Effects of fixed costs in two-person sequential bargaining

Rubinstein (1982) considered the problem of dividing a given surplus between two players sequentially, and then proposed a model in which the two players alternately make and respond to each other's offers through time. He further characterized the perfect equilibrium outcomes, which depend on the players time preferences and order of moves. Using both equal and unequal bargaining cost conditions and an unlimited number of rounds, two experiments were designed to compare the perfect equilibrium model to alternative models based on norms of fairness. We report analyses of final agreements, first offers, and number of bargaining rounds, which provide limited support to the perfect equilibrium model, and then conclude by recommending a shift in focus from model testing to specification of the conditions favoring one model over another.     

Voting power measurement: a story of misreinvention

In this account of the history of voting-power measurement, we confine ourselves to the concept of a priori voting power. We show how the concept was re-invented several times and how the circumstances in which it was reinvented led to conceptual confusion as to the true meaning of what is being measured. In particular, power-as-influence was conflated with value in the sense of transferable utility cooperative game theory (power as share in constant total payoff). Influence was treated, improperly, as though it were transferable utility, and hence an additive and distributive quantity. We provide examples of the resulting misunderstanding and mis-directed criticism.     

Sequential voting by veto: Making the Mueller-Moulin algorithm more versatile

This paper shows that a relatively easy algorithm for computing the (unique) outcome of a sophisticated voting procedure called sequential voting by veto (SVV) applies to a more general situation than considered hitherto. According to this procedure a sequence of n voters must select s out of m + s options (s > 0, m 3 n 3 2). The ith voter, when his turn comes, vetoes k _i options (k _i 1, k _i = m). The s remaining non-vetoed options are selected. Every voter is assumed to be fully informed of all other voters total (linear) preference orderings among the competing options, as well as of the order in which the veto votes are cast. This algorithm was proposed by Mueller (1978) for the special case where s and the k _i are all equal to 1, and extended by Moulin (1983) to the somewhat more general case where the k _i are arbitrary but s is still 1. Some theoretical and practical issues of voting by veto are discussed.     

Annexations and alliances: When are blocs advantageous a priori?

For any simple voting game (SVG), we consider the question posed in the title from two different points of view as to what voting power means. We also distinguish between blocs imposed by annexation and those formed voluntarily, and present some general theoretical results concerning these notions. We illustrate our theoretical findings with examples using both toy SVGs and the Qualified Majority Voting rule of the Council of Ministers of the European Community (CMEC). We show that when voting power is understood as influence (I-power), forming a voluntary bloc may be advantageous even if its voting power is smaller than the sum of the original powers of its members; and it may be disadvantageous even if its voting power is greater than that sum. Received: 9 February 2000/Accepted: 19 October 2000     

The Bicameral Postulates and Indices of a Priori Voting Power

If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used as a tool for analysing abstract, uninhabited decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x less powerful than y when measured in the simple voting game G₁ , but more powerful than y when G₁ is bicamerally joined with a second chamber G₂ . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power).     

The Treaty of Nice and qualified majority voting

We analyse and evaluate three decision rules for the Council of Ministers of the EU, which are prescribed by the `definitive form' of the Treaty of Nice. The first will apply from 2005 to the present 15-member EU, if it will not have been enlarged by then. The second or third will apply to an enlarged 27-member EU. We conclude that the first of these is an improvement on the current decision rule; but the other two have extremely undesirable features. Received: 23 February 2001/Accepted: 26 March 2001     

Varieties of failure of monotonicity and participation under five voting methods

In voting theory, monotonicity is the axiom that an improvement in the ranking of a candidate by voters cannot cause a candidate who would otherwise win to lose. The participation axiom states that the sincere report of a voter’s preferences cannot cause an outcome that the voter regards as less attractive than the one that would result from the voter’s non-participation. This article identifies three binary distinctions in the types of circumstances in which failures of monotonicity or participation can occur. Two of the three distinctions apply to monotonicity, while one of those and the third apply to participation. The distinction that is unique to monotonicity is whether the voters whose changed rankings demonstrate non-monotonicity are better off or worse off. The distinction that is unique to participation is whether the marginally participating voter causes his first choice to lose or his last choice to win. The overlapping distinction is whether the profile of voters’ rankings has a Condorcet winner or a cycle at the top. This article traces the occurrence of all of the resulting combination of characteristics in the voting methods that can exhibit failures of monotonicity.     

Microcosms and macrocosms: Seat allocation in proportional representation systems

Three alternative methods are proposed to determine a normative standard concerning the fair proportion of seats a party ought to receive in a representative assembly as a function of the voters' preference orderings. The methods differ from one another in their treatment of indifference relations and the assumptions they make about the type of scale underlying voters' preferences. Common to all three methods is the basic idea that the ratio between the number of voters preferring party i over j to the number of voters preferring party j over i can be tested for consistency, in a precisely defined sense, and if sufficiently consistent, can be appropriately scaled to determine the proportion of seats each party ought to receive. The proposed solutions are shown to satisfy several desiderata when the matrix of preference ratios is consistent. When there are cyclical majorities of equal size, the matrix of preference ratios is inconsistent. The main application of the proposed scheme is as a normative benchmark against which actual or proposed voting procedures can be evaluated in proportional representation systems. The theoretical implications of these solutions are briefly discussed.This research was supported by the United States - Israel Binational Science Foundation (BSF), Jerusalem, Israel. We wish to thank Steven Brams, David V. Budescu, and Joseph Greenberg for many helpful comments.     

Postulates and paradoxes of relative voting power — A critical re-appraisal

This paper re-evaluates the problem of measuring thea priori relative voting power of a voter in an assembly. We propose several new intuitively compelling postualtes that any reasonable index of voting power ought to satisfy. At the same time we argue that most of the paradoxes of voting power discussed in the literature are paradoxical only in a weak sense, if at all. This leaves three crippling paradoxes — the well-known paradox ofweighted voting, and two new ones presented here: thebloc anddonation paradoxes. We evaluate the four main relative power indices discussed in the literature with respect to these three severe paradoxes. The Shapley-Shubik index is seen to be immune to all three paradoxes, while the Deegan-Packel index is vulnerable to all three. The Banzhaf and the Johnston indices are demonstrably immune to the paradox of weighted voting. However, they are shown to suffer from both the bloc and the donation paradoxes. We argue that this seriously undermines these indices in a hitherto unsuspected way. Several other theoretical issues relating to voting power are discussed.