This is a classic problem in game theory: dividing the cake. How can I divide it to satisfy both children? Game theory can help us break the game.
Let's put aside the problem of sharing cakes for the time being, and first meet two game theory masters-von Neumann and johnf nash.
Von Neumann (hereinafter referred to as Feng) is the originator of two fields. He is called "the father of computers". The prototype of modern computer was designed by Feng, and it has been used ever since. He is also known as the "father of game theory" because he first made an in-depth study of zero-sum game and put forward the "minimax principle".
Johnf nash Beavon was born more than 20 years later. He is young and promising, and put forward the famous "Nash equilibrium" theory in his doctoral thesis. Unfortunately, God envies talents, and Nash's paranoia becomes more and more serious with his age. However, his wife never abandoned him and accompanied Nash until the last moment of his life, only to get the shocking film A Beautiful Mind.
Back to the problem of cake sharing, we invited two masters, Feng and Nash, to solve the problem of cake sharing.
First of all, we need to turn the problem of sharing cakes into a game between two children. The rules of the game are: two children divide the cake, one cuts the cake and the other chooses the cake first.
The goal of game theory is to find a rational solution to the problem-analyze the answer from a rational point of view without considering emotional factors.
Let's make a table of the strategies of the two children and the corresponding results. Remember that the child who cut the cake is A, and the child who chose the cake is B. The result of dividing the cake is expressed by "the size of the cake obtained by A and the size of the cake obtained by B".
| B Choose the big one | B Choose the small one.
: - :|: - :|: - :
Cut into two pieces at a time, as big as | half, half | half, half.
A cut into two pieces of different sizes | small pieces, large pieces | large pieces, small pieces.
Let Feng cut the cake first, that is, Feng is A, so it is natural to apply the minimax principle.
"Minimal" means that B will definitely choose a large block, so it must be a small block for itself, that is, the left column in the table;
"Great" means that A should make his cake as big as possible;
The combination of "minimum and maximum" means that A knows that B will choose a large piece, so it will cut the small piece bigger. For A, the best result is "half, half" in the upper left corner of the table, that is, two people will each get half a cake, which is a rational solution to this problem.
This is the minimax principle, isn't it simple?
Nash equilibrium is not difficult!
It's Nash's turn to cut the cake this time (that is, Nash is a), so he naturally wants to use Nash equilibrium to find a rational solution. A suppose you cut yourself into two pieces of different sizes, and b will naturally choose the big one, which is the lower left corner of the table.
At this time, A will ask B and himself a question: Do you regret it?
B thought: I got a big piece, and I don't regret it!
An idea: if I cut it into two pieces of the same size, I can get more, and I regret it!
So A changed her strategy and cut it into two pieces of the same size, corresponding to the upper left corner of the table. Do you regret repeating that question just now?
B thought: since two cakes are the same size, it's no use regretting. I have no regrets!
A thought: since b chose a big cake, it would be the best result if I could get half a cake. I don't regret it!
When two people don't regret it, Nash equilibrium is reached!
Looking for Nash equilibrium point, we must pay attention to: "regret" is the choice made by the other party without changing the strategy. This is very similar to the mood of fans watching the ball. Whenever I see an empty door, I always feel: no way! I could have scored this goal!
From the example of sharing cake alone, the two theories get the same answer. The difference between them lies in the scope of application. The minimax principle can only be used to analyze zero-sum games-games in which both sides have the same sum of interests. Nash equilibrium is suitable for both zero-sum games and non-zero-sum games, which is also the place where Nash equilibrium is powerful. But the purpose of Nash equilibrium is to find a "rational solution that neither side regrets", and this rational solution is not necessarily the maximization of individual or collective interests in the game.
At this point, we met two masters-von Neumann and johnf nash, and learned two principles-minimax and Nash equilibrium.
Next, let's learn a familiar and unfamiliar game problem-prisoner's dilemma.
Oh, the second simplest game theory: you and I are both prisoners.