Three extremely selfish people share a cake. What strategy can make all three people feel fair?

This is the famous problem of cutting cakes. Fair distribution

The so-called "three-person satisfaction" has many possible meanings in mathematics, and the two commonly used ones are:

Fairness: All three people think their share is not less than 1/3.

No complaints: they don't think others are less jealous than themselves.

It must be fair not to complain, but fairness does not necessarily mean not to complain.

Daniel's answer, if the above two conditions are not met, it will only lead to self-blame, dissatisfaction/unfairness, and mistakes.

Their situation is simple: I cut, you choose.

The situation of three people has not been solved for a long time. /kloc-a fair procedure was found in the 1940s, and1a non-appeal procedure was announced in the 1980s.

Many people complain that the method of cutting without a door has not been satisfactorily solved.

Daniel's answer is "moving knife program". See Stromquist moving tool program, which was put forward by Stromquist in 1980s.

Need a referee, walk from left to right, and three people stand on the right side of the referee with knives, keeping the position of dividing the cake equally on the right side (according to their own standards). Once one of the three people shouted "Stop", that person got the cake on the referee's left. Then the one (b) among the three people cut the knife. Two people without cakes, the one near the referee takes the middle piece and the one far away takes the right piece.

It is easy to prove that all three people think they have the largest share.

The disadvantage of feed program is continuity, assuming that the probability of two people stopping at the same time is zero, assuming that the cake is infinitely separable, it is not easy to operate in reality.

Discrete program was put forward by selfridge in 1960s, and was independently put forward and published by Conway in 1990s.

A cut the cake into three pieces according to your own standards.

If B thinks that the biggest two pieces are the same size, then choose the cakes in the order of C, B and A and finish.

If B thinks M is the largest, he cuts a small piece of R from M to make it as big as the second big piece, and puts R aside.

C choose first. If C doesn't choose M, then B must choose M, otherwise everything is normal, and A takes the last piece.

The one who doesn't get M in B and C divides R into three parts. Let the one who gets M in B and C choose one first, then A chooses one and keeps the last one for himself. End.

It can be proved that all three people think they have the largest share, which can be found on the wiki page.

The four-person complaint-free segmentation scheme was proposed by Brams, Taylor and Zwicker in 1997. The discrete program of multi-person non-litigation segmentation was put forward by Brams and Taylor in 1995, but the number of cuts may be infinite, so it should be said that it has not been satisfactorily solved.

The above is the cut of "don't complain". "Fair" cutting is simpler. Here is a very popular introduction: mathematics in Europe, Polish mathematicians have made great contributions. The general fair procedure for N people is as follows (proposed by Barnah and Knaster):

Put it in order first.

The first person cuts out what he thinks is1/n.

In order, let's judge whether this is too big. If yes, cut off a little and replace the original cake, if not, skip it.

After everyone has finished the evaluation, this piece will be given to the person who cut the cake last; If no one peels the cake, this piece will be given to the first person.

Repeat 2-4 until there are only two people left, depending on the way I cut you.

The simplified program of n=3 was put forward by steinhaus in 1943. The answer of @ Park III is an oversimplified version of steinhaus's scheme, which is wrong. The problem is that A chooses first and B chooses second. If B chooses the cup that A thinks is not the least, then the whole process is unfair.

= = = = Supplement = = = =

Why is fairness not necessarily without resentment? This is of course based on the mathematical definition, and its expression has pointed out this logical relationship.

The practical significance of these two concepts is because the same cake has different values for everyone.

For example, the following is an exaggerated example:

Imagine a cake with different flavors, such as chocolate, cream, strawberry, etc. People who share cakes have different tastes, so different parts are given different values. Geometrically simple average distribution can't solve the problem here, and fair distribution may not be satisfactory. This is the problem to be solved in this math problem.

It is in this sense that many people insist on "acting before acting", whether it is the ultra-short version of @ Wang Cheng or the redundant "rigorous" version of @ Chen Qihang, it is wrong. The former does not even have a complete algorithm. The first person who cuts will try to divide it equally according to his own standards, but this is not necessarily the standard of the other two, which may lead to unfairness between the other two.

For example, A-B cuts the "strategy" chosen by C-B-A B-A. The following is unfair:

A slice 1/3 and 2/3 are divided according to size, but in BC's view, because there are many small pieces of chocolate, the value is 3/7 and 4/7 respectively. At this time, B's best strategy is to cut off the self-righteous 3/7, 3/7, 1/7. C has the same vision, but in A's view, they are 1/3, 1/2, 1/6 respectively. The second piece is bigger, but there are not many chocolates. If you choose in the order of C-B-A, then A can only get 1/6 in his eyes and 1/7 in BC's eyes.