for every agenda and partial list of winners.8 There will typically be a large number of subgame perfect equilibria in the voting subgames, due to the fact that, if every agent votes for x over y, then no one agent’s vote can change the result (unless one agent is a “dictator,” i.e., C ∈ D if and only if i ∈ C). Thus, typically, every alternative on an agenda can be supported as a subgame perfect equilibrium of the voting subgame.

We eliminate many implausible equilibria by requiring that the agents’ voting strategies survive the iterative elimination of weakly dominated strategies (IEWDS) for a particular order of elimination. If each ui is one-to-one, so that preferences between any two alternatives are strict, and if D is strong, then no procedural or preferential ties can occur, and the outcomes of voting strategies surviving IEWDS are well-understood: in a vote between two alternatives, the unique IEWDS outcome is the unique alternative preferred by all members of a decisive coalition. For binary voting agendas, such as amendment agendas, McKelvey and Niemi (1978) have deﬁned the “multistage sophisticated solution,” which builds on this insight to give the unique IEWDS outcome as the solution to a straightforward backward induction algorithm. In Figure 1, for example, we give the “voting tree” representation of an amendment agenda, the top branches indicating a vote between x3 and x2, the winner then being matched against x1, with the winner of that vote against x0.

Suppose there are three agents with utilities below, and let D be majority rule.

u1(x0) > u1(x2) > u1(x1) > u1(x3) u2(x3) > u2(x1) > u2(x0) > u2(x2) u3(x2) > u3(x3) > u3(x1) > u3(x0)

In the votes between x0 and x1, since a majority of agents prefer x1, it is the unique outcome of IEWDS. Similarly, x0 beats x2, and x3 beats x0. Thus, we have solved all of the terminal votes. To solve the penultimate votes, say x2 vs. x1, it is necessary to consider the final outcome when either of these alternatives is passed: if x1 passes, then it will pass the final vote and be the outcome; in contrast, if x2 passes, then x0 will pass the final vote and be the outcome. Thus, x1 will receive a majority of votes against x2, even though two out of three voters would prefer x2. These final outcomes, called “sophisticated equivalents,” are indicated in Figure 1 with circles. We continue in this way until we have a sophisticated equivalent of the first vote, which will be the outcome of IEWDS in the agenda. In the above example, we see that x3 is the unique outcome.

The process of IEWDS is less well-understood in the presence of ties. If only procedural

8There could conceivably be equilibria in voting strategies that condition on more than which alternatives have won, namely, how particular voters voted. Our focus could be viewed as an equilibrium reﬁnement. Alternatively, we could assume votes are by secret ballot.

6 x 3 ...... ...... ...... x3 x3 ...... x2 x1 ...... ...... ...... ...... ...... x3 x3 ...... x1 x1 x0 x2 ...... x1 x1 ...... ...... ...... x3 ...... x0 x1 ...... x0 x2 ...... x0 x1 ...... x0 ......

Figure 1: An Amendment Agenda ties are possible and the tie-breaking convention in favor of the alternative proposed earlier is assumed, then little is changed. The multistage sophisticated solution, though not actually defined in such environments, is easily extended. When preferential ties are possible, however, IEWDS may lead to multiple outcomes of voting. Consider the final vote between x0 and some alternative, say x. For agents who strictly prefer x to x0, voting for x0 is weakly dominated, and we eliminate that strategy. Such voters then vote for x. For agents who strictly prefer x0, an analogous argument shows that they will vote for x0. Indifferent agents may vote either way. Thus, the outcomes of IEWDS are as follows:

• If all members of some decisive coalition strictly prefer x, then all will vote for it, so x is the unique outcome.

• If all members of some blocking coalition strictly prefer x0, then all will vote for it; and since we break procedural ties in favor of the lower indexed alternative, x0 in this case, it is the unique outcome.

• If all members of some decisive coalition weakly prefer x, then all members may vote for it, so x is an outcome of undominated strategies.

• If all members of some blocking coalition weakly prefer x0, then all members may vote for it; and by our tie-breaking assumption for procedural ties, x0 is an outcome of undominated strategies.

In contrast to the case of no indiﬀerences, we can now easily have multiple IEWDS outcomes:

7 for a trivial example, suppose all agents are indiﬀerent between x0 and x.

The extension of the multistage sophisticated solution to this case is still conceptually straightforward. When there are two possible outcomes of IEWDS at a vote, we arbitrarily choose one and proceed with the algorithm until we have a sophisticated equivalent of the initial vote between xk and xk−1. We then return to an instance of multiple outcomes, choose the other alternative, continue until an outcome is determined, and repeat until all possible combinations of winners at diﬀerent nodes have been covered. Now modify the example in Figure 1 to give agent 2 one indiﬀerence, i.e.,

u2(x3) > u2(x1) > u2(x0) = u2(x2).

The final vote between x2 and x0 then has two possible outcomes, so we arbitrarily pick one. Choosing x0 and continuing, we arrive at the same outcome, x3, as we did without indifference. Now choosing x2 as the sophisticated equivalent of this vote, we see that a majority of agents prefer x2 to x1 (the sophisticated equivalent of the x1 vs. x0 vote), so that x2 is the unique winner in the x1 vs. x2 vote after the next round of elimination. Similarly, a majority prefer x2 to x3, and x2 is the sophisticated equivalent of the first vote and, therefore, the outcome after the last round of elimination.

The application of the multistage sophisticated solution described above can be time consuming, even when ties are not an issue. For the special case of amendment agendas, which we consider here, Shepsle and Weingast (1984) deﬁne a simpler algorithm that yields the same outcomes in the absence of ties. To extend their algorithm to the current setting, where we allow ties, let Ri(x) be the alternatives that give i utility at least ui(x), and let Pi(x) be the alternatives that give i utility strictly greater than ui(x), i.e.,

Ri(x) = {z ∈ X | ui(z) ≥ ui(x)} and Pi(x) = {z ∈ X | ui(z) > ui(x)}.

Then deﬁne [ \ [ \ RD(x) = Ri(x) and PD(x) = Pi(x). C∈D i∈C C∈D i∈C

Thus, y ∈ RD(x) if all members of some decisive coalition weakly prefer y to x, and y ∈ PD(x) if all members of some decisive coalition strictly prefer y to x. The relation PD is often given the interpretation of “social” strict preference. If D is strong, then B = D, and

RD is the corresponding “social” weak preference. By continuity of each ui, each Ri has closed graph, and therefore so does RD. Similarly, each Pi has open graph, and therefore so does PD. Given alternatives z1, . . . , zh, also deﬁne \h \h RD(z1, . . . , zh) = RD(zj) and PD(z1, . . . , zh) = PD(zj). j=1 j=1

8 By the above, it follows that these correspondences are closed and open, respectively.

Our method of computing outcomes of the multistage sophisticated voting solution in voting subgames is the following generalization of the Shepsle-Weingast algorithm. Let

(x1, . . . , xk) be an arbitrary agenda, let y0 = x0 be the status quo, and call (y0, . . . , yh) a provisional selection of order h if, for each j = 1, . . . , h, three conditions hold:

• yj ∈ {xj, yj−1},

• yj = xj if xj ∈ PD(y0, . . . , yj−1),

• yj = yj−1 if xj ∈/ RD(y0, . . . , yj−1).

Thus, yj is equal to either xj or yj−1. When xj 6= yj−1, the jth provisionally selected alternative may equal xj if and only if, for every previously selected alternative, there is some decisive coalition whose members weakly prefer xj to the earlier alternative; and it may equal yj−1 if and only if there exists a previously selected alternative such that, for every decisive coalition C, some member of C weakly prefers the earlier alternative to xj.

An implication is that if the ﬁrst h − 1 provisionally selected alternatives are y0, . . . , yh−1, independently of the proposal of agent h, then the agent can ensure that his/her proposal is provisionally selected if and only if

xh ∈ PD(y0, . . . , yh−1) ∪ {yh−1}. We refer to a provisional selection of order k simply as a provisional selection.

Our solution for voting subgames is then the set  ¯  ¯  ¯ there exist y1, . . . , yk−1 such that  W (x , . . . , x ) = y ∈ X ¯ (y , . . . , y , y ) is a provisional 1 k  k ¯ 0 k−1 k  ¯ selection of alternatives that appear at the end of provisional selections. This actually gives us a refinement of the generalized multistage sophisticated solution defined above, and therefore a subset of outcomes of IEWDS in voting subgames, where selections from multiple IEWDS winners in the amendment agenda are subject to a mild consistency condition.9 In the above modification of the example in Figure 1, there are two provisional selections, depending on the vote of agent 2, who is indifferent between x0 and x2. These are depicted below.

agenda x0 x1 x2 x3 1st selection x0 x1 x1 x3 2nd selection x0 x1 x2 x2

9Speciﬁcally, our solution implicitly requires that whenever there is a vote between two sophisticated equivalents in the tree, say one agent’s proposal xi from round i and another agent’s proposal xj from round j, and we select xi as the outcome of IEWDS, then we must select similarly at every point in the tree where the same two sophisticated equivalents are compared.

9 The value of y2 is not pinned down here, because x2 ∈ RD(x0, x1) and x2 ∈/ PD(x0). The set of possible equilibrium outcomes consists of the terminal alternatives of the provisional selections, namely, x2 and x3.

This characterization allows a parsimonious representation of equilibrium behavior in voting subgames. Rather than specify voting strategies for every agent in every subgame, we specify a voting equilibrium mapping as a function, y, that maps agendas to provisional selections: we use the notation

y(x1, . . . , xk) = (y0(x1, . . . , xk),..., yk(x1, . . . , xk)) for such a mapping. Since voting takes place after the agenda is set, the equilibrium provisional selection could conceivably vary arbitrarily with the agenda: in particular, votes of indiﬀerent agents in the xh−1 vs. xh vote could potentially be a function of alternatives proposed after xh in the amendment game. This kind of dependence signiﬁcantly complicates the proposal problem of agent h, potentially making it impossible for the agent to propose a provisionally selected alternative. A simple voting equilibrium mapping is a voting equilibrium mapping with the added restriction that, for all h = 0, . . . , k, yh(x1, . . . , xk) depends only on x1, . . . , xh. Thus, we write yh as a function of x1, . . . , xh only, as in yh(x1, . . . , xh). A simple equilibrium of the amendment agenda game is a pure strategy subgame perfect equilibrium such that outcomes in voting subgames are given by a simple voting equilibrium mapping.

