Arrow's Impossibility Theorem

Arrow's Impossibility Theorem was first stated and proved by Arrow, one of the winners of the Nobel Prize in Economics in 1972.

In 1951, Kenneth J. Arrow (Kenneth J. Arrow), in his book "Social Choice and Personal Value", which has now become a classic of economics, adopted mathematical axiomatic methods to affect the prevailing voting methods. Whether it is guaranteed to produce a leader that conforms to the wishes of the majority of people, or "synthesize the order of each individual's expression into the order of preference of the entire group" has been studied. As a result, he came to an amazing conclusion: in most cases it is-impossible! A more accurate expression is: when there are at least three candidates and two voters, there are no election rules that satisfy Arrow's axioms. Or it can be said that as the number of candidates and voters increases, "procedural democracy" will inevitably move away from "substantial democracy".

　　 Theorem premise: Suppose there is a very democratic group, or a society that hopes to make all its own decisions on the basis of democracy. For it, the requirements of every member of the group are equally important. Generally, each member of the group has its own preferences for what should be done most. In order to make decisions, it is necessary to establish a fair and consistent procedure that can combine individual preferences and reach a certain consensus. This is to further assume that each member of the group can sort the various choices needed according to their own preferences, and the aggregation of all these sorts is the sort of the group.
The proof of Arrow's Impossibility Theorem is not difficult, but it requires strict mathematical and logical thinking. There is a story with a rather tortuous plot about this theorem.

Arrow was fascinated by mathematical logic during his college years: when he was in the fourth grade, Polish logician Tarski went to Arrow’s university to teach relational calculus for a year, and Arrow came into contact with him Concepts such as transitivity and sorting preceded this. The logic that Arrow is fascinated by him still depends on self-study.

Later, Arrow was admitted to graduate school. Under the guidance of Harold Hotelling (Harold Hotelling), he studied economics and found that logic is very useful in economics. Choosing the most preferred combination in the game is in line with the logical sorting concept. In another example, the theory of the firm always assumes that the firm pursues profit maximization. When considering the time factor, because the future price is unknown, the firm can only maximize the expected profit based on the expected price. We know that companies in the modern economy are generally owned by many shareholders. 100 shareholders may have 100 different expectations for future prices. Accordingly, there are 100 options when making decisions such as investment based on expected profits. So, how to solve the problem? A natural way is for shareholders (according to how many shares they own) to vote, and the solution with the most votes wins. This is another sorting problem. Arrow’s logical training made him naturally respond to this problem. The result of investigating the transitivity of the relationship easily gave a counterexample.

Arrow’s first serious thinking on the issue of social choice thus became a by-product of his study of standard firm theory. The non-satisfying transitive counter-examples aroused Arrow’s great interest, but it also became an obstacle to his further research because of him. I feel that this paradox has never met, but it seems familiar. In fact, this is indeed a very old paradox. It was proposed by the French political philosopher and probability theorist Gondosai in 1785. But Arrow knew nothing about Gondosai and other original materials, so for the time being Give up further research. This is 1947.

The following year, at the Cowles Economic Research Council in Chicago, Arrow developed a keen interest in the politics of choice for some reason: He found that under certain conditions, "the minority obeys the majority" can indeed be a reasonable Voting rules. But a month later, he found in the "Journal of Political Economy" that an article by Black had already been published. This article expressed the same thought and seemed to have to be abandoned again. Arrow did not continue to study. In fact, there is another reason. He has always taken serious economic research as his own responsibility, especially his commitment to using general equilibrium theory to establish a practical model as the basis of econometric analysis. Examining it in other "side-by-side ways" will distract his energy.

