Shapley-Shubik Index Research Articles

The blocker postulates for measures of voting power

A proposed measure of voting power should satisfy two conditions to be plausible: first, it must be conceptually justified, capturing the intuitive meaning of what voting power is; second, it must satisfy reasonable postulates. This paper studies a set of postulates, appropriate for a priori voting power, concerning blockers (or vetoers) in a binary voting game. We specify and motivate five such postulates, namely, two subadditivity blocker postulates, two minimum-power blocker postulates, each in weak and strong versions, and the added-blocker postulate. We then test whether three measures of voting power, namely the classic Penrose–Banzhaf measure, the classic Shapley–Shubik index, and the newly proposed recursive measure, satisfy these postulates. We find that the first measure fails four of the postulates, the second fails two, while the third alone satisfies all five postulates. This work consequently adds to the plausibility of the recursive measure as a reasonable measure of voting power.

Dual Class Model and Shareholder Agreements: An Analysis of Italian Companies

This paper analysed the changes in ownership concentration of the Italian financial market and the recourse to dual class model and shareholder agreements by Italian listed companies in the period 2009-2020. The analysis shows that the control market did not show signs in the period that would lead to presume an increase in the contestability of our companies. The attenuation in ownership concentration, highlighted by the reduction in the value of the Shapley-Shubik index, and the increase in the average market participation did not produce an increase in the contestability of Italian listed companies since the high concentration and limited contestability of control continue to characterize their ownership structures. Findings also show less recourse by the Italian companies to the instruments of separation between ownership and control in the considered period. The reduction in the number of companies that resort to the issue of shares without voting rights and the shareholders' agreements is also reflected in the lower incidence of the capitalization of these companies compared to the market capitalization.

Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements.

Evaluation of decision power in multi-dimensional rules

This work deals with the evaluation of decision power in multi-dimensional rules. Courtin and Laruelle (2020) introduced a decision process that specifies the collective acceptance or rejection of a proposal with several dimensions. The decision process is modeled as follows: (i) There are several individuals. (ii) There are several dimensions. (iii) Each of the individuals expresses a binary choice (“Yes” or “No”) on each dimension. (iv) A decision process maps each choice to a final binary decision (“Yes” or “No”). We extend and characterize six well-known power indices within this context: the Shapley–Shubik index (Shapley and Shubik, 1954), the Banzhaf index (Banzhaf, 1965), the Public good index (Holler, 1982), the Null individual free index (Alonso-Meijide et al., 2011), the Shift index (Alonso-Meijide and Freixas, 2010) and the Deegan–Packel index (Deegan and Packel, 1978).

Simple Voting Games and Cartel Damage Proportioning

Individual contributions by infringing firms to the compensation of cartel victims must reflect their “relative responsibility for the harm caused” according to EU legislation. Several studies have argued that the theoretically best way to operationalize this norm is to apply the Shapley value to an equilibrium model of cartel prices. Because calibrating such a model is demanding, legal practitioners prefer workarounds based on market shares. Relative sales, revenues, and profits however fail to reflect causal links between individual behavior and prices. We develop a pragmatic alternative: use simple voting games to describe which cartel configurations can(not) cause significant price increases in an approximate, dichotomous way; then compute the Shapley-Shubik index. Simulations for a variety of market scenarios document that this captures relative responsibility better than market share heuristics can.

The power of agents in a dispersed system – The Shapley-Shubik power index

In this paper, dispersed knowledge – accumulated in several decision tables is considered. Dispersion of knowledge is not a part of the work of the system. We assume that the knowledge is already in the dispersed form when it provides to the system. An advanced process of detecting the relations between the decision tables and constructing coalitions is used. The purpose of this paper is to use the measure to determine the strength of the coalition. With this method, a simple method of combining vectors of decisions generated based on local decision tables was applied. The purpose of using the Shapley-Shubik index was to reduce the computational complexity compared to the approach proposed in the earlier papers. In this paper, the results of experiments are presented, and the two approaches are compared. Based on these results, some conclusions have been drawn.

