A Priori Voting Power: What Is It All About?

We explain the meaning of a priori voting power and outline how it is measured. We distinguish two intuitive notions as to what voting power means, leading to two approaches to measuring it. One conception, I-power, focuses on a voter's potential influence over the outcome of decisions by a voting body. The second conception, P-power, focuses on voters' payoff, their expected share of a fixed winning 'prize'. We discuss and rebut some philosophical and pragmatic objections, according to which a priori (as distinct from actual) voting power is worthless or inapplicable.     

The product paradox of voting power

We consider a system made up of two committees, with overlapping membership but with different rules for passing decisions; the passage of a joint resolution requires separate approval by each committee. We show that it is possible for one voter a to be stronger (have greater voting power) than another voter b in both of the two committees, while in the two-committee system (considered as a single composite entity) b is stronger than a. We argue that this apparently paradoxical phenomenon is inherent in the structure of the decision-making system in question, rather than being an artefact of any particular index used to measure voting power.     

Sincere and strategic voters: An Israeli study

The theoretical literature on voting behavior has shown that a rational voter may sometimes decide to vote for a candidate or party that does not constitute his or her first preference. Such voters are traditionally called strategic voters, in contrast to voters who act sincerely, i.e., those who always vote for their first preference regardless of how others are likely to vote. After discussing some of the problems associated with the definition of these two types of voters and suggesting a new operational definition, some attitudes and characteristics of these two types of voters are investigated. It was found that strategic voters constitute a very small percentage of the entire electorate, that their education level is significantly higher than that of sincere voters, that they tend more often to believe that polls influence voters' decisions and hence tend to delay their own final voting decision, that they tend more often than sincere voters to support small parties but do not tend more often than sincere voters to switch the party they decide to support from one election to the next, and that there is no significant difference between them and sincere voters regarding which governing coalition should/will form following an election.     

Weak Condorcet winner(s) revisited

A strong Condorcet winner (SCW) is an alternative, x, that a majority of voters rank higher than z, for every other alternative, z. A weak Condorcet winner (WCW) is an alternative, y, that no majority of voters rank below any other alternative, z, but is not a SCW. There has been some confusion in the voting/social choice literature as to whether particular voting rules that are SCW-consistent are also WCW-consistent. The purpose of this paper is to revisit this issue, clear up the confusion that has developed, and determine whether three additional SCW-consistent voting rules—that as far as we know have not been investigated to date regarding their possible WCW consistency—are indeed WCW-consistent.     

Alternative forms of the Shapley value and the Shapley-Shubik index

We prove the validity of an alternative representation of the Shapley-Shubik (1954) index of voting power, based on the following model. Voting in an assembly consisting of n voters is conducted by roll-call. Every voter is assumed to vote "yea" or "nay" with equal probability, and all n! possible orders in which the voters may be called are also assumed to be equiprobable. Thus there are altogether 2 ⁿ n! distinct roll-call patterns. Given a simple voting game (a decision rule), the pivotal voter in a roll-call is the one whose vote finally decides the outcome, so that the votes of all those called subsequently no longer make any difference. The main result, stated without proof by Mann and Shapley (1964), is that the Shapley-Shubik index of voter a in a simple voting game is equal to the probability of a being pivotal. We believe this representation of the index is much less artificial than the original one, which considers only the n! roll-calls in which all voters say "yea" (or all say "nay"). The proof of this result proceeds by generalizing the representation so that one obtains a value for each player in any coalitional game, which is easily seen to satisfy Shapley's (1953) three axioms. Thus the generalization turns out to be an alternative representation of the Shapley value. This result implies a non-trivial combinatorial theorem.     

Setting the record straight: A note on sophisticated voting under Borda's method

The purpose of this note is to correct some inaccuracies in the literature regarding sophisticated voting under Borda's method. It is shown that if a single candidate must be elected and voters vote sophisticatedly under Borda's method, then: (1) Contrary to both Black's (1976) and Ludwin's (1978) claims, a voter's undominated voting strategy may require him not to give top ranking to his most preferred candidate; (2) Contrary to Black's (1976) claim, an undominated strategy may be such that all candidates except the most preferred one are ranked last; (3) Whereas a candidate who constitutes the true bottom preference of an absolute majority of the voters will never be elected if voters vote sincerely, this candidate may be elected if voters vote strategically; (4) The election of a candidate who constitutes the true top preference of an absolute majority of the voters is not systematic: ceteris paribus this candidate may be definitely elected when voters vote sincerely but not when they vote strategically, as well as vice versa.     

Proportional representation: An empirical evaluation of single-stage,non-ranked voting procedures

Rapoport, Felsenthal and Maoz (1988) have proposed three alternative methods to discern the fair proportion of seats that a party in a representative assembly ought to receive as a function of voters' preference orderings. All three methods assume 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, and, if sufficiently consistent, can be appropriately scaled to discover the proportion of seats each party ought to receive. Using these methods as standards, we use exit-poll data gathered during the 1985 elections to the general convention of the Israeli General Federation of Labor (Histadrut) to examine the extent to which plurality- and approval-voting procedures provide a fair allocation of seats. The findings indicate that: (a) all three methods yield sufficiently consistent matrices of preference ratios; (b) the plurality- and the approval-voting procedures yielded significantly different proportional representations; (c) the proposed proportion of seats according to the three aggregation methods fall midway between the proportion of seats that the plurality and the approval procedures allocate. We discuss practical implications of these findings. Requests for reprints should be sent to: Professor Amnon Rapoport, Department of Psychology, University of North Carolina, Davie Hall 013A, Chapel Hill, NC 27514.     

Proportional representation under three voting procedures: An Israeli study

This paper attempts to evaluate, by means of a public opinion survey, the likely effects on the Israeli political system of replacing the extant electoral procedure of proportional SV (where every voter can vote just for one party) with either proportional AV (where voters can vote in favor of as many parties as they like), or proportional CAD (where voters can vote in favor, as well as against, as many parties as they wish). It also shows how the results of proportional AV and CAD enables one to measure the extent of affinity or rivalry existing between individual parties or party bloes. The main results are: (1) These three alternative proportional representation schemes would have considerably different effects in terms of the number and type of parties gaining representation, as well as in terms of governmental coalitions. (2) The extent of affinity (or rivalry) existing between blocs of parties is usually not symmetrical—one bloc can approve (or disapprove) of another bloc significantly more than vice versa. (3) Voters tend to vote sincerely (rather than strategically) under the proportional SV procedure. Theoretical issues and practical implications of these procedures are discussed.