Note that there always exists a simple voting equilibrium mapping: set y0 = x0 and, for each h and x1, . . . , xh, inductively define ½ xh if xh ∈ RD(y0,..., yh−1) yh(x1, . . . , xh) = yh−1 else, where the arguments of y1,..., yh−1 are suppressed in this expression. This is the mapping that, for every agenda, resolves procedural ties in favor of the alternative proposed earlier (appearing later in the agenda) and, in the case of preferential ties, essentially has indifferent agents vote for alternatives proposed later. Such a provisional selection, which we refer to as a weak provisional selection, treats preferential ties in the same way that the “type 1 trajectories” of Banks and Bordes (1988) do. And when D is strong, they are equivalent to the type 1 trajectories. We could define another simple voting equilibrium mapping by ½ xh if xh ∈ PD(y0,..., yh−1) yh(x1, . . . , xh) = yh−1 else, which selects Banks and Bordes’ (1988) “type 2 trajectories” and resolves all ties in favor of the alternative proposed earlier.

10 Austen-Smith (1987) notes that for some agendas and provisional selections, sophisticated voting, which we consider here, is observationally equivalent to myopic, or “sincere,” voting. For example, suppose that the relation RD, restricted to the agenda, is transitive with later proposals weakly preferred to earlier ones, i.e., xh ∈ RD(xl) whenever h ≥ l. Then the weak provisional selection is just (x1, . . . , xk) itself, which corresponds to xk win- ning each vote along the path of play. Since xk is socially weakly preferred to every other alternative on the agenda, myopic voters would generate the same winners along the path of play. If a simple equilibrium is such that the agenda and provisional selection along the path of play exhibit this property, we follow Austen-Smith in saying that it exhibits “sophisticated sincerity.” We will see that every simple equilibrium outcome can be supported by a “sophisticatedly sincere” simple equilibrium.

3 The Diﬃculty

As discussed in the previous section, there exists a simple voting equilibrium mapping, so the existence problem reduces to finding one for which the agenda formation stage has a subgame perfect equilibrium. If X is finite, then, given any simple voting equilibrium mapping, the agenda formation stage is a finite extensive form game of perfect information and, therefore, possesses a pure strategy subgame perfect equilibrium. In fact, if agents’ preferences are “linear,” in the sense that ui(x) = ui(y) implies x = y, then there is a unique subgame perfect equilibrium outcome of the game. To capture the broad scope for policy- making in real-world politics, however, it is natural to suppose the set X of alternatives is infinite, often a convex subset of Euclidean space. In that case, even though the preferences of agents are assumed to be continuous, the amendment agenda game becomes a complex extensive form game with continuous action spaces, and existence of best responses becomes problematic in some subgames.

A well-known issue in such games is that of discontinuities in an agent’s payoff function induced by the choices of later agents. In a perfect information extensive form game in which the agents move in a fixed order, consider the set of payoff vectors, say Bî+1(ai), that 10 can be supported in an equilibrium of the subgame following agent i’s choice of ai. If we consider varying agent i’s action over a feasible set Ai, the possibility of a best response for that agent will depend on the existence of a selection (as a function of i’s actions) from

Bî+1 satisfying certain properties. It is sufficient, of course, if the agent’s action set is compact and there is a selection continuous in i’s payoff. And a sufficient condition for this, via Michael’s Selection Theorem, is that the correspondence Bî+1 have non-empty, closed, convex values and be lower hemicontinuous in i’s payoff (as a function of i’s actions). These

10We suppress the actions of earlier agents, considering them ﬁxed for now.

11 conditions may be reasonable for the last agent to move, since his/her action determines the final payoff vector, and they are satisfied in the agenda formation stage by the last proposer: if we restrict the last agent’s actions to the set

RD(y0,..., yk−1(x1, . . . , xk−1)), and if we provisionally select all choices from in this set, then agent k’s proposal is the ﬁnal outcome, and the conditions are satisﬁed. The problem is to ensure that these properties are inherited by the decision problems of earlier agents.

Suppose that the correspondence Bî+1 has non-empty, closed, and convex values, and that it is actually continuous (not just lower hemicontinuous) as a function of the actions of earlier agents. Let Bî(ai−1) denote the set of payoff vectors that can be supported in an equilibrium of the subgame following agent i − 1’s choice choice of ai−1. These payoff vectors are generated by best responses for agent i supported by equilibrium outcomes in the continuation game following i’s action, and we assume that these outcomes are given by a continuous selection (as a function of i − 1’s and i’s actions) from Bî+1. If agent i’s action set Ai(ai−1) is compact and varies continuously as a function of agent i − 1’s actions, then, by the Theorem of the Maximum, the correspondence Bî will have non-empty, closed values and will be upper hemicontinuous in i − 1’s payoff (as a function of i − 1’s action). Unless agent i’s optimal proposal is unique for every a−i, however, Bî need not be convex-valued or lower hemicontinuous in agent i − 1’s payoff (as a function of i − 1’s action). Thus, the correspondence for the agent who moves before i may not inherit the properties of Bî+1.

To illustrate this problem in the context of the amendment agenda game, suppose there are three agents with indifference maps over two-dimensional space as depicted in Figure 2. Suppose for simplicity that the status quo and agent 1’s proposal are far-removed from the ideal points of the agents, so agents 2 and 3 can essentially ignore these alternatives: agent 2 proposes y, and agent 3 then proposes z. Suppose that outcomes of voting subgames are given by the weak provisional selections, so that agent 3 can obtain any alternative in RD(y), a compact set that varies continuously in y. Thus, the desired conditions are satisfied for agent 3. Note, however, that agent 2’s payoffs are not convex-valued or lower hemicontinuous in y: for example, at the proposal y0, agent 3 has two optimal proposals, z0 and z00; but as agent 2 moves his/her proposal from y through y0, agent 3’s optimal proposal moves from z to z0, then z0 and z00 are both optimal, and then it leaves through z00. Furthermore, agent 2 strictly prefers z0 to z00, and there is no selection continuous in 2’s payoffs.

Existence of a continuous selection for agent 2 is sufficient for the agent to have an optimal proposal, but not necessary. It would be enough to find a selection that is upper semicontinuous in 2’s payoff, and this is possible in the preceding example: when agent 2

12 ...... 3...... •...... z ...... •...... z00 ...... •...... 0 ...•...... z ... 0 ...... y ...... y.....•...... 2...... 1 ......

Figure 2: Optimal Proposals proposes y0, for instance, we may specify that agent 3 propose z0, which agent 2 prefers to 3’s other optimal proposal. More generally, let

Bˆ3(x1, x2) = arg max{u3(x) | x ∈ RD(x0, x1, x2)}, denote agent 3’s optimal proposals after agents 1 and 2 propose x1 and x2. The approach is to require that agent 3 solve

max{u2(x) | x ∈ Bˆ3(x1, x2)}, (1) i.e., that the agent propose optimally but break indiﬀerences in agent 2’s favor. The correspondence RD has nonempty and compact values, so Bˆ3 has nonempty and closed values. Since X is compact, the optimization problem in (1) indeed has a solution. If the correspondence RD is continuous in x2, then the Theorem of the Maximum implies that Bˆ3 is upper hemicontinuous in x2, and then agent 2’s maximized utility in (1), denoted ˆb2(x1, x2), is upper semicontinuous in x2. Therefore, the maximization problem

max{ˆb2(x1, z) | z ∈ RD(x0, x1)} for agent 2 has a solution, and we have established a selection from Bˆ3 that gives agent 2 an optimal proposal.

Unfortunately, this fails to address the discontinuity problems faced by agents making earlier proposals, agent 1 here. For every proposal x1, we have established that agent 2 has some optimal proposal, say z, and there is some x ∈ Bˆ3(x1, z) that maximizes u2 over

13 Bˆ3(x1, z), i.e., u2(x) = ˆb2(x1, z). In other words, the set of outcomes that can be supported in equilibrium after agent 1’s choice of proposal of x1,

Bˆ2(x1) = {x ∈ X | there exists z ∈ RD(x0, x1) such that

x ∈ Bˆ3(x1, z) and u2(x) = ˆb2(x1, z)}, is non-empty. Continuing the above approach, specifying that agents 2 and 3 break indif- ferences in favor of agent 1, we require that the ﬁnal outcome solve

max{u1(x) | x ∈ Bˆ2(x1)} for every proposal x1. But Bˆ2 need not be closed, and this optimization problem may not have a solution. This is demonstrated explicitly in the distributive politics example of Section 6 (Case 1 in Appendix B), where, if we specify that agent 3 always break indiﬀerence in favor of agent 2, then agent 1 does not have an optimal proposal.11

In short, while the approach outlined above solves the discontinuities faced by agents 2 and 3 (more generally, k −1 and k), it cannot be used recursively to solve the discontinuities facing agents who propose earlier. But there is a further difficulty that has been neglected in the above discussion. If the agents’ utility functions are continuous and strictly quasi- concave, as drawn in Figure 2, then it is known that the correspondence RD(x1) is continuous as a function of x1 (cf. Banks and Duggan (2000), Theorem 7), but the correspondence RD(x1, x2) need not be continuous: intersections of lower hemicontinuous correspondences do not generally inherit lower hemicontinuity, and the typical convexity conditions on agents’ preferences do not solve the problem here. This means that Bˆ3(x1, x2), defined above, may not be upper hemicontinuous in x2, and the maximization problem in (1) may not have a solution after all. Thus, the approach of breaking 3’s indifference in 2’s favor is problematic from the start, if we resolve preferential ties in voting subgames in favor of the alternative proposed later. In order to solve the equilibrium existence problem, we must repair the violations of lower hemicontinuity exhibited by RD, and to do this we must take advantage of the greater flexibility available in selecting equilibria in voting subgames. This is demonstrated explicitly in the distributive politics example of Section 6 (Case 6 in Appendix B), where the standard tie-breaking assumption for preferential ties — provisionally selecting the alternative proposed later (earlier in the agenda) — is inconsistent with the existence of a simple equilibrium.

The approach of the next section is to focus on the sets of possible continuation equilibrium outcomes for the agents, but explicitly allowing for arbitrary simple voting equilibrium selections (rather than just weak provisional selections) and using arbitrary selections from

11See Hellwig, Leininger, Reny, and Robson (1990) for another explicit example of the failure of this approach.

14 best responses of later agents (rather than having agent k break indiﬀerences in favor of agent k − 1).

4 Existence and Continuity

In this section, we prove existence of simple equilibria and provide a characterization of the simple equilibrium outcomes of the amendment agenda game. We then use this to prove that the simple equilibrium outcome correspondence is upper hemicontinuous in the status quo and in the preferences of the agents.

