Voting power and Qualified Majority Voting with a 'no vote' option

📝 Abstract

In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish government suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the “Jagiellonian Compromise”). The Penrose definition of voting power is derived from the citizens’ freedom to vote either “yes” or “no”. This paper defines a corresponding voting power based on “yes”, “no” and “abstain” options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme.

💡 Analysis

In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish government suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the “Jagiellonian Compromise”). The Penrose definition of voting power is derived from the citizens’ freedom to vote either “yes” or “no”. This paper defines a corresponding voting power based on “yes”, “no” and “abstain” options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme.

📄 Content

arXiv:0805.3251v2 [math.GM] 11 Sep 2008 Voting power and Qualified Majority Voting with a “no vote” option Martin Kurth School of Mathematical Sciences, University of Nottingham, UK. † Summary. In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish gov- ernment suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the “Jagiel- lonian Compromise”). The Penrose definition of voting power is derived from the citizens’ freedom to vote either “yes” or “no”. This paper defines a corresponding voting power based on “yes”, “no” and “abstain” options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme. Keywords: Penrose voting; Qualified Majority Voting; Square root voting; Voting power; Voting systems 1. Introduction Following the failure of the draft EU constitution (European Union, 2004) in referenda in France and the Netherlands, and the subsequent negotiations on a Reform Treaty (European Union, 2007), the voting arrangements in the Council of the EU have received a significant amount of public attention, not least through the Polish proposal of a voting system that gives every member state a voting weight proportional to the square root of its population (EUobserver, 2007). The idea of this voting scheme is to give every EU citizen the same in- fluence on decisions in the Council, based on an analysis of their voting power. The concept of voting power used in this context was first introduced by Penrose (1946), and adapted to the EU framework by Zyczkowski and Slomczynski (2004), Zyczkowski and Slomczynski (2006). In this scheme for Qualified Majority Voting, the threshold for a motion to pass in the Coun- cil of the EU is then set according to an optimality condition, see Zyczkowski and Slomczynski (2004), Zyczkowski and Slomczynski (2006), Kurth (2007) for details. The Penrose definition of voting power assumes that all citizens have the freedom to vote either “yes” or “no” in elections or referenda. However, this definition of voting power ignores the freedom not to vote at all, which is a freedom very frequently used by the citizens of EU member states, as voter turnout in elections is generally much less than 100%. The purpose of this paper is to provide an analogous definition of voting power which includes this “no vote” option. In section 2.1, a brief introduction to the Penrose voting power scheme is given, while in section 2.2 it is generalised to the case where citizens have the possibility to abstain from voting. A short conclusion summarises the result. †Address for correspondence: Martin Kurth, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. E-mail: martin.kurth@nottingham.ac.uk 2 Martin Kurth 2. Voting power Wherever constituencies with different numbers of voters elect representatives for a council where the representatives for each constituency have to cast a joint vote, the question of allocation of voting weights to the constituencies arises. This is, for example, the case in the European Union, where population numbers vary between about 400,000 (Malta) and 82,300,000 (Germany), see EuroStat (2007). While it may be intuitive to allocate voting weights proportional to the population numbers, there is actually no sound mathematical foundation for such a scheme. What is needed is a rigorous definition of citizens’ voting power. 2.1. Penrose voting power Penrose defined voting power as the probability that the vote of a single voter is decisive, provided all other voters vote randomly, in an election where voters choose between “yes” and “no”. More precisely, let us assume there are N + 1 voters in the country, and a motion is successful if it receives more “yes” than “no” votes. Whether or not the vote of a single voter is decisive depends on how the remaining N voters have voted. For a single voter to be decisive, the motion must be successful if they vote “yes”, and fail if they vote “no”. If N is even, this is the case when the N votes are evenly split between “yes” and “no”, if N is odd, the individual voter will be decisive if there is one “yes” vote more than there are “no” votes. Penrose’s fundamental assumption is that these N voters vote randomly, ie all 2N pos- sible voting outcomes are equally likely. The number of voting outcomes where the remaining voter is decisive is given by the binomial coefficients  N N/2  (N even),  N (N −1)/2  (N odd). (1) This means that the probability for the remaining voter to be decisive, ie their voting power PN, is PN = 1 2N  N ⌊N/2⌋  , (2) where ⌊N/2⌋denotes the floor function, ie the largest integer not exce

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