　　 In the summer of 1949, Arrow served as a consultant for Rand. This company, which was established to provide consulting to the US Air Force, had a wide range of research at that time, including the little-known strategy theory at the time. Among the staff, a philosopher named Helmer tried to apply game theory to the study of state relations, but there was a problem that made him very difficult: When interpreting the player as a state, despite personal preference It is clear enough, but how is the preference of a collective composed of individuals defined? Arrow told him that economists have considered this question, and a proper formal description has been written by Bergson in 1938 The year is given. Bergson used a mapping called the social welfare function to describe the problem of converging individual preferences into social preferences. It transforms the vector of individual utility into a social utility. Although Bergson’s narrative is based on the concept of cardinal utility, But Arrow told Hermer that it is not difficult to rephrase it with the concept of ordinal utility. So Hermo followed suit and asked Arrow to write a detailed explanation for him. When Alroy asked to do it, he immediately realized that this problem was actually the same as the problem that had been plagued him for two years. Now that it is known that "the minority obeys the majority" generally cannot integrate personal preferences into social preferences, Arrow guesses that there may be other methods. After a few days of trial and error hit a wall, Arrow suspected that this problem would have an improbable result. Sure enough, he quickly discovered such a result; a few weeks later, he further strengthened the result.

　　 Arrow's Impossibility Theorem just fell to the ground.

　　 From the germination in 1947 to the flowering and fruiting in 1950, Arrow’s Impossibility Theorem came out with twists and turns, and it started out after a thousand calls, and it seemed a bit unintentional. However, it is the persevering pursuit of science behind this unintentional that makes logic in the foreign land of social sciences have a wonderful flower that lasts forever. This cannot but be said to be intriguing.
Arrow's Impossibility Theorem states that democratic voting by reflecting the preferences of all individuals in a society cannot produce a social welfare function. Arrow believes that any public decision-making mechanism based on personal preferences must meet some basic requirements:

　　 One is collective rationality. That is, if all individual preferences are complete, transitive, and reflexive, the collective preferences derived from any decision-making mechanism must also have these characteristics.

　　 Two is unlimited. The public decision-making mechanism must not exclude any form of personal preference, as long as the preference is complete, transitive and reflexive.

　　 The third is Pareto superiority. If everyone thinks that Option A is superior to Option B, then the collective preference must also consider Option A to be superior to Option B.

　　Fourth is preference independence. The ranking of the collective preference for options A and B only depends on people's ranking between these two options, and has nothing to do with people's ranking of other options.

Arrow’s Impossibility Theorem states that a public decision that fully satisfies the above conditions must be an autocratic decision, that is, replacing all social preferences with one’s preference order, which is contrary to the public decision-making mechanism based on personal preference. of. Therefore, there is no public decision-making mechanism that meets the above four conditions.

　　 Now, Western economists have realized that although Arrow's Impossibility Theorem has shrouded welfare economics in a pessimistic atmosphere, its role in the development of welfare economics is very important. Because in the 18th and 19th centuries, some people have noticed that collective decision-making may lead to conflicting results, but it was not until Arrow gave a general conclusion on this issue, which saved people from a lot of unnecessary research. More importantly, Arrow's impossibility theorem enables Western economists to conduct in-depth research on the issue of social choice and try to find ways to avoid the pessimistic conclusion of impossibility. Economists’ research has shown that Arrow’s Impossibility Theorem only applies to voting-style collective selection rules and does not have universal significance.
In our minds, the meaning of elections is probably that everyone chooses the people we love or trust the most based on the majority vote principle. However, can this goal be achieved through elections? In 1972, the winner of the Nobel Prize in Economics and the American economist Arrow used the axiomatic method in mathematics to study this problem in depth in 1951 and found that most A negative conclusion under the circumstances is the famous "Arrow Impossibility Theorem". Arrow's Impossibility Theorem means that under normal circumstances, it is impossible to derive a unified order of social preferences from various known personal preference orders. We prove this.

