Voting Game Research Articles

Mathematical structures of simple voting games

We address simple voting games (SVGs) as mathematical objects in their own right, and study structures made up of these objects, rather than focusing on SVGs primarily as co-operative games. To this end it is convenient to employ the conceptual framework and language of category theory. This enables us to uncover the underlying unity of the basic operations involving SVGs.

Open primaries and crossover voting

We examine strategic voting in open primary elections by developing a Poisson voting game. In the model, two parties simultaneously hold primary elections, with two candidates competing in each party. Each voter chooses to vote for one of the four candidates without knowing how many other voters participate in each primary. Analyzing the model, we investigate what types of strategic crossover voting occur in equilibrium and under what circumstances they occur. In particular, we focus on two types of crossover voting: hedging (voting for the moderate candidate of the opposite party) and raiding (voting for the extreme candidate of the opposite party). We show that the pattern of strategic voting in equilibrium critically depends on candidate positions and uncertainty about the outcome in the general election.

Spatial voting games, relation algebra and RelView

We present a relation-algebraic approach to spatial voting games. We give relation-algebraic specifications of some important solution concepts of spatial voting games, such as the uncovered set, the majority core, the Pareto set, the win set, and the loss set. These specifications are relation-algebraic expressions and can be evaluated with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. To give an impression of the tool's possibilities, we present some concrete applications.

Examining Monotonicity and Saliency Using Level-k Reasoning in a Voting Game

This paper presents an experiment that evaluates the effect of financial incentives and complexity in political science voting experiments. To evaluate the effect of complexity we adopt a level-k reasoning model concept. This model by Nagel [1] postulates that players might be of different types, each corresponding to the level of reasoning in which they engage. Furthermore, to postulate the effect of financial incentives on subjects’ choice, we used the Quantal Response Equilibrium (QRE) concept. In a QRE, players’ decisions are noisy, with the probability of playing a given strategy increasing in its expected payoff. Hence, the choice probability is a function of the magnitude of the financial incentives. Our results show that low complexity promotes the highest degree of level-k strategic reasoning in every payment treatment. Standard financial incentives are enough to induce equilibrium behavior, and the marginal effect of extra incentives on equilibrium behavior seems to be negligible. High complexity, instead, decreases the rate of convergence to equilibrium play. With a sufficiently high complexity, increasing payoff amounts does promote more strategic behavior in a significant manner. Our results show with complex voting games, higher financial incentives are required for the subjects to exert the effort needed to complete the task.

Coalition Formation in a Legislative Voting Game

We experimentally investigate the Jackson and Moselle (2002) model where legislators bargain over policy proposals and the allocation of private goods. Key comparative static predictions of the model hold with the introduction of private goods, including “strange bedfellow” coalitions. Private goods help to secure legislative compromise and increase the likelihood of proposals passing, an outcome not predicted by the theory but a staple of the applied political economy literature. Coalition formation is better characterized by an “efficient equal split” between coalition partners than the subgame perfect equilibrium prediction, which has implications for stable political party formation. (JEL C78, D72, H41)

Compulsory versus voluntary voting: An experimental study

We report on an experiment comparing compulsory and voluntary voting institutions in a voting game with common preferences. Rational choice theory predicts sharp differences in voter behavior between these two institutions. If voting is compulsory, then voters may find it rational to vote insincerely, i.e., against their private information. If voting is voluntary so that abstention is allowed, then sincere voting in accordance with a voter's private information is always rational while participation may become strategic. We find strong support for these theoretical predictions in our experimental data. Moreover, voters adapt their decisions to the voting institution in place in such a way as to make the group decision accuracy differences between the two voting institutions negligible. The latter finding may serve to rationalize the co-existence of compulsory and voluntary voting institutions in nature.

Pivotality and Responsibility Attribution in Sequential Voting

This paper analyzes responsibility attributions for outcomes of collective decision making processes. In particular, we ask if decision makers are blamed for being pivotal if they implement an unpopular outcome in a sequential voting process. We conduct an experimental voting game in which decision makers vote about the allocation of money between themselves and recipients without voting rights. We measure responsibility attributions for voting decisions by eliciting the monetary punishment that recipients assign to individual decision makers. We find that pivotal decision makers are punished significantly more for an unpopular voting outcome than non-pivotal decision makers. Our data also suggest that some voters avoid being pivotal by voting strategically in order to delegate the pivotal vote to subsequent decision makers.

Old-age Social Security versus Forward Intergenerational Public Goods Provision

This paper introduces an overlapping-generations model with earnings heterogeneity and borrowing constraints. The labour income tax and the allocation of tax revenue between social security and forward intergenerational public goods are determined in a bidimensional majoritarian voting game played by successive generations. The political equilibrium is characterized by an ends-against-the-middle equilibrium where low-income and high-income individuals form a coalition in favour of a lower tax rate and less social security while middle-income individuals favour a higher tax rate and greater social security. Government spending then shifts from social security to public goods provision if higher wage inequality is associated with a borrowing constraint and a high elasticity of marginal utility of youth consumption.