Theorem 1 There exists a simple equilibrium of the amendment agenda game.

The first part of the proof puts in place some mathematical machinery used later to construct a non-empty set of simple equilibria of the amendment agenda game. We define, given a provisional selection of order k−1, the non-empty set of outcomes of possible optimal proposals by agent k, each of these proposals becoming optimal when the kth provisionally selected alternative is specified appropriately as a function of agent k’s proposal. We then define, given a provisional selection of order k − 2, the non-empty set of outcomes induced by possible optimal proposals by agent k −1, each of these proposals now becoming optimal when the k − 1th provisionally selected alternative is appropriately specified and when we select appropriately from agent k’s set of possible optimal proposals. Lemma 1, in Appendix A, shows that this approach can be used recursively to construct a non-empty set of outcomes of possible optimal proposals for every agent, these proposals being made optimal by appropriate specifications of voting behavior and proposals of later agents. We then show that the set of outcomes of possible optimal proposals for agent 1 consists of simple equilibrium outcomes by constructing, for each element x of this set, a simple voting equilibrium mapping and proposal strategies for the agents that form a simple equilibrium and yield outcome x.

Suppose agent k proposes under the assumption that, in subsequent voting games, the provisional selection of order k − 1 will be (y0, . . . , yk−1). Then xk will necessarily be provisionally selected, and will therefore be the ﬁnal outcome, if and only if xk ∈ PD(y0, . . . , yk−1) or xk = yk−1. Then deﬁne agent k’s “security level” as

bk(y0, . . . , yk−1) = sup{uk(x) | x ∈ PD(y0, . . . , yk−1) or x = yk−1}, the greatest payoﬀ that agent k can ensure by choice of xk. Since PD is open, it follows that bk is lower semi-continuous. Now deﬁne the correspondence

Bk(y0, . . . , yk−1)

15 = {x ∈ RD(y0, . . . , yk−1) ∪ {yk−1} | uk(x) ≥ bk(y1, . . . , yk−1)}, which will consist of the outcomes of possible optimal proposals by agent k.12 Note that, since X is compact, since RD has closed graph, since uk is continuous, and since bk is lower semi-continuous, Bk has non-empty, compact values and closed graph. To make x ∈ Bk(y0, . . . , yk−1) an optimal proposal for agent k, we must define the kth provisionally selected alternative appropriately as a function of k’s proposal. Define this alternative as ½ x xk if xk ∈ PD(y0, . . . , yk−1) ∪ {x} sk(xk|y0, . . . , yk−1) = yk−1 else, so the kth provisionally selected alternative is agent k’s proposal if it defeats all previously selected alternatives regardless of indifference, or if k proposes x. Otherwise, it remains yk−1. Thus, proposing x leads to x as the final outcome, and, since x gives agent k a payoff at least as great as the payoff from yk−1 or any alternative in PD(y0, . . . , yk−1), proposing x is optimal for agent k.

We deﬁne agent k−1’s security level, given (y0, . . . , yk−2), now incorporating the decision of agent k, as

supz minx uk−1(x) bk−1(y0, . . . , yk−2) = s.t. x ∈ Bk(y0, . . . , yk−2, z) z ∈ PD(y0, . . . , yk−2) ∪ {yk−2},

13 where the min operation is well-defined, since Bk has compact values. Thus, in defining this security level, we recognize that any alternative in PD(y0, . . . , yk−2) proposed by agent k−1 must be provisionally selected. After that, however, agent k’s proposal is not controlled by agent k − 1, and hence the minimization. That is restricted, of course, to proposals by agent k that can be made optimal, as discussed above. Define the set

Bk−1(y0, . . . , yk−2)

= {x ∈ X | there exists z ∈ RD(y0, . . . , yk−2) ∪ {yk−2} such that

x ∈ Bk(y0, . . . , yk−2, z) and uk−1(x) ≥ bk−1(y0, . . . , yk−2)}, which will consist of the outcomes induced by possible optimal proposals by agent k − 14 1. Note that, in order to achieve x ∈ Bk−1(y0, . . . , yk−2) as an outcome, it is necessary that agent k − 1 propose an alternative in RD(y0, . . . , yk−2) ∪ {yk−2} such that x is a x possible optimal outcome for agent k following that proposal. Let p = pk−1(y0, . . . , yk−2) ∈

12 This corresponds to the correspondence Bˆ3 in the previous section. 13 Note that this is distinct from the idea of ˆb2 in the previous section: whereas ˆb2 relied on the “optimistic” selection of agent 2’s best outcomes from agent 3’s maximizers, bk−1 assumes the “pessimistic” selection of agent k − 1’s worst outcomes from agent k’s maximizers. 14 This is similar to Bˆ2 in the previous section, except that we use the pessimistic selection bk−1.

16 RD(y0, . . . , yk−2) ∪ {yk−2} be an alternative such that x ∈ Bk(y0, . . . , yk−2, p). Later, we will refer to p as the agent’s “target proposal” to obtain the “target outcome” x. To make p an optimal proposal for agent k − 1, we must define the k − 1th provisionally selected alternative and the final outcome appropriately, as a function of k − 1’s proposal. Define ½ x xk−1 if xk−1 ∈ PD(y0, . . . , yk−2) ∪ {p} sk−1(xk−1|y0, . . . , yk−2) = yk−2 else, so the k − 1th provisionally selected alternative is agent k − 1’s proposal if it defeats all previously selected alternatives regardless of indifference, or if k − 1 proposes p. Otherwise, it remains yk−2.

To define the final outcome, note that, for every z ∈ PD(y0, . . . , yk−2) ∪ {yk−2}, there is 0 an alternative in Bk(y0, . . . , yk−2, z) that agent k − 1 weakly prefers x to. Letting z denote 0 0 15 this alternative, we therefore have z ∈ Bk(y0, . . . , yk−2, z) and uk−1(z ) ≤ uk−1(x). We now define final outcomes as a function of agent k − 1’s proposal as   x if xk−1 = p x 0 ok−1(xk−1|y0, . . . , yk−2) = xk−1 if xk−1 ∈ PD(y0, . . . , yk−2) \{p}  0 yk−2 else. x Note that, according to sk−1, agent k−1 can determine the k−1th element of the provisional selection as any element in PD(y0, . . . , yk−2)∪{p} by choosing xk−1 in this set. In that case, x ok−1(xk−1|y0, . . . , yk−2) is indeed an element of Bk(y0, . . . , yk−2, yk−1), one that gives agent k − 1 a utility no higher than does x. If xk−1 is not in this set, then the k − 1th element x of the provisional selection is yk−1 = yk−2, and again ok−1(xk−1|y0, . . . , yk−2) lies in the set of possible optimal outcomes for agent k given the provisional selection (y0, . . . , yk−2, yk−1) x of order k − 1. Again, ok−1 is defined so that this outcome is no better for k − 1 than x x is x. Thus, with sk−1 determining the k − 1th provisionally selected alternative and ok−1 determining the final outcome, proposing p leads to x as the final outcome and is indeed optimal for agent k − 1.

Given (y0, . . . , yh−1), inductively deﬁne the security level of agent h < k as

supz infx uh(x) bh(y0, . . . , yh−1) = s.t. x ∈ Bh+1(y0, . . . , yh−1, z) z ∈ PD(y0, . . . , yh−1) ∪ {yh−1}, and deﬁne

Bh(y0, . . . , yh−1)

= {x ∈ X | there exists z ∈ RD(y0, . . . , yh−1) ∪ {yh−1} such that

x ∈ Bh+1(y0, . . . , yh−1, z) and uh(x) ≥ bh(y0, . . . , yh−1)},

15 This is meant to deﬁne a notational convention. Given any alternatives x, y, or z in PD(y0, . . . , yk−2) ∪ 0 0 0 {yk−2}, we use x , y , or z to denote an alternative in Bk(y0, . . . , yk−2, ·) that x, y, or z is weakly preferred to.

17 which will consist of the outcomes induced by possible optimal proposals by h. For every x x ∈ Bh(y0, . . . , yh−1), there exists p = ph(y0, . . . , yh−1) ∈ RD(y0, . . . , yh−1) ∪ {yh−1} such that x ∈ Bh+1(y0, . . . , yh−1, p). Deﬁne ½ x xh if xh ∈ PD(y0, . . . , yh−1) ∪ {p} sh(xh|y0, . . . , yh−1) = yh−1 else, which will give the hth element of the provisional selection as a function of agent h’s proposal.

From Lemma 1, in Appendix A, it follows that Bh has nonempty values and closed graph. As a consequence, with compactness of X, we may replace the inﬁmum operation in the deﬁnition of agent h’s security value with the minimum operation, as in

supz minx uh(x) bh(y0, . . . , yh−1) = s.t. x ∈ Bh+1(y0, . . . , yh−1, z) z ∈ PD(y0, . . . , yh−1) ∪ {yh−1}.

0 Therefore, for every z ∈ PD(y0, . . . , yh−1)∪{yh−1}, there exists z ∈ Bk(y0, . . . , yh−1, z) such 0 that uh(z ) ≤ uh(x). Define   x if xh = p x 0 oh(xh|y0, . . . , yh−1) = xh if xh ∈ PD(y0, . . . , yh−1) \{p}  0 yh−1 else, which will be the final outcome of voting as a function of agent h’s proposal. As argued x x for agent k − 1, with sh determining the hth provisionally selected alternative and oh determining the final outcome, proposing p leads to x as the final outcome and is indeed optimal for agent h.

As a consequence of Lemma 1, the set of outcomes induced by the possible optimal ∗ proposals of agent 1 is non-empty. We now construct, for each x ∈ B1(x0) a simple equilibrium for which x∗ is the final outcome. We first define an appropriate simple voting equilibrium mapping and then subgame perfect proposal strategies. To define the voting equilibrium mapping, pick an arbitrary agenda (x1, . . . , xk). We specify provisionally selected alternatives by working forward through these proposals. Recall, from the construction above, that the hth provisionally selected alternative depends on three things: (i) the previously provisionally selected alternatives, y0, . . . , yh−1, (ii) the “target outcome,” say x, x from Bh(y0, . . . , yh−1), (iii) and the “target proposal” by agent h, namely ph(y0, . . . , yh−1), in order to obtain that outcome. In order to define each provisional selection, we must specify these three items for each h = 1, . . . , k. We do so inductively, starting with h = 1.

