Recently, some fresh food stores have appeared in some cities in China that offer discounts and promotions on time. Take the Q shop as an example. The shop starts to offer a 10% discount every night at 19:00, and a 10% discount every half an hour. There is a Q store next to the community where Xiao Wu lives in Zhuzhou, Hunan. After learning about this discount model, he felt that it was profitable, because according to the store’s regulations, all goods in the store were given away for free at 11:30 in the evening. One night, Xiao Wu planned to go to the Q store to buy some discounted meat. But what he didn’t expect was that it wasn’t until 9 o’clock in the evening (at 50% off) that the vegetables in the store had been sold out, and there were only a few pieces of very fatty pork belly left at the meat counter. After hesitating again and again, he picked a piece of it, the original price was 35 yuan, and the discount was 17.5 yuan.
John Nash who proposed the “Nash Equilibrium”.
Although I bought a piece of pork belly at half price, it was picked and left, so Xiao Wu didn’t feel much sense of accomplishment. He was very puzzled: Why was the goods in the store almost sold out until 9 o’clock in the evening? Why don’t you wait until 11:30 in the evening to give away free merchandise? In fact, no consumer can receive the remaining goods in the store for free at 11:30 in the evening, and this is an inevitable outcome predicted by game theory. American mathematician John Nash called this outcome the “Nash Equilibrium”, and he also won the 1994 Nobel Prize in Economics for proposing the “Nash Equilibrium” model.
Game: reach equilibrium through competition
What is game theory? Game is competition or confrontation. It ranges from guessing boxing, chess and bargaining to financial transactions and international relations. In fact, the entire human society and the natural world are full of games between all parties. Game theory studies what the parties to the game think and do. The game thought has existed in ancient times, and “The Art of War by Sun Tzu” is one of the ancient works explaining the game thought. In modern times, mathematicians began to systematically study game theory. In 1928, von Neumann, known as the “Father of Computers”, proved the basic mathematical principles of game theory and declared game theory to become a formal discipline. The core of game theory is that only when a person or an animal predicts the behavior of other participants while considering itself, can it choose the most beneficial plan for itself.
What is Nash Equilibrium? Let’s take the example of community residents going to Q store to buy vegetables. Before the discount period, there are always people who will come into the store to buy groceries. The Q store has a fixed amount of vegetables and meat on the shelves every day, and will not be restocked after it is sold out. This means that consumers who go early can choose the best quality vegetables or meat, and the later they enter the store, the more they choose. Few, or even no dishes to choose from. Therefore, in the competition involving the price and quality of fresh products, there will never be an end where consumers buy good food at 0 yuan, but there will only be a balanced ending where some people buy good food at the original price and others buy inferior food at a low price. This is a kind of Nash equilibrium.
Another example of Nash equilibrium is price wars. Although the price war goes against the basic logic of businessmen (the smallest investment in exchange for the largest profit), it is the safest way to survive. