Voting paradox

The Condorcet paradox (also called " problem of cyclical majorities ," Circle of Preferences, " Ching Chang Chong " principle called ) is a named after Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet paradox of voting procedures, which mainly engaged in pairwise votes and elections ( Condorcet method ) effect. The so-called paradox is the following: the vote or the collective preference / decision is cyclical, that is not transitive, although individual preferences are transitive.

Basic statement: It is possible that a majority preferred option A over a B option at the same time a majority of the Option B preferred over a C option, yet a majority preferred option C over option A.

This is possible because each voter has its own order of preference. Parts, however, the choices into two opposing camps to go its choice only stronger or weaker in this direction, this phenomenon does not occur.

Explanation

We assume that there are three persons: x, y and z x has thereby prefer option A, your second choice option B and least liked option C. y has prefer option B, then Option C and finally A. Person z finally has to wish list C, A, B.

In tabular form:

Two of the three (x and z), the preferred option A before the switch B. Two of the three ( x and y), the preferred option B in front of the option C. However, there are also two ( Y and Z), the option C option A prefer. To establish a common rank list according to the Condorcet method, one would therefore have both A before B and B before C and C arranged in front of A, because in a direct comparison, A before B, B before C and C before A is the majority. However, such a ranking is not possible.

This is also true if x, y and z represent not just one person, but ( approximately) equal size groups. More specifically, each group only has to be smaller than the other two together.

The result is therefore dependent on the tuning head and the choice of the order of the electoral process: It should be given the above situation, and it is recognized by the voting manager. Then he may, if he prefers even Alternative A, let first vote between B and C: here wins B. He explains C for excreted and can be tuned between A and B, where A wins now. It now looks as if an overwhelming majority was behind A, after all that has clearly triumphed over B and B over C clearly. A reconciliation between A and C, which would have shown that the preference is not clear, has not taken place.

Importance

The social choice theory examines the Condorcet paradox and other aggregation problems in votes and elections. The Condorcet paradox is a simple example that can be of several individual transitive preference lists without arbitrary preference create not always transitive collective preference lists. In particular, it is a special case of the impossibility theorem of Arrow, which proves the fundamental impossibility of ever-present "democratic " collective preference list. This raises some questions in the theory of democracy; in particular, it shows the view of some, after that a democratization of economic or political decisions does not always lead to optimal results. But how often appear circular preferences?

If we replace the abstract variables in the table by specific options in a decision on the merits: A panel with three members ( Xavier, Yoshi, Zelda) discusses the speed limit on a road.

A = lower speed B = the current speed C = higher speed If we read the table: Xaver wants most likely the lower speed and least of all the higher. Yoshi wants the current best compromise. Zelda might most likely the highest speed, second choice has the lowest speed. The preferences of the committee member's Zelda are strange. It can always happen that preferences are not transitive. We might think that circular majorities in one-dimensional decisions practically do not show up. But this is wrong. So Zelda might think, to have recognized that it can be easily slowed down at low speed and at high speed a hormone would be distributed, which would increase the vigilance. May be on the road, a set of traffic lights, and only at a higher or lower speed, the green phases can be exploited. Only at normal speed there is no advantage. It follows that cyclical preferences are not at all uncommon.

Discovery

Presumably, as first described Condorcet paradox in his Essai sur l' application de l' analyze à la probabilité of Decisions rendues à la voix pluralité of (Paris, 1785). It became virtually forgotten until Charles Lutwidge Dodgson and Edward John Nanson it again independently discovered in the 1870s. Then it came again into oblivion, to Duncan Black and Kenneth Arrow again independently discovered in the 1940s in their investigations.