∗ To begin, let y0 = x0 denote the ﬁrst component of the voting equilibrium mapping, ∗ ∗ let x0 = x be the target outcome starting from the initial node of the amendment agenda

18 ∗ x0 ∗ game, so p1 (y0) is agent 1’s target proposal. After agent 1 proposes x1 (which may or may not be the target proposal), we deﬁne the ﬁrst provisionally selected alternative as

∗ ∗ x0 ∗ y1(x1) = s1 (x1|y0).

The target outcome following proposal x1 is deﬁned as

∗ ∗ x0 ∗ x1(x1) = o1 (x1|y0),

∗ x1(x1) ∗ ∗ and the target proposal for agent 2 is then p2 (y0, y1(x1)). After agent 2 proposes x2, the second provisionally selected alternative is deﬁned as

∗ ∗ x1(x1) ∗ ∗ y2(x1, x2) = s2 (x2|y0, y1(x1)).

Generally, for j = 2, . . . , k, the target outcome after proposals (x1, . . . , xj) by the ﬁrst j agents is deﬁned as

∗ ∗ xj−1(x1,...,xj−1) ∗ ∗ xj (x1, . . . , xj) = oj (xj|y0,..., yj−1(x1, . . . , xj−1)),

∗ xj (x1,...,xj ) ∗ ∗ the target proposal for agent j is pj (y0,..., yj−1(x1, . . . , xj−1)), and the jth provisionally selected alternative is

∗ ∗ xj−1(x1,...,xj−1) ∗ ∗ yj (x1, . . . , xj) = sj (xj|y0,..., yj−1(x1, . . . , xj−1)).

∗ ∗ ∗ ∗ This defines a voting equilibrium mapping y = (y0,..., yk), where each yh depends only on the first h proposals, so the voting equilibrium mapping is simple. To construct subgame perfect equilibrium proposal strategies, we define the proposal of agent h = 1, . . . , k as a ∗ function ph of (x1, . . . , xh−1) as

∗ ∗ xh−1(x1,...,xh−1) ∗ ∗ ph(x1, . . . , xh−1) = ph (y0,..., yh−1(x1, . . . , xh−1)).

That is, agent h makes the target proposal after (x1, . . . , xh−1) in order to obtain the ∗ target outcome xh−1(x1, . . . , xh−1). The simple voting equilibrium mapping speciﬁed here is consistent with the assumptions mentioned above in the construction of the correspondences

Bh, and we have veriﬁed that, under these expectations, the latter proposals are optimal after every history of proposals. This completes the proof.

In fact, the above arguments for Theorem 1 deliver a characterization of the simple equilibrium outcomes in terms of the set B1(x0). We have actually shown that every alternative in this set is supported by a simple equilibrium exhibiting sophisticated sincerity, so Theorem 2 has an interesting implication: every simple equilibrium outcome is the outcome of a simple equilibrium exhibiting sophisticated sincerity.

19 Theorem 2 The alternative x is the outcome of a simple equilibrium of the amendment agenda game if and only if x ∈ B1(x0).

We have already shown that every alternative in B1(x0) can be supported as a simple equilibrium outcome. Consider any simple equilibrium, with associated simple voting equilibrium mapping y. It is clear that, given proposals x1, . . . , xk−1 and the provisional selection (y0,..., yk−1(x1, . . . , xk−1)) of order k − 1, the outcome following agent k’s equilibrium proposal must lie in Bk(y0,..., yk−1(x1, . . . , xk−1)). For an induction argument, assume this is true for agent h + 1, and consider agent h. Given proposals x1, . . . , xh and the provisional selection (y0,..., yh−1(x1, . . . , xh−1)) of order h − 1, the next provisional selection will by deﬁnition lie in the set

RD(y0,..., yh−1(x1, . . . , xh−1)) ∪ {yh−1}.

And it follows directly from subgame perfection that the utility from the outcome, say x, following agent h’s equilibrium proposal satisﬁes

uh(x) ≥ bh(y0,..., yh−1(x1, . . . , xh−1)).

Thus, x ∈ Bh(y0,..., yh−1(x1, . . . , xh−1)), completing the proof.

The continuity properties of the simple equilibrium outcomes can then be analyzed through the correspondence B1. For the next result, we parameterize the utility functions of the agents by λ, where λ lies in an arbitrary metric space Λ and each ui: X × Λ → R is jointly continuous. We then deﬁne B1: X × Λ ⇒ X as above, now using ui(x, λ) in the deﬁnition of each Bh, h = 1, . . . , k. The next result shows that the simple equilibrium outcomes are upper hemicontinuous in the status quo and preference parameters of the amendment agenda game. The proof follows directly from Lemma 1, in Appendix A.

Theorem 3 The correspondence B1: X × Λ ⇒ X has closed graph.

We end this section with a brief discussion on the procedural tie-breaking assumption not used in this paper, namely, resolving such ties in favor of the alternative proposed later.

In this case, only the votes of a blocking coalition are needed against each yj, j = 0, . . . , h−1, for the proposal of agent h to be provisionally selected. We cannot, however, simply change decisive coalitions to blocking coalitions in our deﬁnition of provisional selection. Since blocking coalitions may be disjoint, there is now the possibility of a provisional selection of order h − 1, say (y0, . . . , yh), such that (i) xh 6= yh−1, (ii) xh = yl for some l = 0, . . . , h − 2, and (iii) xh is strictly preferred by the members of a blocking coalition against each yj, j ∈ {0, . . . , h−1}\{l}. Under these conditions, xh must be provisionally selected, motivating

20 + + a modification of our above definition. First, let Pi (x) = Pi(x) ∪ {x}, and define PB and + RB using the conventions in Section 2. In particular, x ∈ PB (z) if x = z or there is a blocking coalition the members of which strictly prefer x to z.

We would then deﬁne (y0, . . . , yh) as a provisional selection of order h if y0 = x0 and, for all j = 1, . . . , h,

• yj ∈ {xj, yj−1},

+ • yj = xj if xj ∈ PB (y0, . . . , yj−1),

• yj = yj−1 if xj ∈/ RB(y0, . . . , yj−1).

The approach of the proof of Theorem 1 cannot be applied under this procedural tie-breaking + assumption, because the correspondence PB (z0, . . . , zh) ∪ {zh} is not generally lower hemicontinuous, even when restricted to provisional selections of order h. This property is essential in establishing that the correspondences Bh in the proof have nonempty values and closed graph. We avoid this problem by resolving procedural ties in favor of the alternative proposed earlier, which precludes the existence of provisional selections satisfying (i), (ii), and (iii) (changing “blocking” to “decisive”) and allows us to use the correspondence

PD in the proof of Theorem 1.

5 Pareto Optimality and the Core

Because of the complexity of the amendment agenda game, characterizing simple equilibrium outcomes is a diﬃcult task. In this section, we give some elementary results in connection with the Pareto optimal alternatives and the social choice concept of the core. We will see that simple equilibrium outcomes lie in the set of Pareto optimals, and when the core is nonempty and satisﬁes an external stability property, the core is the unique simple equilibrium outcome. With Theorem 3, this shows that when the agents’ preferences are such that the core is close to nonempty and externally stable, the simple equilibrium outcomes of the amendment agenda game must be close to being in the core. In one dimension with single-peaked preferences, we characterize the (essentially) unique simple equilibrium outcome, regardless of the stability properties of the core.

We say an alternative x is Pareto optimal if there is no other alternative that every agent strictly prefers to x, and we let PO denote the set of Pareto optimals, i.e., \ [ PO = Ri(z). z∈X i∈N

21 While this is typically a large set, it is of interest that equilibrium agenda formation does indeed lead to Pareto optimal outcomes.

Theorem 4 If x is a simple equilibrium outcome of the amendment agenda game, then x ∈ PO.

To prove the theorem, suppose x is the outcome of a simple equilibrium but, for some ∗ z ∈ X and all i ∈ N, ui(z) > ui(x). Let y be the associated simple voting equilibrium ∗ ∗ mapping, and let x1, . . . , xk be the proposals made in equilibrium with provisional selection ∗ ∗ ∗ ∗ (y0, . . . , yk). Note that x = yk ∈ RD(yh) for all h = 0, . . . , k − 1. Now consider the ∗ ∗ deviation for agent k in which, following proposals x1, . . . , xk−1, the agent proposes z. Since ∗ ∗ ∗ x ∈ RD(y0, . . . , yk−1), it must be that {i ∈ N | ui(x) ≥ ui(yh)} ∈ D for all h = 0, . . . , k − 1. ∗ Since ui(z) > ui(x) for all i ∈ N, this implies {i ∈ N | ui(z) > ui(yh)} ∈ D for all h = ∗ ∗ ∗ 0, . . . , k − 1, i.e., z ∈ PD(y0, . . . , yk−1). Since the voting equilibrium mapping y is simple, ∗ ∗ ∗ the provisional selection of order k − 1 after the deviation remains (y0, y1, . . . , yk−1), and ∗ ∗ ∗ therefore the ﬁnal outcome of the deviation is yk(x1, . . . , xk−1, z) = z. Since uk(z) > uk(x), the deviation is proﬁtable, a contradiction. This completes the proof.

Deﬁne the core, denoted K, to consist of the alternatives that are socially at least as good as all others according to the voting rule, i.e.,16

K = {x ∈ X | for all z ∈ X \{x}, z∈ / PD(x)}.

Deﬁne the strong core, denoted K∗, to consist of the alternatives that are strictly preferred all others, i.e.,

∗ K = {x ∈ X | for all z ∈ X \{x}, x ∈ PD(z)}.

Clearly, the strong core consists of at most one element, say x∗, and this alternative is then the unique core point. If D is strong and agents have linear preferences, then K = K∗. This equivalence also follows if D is strong, X is a convex subset of ﬁnite-dimensional Euclidean space, and agents have strictly convex preferences, i.e., each ui is strictly quasi-concave. It is known that, for most voting rules, the core is generically empty in multiple dimensions, but when X is a convex subset of the real line and agents’ utility functions are strictly quasi-concave, the core is non-empty.

Thus, the next result gives us a characterization of the simple equilibrium outcomes of the amendment agenda game in the interesting, if special, class of single-peaked environments. With Theorem 3, it has the following implication if X is multidimensional: if the

16 Equivalently, K = {x ∈ X | for all z ∈ X \{x}, x ∈ RB(z)}. An element x of the core will not generally satisfy x ∈ RD(z) for all z ∈ X \{x}, because we deﬁne the relation RD in terms of decisive coalitions, rather than blocking ones.

