However, on closer examination, this method is not completely fair. For those who divide the cake, the value of the two cakes is equal, but for those who choose the cake, the value of the two cakes may be very different. Therefore, people who choose cakes can often get more value than 1/2. A simple example is that the surface of the cake is half strawberry and half chocolate. The person who divides the cake is only interested in the cake volume, so he divides the strawberry part into one piece and the chocolate part into one piece; But he doesn't know that people who choose cakes prefer chocolate. Therefore, the person who chooses the cake can get more than half of the total value of the cake, while the person who divides the cake can only get exactly half of the value. In fact, it is fairer for the former person to get all the strawberry parts and a small chocolate part, and the latter person will get the rest of the chocolate part. This will ensure that both people can get a little more than half the value.
However, in order to achieve the ideal division mentioned above, both parties need to fully disclose their information and be able to fully trust each other. However, in real life, this is very difficult to do. Considering the possibility of cheating between the two sides of the cake, it is almost impossible to achieve absolute fairness. Therefore, we can only settle for the second best and give a generally acceptable definition of "fairness". In the issue of fair division, there is a fundamental principle of fairness called "proportional division". It means that if there are n people sharing the cake, everyone thinks that they have got at least 1/n value of the whole cake. From this perspective, "you
The scheme of "let me choose" is fair-in the case of asymmetric information, getting half of the total value is already a very satisfactory result.
If there are more people sharing the cake, balanced segmentation can also be achieved, and there are more than one way to achieve it. One of the simple methods is that everyone who has been given the cake will divide the cake into smaller equal parts and let the next person who has not been given the cake choose. Specifically, let two of them divide the cake into two pieces by the method of "you divide it and I choose it"; Then, everyone divided the cake in their hands into three parts, and asked the third person to pick out one from each hand; Then, everyone divided the cake in their hands into four parts, and let the fourth person choose one from each of the three people; Keep going like this until the last person chooses his own cake. As long as everyone can divide the cake equally, no matter which piece is picked, he will not suffer; And the nth person gets at least 1/n pieces in everyone's hands, which together will naturally not be less than 1/n of the total cake value. Although the cake may be divided into bits and pieces in this way, it can ensure that the cake in everyone's hand is not less than the total value of the cake 1/n in his own opinion.
There is also a completely different segmentation scheme called "the last reduction algorithm", which can also achieve balanced segmentation. We still use the letter n to represent the total number of people. First, the first person cuts out what he thinks 1/n from the cake, and then passes this small piece to the second person. The second person can choose to give the cake directly to the third person, or he can choose to cut a small piece from it (if the cake is bigger than 1/n in his opinion) and give it to the third person. By analogy, everyone gets a chance to "trim" the cake and then hand it over to the next person. It is stipulated that the last person who changes the size of the cake will get the cake, and the remaining n-1 individuals will repeat the previous process from the beginning and divide the remaining cakes. Every time a process is completed, one person will get a cake that satisfies him, and the number of people who repeat the process will be reduced by one person next time. continuously
Do this until everyone gets a piece of cake.
After the first round of the process, the person who gets the cake can guarantee that the cake in his hand is worth the whole cake1/n. For everyone who didn't get the cake, because after he passed it on, the people behind him could only reduce the cake without adding it, so in his opinion, the part of the cake that was taken away must not be 1/n, and the rest of the cake is still enough for him. In the next process, a similar truth is also established. What's more, under the rules of this game, everyone will consciously trim the cake in their hands into what they think is 1/n, and cheating will not bring him any benefits. The person who divides the cake never dares to cut it smaller, otherwise it may be him who gets the cake; And if he gives a cake larger than 1/n to someone else, in his eyes, the rest of the cake is not enough, and his final share is probably far less than1/n.
In this way, the problem of balanced segmentation is perfectly solved. However, as we said before, the equilibrium condition is only a minimum requirement. In life, people still have many more difficult formal understandings of the concept of "fairness". If the requirements for fairness are slightly modified, the defects of the above scheme will be exposed. Let's look at this situation: if n people share the cake, everyone thinks they have got at least 1/n of the cake, but two of them still fight, what may be the reason? Because different people have different criteria for judging the value of each part of the cake, it is entirely possible that although he has been assigned at least 1/n copies, in his opinion, there is someone who has more cakes in his hand than he does. It seems that what we usually call fairness has at least one meaning-everyone thinks that other people's cakes are not as good as mine. In the theory of fair division, we call the cake division scheme that meets this condition envy-free division.
Jealousy-free segmentation is a stronger requirement than balanced segmentation. If everyone's cake is not as much as mine, then my cake has at least 1/n, which means that the segmentation that meets the jealousy-free condition must meet the balanced condition. On the other hand, the segmentation that meets the equilibrium condition is not necessarily jealousy-free. For example, A, B and C share the cake, but A only cares about the size of the cake, B only cares about the number of strawberries on the cake, and C only cares about the number of chocolate bars on the cake. The final result is that the cakes of A, B and C are equal in size, but there is nothing on A's cake. There is a strawberry and two chocolates on B's cake, and there are two strawberries and one chocolate on C's cake. Therefore, everyone gets the value of the whole cake exactly 1/3 from their own point of view, but this division is obviously unscientific-B and C will be jealous of each other.
The two balanced segmentation schemes we introduced before are not satisfied with jealousy-free. Take the first plan, for example. If there are three people sharing the cake, according to the rules, the first person should be divided into the second person, and then each person should cut his cake into three equal parts, so that the third person can choose one from each person. This dividing method can ensure that everyone can get at least13 of the cake, but it may happen that the part selected by the third person from the second person is exactly what the first person wants very much. In this way, the first person will feel that the cake in the third person's hand is better, and this division is not harmonious.