William S. Zwicker

Weighted voting, abstention, and multiple levels of approval

In this paper we introduce the class of simple games with several ordered levels of approval in the input and in the output – the ( j,k) simple games – and propose a definition for weighted games in this context. Abstention is treated as a level of input approval intermediate to votes of yes and no. Our main theorem provides a combinatorial characterization, in terms of what we call grade trade robustness, of weighted ( j,k) games within the class of all ( j,k) simple games. We also introduce other subclasses of ( j,k) simple games and classify several examples. For example, we show the existence of a weighted representation for the UNSC, seen as a voting system in which abstention is permitted.

A characterization of weighted voting

A simple game is a structure G=(N, W) where N={1, ..., n} and W is an arbitrary collection of subsets of N. Sets in W are called winning coalitions and sets not in W are called losing coalitions. G is said to be a weighted voting system if there is a function w : N→R and a «quota» q∈R so that X∈W iff Σ{w(x) : x∈X}≥q. Weighted voting systems are the hypergraph analogue of threshold graphs. We show here that a simple game is a weighted voting system iff it never turns out that a series of trades among (fewer than 2 a , a=2 n not necessarily distinct) winning coalitions can simultaneously render all of them losing. The proof is a self-contained combinatorial argument that makes no appeal to the separating of convex sets in R n or its algebraic analogue known as the Theorem of the Alternative

Coalition formation games with separable preferences

Abstract We provide sufficient conditions for the existence of stable coalitional structures in a purely hedonic game, that is in a coalition formation game such that players’ preferences over coalitions are completely determined by the members of the coalition to which they belong. First, we show that the existence of core stable and Nash stable solutions for the game depends on certain vector decompositions of the utility functions representing additively separable and symmetric preferences. Then, we generalize the results obtained and show that equilibria with the same qualitative features exist under much weaker restrictions on agents’ preferences. Finally, we examine the relationships between the properties we introduce and other conditions, already known in the literature, that guarantee the existence of stable partitions.

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).

A Moving-Knife Solution to the Four-Person Envy-Free Cake-Division Problem

We present a moving-knife procedure, requiring only 11 cuts, that produces an envy-free allocation of a cake among four players and discuss possible extensions to five players.

Voting on Referenda: the Separability Problem and Possible Solutions

Abstract Assume that voters choose between yes (Y) and no (N) on two related propositions in a referendum, where YN, for example, signifies voting Y on the first and N on the second. If a voters preference order for the four possible combinations is, say, YY>NN>YN>NY, then this voters preferences are nonseparable, because whether he or she will prefer Y or N on either proposition depends on whether Y or N is the outcome selected on the other. Since voters must make simultaneous choices in a referendum, nonseparability forces voters to make choices that they may come to regret after the fact. The usual procedure for conducting multiple referenda, which we call ‘standard aggregation’, can be interpreted as a scoring system in which each voters ballot adds to (or substracts from) the score of each possible combination of Ys and Ns; the combination with the greatest score is the winner. Viewing voting on multiple referenda as voting for Y-N combinations in a multicandidate, single-winner election suggests that other voting procedures, such as approval voting or the Borda count, would be superior in finding consensus choices. In the absence of ballot data to test the effects of these alternative procedures on possible outcomes, we analyzed two variants of the plurality procedure, called ‘approval aggregation’ and ‘split aggregation’, that count abstentions as supportive of both sides, but in different ways. Either of these alternatives would have produced a different winning combination from that of standard aggregation on three related environmental propositions in the 1990 California general election, based on the voting behavior of the 1.7 million Los Angeles County voters. These alternative aggregation methods seem better at finding strongly supported winning combinations than standard aggregation, which produced a ‘weak’ compromise in the 1990 election. But they severely limit the ability of voters with nonseparable preferences to express themselves, which approval voting or the Borda count would better equip them to do.

Old and new moving-knife schemes

ConclusionsIn general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?

Simple games and magic squares

Abstract For each integer k ⩾ 3, we introduce a simple game ⊟k built from a k × k “strongly rigid” magic square. These games are not weighted, yet come very close to being weighted, and thus they provide a uniform sequence of counterexamples to several conjectures that have arisen over the past three decades in the fields of threshold logic, hypergraphs, reliability systems, and simple games. In particular, we show that ⊟k is k − 1 asummable but not k asummable (thus strengthening and simplifying an often-referenced, but unpublished, result of R. O. Winder) and that a certain variant of ⊟k is monotonic, strong, proper, has an acyclic “group” desirability relation, and yet is not weighted (thus strengthening a result of E. Einy and answering a question of B. Peleg).

Analysis of binary voting algorithms for use in fault-tolerant and secure computing

We examine three binary voting algorithms used with computer replication for fault tolerance and separately observe the resultant reliability and security. We offer insights to answer the question: Can a voting algorithm provide a system with both security and reliability? We show that while random dictator (i.e., randomly choosing one of the replicas) provides good security and majority rule yields good fault tolerance neither is effective in both. We present the random troika (a subset of 3 replicas) as an effective combination of fault-tolerant and secure computing.

One-way monotonicity as a form of strategy-proofness

Suppose that a vote consists of a linear ranking of alternatives, and that in a certain profile some single pivotal voter v is able to change the outcome of an election from s alone to t alone, by changing her vote from Pv to