22 strong core is nonempty for one specification of agents’ preferences, then the unique simple equilibrium outcome is the strong core point; and if preferences are perturbed slightly, lead- ing to an empty core, then the simple equilibrium outcomes will be close to the strong core point for the original specification. We use the assumption that for every alternative, there is a proposer who strictly prefers the strong core point to that alternative. This assumption is quite weak: since D is proper, it is sufficient that {1, . . . , k} ∈ D. If D is strong and each 17 ui is strictly pseudo-concave, then it is straightforward to show that the strong core point must uniquely maximize the utility of some agent, say i∗, in which case it is sufficient that i∗ ≤ k.

Theorem 5 Assume K = {x∗}, and for all z ∈ X \{x∗}, there exists i ≤ k such that ∗ ∗ ui(x ) > ui(z). Then x is the unique simple equilibrium outcome.

To prove the theorem, suppose there is a simple equilibrium with outcome z 6= x∗. ∗ By assumption, we have uh(x ) > uh(z) for some agent h ≤ k. But regardless of the particular simple voting equilibrium mapping, say y∗, and regardless of later proposals, if ∗ ∗ ∗ ∗ ∗ ∗ agent h proposes x , then the ﬁnal outcome will be y (x1, . . . , xh−1, x , xh+1, . . . , xk) = x , a contradiction. This completes the proof.

Theorem 5 does not apply when the core is nonempty but not strong, raising two general questions in connection to the core. First, when the core is nonempty, can there be simple equilibrium outcomes outside the core? Second, can every alternative in the core be supported as a simple equilibrium outcome? Assuming a one-dimensional set of alternatives and single-peaked preferences, the next result answers both questions by giving an exact characterization of simple equilibrium outcomes. Assuming convexity of the set of alternatives, there is a unique simple equilibrium outcome, even when the core consists of multiple alternatives, and this outcome is in the core. Thus, simple equilibrium outcomes always lie in the core, providing an “endogenous agenda” foundation for Black’s (1958) median voter theorem, and it is not the case that every alternative in the core can generally be obtained as an equilibrium outcome.

Formally, given X ⊆ R, we say ui is single-peaked if (i) it has a unique maximizer, i i i x˜ , (ii) for all x, z ∈ X with x < z < x˜ , we have ui(x) < ui(z) < ui(˜x ), and (iii) for i i all x, z ∈ X withx ˜ < x < z, we have ui(˜x ) > ui(x) > ui(z). Clearly, this generalizes the assumptions that X is convex and ui is strictly quasi-concave. Our result uses the following characterization of the core in single-peaked environments. Lettingx ˜i denote the

17A differentiable function f: f(x), we have ∇f(x) · (y − x) > 0. A weaker sufficient condition on preferences is that each ui is differentiable and has a zero gradient only at its unique maximizer.

23 unique utility-maximizing alternative, or “ideal point,” for agent i, the core is the interval K = [˜xl, x˜r], where l and r solve

min x˜i max x˜i i∈N and i∈N s.t. {j ∈ N | x˜j > x˜i} ∈/ D s.t. {j ∈ N | x˜j < x˜i} ∈/ D respectively. When D is strong, we havex ˜l =x ˜r, and the core is strong. Thus, Theorem 5 applies.

The next result covers the case in which the collection of decisive coalitions is not strong. We assume only that there is at least one proposer from “each side of the core,” i.e., there is at least one proposer with an ideal point weakly to the left ofx ˜r and at least one proposer with an ideal point weakly to the right ofx ˜l. Then the simple equilibrium outcome is unique in all but one case, where the outcome may either be the best core alternative for agent 1 weakly socially preferred to the status quo or the alternative “next to” it, depending on behavior in voting subgames. If X is convex, then existence of a best response for agent 1 precludes the latter case, and we gain uniqueness.

Theorem 6 Assume X ⊆ R, each ui is single-peaked, and there exist i, j ≤ k such that x˜l ≤ x˜i and x˜j ≤ x˜r. If xˆ is a simple equilibrium outcome of the amendment agenda game, then xˆ ∈ K. Furthermore, if x0 ∈ K, then xˆ = x0; and if x0 ∈/ K, then

u1(ˆx) ≥ sup{u1(z) | z ∈ PD(x0) ∩ K} (2)

u1(ˆx) ≤ max{u1(z) | z ∈ RD(x0) ∩ K}. (3)

Thus, if X is convex, then there is a unique simple equilibrium outcome, which is given by ½ x if x ∈ K xˆ = 0 0 arg max{u1(z) | z ∈ RD(x0) ∩ K} else.

The proof of the theorem uses three observations. First, for all x ∈ K and all z ∈ X, h l we have z∈ / RD(x): if x < z, for example, then uh(x) > uh(z) for all h withx ˜ ≤ x˜ , which l l l is a blocking coalition. Second, if z < x˜ , thenx ˜ ∈ PD(z): then uh(˜x ) > uh(z) for all h withx ˜h ≥ x˜l, which is a decisive coalition. Of course, a similar claim holds for z > x˜r. l l Third, if z < x˜ < w and z ∈ RD(w), thenx ˜ ∈ PD(w). To see this, take any agent i i l i i l such that ui(z) ≥ ui(w). Ifx ˜ ≤ x˜ , then ui(˜x ) > ui(w) by single-peakedness. Ifx ˜ > x˜ , l l then ui(˜x ) > ui(z) by single-peakedness, and ui(˜x ) > ui(w) follows. By assumption, we have ui(z) ≥ ui(w) for all members of some decisive coalition, and we therefore have l ui(˜x ) > ui(w) for that same decisive coalition.

∗ ∗ Take any simple equilibrium with some outcomex ˆ and equilibrium proposals x1, . . . , xk. ∗ ∗ ∗ Let y be the associated simple voting equilibrium mapping, and let (y0, . . . , yk) be the

24 ∗ equilibrium provisional selection, so that yk =x ˆ. If x0 ∈ K, then, by our ﬁrst observation, ∗ ∗ we have yk(x1, . . . , xk) = y0 = x0 for all proposals of the agents, sox ˆ = x0. So suppose l x0 ∈/ K, and without loss of generality assume that x0 < x˜ . We consider four cases.

Case 1: Supposex ˜1 < x˜l. Suppose that agent 1 proposesx ˜l. By the second observation ∗ l l ∗ l l above, we have y1(˜x ) =x ˜ . By the ﬁrst observation, we have yk(˜x , x2 . . . , xk) =x ˜ for all proposals of the later agents. Since agent 1 can obtainx ˜l, and since the agent’s equilibrium proposal must be a best response, it follows from single-peakedness thatx ˆ ≤ x˜l. Now suppose thatx ˆ < x˜l, and consider the deviation in which agent i proposesx ˜l after proposals ∗ ∗ x1, . . . , xi−1. We claim that l ∗ ∗ x˜ ∈ PD(y0, . . . , yi−1). ∗ l l ∗ To see this, note that for j = 0, . . . , i − 1 such that yj < x˜ , we havex ˜ ∈ PD(yj ) from the ∗ ∗ l second observation above. Fromx ˆ ∈ RD(y0, . . . , yk) and the supposition thatx ˆ < x˜ , the ∗ l second observation also implies that we cannot have yj =x ˜ for any j. For j = 0, . . . , i − 1 ∗ l l ∗ ∗ l such that yj > x˜ , note thatx ˜ ∈ PD(yj ) follows fromx ˆ ∈ RD(yj ), fromx ˆ < x˜ , and from the third observation above. Thus, we obtain the claim, and it follows that

∗ ∗ ∗ l l yi (x0, . . . , xi−1, x˜ ) =x ˜ .

By the ﬁrst observation above, we then have

∗ ∗ ∗ l l yk(x0, . . . , xi−1, x˜ , xi+1, . . . , xk) =x ˜ for all proposals of later agents, so agent i can obtainx ˜l by this deviation. Since the agent’s equilibrium proposal must be a best response, it follows from single-peakedness thatx ˆ ≥ x˜l, a contradiction. We conclude thatx ˆ =x ˜l, fulﬁlling (2) and (3), as required.

1 Case 2: Supposex ˜ ∈ RD(x0) ∩ K. By the above argument, agent 1 can obtain his/her ideal point by proposing it. Since the agent’s equilibrium proposal must be a best response, it follows thatx ˆ =x ˜1, as required.

1 Case 3: Supposex ˜ ∈ K \ RD(x0). If we had u1(ˆx) < sup{u1(z) | z ∈ PD(x0) ∩ K}, then there would exist z ∈ PD(x0) ∩ K with u1(ˆx) < u1(z), but then agent 1 could propose ∗ ∗ z. By the above arguments, we would have y2(z) = z and yk(z, x2, . . . , xk) = z for all proposals of later agents. But then this would be a proﬁtable deviation for agent 1, a contradiction. Therefore, (2) follows. We havex ˆ ∈ RD(x0) by deﬁnition of a provisional 1 selection, and moreover we claim thatx ˆ ∈ K. To see this, note thatx ˜ ∈ K \ RD(x0) and 1 r single-peakedness imply that max RD(x0) < x˜ ≤ x˜ . Thus,x ˆ ∈ RD(x0) \ K implies that l l xˆ < x˜ . But, because our second observation shows thatx ˜ ∈ PD(x0), this contradicts (2). Therefore,x ˆ ∈ RD(x0) ∩ K, and (3) follows, as required.

Case 4: Supposex ˜1 > x˜r. The argument in this case is exactly that for Case 3.

25 l Finally, return to the assumption that x0 < x˜ , and now assume X is convex. Note that single-peakedness implies RD(x0) is convex, so u1 is indeed uniquely maximized over RD(x0) ∩ K by, say, z. Suppose that u1(ˆx) < u1(z). By convexity and single-peakedness, we then have x = (1/2)ˆx + (1/2)z ∈ PD(x0) ∩ K and u1(x) > u1(ˆx), contradicting (2). This completes the proof.

If we do not restrict indiﬀerences in some way, then alternatives outside the core can be supported as simple equilibrium outcomes. As the following example shows, this is true even if when majority rule is used, X is one-dimensional, and all but one agent have single- peaked preferences with no indiﬀerences. Suppose there are four alternatives, w, x, y, and z, with status quo x0 = z, D is majority rule, and three agents have utilities below.

u1(x) = u1(y) = u1(z) = u1(w) u2(y) > u2(z) > u2(x) > u2(w) u3(x) > u3(w) > u3(y) > u3(z)

Note that the core here consists of x and y, and that the preferences of agents 2 and 3 are single-peaked with respect to the ordering w < x < y < z of alternatives, while agent 1’s preferences are not.

