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Register a new userOn influence and compromise in two-tier voting systems
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Authors
Geoffrey R. Grimmett
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We examine two aspects of the mathematical basis for two-tier voting systems, such as that of the Council of the European Union. These aspects concern the use of square-root weights and the choice of quota. Square-root weights originate in the Penrose square-root system, which assumes that votes are cast independently and uniformly at random, and is based around the concept of equality of influence of the voters across the Union. There are (at least) two distinct definitions of influence in current use in probability theory, namely, absolute and conditional influence. These are in agreement when the underlying random variables are independent, but not generally. We review their possible implications for two-tier voting systems, especially in the context of the so-called collective bias model. We show that the two square-root laws invoked by Penrose are unified through the use of conditional influence. In an elaboration of the square-root system, Slomczynski and Zyczkowski have proposed an exact value for the quota $q=q^*$ to be achieved in a successful vote of a two-tier system, and they have presented numerical and theoretical evidence in its support. We indicate some numerical and mathematical issues arising in the use of a Gaussian (or normal) approximation in this context, and we propose that other values of $q$ may be as good if not better than $q^*$. We discuss certain aspects of the relationship between theoreticians and politicians in the design of a two-tier voting system, and we reach the conclusion that the choice of quota in the square-root system is an issue for politicians informed by theory.
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We analyse optimal voting weights in two-tier voting systems. In our model, the overall population (or union) is split in groups (or member states) of different sizes. The individuals comprising the overall population constitute the first tier, and the council is the second tier. Each group has a representative in the council that casts votes on their behalf. By `optimal weights, we mean voting weights in the council which minimise the democracy deficit, i.e. the expected deviation of the council vote from a (hypothetical) popular vote. We assume that the voters within each group interact via what we call a local collective bias or common belief (through tradition, common values, strong religious beliefs, etc.). We allow in addition an interaction across group borders via a global bias. Thus, the voting behaviour of each voter depends on the behaviour of all other voters. This correlation is stronger between voters in the same group, but in general not zero for voters in different groups. We call the respective voting measure a Collective Bias Model (CBM). The `simple CBM introduced by Kirsch (2007) and in particular the Impartial Culture and the Impartial Anonymous Culture are special cases of our general model. We compute the optimal weights for large groups rather explicitly. Those optimal weights are unique as long as there is no `complete correlation between the groups. If the correlation between voters in different groups is extremely strong, then the optimal weights are not unique at all. In fact, in this case, the weights are essentially arbitrary.
We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.
Two-tier networks, comprising a conventional cellular network overlaid with shorter range hotspots (e.g. femtocells, distributed antennas, or wired relays), offer an economically viable way to improve cellular system capacity. The capacity-limiting factor in such networks is interference. The cross-tier interference between macrocells and femtocells can suffocate the capacity due to the near-far problem, so in practice hotspots should use a different frequency channel than the potentially nearby high-power macrocell users. Centralized or coordinated frequency planning, which is difficult and inefficient even in conventional cellular networks, is all but impossible in a two-tier network. This paper proposes and analyzes an optimum decentralized spectrum allocation policy for two-tier networks that employ frequency division multiple access (including OFDMA). The proposed allocation is optimal in terms of Area Spectral Efficiency (ASE), and is subjected to a sensible Quality of Service (QoS) requirement, which guarantees that both macrocell and femtocell users attain at least a prescribed data rate. Results show the dependence of this allocation on the QoS requirement, hotspot density and the co-channel interference from the macrocell and surrounding femtocells. Design interpretations of this result are provided.
The reader is reminded of several puzzles involving randomness. These may be ill-posed, and if well-posed there is sometimes a solution that uses probabilistic intuition in a special way. Various examples are presented including the well known problem of the lost boarding pass: what is the probability that the last passenger boarding a fully booked plane sits in the assigned seat if the first passenger has occupied a randomly chosen seat? This problem, and its striking answer of $frac12$, has attracted a good deal of attention since around 2000. We review elementary solutions to this, and to the more general problem of finding the probability the $m$th passenger sits in the assigned seat when in the presence of some number $k$ of passengers with lost boarding passes. A simple proof is presented of the independence of the occupancy status of different seats, and a connection to the Poisson--Dirichlet distribution is mentioned.
We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics. A complete bibliography is included of his publications.
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