Degressive proportionality - source, findings and discussion of the Cambridge Compromise. Mathematical Economics, 2011, Nr 7 (14), s. 39-50
The principle of degressive proportionality in relation to the division of indivisible goods in the past few years – especially since the signing of the Treaty of Lisbon – has been one of the topics frequently appearing in studies of economists, mathematicians, lawyers and people involved in politics. It combines a purely theoretical and easily definable question with a very important practical problem that concerns primarily members of the European Union and countries aspiring to become one. The article shall include a debate and discussion of the findings of the Cambridge Compromise concerning proposals to solve the problem of allocating seats in the European Parliament.
A numerical method to determine a degressive proportional distribution of seats in the European Parliament
Distribution of seats in the European Parliament postulated by the Treaty of Lisbon should be degressively proportional. We propose a numerical method which determines uniquely a distribution of seats in the European Parliament which fulfil the requirements of degressive proportionality. More generally, let li, 1≤i≤n, be any non-increasing sequence of real positive numbers. We say that a sequence of positive integers mi is degressively proportional with respect to the sequence li, if mi and limi are non-increasing sequences. For fixed positive integer M we determine inductively a sequence Mi degressively proportional with respect to li which is smaller, then any other such sequence with the first element ≥M. Moreover, for every integer ∑i=1nMi≤Y≤nM we determine a degressively proportional sequence mi with respect to li, which has the first element m1=M and satisfies inequalities Y−(n−2)≤∑i=1nmi≤Y.
Strategy and Manipulation in Medieval Elections
There are many goals in developing electoral protocols, including a desire for a system which is transparent, in that it is clear what the rule or procedure to follow is; nonmanipulable, in that it is not in a person’s best interest to misrepresent their preferences; honest, in the sense that it elects the ‘right’ candidate; and not open to strategizing, i.e., bribery or collusion. However, these desiderata are in tension with each other: Often, transparent electoral procedures are the least strategy resistant, and many honest procedures encourage manipulation. Thus a balance between these different goals must be sought. In modern times, since the seminal result on vote manipulation, the Gibbard-Satterthwaite Theorem [5, 6], much attention has been devoted to developing voting rules where manipulation is never in the best interest of the voters [4] or which are computationally too complex for the average bounded agent to be able to manipulate [1]. This focus on computational aspects of electoral methods is one of the hallmarks of modern studies on voting. But pursuit of these goals is not restricted to modern times: Those participating in elections in the Middle Ages also sought transparency, non-manipulability, honesty, and strategyproofness in so far as these properties can be consistently expressed in a single procedure. However, given the lack of computational sophistication in the Middle Ages, alternate approaches were needed in order to promote honesty, discourage strategizing, etc. These approaches can be classified as either external (constraints introduced outside of the electoral procedure, such as incentives for coming to consensus quickly) or internal (constraints introduced within the electoral procedure, such as voting rules which cannot be manipulated without adverse effects, or which are too difficult for the average bounded agent to manipulate). Surveying examples of both approaches in the context of medieval ecclesiastical and secular elections provides an interesting comparison to modern electoral procedures. Elections in the Middle Ages were used for the same reasons that they are today: To select suitable candidate(s) for a particular office, duty, or obligation. However, it is important to note that the term electio was used in the Middle Ages in a broader sense than our modern ‘election’. Its primary sense was ‘selection’ or ‘choice’, and only secondarily ‘election’ in the modern sense. Thus, many records which purportedly discuss elections are not discussing elections of the type which interests us. We can identify four categories of medieval electoral processes: (1) Election by an external authority having no direct interest in the election; (2) Indirect election, where electors name other electors who then select or elect the officials; (3) Election by lot; and (4) Election by ballot. Elections of the first and third types are generally computationally uninteresting; the first type corresponds to dictatorial voting rules, and the third type collapses to probability theory. In general, interesting voting methods are found only in the fourth type, election by ballot, though they can also occur in indirect election.