Shapley-Shubik Index incorporated debottlenecking framework for sustainable food-energy-water nexus optimised palm oil-based complex

Palm oil mill effluent (POME) elimination has been considered recently as the alternative to expensive biogas recovery to conquer environmental criticisms in the palm oil industry involving the release of biogas (i.e., a primary contributor towards climate change), during POME treatment. With POME elimination and mill-refinery integration considered, the integrated palm oil-based complex (POBC) serves as an appropriate system for sustainable food-energy-water (FEW) nexus evaluations. In this respect, multi-objective optimisation and debottlenecking of POBC has been suggested to address the economic and environmental trade-offs in palm oil mill (POM) for cleaner production. In this study, a stepwise debottlenecking framework was proposed to include previous fuzzy multi-objective optimisation approach and a new debottlenecking method to improve the fuzzy optimal POBC design and evaluate the impact of process on FEW nexus contributions. Each process creates distinctive impact or influence on the POBC performance. Shapley-Shubik Power Index (SSI) has been applied in the notion of power for yes-no voting systems. By evaluating the operate-fail possibilities of internal processes, SSI can be utilised to allocate the power of each process in achieving or failing the POBC performance target, prior to identifying the system bottleneck (SB) in terms of process stage. Therefore, a novel process-level debottlenecking approach with SSI incorporation for a sustainable FEW system was proposed. The SB of greatest impact on overall POBC performance within the fuzzy optimal flowsheet was identified based on SSI allocation and weightage assignment considering decision-maker's interest in multiple goals. Subsequently, the debottlenecked POBC flowsheet with improvement in SB process parameter was generated subjected to profit maximisation and environmental constraints. In this study, the nut/kernel separation and biomass cogeneration system was identified as the SB of fuzzy optimal POBC. The debottlenecked results show 0.8% and 51% of profit and net energy improvement upon 30% increment of boiler efficiency. The debottlenecked flowsheet is validated in the benefit-to-drawback ratio analysis with a feasible score of 3.9. The POME-eliminated POBC generated via the integrated multi-objective optimisation and debottlenecking framework is applicable for POM retrofit to secure 13%, 97% and 47% of improvements in terms of economic potential, greenhouse gas emissions and water footprint compared to the status quo biogas utilisation scenario. The results of the study could aid palm oil holders in real-life planning of budget distribution and maintenance schedule for process stages in the retrofitted POM.

Who Controls Multilateral Development Finance?

Abstract Multilateral development banks (MDB s) are a growing source of development finance, with nearly two trillion dollars in assets. They have developed a wide array of governance structures, with implications for the distribution of members’ control over those assets. This paper measures that power distribution in 28 MDB s using Penrose-Banzhaf and Shapley-Shubik power indices and members’ relative voting power on MDB governance boards. It uses these calculations to create a typology of MDB governance structures: creditor-led MDB s distribute power among non-borrowers, core borrower-led MDB s distribute power among a few central borrowers, and mutual aid-oriented MDB s distribute power among a wide group of borrowers. Finally, it explores the impact of the creation of two new MDB s, the Asian Infrastructure Investment Bank and the New Development Bank, and finds that while they do not dramatically alter the global landscape they do allow significantly greater access to capital for some borrowers.

Computing power indices for weighted voting games via dynamic programming

We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of smallest size, as well as a very first method for computing Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.

DanceSport and Power Values

DanceSport is a competitive form of ballroom dancing. At a DanceSport event, couples perform multiple dances in front of judges. This paper shows how a goal for a couple and the judges' evaluations of the couple's dance performances can be used to formulate a weighted simple game. We explain why couples and their coaches may consider a variety of goals. We also show how prominent power values can be used to measure the contributions of dance performances to achieving certain goals. As part of our analysis, we develop novel visual representations of the Banzhaf and Shapley-Shubik index profiles for different thresholds. In addition, we show that the "quota paradox" is relevant for DanceSport events.

