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Again the quantity to watch is the ‘seat product’. The inverse square law of cabinet durations says that the mean cabinet duration in a country is approximately forty-two years divided by the square of the effective number of parties. In institutional terms, mean cabinet duration is approximately the inverse of the cube root of the seat product. This quantitatively predictive logical model agrees with the world averages. The results are approximate, because other factors enter.

Resistance to simplifying a complex electoral system is at its least when the existing number and size of parties are not altered. The existing effective number of parties is most likely to be maintained, when the district magnitude used in the new simple electoral system is taken as the sixth power of the effective number of parties, divided by assembly size. The task remains risky, especially when one party is very large and the others very small.

Using nothing but the product of assembly size and district magnitude, a quantitatively predictive logical model allows us to estimate the average effective number of parties based on votes. Average deviation from proportional representation can also be estimated from this seat product, within plus or minus four percentage points. When estimating the likely effect of changes in electoral laws on the party system and deviation from PR, the past tendencies in the given country also must be taken into account.

The nationwide threshold of minimal representation is the average vote share needed to win one seat in the assembly. It is close to 38% divided by the square root of the seat product (assembly size times district magnitude). If greater inclusion of political minorities is desired, this threshold can be lowered by increasing district magnitude and/or assembly size. The number of serious or pertinent parties might be close to square root plus fourth root of district magnitude, plus one.

To allocate seats to candidates or parties, even a simple electoral system must specify at least the following: the total number of seats in the assembly (assembly size); the number of seats allocated in each electoral district (district magnitude); how these seats are allocated (seat allocation formula); and how a voter can express her/his preferences (ballot structure). The simplest seat allocation formulas are d'Hondt and Sainte-Laguë divisors, and Hare quota plus largest remainders. For single-seat districts, these proportional representation formulas are reduced to First-Past-The-Post, where the candidate with the most votes wins. Complex electoral systems may offer advantages, but the ability to predict the number of parties and the average proportionality of seats to votes is lost.

The micro-mega rule says that for representation of small parties, it helps to have large assembly sizes, large district magnitudes, and large quotas or large gaps between divisors in seat allocation formulas. Conversely, large parties would prefer small assemblies, magnitudes and quotas — but only if they are absolutely certain to stay large. Worldwide tendency has been to play it safe and move toward more inclusive representation. The number of parties increases with increasing ‘seat product’ — the number of seats in the assembly times the number of seats in the average district — unless the seats are allocated by plurality in multi-seat districts.

Starting with the characteristics of the electoral systems, the ‘Duvergerian agenda’ aims at predicting the average seat and vote share distributions of parties, the effective number of parties, and deviation from proportional representation. In the case of simple electoral systems, prediction of seat shares of parties and cabinet duration has become possible. Simple electoral systems are those that use a usual proportional representation formula or First-Past-The-Post, so that assembly size and district magnitude tell most of the story.

The quantity to watch is the ‘seat product’ — the product of the assembly size and district magnitude. The number of parties represented in the assembly is approximately the fourth root of the seat product. The fractional seat share of the largest party is approximately the inverse of the eighth root of the seat product. This quantitatively predictive logical model agrees with the world averages. The results are approximate, because other factors are present.

As with Chapter 8, the ‘seat product’ is the important quantity to watch. The effective number of parties represented in the assembly is approximately the sixth root of the seat product. The average seat shares of second-largest and third-largest parties also can be calculated from the seat product. This quantitatively predictive logical model agrees with the world averages. The results are approximate, because other factors enter.

This chapter establishes the importance of the electoral system as the central political institution in representative democracy. It discusses the dimensions of electoral systems, such as district magnitude, ballot structure, the extent of voter choice for individual candidates, levels of seat allocation, thresholds, and malapportionment. It identifies the most important questions to ask about the origins and effects of electoral systems.

Chile’s binomial open-list PR system was selected by the Pinochet regime as the most likely to optimise the parliamentary strength of the authoritarian right given its level of electoral support. The system was intended to lead to the fundamental reshaping of the party system, but instead, two broad alliances have arisen, incorporating many elements of the pre-Pinochet party system. The small district magnitude puts a premium on the need for parties to form alliances, and leads to extensive intra-alliance negotiations regarding candidate selection. The electoral system is unpopular with parties of the centre and left, but institutional barriers to electoral reform are high.

Why do more women get elected to Latin American legislatures today than 30 years ago and why are more women elected in some countries than others? This chapter examines how formal representation affects descriptive representation of women focusing on the proportionality of electoral rules and electoral gender quotas. It presents a statistical analysis with data on 18 Latin American democracies from 1974 (or the year of democratic transition) through 2007. It shows that large district magnitudes, large party magnitudes, and gender quotas are the most important explanations for the varying percentage of women in office. Further, it shows that the design of gender quota laws matters—quotas that require larger percentages of positions on party ballots to be female, that mandate that women be placed in winnable positions on party lists, and that provide mechanisms for enforcing quotas get more women into office than weaker quota laws.

This chapter begins by summarizing the evidence regarding functional specialization of areas of the cerebral cortex for specific perceptual, motor, and cognitive functions. It reviews evidence that the cerebral cortex is highly plastic, that is, the nature and organization of representations in the cortex can dramatically change following damage during development. In addition, the role of experience in shaping the responses of neurons in various cortical areas is reviewed, showing both short term plasticity and lasting changes in the firing patterns of cortical neurons in animals performing memory tasks. These findings reveal striking commonalities in the forms of plasticity observed across cortical areas and among different types of experiential modifications. Cortical neurons alter the magnitude or bias of response towards particular stimuli, and they can sustain or reactivate responses to remembered stimuli.

This chapter lays out the evidence for two distinct systems of core cognition with numerical content—analog magnitude representations of number and parallel individuation of small sets— contrasting the two. The analog magnitude number system and the system of parallel individuation each take sets of individuals as input and create representations that support quantitative computations. However, their qualitatively different processing signatures provide evidence for radically different formats of representation, with mental symbols with very different content.