Suppose agent 1 proposes y. Because agent 1 is indiﬀerent over all alternatives, this is an optimal proposal regardless of the other agents’ strategies, so we only consider the subgame beginning with x1 = y. Following this, let agent 2 propose w. Let agent 3 propose w, regardless of agent 2’s proposal. Consider any simple voting equilibrium mapping satisfying the following: y0 = z, y1(y) = y,

y2(y, x) = y2(y, y) = y2(y, z) = y2(y, w) = y, and ½ w if x = w y (y, x , x ) = 3 3 2 3 y else.

Thus, after agent 1 proposes y, the ﬁnal outcome is w if agent 3 proposes it, otherwise y. Clearly, it is optimal for agent 3 to always propose w and, because agent 2’s proposal is inconsequential, it is optimal for agent 2 to propose w. Thus, the above strategies form a simple equilibrium with outcome w∈ / K.

6 Application to Distributive Politics

This section examines the special case of pure distribution, where the agents must “divide a dollar” among themselves, and each agent cares only about the amount of the dollar

26 allocated to him/herself. Formally, we assume the number of agents is three, the set of alternatives is the unit simplex in R3, i.e.,

3 X = {(a1, a2, a3) ∈ R+ | a1 + a2 + a3 ≤ 1}, and ui(a1, a2, a3) = ai for each agent. The status quo allocation is x0 = (0, 0, 0), and voting is by majority rule. To avoid confusion, we denote the proposals of agents 1, 2, and 3 by a, b, and c, respectively. Then a = (a1, a2, a3) is an allocation of the dollar across the agents, and a2, for example, is the amount received by agent 2 in alternative a (rather than a proposal by agent 2).

In Appendix B, we find simple equilibria of the amendment agenda game supporting ∗ 1 2 two paths of play: in the first, agent 1 proposes a = ( 3 , 3 , 0), then agent 2 proposes ∗ 2 1 ∗ 1 1 1 b = ( 3 , 3 , 0), and finally agent 3 proposes c = ( 3 , 3 , 3 ), which passes in the voting ∗∗ 1 2 ∗ subgame; in the second, agent 1 proposes a = ( 3 , 0, 3 ), then agent 2 again proposes b , and finally agent 3 again proposes c∗, which passes in the voting subgame. n a model with Euclidean preferences, Banks and Gasmi (1987) find corresponding equilibrium paths of play, and they note informally that these translate to the Euclidean setting. We conclude that endogenous agenda formation leads to equal division in the distributive politics setting.

In solving for a simple equilibrium, if agent 1 has proposed a ∈ X, then we may without loss of generality restrict agent 2’s proposal, b, to RD(a). This set is formed by the union of the Ri(a) ∩ Rj(a)(i 6= j) sets, each one a triangle bounded by the indiﬀerence “curves” (actually straight lines, in this model) of agents i and j. Given a and b, we may restrict agent 3’s proposal, c, to RD(a, b), which, depending on agent 2’s proposal, may have multiple “peaks.” In order to give agent 3 an optimal proposal, unless the highest of these peaks is isolated from PD(a, b), we must specify a simple voting equilibrium mapping which makes it possible for the agent to obtain the highest of these peaks by proposing it: otherwise, the agent could propose an alternative in PD(a, b) arbitrarily close to it and make that the ﬁnal outcome.

If the peaks are at the same height and each would pass in the voting subgame, then in all but two cases we specify that agent 3 propose the peak closest to agent 2’s ideal point (which will be the left peak), breaking indifference in 2’s favor. Of those two cases, one in particular is critical in supporting the first path of play described above: there is no simple equilibrium in which agent 1 proposes a∗ and agent 3 always breaks indifference over proposals in 2’s favor.18 The exceptional case (Case 1 in Appendix B) occurs when agent 1 proposes a∗ = (1/3, 2/3, 0) and agent 2 proposes b∗ = (2/3, 1/3, 0), creating three

18In the other case, we could specify the simple voting equilibrium mapping so that agent 3 has a strict preference for agent 2’s worst peak, even though it is the same height as 2’s preferred one: voting strategies would be such that the latter would not be provisionally selected. Thus, our convention for breaking indiﬀerence for agent 3 is not critical here.

27 3 ...... c∗...... t...... ∗...... ∗...... a...... b...... N...... 12

Figure 3: Breaking Indifference Between Proposals peaks of equal height. See Figure 3, where R(a∗, b∗) is shaded. In equilibrium, agent 3 must propose c∗ = (1/3, 1/3, 1/3), the middle peak. If agent 3 chose the left peak, giving agent 1 nothing, then agent 1 could move a∗ slightly to the left, increasing his/her final amount of the dollar: the best agent 2 could do in that case would be to propose slightly to the right of b∗, and agent 3 would then choose either the middle or right peak, giving agent 1 a positive payoff. If agent 3 chose the right peak, then agent 2 could move b∗ slightly to the right, increasing his/her payoff. Thus, restricting agent 3 to break indifference in favor of either 1 or 2 leads to non-existence of a simple equilibrium in which agent 1 proposes a∗.

Since the correspondence RD is lower hemicontinuous in this case, this is solely an instance of the ﬁrst technical diﬃculty discussed in Section 3.

Given proposals a and b by agents 1 and 2, every proposal in the set RD(a, b) will be the outcome of some provisional selection, but we cannot assume that every such proposal is provisionally selected in equilibrium: contrary to Austen-Smith (1987) and Banks and Gasmi (1987), the convention of resolving ties in voting subgames in favor of the alternative proposed later leads to the absence of a best response in some subgames, and therefore to the non-existence of equilibrium. The subtlety in constructing the appropriate simple voting equilibrium mapping arises in a case (Case 6 in Appendix B) where RD(a, b) has two peaks. In Figure 4, this is the shaded area together with the open circle. That “peak,” denoted c0, is higher than the left, making it more attractive to agent 3 and less to agent 2. Note that it is isolated from R(a, b). If we specify voting strategies so that c0 is provisionally selected, then proposing it is agent 3’s unique best response. Moving b slightly to the right to b0, however, c0 is no longer available to agent 3, and 3’s optimal proposal is c00, yielding a payoﬀ to agent 2 arbitrarily close to that from c. Thus, agent 2 has no optimal proposal in this

28 3 ...... a .... c0 ...... N...... d...... t 00...... t....c ...... c...... 12 b b0

Figure 4: Convention for Ties Implies No Best Response subgame. We must specify equilibrium strategies so that, following proposals a, b, and c0, the provisional selection is (a, b, b); and following proposals a, b, and c, the provisional selection is indeed (a, b, c), making c an optimal proposal for agent 3 and solving agent 2’s best response problem. Driving this problem is the fact that RD violates lower hemicontinuity at (a, b). This case does not involve indiﬀerence on the part of a proposer and is solely an instance of the second technical diﬃculty discussed at the end of Section 3.

7 An Extension

A more general model would allow for the selection of proposers to be random, as in Austen- Smith (1987), rather than given by a fixed sequence. We can model the proposer selection process by a probability distribution µ on the sequences σ of agents bounded in length by some finite K, where ih(σ) is the hth component of sequence σ and the length of σ is k(σ) ≤ K. Thus, given the selection (i1, . . . , ih) of agents and given proposals by each of them, agent j is selected as the next proposer with probability P {µ(σ) | (i1(σ), . . . , ih+1(σ)) = (i1, . . . , ih, j)} µ(j|i1, . . . , ih) = P , {µ(σ) | (i1(σ), . . . , ih(σ)) = (i1, . . . , ih)} whenever this is well-defined. The existence proof carries over to this model using essentially the same idea but with more complex notation. If there is zero probability that another agent is selected after (i1, . . . , ih), then agent ih’s problem is identical to agent k’s in Sec- tion 4. For every such sequence, i.e., for every sequence that is terminal with probability one, we define the correspondence Bih (y0, . . . , yih−1|i1, . . . , ih−1) as in Section 4, but now

29 conditioning on the sequence of prior proposers. As before, each of these correspondences will have nonempty values and closed graph in (y0, . . . , yih−1).

We then consider every sequence (i1, . . . , ih−1) such that, for all j ∈ N, either µ(j|i1,..., ih−1) = 0 or (i1, . . . , ih−1, j) is terminal with probability one. Here, agent ih−1’s problem is somewhat more complex than agent k − 1’s in Section 4, because the agent may not know who will be selected next. But we again deﬁne agent ih−1’s security level, now minimizing with respect to a vector (x1, . . . , xn), rather than x, and using the agent’s expected utility, i.e.,

bi (y0, . . . , yi |i1, . . . , ih−2) h−1 h−2 P supz min(x1,...,xn) j∈N µ(j|i1, . . . , ih−1)uih−1 (xj)

= s.t. xj ∈ Bj(y0, . . . , yih−2 , z|i1, . . . , ih−1), j = 1, . . . , n

z ∈ PD(y0, . . . , yih−2 ) ∪ {yih−2 }.

We then deﬁne the correspondence Bih (y0, . . . , yih−2 |i1, . . . , ih−2) as in Section 4, but now the elements of this set are vectors (x1, . . . , xn). This correspondence will inherit the properties of nonempty values and closed graph, and again this leads to an induction proof that, for every agent j selected to propose ﬁrst, Bj(y0) has nonempty values and closed graph.

The characterization results carry over with few changes. The argument for Theorem 4 applies in the extended model to the last agent of a terminal sequence, as long as the distribution µ is such that the last agent of every terminal sequence (i1, . . . , ih) knows he is last.19 Theorem 5 holds if we assume: (i) the strong core alternative is the ideal point of an agent, say i∗, who is chosen to propose with probability one, (ii) X is convex, and (iii) each ∗ ui is strictly concave. When agent i is selected, each proposal by the agent determines a lottery over alternatives; the agent can propose the strong core point x∗ and, using strict concavity, obtain it with probability one, so the equilibrium lottery following agent i∗’s proposal must be at least as good for the agent as x∗; since x∗ is the agent’s ideal point, the equilibrium lottery must therefore be degenerate on x∗, as required. The proof of Theorem 6 can be adapted, assuming (ii), (iii), and, with probability one, at least one agent with an ideal point weakly to the left ofx ˜r is selected and at least one agent with an ideal point weakly to the right ofx ˜l is selected, though now of course the identity of the ﬁrst proposer in the theorem is stochastic.

