Glamor Economics

Chapter 141

Chapter 18 Section 3 Subordination of the Minority to the Majority is Not Necessarily a Democracy——Arrow's Law

Is democracy necessarily achieved when the minority obeys the majority?Let's first look at a case of Beijing's 1992 Olympic bid.

Beijing began applying to host the 1992 Olympic Games in 2000.The voting rules for bidding to host the Olympic Games are a gradual elimination system. Members with voting rights vote in the cities participating in the application, and the city with the fewest votes will be eliminated.Beijing has been leading in the first two rounds of voting.After two rounds of voting, three cities remained: Berlin in Germany, Sydney in Australia and Beijing in China.In the third round of voting, Beijing received the most votes, with Sydney second and Berlin third.

After this round of voting, Berlin was eliminated.If there is only one vote, Beijing will win, but the problem is that another vote must be cast.When it comes to Beijing vs Sydney, is Beijing sure to win again?
The truth is, Beijing lost and Sydney won the right to host the 2000 Olympics.Why is this happening?Most of the voters who originally supported Berlin switched to Sydney.That's why Sydney won.

From this point of view, democratic voting cannot produce a unique result, and the election result depends on the procedural arrangement of democratic voting and the number of candidates determined each time, that is, the voting rules.Different voting rules will lead to different election results.That said, democratic voting has inherent flaws.We will use the "impossibility theorem" proposed by the famous economist Arrow to illustrate that there are defects in the democratic system.

When seeing all the people rushing to find the "optimal principle of public choice" but got nothing, Stanford University professor Kenneth?Arrow conducted painstaking research and proposed an ideal election experiment in his book "Social Choice and Individual Value" published in 1951.

The first step in Arrow's ideal election is that voters cannot be oppressed or coerced by specific external forces, and have normal intelligence and rationality.There is no doubt that these demands on voters are not excessive at all.

The second step of Arrow's ideal election is to regard election as a rule, which can synthesize the preference order expressed by individuals into the preference order of the whole group, and at the same time meet the requirements of "Arrow's theorem".The so-called "Arrow's theorem" is:
1. Any ordering relationship that all voters have in mind for the alternatives is actually possible.That is to say, every voter is free, and they can vote independently according to their own wishes without being persecuted.

2. For any pair of alternatives A or B, if A is better than B for any voter, it should be determined that A is selected according to the election rules.This actually means: the unanimous wish of all voters must be respected.

However, the actual situation is often not the case. For example, two schemes A and B are chosen by two voters C and D.For C, plan A is certainly better, but plan B has no major loss; but for D, plan A may mean survival, and plan B means death, so let C and D each have one person, one vote, of course It is not fair and equal.

3. For any pair of alternatives A and B, if A is better than B in the result of a certain vote, then in another vote, if the position of each voter remains unchanged or advances, then The final result obtained according to the same election rules should also include that A is better than B.That is to say: if all voters like a certain candidate, relative to other candidates, there is no ranking lower, then the position of the candidate in the election results will not change.

This is a basic guarantee for the fairness of elections.For example, when a housewife decides whether she should have good, cheap pork or bad, expensive pork for lunch, we know that it is unlikely that her liking for good pork or bad pork will change—yet This time she bought stale pork.This must show that there are some bad factors involved in the "election" of housewives against pork.Of course, if the reason is that the market is [-]% stale pork, it means that "elections" no longer exist, and housewives have been "dictated" by stale pork.That's beyond the scope of our discussion.

4. If the order of the elements in the subset of the alternative set does not change during the two voting processes, then the order of the elements in the subset also remains unchanged in the final results of the two elections.

For example, the housewife who buys pork has to choose the staple food for her family’s lunch. There are three “candidates”: good flour for one yuan per catty, moldy flour for one yuan per catty, and quicklime for one yuan per catty.Regardless of the order of housewives' choices, it is clear.However, the current situation is: after quicklime was out, the housewife actually chose mildew flour.This must mean that factors other than this "election" have strongly intervened.For example, the head of the housewife's unit is the brother-in-law of the owner of the mildew flour factory.

The combination of Arrow's Theorem 3 and 4 also means that the electoral performance of candidates depends only on the independent and non-interference evaluation made by voters.

5. There is no such voter, so that for any pair of alternatives A and B, as long as the voter determines that A is better than B in the election, the election rules determine that A is better than B.In other words, no voter can rely on his personal wishes to determine the final result of the election.

These five rules are undoubtedly the most basic requirements for a fair and reasonable election.

However, Arrow found that: when there are at least three candidates and two voters, there is no election rule that satisfies Arrow's theorem, that is, "Arrow's Impossibility Theorem".Even under the absolute ideal situation where voters have clear, known preferences free from external intervention, and there are no negative factors in real politics, it is also impossible to derive the order of social preferences from the order of personal preferences through a certain method. It is impossible to accurately express the personal preferences of all members of the society or achieve consensus public decision-making through certain procedures.

The minimum fair and reasonable election that people pursue and expect that meets the five requirements of Arrow's theorem is actually impossible to exist.This is undoubtedly the most fundamental blow to the voting system.With the increase of candidates and voters, "formal democracy" will inevitably become more and more far away from "substantive democracy".

[links to related words]

Public choice is an important branch of modern microeconomics.Its core content is the behavior characteristics of public selectors (officials, political parties, governments, voters, etc.); the possible results of political decisions under different rules; A charter for improving the efficiency of public decision-making.

(End of this chapter)