Simple Voting Game Research Articles

Luxembourg in the Early Days of the EEC: Null Player or Not?

Voters whose yes-or-no decision never makes a difference to the outcome in a simple voting game are known as “null players”. Luxembourg’s role in the Council of Ministers during the first period of the European Economic Community (EEC) is often cited as a real-world case. The paper contrasts the textbook claim that Luxembourg was a null player with a more comprehensive picture of Luxembourg’s role in EEC’s voting system. The assessment of Luxembourg’s voting power is sensitive to the role played by the European Commission in the decision-making procedure and the measurement concepts underlying power evaluations.

Optimal Provision of a Non-Excludable Public Good

In this paper, we consider the problem that a benevolent designer wants to provide a non-excludable public good with a fixed cost to agents with privately known valuations. Adopting utilitarian's point of view, the designer maximizes the ex-ante total utility of all agents. The impossibility result (e.g., Green and Laffont, 1977) shows that among all strategy-proof mechanisms, none can achieve both allocation efficiency and budget-balanced-ness at the same time. For a canonical model with two agents, we characterize the optimal mechanisms. The optimal mechanisms are always budget-balanced under a large class of well-behaved and commonly used distributions, within all feasible strategy-proof mechanisms, including popular mechanisms like V-C-G mechanisms and budget-balanced mechanisms and beyond. Moreover, the result can be extended to the multi-agent model by imposing one additional fairness condition. In particular, under the optimal mechanism agents share the cost equally. We further show that such equal-cost-sharing mechanism can be implemented by a simple voting game where the voting rule is increasing in the cost.

Mechanisms for Efficient Voting with Private Information about Preferences

We experimentally study behavior in a simple voting game where players have private information about their preferences. With random matching, subjects overwhelmingly follow the dominant strategy to exaggerate their preferences, which leads to ineciency. We analyze an exogenous linking mechanism suggested by Jackson and Sonnenschein (2007) as well as repeated interaction in dierent settings, which could allow endogenous linking mechanisms to evolve. We find that applying the exogenous mechanism captures nearly all achievable eciency gains, whereas repeated interaction leads to significant gains in truthful representation and eciency only in

The inverse Banzhaf problem

Let \({\mathcal{F}}\) be a family of subsets of the ground set [n] = {1, 2, . . . , n}. For each \({i \in [n]}\) we let \({p(\mathcal{F},i)}\) be the number of pairs of subsets that differ in the element i and exactly one of them is in \({\mathcal{F}}\). We interpret \({p(\mathcal{F},i)}\) as the influence of that element. The normalized Banzhaf vector of \({\mathcal{F}}\), denoted \({B(\mathcal{F})}\), is the vector \({(B(\mathcal{F},1),\dots,B(\mathcal{F},n))}\), where \({B(\mathcal{F},i)=\frac{p(\mathcal{F},i)}{p(\mathcal{F})}}\) and \({p(\mathcal{F})}\) is the sum of all \({p(\mathcal{F},i)}\). The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n − k dummy voters.

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.

A Special Case of Penrose’s Limit Theorem When Abstention is Allowed

In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: ‘yes’ or ‘no’. Felsenthal and Machover [Felsenthal, D.S. and Machover, M. (1997), International Journal of Game Theory 26, 335–351.] introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley–Shubik and Banzhaf indices in TVGs. Braham and Steffen [Braham, M. and Steffen, F. (2002), in Holler, et al. (eds.), Power and Fairness, Jahrbuch fur Neue Politische Okonomie 20, Mohr Siebeck, pp. 333–348.] argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between ‘yes’ or ‘no’. The purpose of this paper is two-fold. First, we compare the two approaches and show how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose’s Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.

Overcoming Incentive Constraints? The (In-)Effectiveness of Social Interaction

We experimentally study behavior in a simple voting game where players have private information about their preferences. With random matching, subjects overwhelmingly follow the dominant strategy to exaggerate their preferences. Applying the linking mechanism suggested by Jackson and Sonnenschein (2005) captures nearly all achievable e‐ciency gains. Repeated interaction leads to signiflcant gains in truthful representation and e‐ciency only if players can choose their partners.

