Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation. (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft

Some conjectures on the two main power indices

The purpose of this paper is to present a structural specification of the Shapley- Shubik and Banzhaf power indices in a weighted voting rule. We compare them in term of the cardinality of the sets of power vectors (PV). This is done in different situations where the quota or the number of seats are fixed or not.Shapley-Shubik, Banzhaf, power index, power vectors.

Political Influence in Multi-Choice Institutions: Cyclicity, Anonymity and Transitivity

We study political influence in institutions where members choose from among several options their levels of support to a collective goal, these individual choices determining the degree to which the goal is reached. Influence is assessed by newly defined binary relations, each of which compares any two individuals on the basis of their relative performance at a corresponding level of participation. For institutions with three levels of support (e.g., voting games in which each voter may vote "yes", "abstain", or vote "no"), we obtain three influence relations, and show that the strict component of each of them may be cyclical. The cyclicity of these relations contrasts with the transitivity of the unique influence relation of binary voting games. Weak conditions of anonymity are sufficient for each of them to be transitive. We also obtain a necessary and sufficient condition for each of them to be complete. Further, we characterize institutions for which the rankings induced by these relations, and the Banzhaf-Coleman and Shapley-Shubik power indices coincide. We argue that the extension of these relations to firms would be useful in efficiently allocating workers to different units of production. Applications to various forms of political and economic organizations are provided.Level-based influence relations, Multi-choice institutions, cyclicity, anonymity, transitivity

Power theories for multi-choice organizations and political rules: Rank-order equivalence

AbstractVoting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options

Stable Allocations of Vaccines in a Political Economy

We develop a theory that addresses the problem of the existence of stable vaccine allocations in a political economy. These are allocation policies that a political leader can enforce without losing their popularity. Our analysis distinguishes between contexts where vaccination has positive externalities and contexts where it does not. We show that a stable allocation may not exist if vaccine supply is sufficiently low relative to the number of individuals eligible to receive a dose. We then fully characterize the minimum number of vaccine doses that guarantees the existence of a stable vaccine allocation, regardless of society\u27s preference heterogeneity level. The minimum dose number depends only on a society\u27s influence structure or voting rule. When individuals have unequal voting rights, stable allocations favor those with greater voting power. We generalize our main characterization result to economies where spatial proximity between individuals varies and preferences are unselfish due to positive vaccine externalities. Applying the theory, we find that a political leader can enforce stable vaccine allocation policies that are minority-inclusive only when the supply of vaccines is sufficiently high

A parameterization for a class of complete games with abstention

Voting games with abstention are voting systems in which players can cast not only yes and no vote, but are allowed to abstain. This paper centers on the structure of a class of complete games with abstention. We obtain, a parameterization that can be useful for enumerating these games, up to isomorphism. Indeed, any I-complete game is determined by a vector of matrices with non-negative integers entries. It also allows us determining whether a complete game with abstention is a strongly weighted (3, 2) game or not, and for other purposes of interest in game theory.Peer ReviewedPostprint (author's final draft

Trial-Based Tournament: Rank and Earnings

Trial-based tournament is a widespread hiring mechanism in organizations. Upon a job opening, an applicant is tried out at the job, then swaps with another competing applicant, and so on, with each non-competing worker holding the same position across trials. The job is offered to the applicant whose trial has had the most positive effect on the organization's output. We formalize this tournament model, deriving measures of relative performance that can be used to rank workers for each job and assess their comparative advantage when absolute performance cannot be observed. As a second goal, we study the relationship between tournament rank and earnings as determined by marginal productivity. We show that pay is a weakly increasing function of tournament rank, and we characterize organizations for which pay strictly reflects tournament rank and vice-versa. These organizations are linear and top-down biased, and they strictly include the popular class of von Neumann-Morgenstern organizations. The analysis implies that hierarchical organizations that promote fairness in pay should not have too many layers

Fraudulent Democracy: A Dynamic Ordinal Game Approach

We propose a model of political competition and stability in nominally democratic societies characterized by fraudulent elections. In each election, an opposition leader is pitted against the leader in power. If the latter wins, he remains in power, which automatically makes him the incumbent candidate in the next election as there are no term limits. If he loses, there is an exogenously positive probability that he will steal the election. We model voter forward-looking behavior, defining a new solution concept. We then examine the existence, popularity, and welfare properties of equilibrium leaders, these being leaders who would remain in power indefinitely without stealing elections. We find that equilibrium leaders always exist. However, they are generally unpopular, and may be inefficient. We identify three types of conditions under which equilibrium leaders are efficient. First, efficiency is achieved under any constitutional arrangement if and only if there are at most four competing leaders. Second, when there are more than four competing leaders, efficiency is achieved if and only if the prevailing political system is an oligarchy, which means that political power rests with a unique minimal coalition. Third, for a very large class of preferences that strictly includes the class of single-peaked preferences, equilibrium leaders are always efficient and popular regardless of the level of political competition. The analysis implies that an excessive number of competing politicians, perhaps due to a high level of ethnic fragmentation, may lead to political failure by favoring the emergence of a ruling leader who is able to persist in power forever without stealing elections, despite being inefficient and unpopular

Fraudulent Democracy: A Dynamic Ordinal Game Approach

We propose a model of political competition and stability in nominally democratic societies characterized by fraudulent elections. In each election, an opposition leader is pitted against the leader in power. If the latter wins, he remains in power, which automatically makes him the incumbent candidate in the next election as there are no term limits. If he loses, there is an exogenously positive probability that he will steal the election. We model voter forward-looking behavior, defining a new solution concept. We then examine the existence, popularity, and welfare properties of equilibrium leaders, these being leaders who would remain in power indefinitely without stealing elections. We find that equilibrium leaders always exist. However, they are generally unpopular, and may be inefficient. We identify three types of conditions under which equilibrium leaders are efficient. First, efficiency is achieved under any constitutional arrangement if and only if there are at most four competing leaders. Second, when there are more than four competing leaders, efficiency is achieved if and only if the prevailing political system is an oligarchy, which means that political power rests with a unique minimal coalition. Third, for a very large class of preferences that strictly includes the class of single-peaked preferences, equilibrium leaders are always efficient and popular regardless of the level of political competition. The analysis implies that an excessive number of competing politicians, perhaps due to a high level of ethnic fragmentation, may lead to political failure by favoring the emergence of a ruling leader who is able to persist in power forever without stealing elections, despite being inefficient and unpopular