Ultimate Control Rights and Corporate Fraud: Evidence from China

ABSTRACT This study uses the Shapley-Shubik Power Index to measure the ultimate control rights of companies listed on China’s main board market and empirically tests the effect of ultimate control rights on corporate fraud. We confirm that ultimate control rights have a significant restraining effect on corporate fraud. Specifically, the greater the ultimate control rights, the lower the probability of fraud and the fewer the number of frauds. Furthermore, we find that under different fraud subjects, fraud types, and fraud severity, this restraining effect still exists. In addition, we also find the ultimate control rights are negatively associated with fraud propensity, and are positively associated with fraud detection.

The Public Good Spatial Power Index in political games

We propose a new spatial index called the Public Good spatial index, which is the spatial version of the standard Public Good index under independence behaviors (PGI). However, we show that the spatial model used, is not well suited for both, the standard Shapley Shubik index and the standard Public Good index under Homogeneity behaviors (PGH); and consequently they do not have a consistent spatial version with respect to our model. By contrast, the same spatial model is appropriated for the Banzhaf index and the aforementioned PGI index, concomitantly, allowing their spatial versions in political games. We also argue that those two well convenient spatial indices are only understandable under behavioral descriptions with independence assumptions on political issues. Finally, the paper details our findings by means of examples, comparisons, and it also provides relevant ways of computing spatial power indices in real case studies when it comes to lower dimensions.

Axiomatizations for the Shapley–Shubik power index for games with several levels of approval in the input and output

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called $(j,k)$ simple games. Here we present a new axiomatization for the Shapley-Shubik index for $(j,k)$ simple games as well as for a continuous variant, which may be considered as the limit case.

Dynamic programming algorithms for computing power indices in weighted multi-tier games

In weighted games each voter has a weight assigned and “yes”-voters win if the sum of each weight is greater than or equal to the quota. In weighted multi-tier games, we have several weighted games (tiers) over the same set of voters. In this article, algorithms for calculating Banzhaf and Shapley–Shubik indices for three different type of games are proposed: weighted AND-games, weighted OR-games and games where the tiers have conjunctive normal form. The presented algorithms are generalizations of known computational methods using dynamic programming technique. Finally, some applications and experiments are carried out and these algorithms are compared to a fairly new method based on binary decision diagrams.

Financial Safety Nets in East Asia and Europe: A Political Economy Assessment

This article aims to measure and compare the voting power of the member states of two financial nets: the ASEAN Plus Three Macroeconomic Research Office (AMRO) - Chiang Mai Initiative Multilateralized (CMIM) and the European Stability Mechanism (ESM). Furthermore, the study observes the changes to the CMIM before and after the increase of its resources in 2012. The literature on the comparison between regional safety nets lacks proper evaluations from a political economy perspective. This work fills the gap in the literature by placing two of the most important and recent regional financial safety nets under scrutiny. The article employs empirical analyses using two typical measurements of voting systems such as the Shapley-Shubik and Banzhaf indices. The article shows that the small ASEAN countries, contrary to assumptions in the literature, have been penalized after changes in 2012. By observing simple voting weights only, these effects are not visible. However, based on the results obtained from Shapley-Shubik and the Banzhaf power measurements, we argue that the voting powers of big countries, such as Japan and China, have increased after the changes in the system in 2012. In contrast to the ASEAN example, results in the case of the ESM show that there are no substantial differences in the voting powers of member states based on the Banzhaf index and Shapley-Shubik Index. Based on the empirical results of the article, the authors suggest that AMRO-CMIM should take into account the ESM experience regarding the voting mechanism.

