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Chapter 4 Binary Agendas In Chapter 2 we argued that incentives for strategic misrepresentation of preferences are inherent in nondictorial collective choice. A positive response to this fact is to design institutions for collective decision-making that internalize such strategic incentives and yield the desired outcomes irrespective of individual efforts at manipulation. Chapter 3 pursued this idea and studied the relationship between various collective choice rules and the mechanisms (in effect, institutions) that implement those rules. And it turns out that these relationships typically depend in important ways on the details of individual behavior and institutional design. In this chapter, we essentially turn the issue around: rather than beginning with a choice rule and exploring what sort of mechanism might implement it, we first specify a mechanism and explore what sort of rule it might support. Specifically, we focus on an empirically important class of institution, or mechanism, for voting over a given set of mutually exclusive alternatives when the number of individuals eligible to vote is relatively small, as is the case, for example, in legislative and judicial committees, parliamentary cabinets and so forth. With a slight abuse of conventional understanding, we refer to such small-number voting bodies generically as "committees". For purposes of this chapter, the salient feature of voting in committees is that it is more often than not governed by fixed rules. Widely used committee voting rules include amendment agendas (section 2.2), successive elimination agendas and issue-by-issue voting (section 2.7): these are all examples of binary agendas, under which voting is sequential, with every decision in the sequence involving two (possibly composite) alternatives. In view of the prevalence of binary agendas, it is important to understand how they work. In particular, since voting rules themselves are at some stage 113 114 CHAPTER 4. BlliARYAGENDAS subject to collective choice, rational individuals' preferences over rules are, at least in part, induced by their expectations on the likely consequences for committee decision-making of adopting one set of rules rather than another. So, to develop any understanding of why the decision-making institutions we observe are in place, we first need to understand how they might be expected to connect policy preferences to policy choices and this, at least for the class of binary agendas, is the subject of the chapter. There are three principal sorts of question that can be asked of any class of voting institution: the first concerns the structure of voting behavior; the second concerns how institutional details affect the set of achievable outcomes from any given set of feasible alternatives; and the third concerns how, under any institution, the set of feasible alternatives is determined. In this chapter we consider only the first two of these for the class of binary agendas, deferring the third to Chapter 5. Specifically, given a fixed and finite set of feasible alternatives, we examine the structure of Nash equilibrium (that is, mutual best response) voting behavior on arbitrary binary agendas and subsequently characterize the set of equilibrium outcomes for particular sub-classes of binary agenda. In part, this second issue concerns the extent to which those in control of the agenda are able to influence the outcome through the exercise of such control. It turns out that both the extent of agenda-setting power and the qualitative properties of equilibrium outcomes vary with the form of the agenda. Thus the purely procedural details of voting rules are consequential. 4.1 Binary agendas and sophisticated voting Consider a committee of n individuals. Unless explicitly stated otherwise, we assume throughout this and the following two sections that the set of alternatives X is finite, that the committee uses simple majority rule to make decisions and, to avoid complications with ties, that n is odd with no individual indifferent over any pair of alternatives in X. The set of preference profiles is therefore pn with typical element p = (PI, ..., Pn). Although we have not presumed a total absence of indifference hitherto (save as an intermediate step in proving various theorems), when the set of alternatives is exogenously given and finite, the assumption, although certainly a limitation, is not too demanding. One way to think about this is to consider the finite set X as coming from some underlying continuous space; for example, when X is a finite set of budget allocations. Then individuals' preferences can be assumed continuous on the underlying space, 4.1. BINARY AGENDAS AND SOPHISTICATED VOTING 115 in which case any indifference between...

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