The minimum sum representation as an index of voting power

We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley–Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

The Political Economy of Social Security in a Borrowing‐Constrained Economy

AbstractThis paper introduces a three‐income class, overlapping‐generations model with borrowing constraints. The labor income tax for financing pay‐as‐you‐go social security is determined in a majoritarian voting game played by successive generations. When the interest‐rate elasticity of consumption is low, the political equilibrium might be characterized by an equilibrium where the old and the middle‐income young individuals form a coalition in favor of a higher tax rate and greater social security, while the low‐ and the high‐income young individuals favor a lower tax rate and less social security. In this equilibrium, the size of social security is decreased by the mean‐preserving reduction of a decisive voter's wage if he/she is borrowing‐constrained.

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.

Femtocell network power control scheme based on the weighted voting game

Future wireless networks are designed to cope with drastically increasing user demands. However, network resources reach the limits of their capacity to user requirements. Recently, femtocell networks have attracted much attention to enhance the efficiency of wireless resource usage. In this article, a new adaptive power control scheme is developed for the femtocell network system. By using the concept of the weighted voting game, the proposed scheme adaptively adjusts a transmit power level while ensuring relevant tradeoff between system efficiency and fairness. This power control paradigm can provide the ability to practically respond to current communication conditions and suitable for real network operations. Simulation results are presented to verify the effectiveness of the proposed voting game-based scheme in comparison with the existing schemes.

A heuristic approximation method for the Banzhaf index for voting games

The Banzhaf index is a well known and widely used index for measuring the power a player has in a voting game. However, the problem of computing this index is computationally hard. To overcome this problem, a number of approximation methods were developed for one majority voting games. While it may be possible to extend some of these to k-majority games which are generalized versions of one majority games, to date, there has been no performance analysis of these methods in the context of the Banzhaf index for k-majority games. In this paper, we fill this gap, by first presenting an approximation method for the Banzhaf index for k-majority games. This is a heuristic method that uses randomization to estimate an approximate. We then show that this method is computationally feasible. Finally, we evaluate its performance by analyzing its error of approximation, and show how the error varies with k. Specifically, we show that the average percentage error increases from 15% for games with k=1 to 30% for games with k=5.

Why Just Two Parties? A Voting Game to Illustrate Duverger's Law

AbstractWith the 2012 presidential elections approaching, student interest in (and frustration with) the US two-party system is likely to grow. Although many explanations exist for this phenomenon, their number and complexity often overwhelm students. This article describes an iterative game, designed for a single class session, that illustrates one of the major institutional explanations for the US two-party system—Duverger's Law. The game can also spur discussion of some of the major criticisms of the Law.

Scoring Rules: A Game-Theoretical Analysis

We prove two results on the generic determinacy of Nash equilibrium in voting games. The first one is for negative plurality games. The second one is for approval games under the condition that the number of candidates is equal to three. These results are combined with the analogous one obtained in De Sinopoli (2001) for plurality rule to show that, for generic utilities, three of the most well-known scoring rules, plurality, negative plurality and approval, induce finite sets of equilibrium outcomes in their corresponding derived games - at least when the number of candidates is equal to three. This is a necessary requirement for the development of a systematic comparison amongst these three voting rules and a useful aid to compute the stable sets of equilibria (Mertens, 1989) of the induced voting games. To conclude, we provide some examples of voting environments with three candidates where we carry out this comparison.

In quest of the banks set in spatial voting games

The Banks set (1(4):295–306, 1985) is one of the more important concepts in voting theory since it tells us about the sophisticated outcomes of standard amendment voting procedures commonly in use throughout the English speaking world (and elsewhere as well). While the properties of the Banks set for finite voting games have been extensively studied, little is known about how to find members of this set for majority rule spatial voting games involving possibly infinite agendas. We look at this question for two-dimensional games where voters have Euclidean preferences, and offer a variety of new results that delimit areas of the space that can be shown to lie within the Banks set, such as the Schattschneider set, the tri-median set, and the Banks line set—geometric constructs which we show to be nested within one another.

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.

Markets as a structural solution to knowledge-sharing dilemmas

The authors propose a new structural solution to the knowledge-sharing dilemma. They show that simple auction mechanisms, which impose a rigid set of rules designed to standardize interactions and communication among participants, can prevent some of the detrimental effects associated with conflict of interest in freely interacting groups. The authors report results of two experiments that show that transparent conflicts of interest lead to a breakdown of information sharing, learning, and knowledge transfer in freely interacting groups, but not in simple markets and auctions. In these settings, participants were able to identify the best candidate in a voting game and to learn the solution to an intellective reasoning task, allowing participants to successfully transfer their insights to a series of new intellective tasks (tackled up to four weeks later), despite the conflicts of interests among traders. The authors explain their findings within the theoretical framework of collective induction.

An evolutionary analysis of turnout with conformist citizens

We propose an evolutionary analysis of a voting game where citizens have a preference for conformism that adds to the instrumental preference for the electoral outcome. Multiple equilibria arise, and some generate high turnout. Simulations of best response dynamics show that high turnout is asymptotically stable if conformism matters but its likelihood depends on the reference group for conformism: high turnout is more likely when voters care about their own group's choice, as this better overrides the free rider problem of voting games. Comparative statics on the voting cost distribution, and the groups' composition are also done.

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