8 Conclusion

Agendas are prominent in the literature on tournaments, where the set of alternatives is ﬁnite and majority-ties are assumed away, and in the political science literature, where it is

19 That is, µ(i1, . . . , ih) > 0 implies that for all ih+1, µ(i1, . . . , ih, ih+1) = 0.

30 more natural to model the policy space as a convex subset of finite-dimensional Euclidean space. This paper carefully considers the issue of equilibrium existence in a general model that subsumes these — and, in particular, allows for a “continuous” set of alternatives. This class of games presents two non-trivial problems that have not been considered previously. The first has to do with an agent’s selection of an optimal proposal, when not uniquely defined. The second concerns the resolution of ties in voting subgames: the usual convention of resolving ties in favor of the alternative proposed later can lead to the absence of a best response for an agent in some subgames, and therefore to the non-existence of subgame perfect equilibrium.

These difficulties serve to illustrate the complexity of agenda setting games when the set of alternatives is continuous. One solution is to limit ourselves to finite models, bypassing technical problems in establishing existence of equilibria. This approach is lacking on model- ing grounds, however, for the same reason that it is desirable to model policy-making in the spatial setting with a convex set of alternatives: given the broad scope for policy-making in the real world, the assumption that an agent may not propose an alternative “very close” to another is questionable ground on which to build a theory. Furthermore, assuming a convex structure on the set of alternatives allows us to express intuitive restrictions on the preferences of agents. In economic models, such as public good provision, it also allows us to define the set of alternatives in terms of intuitive restrictions on production technology, e.g., a convex total cost function.

Since the complexity of the amendment agenda game derives from the multitude — in fact, continuum — of proposal subgames, another solution is to weaken subgame perfection by ignoring some subgames. In the distributive politics model, for example the resolution of ties was an issue only off the equilibrium path of play (Case 6 in Appendix B), and so it might be argued that agent 2’s lack of a best response should not affect the analysis of the game: it may not be possible for agent 1 to predict agent 2’s behavior in these subgames, but if agent 2 makes proposals that are “close” to optimal, then this will not affect agent 1’s proposal problem. This approach has some promise, but it is not clear that the issue of resolving ties in voting subgames can always be dealt with in this way. In fact, the distributive politics model shows that the other technical issue, involving indifferent proposers, can indeed arise along the equilibrium path of play.

The approach in this paper is to provide a general solution to the existence problem: we inductively deﬁne, for any agent, the set of alternatives that can be supported as equilibrium outcomes in subgames beginning with that agent’s proposal; we show that this correspondence has closed graph and nonempty values; and we show these properties are inherited by the correspondences of earlier proposers. Thus, the set of equilibrium outcomes in the subgame beginning with the ﬁrst agent is nonempty, and therefore there is at least

31 one equilibrium. In proving this, we deliver somewhat more: a characterization of sophisticated voting outcomes in agendas when there are majority-ties; a refinement of subgame perfection in which outcomes of voting subgames depend on agendas in a restricted way; a characterization of simple equilibrium outcomes; and upper hemicontinuity of the simple equilibrium outcome correspondence. We prove Pareto optimality of simple equilibrium outcomes; we establish connections to the core, when nonempty, including a characterization in the one-dimensional model with single-peaked preferences; and we explicitly solve the special case of distributive politics with three agents. Our model subsumes a great variety of other special cases, e.g., tournaments, public good provision, public projects with district-specific benefits, etc., and we leave open the interesting problem of characterizing outcomes of agenda setting in these environments.

A The Lemma

Let Λ be an arbitrary metric space parameterizing the utility functions of the agents, let each ui: X × Λ → R be jointly continuous, and let Ri(z, λ) and Pi(z, λ) be the weak and strict upper sections of ui(·, λ) at z. Then deﬁne RD(z1, . . . , zh, λ) and PD(z1, . . . , zh, λ) as \h [ \ RD(z1, . . . , zh, λ) = Ri(zj, λ) j=1 C∈D i∈C \h [ \ PD(z1, . . . , zh, λ) = Pi(zj, λ). j=1 C∈D i∈C

By continuity of each ui, RD is closed and PD is open. We deﬁne agent k’s security level as

bk(y0, . . . , yk−1, λ) = sup{uk(x, λ) | x ∈ PD(y0, . . . , yk−1, λ) or x = yk−1}.

Since PD is open, it follows that bk is lower semi-continuous. Now deﬁne the correspondence

Bk(y0, . . . , yk−1, λ)

= {x ∈ RD(y0, . . . , yk−1, λ) ∪ {yk−1} | uk(y, λ) ≥ bk(y1, . . . , yk−1, λ)}, and note that this has nonempty values and closed graph. Given (y0, . . . , yh−1), inductively deﬁne the security level of agent h < k as

supz infx uh(x, λ) bh(y0, . . . , yh−1, λ) = s.t. x ∈ Bh+1(y0, . . . , yh−1, z, λ) z ∈ PD(y0, . . . , yh−1, λ) ∪ {yh−1}, and deﬁne

Bh(y0, . . . , yh−1, λ)

32 = {x ∈ X | there exists z ∈ RD(y0, . . . , yh−1, λ) ∪ {yh−1} such that

x ∈ Bh+1(y0, . . . , yh−1, z, λ) and uh(x, λ) ≥ bh(y0, . . . , yh−1, λ)}.

We then have the following result.

Lemma 1 Each correspondence Bh has non-empty values and closed graph.

The proof is inductive. We have noted that the lemma holds for h = k. Now suppose it is true for h ≥ 2, and consider agent h − 1. To see that Bh−1(y1, . . . , yh−2, λ) 6= ∅, take any m m m sequences {x } and {z } such that z ∈ PD(y0, . . . , yh−2, λ)∪{yh−2}, which is non-empty, and m m x ∈ arg min{uh−1(x, λ) | x ∈ Bh(y1, . . . , yh−2, z , λ)} m for all m, and such that bh−1(y1, . . . , yh−2, λ) = limm→∞ uh−1(x , λ). Note that Bh(y1,..., m yh−2, z , λ) is compact by hypothesis, so the min operation is well-deﬁned. By compactness of X, we may suppose, going to a subsequence if needed, that there exist x, z ∈ X such m m that x → x and z → z. Since RD has closed graph, z ∈ RD(y0, . . . , yh−2, λ) ∪ {yh−2}. And since Bh has closed graph by hypothesis, x ∈ Bh(y1, . . . , yh−2, z, λ). Continuity of uh−1 then yields uh−1(x, λ) = bh−1(y1, . . . , yh−2, λ), implying y ∈ Bh−1(y1, . . . , yh−2, λ).

m m m m To show that Bh−1 has closed graph, take any sequences {(y1 , . . . , yh−2, λ )} and {x } m m m m m m m such that x ∈ Bh−1(y1 , . . . , yh−2, λ ) for all m and (y1 , . . . , yh−2, λ ) → (y1, . . . , yh−2, λ) m m m m m m and x → x. For each m, there exists z ∈ RD(y0 , . . . , yh−2, λ ) ∪ {yh−2} such that m m m m m x ∈ Bh(y1 , . . . , yh−2, z , λ ). Since X is compact, we may assume without loss of general- m ity that {z } converges to some z ∈ X. Since RD has closed graph, z ∈ RD(y0, . . . , yh−2, λ)∪ {yh−2}. And since Bh has closed graph by hypothesis, x ∈ Bh(y1, . . . , yh−2, z, λ). Thus, we m m m m must show that uh−1(x, λ) ≥ bh−1(y1, . . . , yh−2, λ). Since uh−1(x , λ ) ≥ bh(y1 , . . . , yh−2, m λ ) for all m, and since uh−1 is continuous, the desired result holds if bh is lower semicontinuous.

Note that, since uh−1 is continuous, since X is compact, and since Bh is closed, lower semicontinuity of

∗ minx uh−1(x, λ) uh−1(y0, . . . , yh−2, z, λ) = s.t. x ∈ Bh(y0, . . . , yh−2, z, λ)

∗ follows from Aliprantis and Border’s (1999) Lemma 16.30. Furthermore, since uh−1 is lower semicontinuous and PD is open, lower semicontinuity of

∗ supz uh−1(y0, . . . , yh−2, z, λ) bh−1(y0, . . . , yh−2, λ) = s.t. z ∈ PD(y0, . . . , yh−2, λ) ∪ {yh−1} follows from Aliprantis and Border’s (1999) Lemma 16.29, as desired.

33 B Analysis of Distributive Politics

The distributive politics amendment agenda game has multiple simple equilibria that generate two diﬀerent paths of play, depending on who agent 1 initially proposes to.

B.1 Agent 1 Proposes to Agent 2

The analysis proceeds by considering the optimal proposals of agents 2 and 3, given a proposal a by agent 1 falling into 8 possible cases:

1. a2 ≥ 1/2 and a2 ≥ 2a1

2. 2a1 > a2 ≥ 1/2

3. 2a2 ≥ a1 ≥ 1/2

4. a1 ≥ 1/2 and a1 ≥ 2a2

5. 1/2 ≥ a1, 1/2 > a2, and 1/2 ≥ a3

6. a3 ≥ 1/2 and 2a2 ≥ a3

7. a3 ≥ 1/2, a3 ≥ 2a1, and a3 ≥ 2a2

8. a3 ≥ 1/2 and 2a1 ≥ a3.

These cases are not defined to be mutually exclusive, for usually the equilibrium behavior at the boundaries of two cases is the same (or differs because agent 2 is indifferent between two proposals that lead to the same final outcome) in both. A more detailed explanation is provided below. This is not true of boundaries involving Case 1, however, and Cases 2 and 5 are therefore explicitly defined to be disjoint from Case 1. See Figure 5 for two representations of these different cases.

∗ 1 2 Agent 1’s strategy is speciﬁed as a = ( 3 , 3 , 0). In what follows, we give solutions for agent 2’s and 3’s equilibrium behavior in subgames falling into each of the above cases. Agent 2’s proposal strategy as a function of a is:

∗ a2 a2 1. b = (a1 + 2 , 2 , a3)

∗ a2 a2 2. b = (a1 + 2 , 2 , a3)

∗ 3. b = (a1 − a2, a1, 1 − 2a1 + a2)

34 3...... 7 ...... 6 ...... 8 ...... 5 ...... 1 ...... 4 ...... 2 ...... 3 ...... 12

1 ...... a2 ...... 1 ...... 2 ...... 1 ...... 2 ...... 1 ...... 56 . 3 ...... 3 ...... 4 ...... 7 ...... 8 ...... 1 1 03 2 1

Figure 5: Cases for Agent 1’s Proposal

35 ∗ a1 a1 4. b = ( 2 , 2 + a2, 1 − a1 − a2)

∗ 1 1 5. b = ( 2 , 2 , 0)

∗ 6. b = (a1 + a2, a3, 0)

∗ a3 a3 7. b = (a1 + 2 , a2 + 2 , 0)

∗ 8. b = (a3, a1 + a2, 0).