Achievable Hierarchies In Voting Games

Previous work by Diffo Lambo and Moulen [Theory and Decision 53, 313–325 (2002)] and Felsenthal and Machover [The Measurement of Voting Power, Edward Elgar Publishing Limited (1998)], shows that all swap preserving measures of voting power are ordinally equivalent on any swap robust simple voting game. Swap preserving measures include the Banzhaf, the Shapley–Shubik and other commonly used measures of a priori voting power. In this paper, we completely characterize the achievable hierarchies for any such measure on a swap robust simple voting game. Each possible hierarchy can be induced by a weighted voting game and we provide a constructive proof of this result. In particular, the strict hierarchy is always achievable as long as there are at least five players.

Voting Games with Abstention: A Probabilistic Characterization of Power and a Special Case of Penrose's Limit Theorem

In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: 'yes' or 'no'. Felsenthal and Machover (1997) introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley-Shubik and Banzhaf indices in TVGs. Braham and Steffen (2002) argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between yes or no. The purpose of this paper is twofold. First, it compares the two approaches and shows how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose's Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.

Testing Explanations of Strategic Voting in Legislatures: A Reexamination of the Compromise of 1790

A difficult yet prevalent problem in legislative politics is how to assess explanations when observable actions may not represent true (and unobserved) legislator preferences. We present a method for analyzing the validity of theoretical/historical accounts that unifies theory, history, and measurement. We argue that approaches to testing accounts of legislative behavior which are theoretically and historically agnostic are not always best and present an approach which: (1) forms an explicit explanation of behavior (here a simple dynamic voting game) that yields estimable parameter constraints, and (2) tests these constraints using a customized empirical model that is as consistent as possible with the explanation. We demonstrate the method using legislative voting data from the first Congress (1789–1791). Using the idea of sophisticated equivalents from voting theory we subject the traditional account of the “Compromise of 1790” to a statistical test and find that there is reason to doubt the claim that legislators of the time believed the specified log roll was taking place. The results suggest that the capital location and assumption issues were resolved independently.

Redistribution and fairness: a note

The introduction of (inequity adverse) fair agents in a simple redistributive voting game reduces the political relevance of the middle class and increases the equilibrium level of redistribution. Interestingly, some of the predictions in Meltzer and Richard [J. Polit. Econ. 89 (1981) 914–927] are affected: a rise in the income inequality between poor and middle class may not decrease redistribution, because of the additional support for redistribution provided by the fair agents.

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.

Minimizing the mean majority deficit: The second square-root rule

Let W be a composite (two-tier) simple voting game (SVG) consisting of a council, making yes/no decisions, whose members are delegates, each voting according to the majority view in his/her district. The council's decision rule is an arbitrary SVG V . The mean majority deficit Δ[ W ] is the mean difference between the size of the majority camp among all citizens and the number of citizens who agree with the council's decision. Minimizing Δ[ W ] is equivalent to maximizing the sum of the voting powers of all the citizens, as measured by the (absolute) Banzhaf index β′. We determine the V which minimize Δ[ W ]. We discuss the difference between majoritarianism and equalization of the voting powers of all citizens.

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).

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 2nn! 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.

Simulation of global and local intransitivities in a simple voting game under majority rule

In this paper we develop a mathematical model of social choice that provides the basis for a computer simulation of a political decisionmaking process. Prior research has demonstrated the nonexistence of an equilibrium for multidimensional voting games, a result that is obtained both globally and locally. Our simulation, while successfully modeling these results, provides a capability for analyzing the systemic effects (voting cycle characteristics) of iterated voting games that has not before been available to researchers. The structure of our computer model also anticipates development of dynamic models of social choice and provides a basic framework for eventual stability analyses of decisionmaking behavior in simulated and actual political environments.