Jacek Mercik

Power Indices Methodology: Decisiveness, Pivots, and Swings

Three most frequently used measures of a priori voting power of members of a committee were proposed by Shapley and Shubik (1954), Penrose (1946) and Banzhaf (1965), and Holler and Packel (1983). We shall refer to them also as SS-power index, PB-power index and HP-power index. There exist also some other well defined power indices, such as Johnston index (1978) and Deegan-Packel index (1979).1

On a Priori Evaluation of Power of Veto

The main goal of the paper is the evaluation of power connected with veto attribute of the decision maker. A special kind of action attributed to some players is the right to veto, i.e. to stop the action of others permanently or temporarily. In certain cases, it is possible to calculate a value of power of veto attributed to the decision maker and to give the exact value of the power index as well. In other cases, it is only possible to compare the situation with and without veto attribute. In this paper we would like to analyse the power of a player with a right to veto, expecting that the difference between the power of player with veto and his power without veto allows us to evaluate directly or indirectly the power of veto itself.

Classification of committees with vetoes and conditions for the stability of power indices

Abstract Decision making by a committee may be modelled by simple games. Some of the committee׳s members are equipped with a veto, i.e. they may stop an action temporarily or permanently (by transforming a winning coalition into a losing coalition). A classification of such committees and of power indices is presented in this paper. Special emphasis is given to particular characteristics of winning coalitions and, in consequence, to a priori power indices and conditions for their stability from the perspective of both axioms and parameters.

The Effect of Brexit on the Balance of Power in the European Union Council: An Approach Based on Pre-coalitions.

This article anlayses the change in the balance of power in the European Union Council due to the United Kingdom leaving (referred to as Brexit). This analysis is based on the concept of power indices in voting games where natural coalitions, called pre-coalitions, occur between various players (or parties). The pre-coalitions in these games are assumed to be formed around the six largest member states (after Brexit, the five largest), where each of the remaining member states joins the pre-coalition based around the large member state which is the most similar according to the subject of the vote. This is illustrated by an example. We consider adaptations of three classical indices: the Shapley-Shubik, Banzhaf-Penrose and Johnston indices based on the concept of a consistent share function (also called quotient index). This approach can be interpreted as a two-level process of distributing power. At the upper level, power is distributed amongst pre-coalitions. At the lower level, power is distributed amongst the members of each pre-coalition. One of the conclusions of the research is that removing the UK from the voting game means that the power indices of small countries actually decrease. This seems somewhat surprising as the voting procedure in the EU council was designed to be robust to changes in the number and size of member states. This conclusion does not correspond to a general result, but does indicate the difficulty of defining voting rules which are robust to changes in the set of players.

On a Simple Game Theoretical Equivalence of Voting Majority Games with Vetoes of First and Second Degrees

Introducing a veto into the process of group decision making (voting, aggregating preferences) drastically changes the position of decision makers and, consequently, it changes their power index. In this paper we derive the Shapley-Shubik and Penrose-Banzhaf indices for a class of voting games with vetoes. We also present a way of constructing a simple voting game which is equivalent to a game with vetoes of first degree. This simplifies the calculation of power indices by allowing us to use standard algorithms which are available online.

On axiomatization of power index of veto

Relations between all constitutional and government organs must be moderated and evaluated depending on their way of decision making. Among their attributes one may find the right to veto. It is known already that a priori veto is rather strengthening the position of beholder. The evaluation of a power to make a decision is directly connected with a way of power measuring, i.e. with power index choice. In the paper we consider axiomatic base for such choice of an index of power evaluation.

Formal a Priori Power Analysis of Elements of a Communication Graph

This paper presents the idea of measuring the formal impact of elements of a communication graph structure consisting of nodes and arcs on its entirety or subparts. Arcs and nodes, depending on the context, can be assigned different interpretations. E.g. in game theory its nodes may represent the players, often referred to as policy makers and arcs symbolize the relationships between them. In another context, however, nodes and arcs of the graph represent elements of technical infrastructure, e.g. a computer. The graph representing the tested relationships is called the communication graph and the influence of the elements on the entire graph (or its subpart) is referred to as power of the element. Taking into account the power of nodes and connections creates so-called incidence-power matrix more completely than the one formerly describing the communication graph.

Identification of Ideological Dimensions under Fuzziness: The Case of Poland

Under the new political conditions (parliamentary democracy) the results of elections to the parliament have to be considered as an unprecise representation of the real political situation. Consensus reached under such conditions is not stable.

A Power-Graph Analysis of Non-fast Information Transmission

Specific types of information (e.g. knowledge, intellectual capital, conversation) require different models in the analysis of transmission, called non-fast transmission model. This paper introduces graphs with logical structure and proposes a method for evaluating the components of such graphs of transmission of information. The method may allow for optimization of transmission. This involves the maximization of the dissemination of non-fast information, and analysis of the impact of the information or its speed to reach various participants of the process of transmission.

A Dynamic Model of a Decision Making Body Where the Power of Veto Can Be Invoked

Classical analysis of the power of individuals or groups in decision making bodies tends to consider processes of coalition formation in isolation from each other. However, the results of attempts to form a winning coalition will affect the dynamics of coalition formation in the future, particularly when certain players are endowed with the power of veto. When a voter invokes their power of veto to block a generally popular motion, this is likely to provoke retaliation from the supporters of the motion. For this reason, even if a player is not in favor of a bill and can block it, it may be favorable for that party to abstain rather than veto in return for support regarding issues to be considered in the future. Hence, players should only use their power of veto if they are very strongly against a bill. In this paper, we present a model of voting in which the results of previous votes can affect the process of coalition formation. We present a model of such a dynamic voting game and present the form of an equilibrium in such a game. This theory is illustrated using an example based on the voting procedures used in the United Nations Security Council.