　　 Suppose there are three people, Zhang San, Li Si, and Wang Wu. They have a dispute over their favorite star. They can't dispute whether Andy Lau, Jacky Cheung, and Aaron Kwok are more popular with the audience. The order of Zhang Sanpai is Andy Lau, Jacky Cheung, and Guo Fucheng. The order of Li Si Pai is Jacky Cheung, Aaron Kwok, and Andy Lau. The order of Wang Wu Pai is Guo Fucheng, Andy Lau, and Jacky Cheung. Who is more popular? There is not a result that everyone agrees. If it is stipulated that each person only casts one vote, the three stars will each get one vote, and the winner cannot be divided. If it is changed to three votes for every two stars and the order is determined according to the principle of minority obeying the majority, the result is again What will happen?

First look at the evaluation of Andy Lau and Jacky Cheung. Since both Zhang San and Wang Wu put Andy Lau in front of Jacky Cheung, both of them will choose Andy Lau and abandon Jacky Cheung. Only Li Si believes that Jacky Cheung’s charm is greater than Andy Lau. According to the principle of the minority obeying the majority, In the first round, Andy Lau won by two to one. Looking at the evaluation of Jacky Cheung and Aaron Kwok, both Zhang San and Li Si believed that Jacky Cheung should be placed in front of Aaron Kwok. Only Wang Wuyi voted for Aaron Kwok. In the second round of the competition, Jacky Cheung naturally won. Next, let’s look at the evaluation of Andy Lau and Aaron Kwok. Both Li Si and Wang Wu think Aaron Kwok is better. Only Zhang San thinks that Andy Lau should be put in front. Of course, the third round is Aaron Kwok won.

　　 Through these three rounds of voting, we found that the evaluation of Andy Lau was greater than Jacky Cheung, Jacky Cheung was more evaluated than Jacky Kwok, and the evaluation of Kwok Fucheng was greater than Andy Lau. It is obvious that we are in a circular situation. This is the "voting paradox", which means that no matter what game rules are adopted, it is impossible to get results that conform to the rules of the game through voting. It would be much easier if the world was limited to the selection of stars. The problem is that some decisions related to the fate of the country often have the aforementioned "voting paradox". Many people have discussed this, but they have not come up with a more convincing way.

　　 When everyone was rushing to find the "optimal public choice principle" without gain, American economist Arrow put forward his impossibility theorem in "Social Choice and Personal Value" published in 1951 after painstaking research. And for this he won the Nobel Prize in Economics in 1972. Arrow’s Impossibility Theorem means, “As long as you give a few preconditions that the selector will inevitably accept, under these preconditions, people cannot find a set of rules (or procedures) in a general or universal sense. Deduced on the basis of personal selection order". It is further derived from this that, in a general or universal sense, it is impossible to find a social state that can guarantee that the welfare of all selectors will only increase and not be damaged.

　　 The conditions that Arro said that several selectors must accept are: universality. There are at least three or more selected schemes for the selector to choose; consistency. Although a certain order of social choices is based on certain personal choices, it must conform to the public's consistent preference; independence. Irrelevant programs are independent; the principle of independent sovereignty. The choice and determination of alternatives should be determined by citizens based solely on personal preferences, and cannot be imposed by society; it is not authoritarian. We cannot let everyone's preferences determine the order of the alternatives in the entire society, and we should adhere to the principles of freedom and democracy.

　　 Arrow believes that each of the above five mutually independent conditions is necessary, but it is impossible to construct a social welfare function that can satisfy these conditions at the same time. The reason for the impossibility is that there are contradictions among the 1-5 conditions, so it is impossible to achieve complete agreement. He drew a seemingly incredible conclusion: there is no solution that can get rid of the shadow of the "voting paradox". When transitioning from personal preference to social preference, it can satisfy social preference and represent a wide range of personal preferences. A sorting method, only coercion and dictatorship. In this way, the effort to find a reasonable social choice mechanism is almost in trouble.

　　 Arrow’s Impossibility Theorem breaks some views that are believed to be truth, and also gives us a new understanding of public choice and democratic systems. Because the social choice method of "the minority obeys the majority" that we admire cannot satisfy the "Arrow's five conditions", just as the market fails, the principle of public choice will also lead to the failure of democracy. Therefore, the rationality of the majority vote principle is limited.