Weighted and Restricted Deegan-Packel Power Indices

In the paper we introduce weighted and restricted versions of the Deegan-Packel power index. We show that the classical Deegan-Packel index, which was proposed as an alternative to the Shapley-Shubik index, in fact coincides with the Shapley value of some specific game determined by the set of minimal winning coalitions, and therefore, it has close affinities with the Shapley-Shubik index. We investigate monotonicity properties of the weighted Deegan-Packel index and introduce easy to check conditions under which it is monotonic with respect to the players' weights. An axiomatic characterization of the weighted Deegan-Packel index is provided. The computations done for three real-life examples from realm of politics demonstrate clearly the coincidence of our theoretical predictions with the reality.

Axiomatizations for the Shapley-Shubik Power Index for Games With Several Levels of Approval in the Input and Output

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called (j,k) simple games. Here we present a new axiomatization for the Shapley-Shubik index for (j,k) simple games as well as for a continuous variant, which may be considered as the limit case.

The vertex cover game: Application to transport networks

As communications are progressing, transport networks need to be monitored, more specifically, camera surveillance of violations is needed. Standards are being developed on how to install cameras, and the question of efficiently distributing surveillance devices across the road network ensue. This task is addressed in this paper by using the methods of the cooperative game theory. The vertex cover game is introduced, and its properties are studied. Since surveillance cameras are to cover all areas of the network, the characteristic function depends on the vertex covers of the graph. The Shapley–Shubik index is used as the measure of centrality. The Shapley–Shubik index is shown to be efficient in a vertex cover game for the allocation of cameras in a transport network. Proceeding from the Shapley–Shubik indices calculated in this study, recommendations were given for the allocation of surveillance cameras in a specific transport network in a district of the City of Petrozavodsk, Russia.

Generalized Coleman-Shapley indices and total-power monotonicity

We introduce a new axiom for power indices, which requires the total (additively aggregated) power of the voters to be nondecreasing in response to an expansion of the set of winning coalitions; the total power is thereby reflecting an increase in the collective power that such an expansion creates. It is shown that total-power monotonic indices that satisfy the standard semivalue axioms are probabilistic mixtures of generalized Coleman-Shapley indices, where the latter concept extends, and is inspired by, the notion introduced in Casajus and Huettner (Public choice, forthcoming, 2019). Generalized Coleman-Shapley indices are based on a version of the random-order pivotality that is behind the Shapley-Shubik index, combined with an assumption of random participation by players.

Using information technology for understanding of collective decision-making processes by the Verkhovna Rada of Ukraine

The article is devoted to the problem of assessing the influence of parliamentary fractions, which allows to determine which political forces are making biggest influence on the state cybernetic security. The method of automated data collection and processing on the basis of open voting results collected from the official parliament web-resource is proposed. The collection and processing of information for the system of monitoring the activities of legislatures of different countries should be based on the principles of Big Data, where the main ones are: data processing speed in accordance with their growth; scalability of solutions; processing both structured and semi-structured data such as data from legislatures web-sites. The method of impact assessments calculation, which takes into account the mutual preferences of the voting participants using the rank of the indexes of the match of positions by voting, is proposed. Software using the capabilities of the Java language have been developed, which allows to calculate the classical and modified Shapley-Shubik indices, split threshold value, position matching index, fraction effectiveness index, and to perform correlation analysis of the results. The corresponding calculations were made for the fractions of the Verkhovna Rada of Ukraine of VIII convocation in period from December 2014 to July 2018. Using correlation analysis, the interaction between fractions and the trends in their fractional discipline were determined. There are trends in the interaction at shorter time intervals, which are not noticeable at longer time intervals. It has been found that some fractions exceed their potential influence in parliament due to the high threshold of the split, which indicates a good fractional discipline. On the contrary, the low threshold of split and the impossibility of realizing winning coalitions negatively influence the effectiveness of the fractions' work. The results allow to assess the coherence and effectiveness of the fractions work, which are a necessary for obtaining agreed solutions in the field of national and cybernetic security of the state.