∗ 2 1 ∗ Thus, on the path of play, agent 2 proposes b = ( 3 , 3 , 0) after agent 1 proposes a = 1 2 ( 3 , 3 , 0). Given proposals by agents 1 and 2, agent 3’s maximization problem is fairly straightfoward: the agent essentially proposes the highest peak of RD(a, b), of which there may be one, two, or three. When there are multiple highest peaks, we specify that agent 3 propose the peak most favorable to agent 2 (the left peak), with three exceptions. First, in Case 1, ∗ a2 a2 when agent 2 proposes b = (a1 + 2 , 2 , a3), we specify that agent 3 proposes the middle ∗ 1−a1−a3 1−a1+a3 peak, which is c = (a1, 2 , 2 ). As discussed in Section 6, this is needed to give the other agents best responses: if 3 proposed the right peak, then agent 2 could gain by moving b∗ slightly to the left; if agent 3 proposed the left peak, then agent 1 could gain by ∗ 1 1 moving a slightly to the right. Second, in Case 1, when agent 1 proposes a = (0, 2 , 2 ) and 1 1 1 1 agent 2 proposes b = ( 2 , 2 , 0), there are in fact only two peaks: RD(a, b) = {a, ( 2 , 0, 2 )}. We will later specify that both alternatives would pass in the voting subgame, so that agent 3 is indifferent between them,20 and we specify that agent 3 propose the right peak, 1 1 ∗ 1 1 1 ( 2 , 0, 2 ). This is needed to make the proposal specified for agent 2, b = ( 4 , 4 , 2 ), optimal, for otherwise proposing b would yield a higher payoff for the agent.21 Technically, this prob- 1 1 lem arises because RD violates lower hemicontinuity at (a, b): b = ( 2 , 2 , 0) is isolated from RD(a, b) if we move a down slightly. Third, in Case 6, when agent 2 makes the equilibrium ∗ ∗ proposal b = (a1 + a2, a3, 0), we specify that agent 3 proposes c = (a1, a3, a2), despite the fact that a better alternative is weakly majority-preferred to the other agents’ propos- ∗ als: (a1 + a2, 0, a3) ∈ RD(a, b ). This will be optimal, given the simple voting equilibrium mapping defined below.

This yields the following proposals for agent 3 when agent 2 makes his/her proposal b∗ as speciﬁed above.

∗ a2 a2 1. c = (a1, 2 , a3 + 2 ) 20Alternatively, if we were to specify the simple voting equilibrium mapping so that the left peak, a, would not be provisionally selected, then agent 3’s unique optimal proposal would be to propose the right peak. 21Another way to handle this problem is to specify the simple voting equilibrium mapping so that, following 1 1 a, a proposal of b = ( 2 , 2 , 0) is not provisionally selected.

36 ∗ 2. c = (0, a2, 1 − a2)

∗ 3. c = (0, a1, 1 − a1)

∗ a1 a1 4. c = (0, 2 + a2, 1 − 2 − a2)

∗ 1 1 5. c = (0, 2 , 2 )

∗ 6. c = (a1, a3, a2)

∗ a3 a3 7. c = (a1, a2 + 2 , 2 )

∗ 8. c = (0, a1 + a2, a3).

∗ 1 2 The path of play in the agenda formation stage is that agent 1 proposes a = ( 3 , 3 , 0), then ∗ 2 1 ∗ 1 1 1 agent 2 proposes b = ( 3 , 3 , 0), and ﬁnally agent 3 proposes c = ( 3 , 3 , 3 ). These strategies in proposal subgames are supported by the following simple voting equilibrium mapping. Given proposals (a, b, c), we assign the weak provisional selection, ∗ except in Case 6 when agent 2 makes the equilibrium proposal b = (a1 + a2, a3, 0): as 0 discussed in Section 6, if agent 3 proposes c = (a1 +a2, 0, a3), then the provisional selection ∗ ∗ ∗ is (a, b , b ). This repairs a violation of lower hemicontinuity of RD at (a, b ), whereby ∗ ∗ (a1 +a2, 0, a3) is isolated from RD(a, b ) when we move b slightly to the right. On the path of play, it follows that agent 3’s proposal, namely, equal division, is the ﬁnal outcome of voting. See Figure 6 for a graphical description of agent 2’s and 3’s equilibrium proposals as a function of agent 1’s proposal, where we denote 1’s, 2’s, and 3’s proposals by N, , and •, respectively.

As mentioned above, most boundary issues not involving Case 1 are not serious. At the boundary between Cases 3 and 5, for example, when a1 = 1/2 and 2a2 ≥ a1 ≥ a2, the above ∗ 1 1 ∗ 1 1 strategies are as follows. In Case 5, we have b = ( 2 , 2 , 0) followed by c = (0, 2 , 2 ), while ∗ ∗ 1 1 in Case 3, we have b = (a1 − a2, a1, 1 − 2a1 + a2) followed by c = (a1, 1 − a1) = (0, 2 , 2 ). This is simply an example where agent 2 is indiﬀerent between two proposals, and the ﬁnal outcome is clearly independent of whether we include a in Case 3 or 5. An implication is that, as should be expected, there are multiple equilibria of the agenda game supporting the same path of play. The other examples where boundary issues can be similarly treated are between Cases 2 and 3, between Cases 3 and 4, and between Cases 2 and 5.

There are only two instances not involving Case 1 where some care must be taken at boundary points. At the boundary between Cases 5 and 6, where a3 = 1/2 and a1 ≤ 1/2, ∗ 1 1 the above strategies specify that agent 2 choose b = ( 2 , 2 , 0), while agent 3’s proposal 1 1 in Case 5 is (0, 2 , 2 ) and in Case 6 is (a1, a3, a2). The latter proposal by agent 3 yields the agent a strictly lower payoﬀ than the former. This inconsistency is due to another

37 3 3 3 ...... t...... t...... t...... N...... N...... N...... 12 ...... 12 ...... 12 Case 1 Case 2 Case 3

3 3 3 ...... not ...... provisionally ...... selected ...... t...... N...... d...... t...... t...... N...... N...... 12 ...... 12 ...... 12 Case 4 Case 5 Case 6

3 3 ...... N...... •...... t ..... N...... t...... 12 ...... 12 Case 7 Case 8

Figure 6: Agent 2’s and 3’s Proposals

38 ∗ violation of lower hemicontinuity: here, (0, a1 + a2, a3) is isolated from RD(a, b ) if we move ∗ 1 1 b = ( 2 , 2 , 0) slightly to the left. There are two solutions to this problem. First, we can 1 1 simply include such proposals by agent 1 in Case 5, so that agent 3 proposes (0, 2 , 2 ), yielding the higher payoﬀ. Second, we may modify the simple voting equilibrium mapping deﬁned above so that, following such a proposal a by agent 1 and following proposal b∗ by 0 ∗ ∗ agent 2 and proposal c = (0, a1 + a2, a3) by agent 3, the provisional selection is (a, b , b ). This makes agent 3’s proposal in Case 6 optimal on the boundary.

The second instance is at the boundary between Cases 7 and 8, where 2a1 = a3 ≥ 1/2 ∗ and a3 ≥ 2a2. The above strategies specify that agent 2 choose b = (a3, a1 + a2, 0), while agent 3’s proposal in Case 7 is (a1, a1 + a2, a1) and in Case 8 is (0, a1 + a2, a3). This inconsistency is also due to a violation of lower hemicontinuity: (0, a1 + a2, a3) is isolated ∗ from RD(a, b ) if we move a into the interior of Case 7. There are two solutions. First, we can include such proposals by agent 1 in Case 8, or we can modify the voting equilibrium mapping deﬁned above so that, following such a proposal a by agent 1 and following proposal ∗ 0 b by agent 2 and proposal c = (0, a1 + a2, a3) by agent 3, the provisional selection is (a, b∗, b∗). This makes agent 3’s proposal in Case 7 optimal.

Once these adjustments have been made, it is straightfoward (if time-consuming) to verify that the strategies specified for agents 2 and 3 are optimal in the subgames corresponding to agent 1’s possible proposals. Once that is done, agent 1’s optimal proposal is easy to calculate. In fact, there are only three cases where agent 1 can receive a positive 1 2 payoff: Cases 1, 6, and 7. In Case 1, agent 1’s optimal proposal is ( 3 , 3 , 0), which yields a final payoff of 1/3 for the agent. In Case 6, the supremum of payoffs attainable by agent 1 is 1/4. Whether the agent has an optimal proposal in this case depends on how the boundary points between Cases 5 and 6 are resolved, but this is immaterial, given agent 1’s higher payoff in Case 1. In Case 7, the supremum of payoffs attainable by agent 1 is 1/3. Whether the agent has an optimal proposal in this case depends on how the boundary points between Cases 7 and 8 are resolved, but this is immaterial, given agent 1’s payoff in ∗ 1 2 Case 1. Therefore, a = ( 3 , 3 , 0) is indeed an optimal proposal for agent 1.

B.2 Agent 1 Proposes to Agent 3

There is a subgame perfect equilibrium of the amendment agenda game with a different path of play, though also with equal division as the final outcome. We now specify agent ∗∗ 1 2 1’s proposal as a = ( 3 , 0, 3 ), and agent 2’s and 3’s proposals are defined as above, with two qualifications. First, our earlier requirement that, in Case 1, agent 3 chose the middle peak may now be relaxed: we may let the agent choose the left peak. This may mean that agent 1 no longer has an optimal proposal in Case 1, but that will be immaterial. Second,

39 it now becomes important to specify strategies at the boundary between Cases 7 and 8 in the appropriate way. We must include the proposal a∗∗ in Case 7, so that agent 2 proposes ∗ 2 1 ∗ 1 1 1 b = ( 3 , 3 , 0) and agent 3 proposes c = ( 3 , 3 , 3 ). To make this proposal by agent 3 optimal, we modify the voting equilibrium mapping deﬁned above so that, following the proposal a∗∗ ∗ 0 2 1 by agent 1, proposal b by agent 2, and proposal c = (0, 3 , 3 ) by agent 3, the provisional selection is (a∗∗, b∗, b∗).

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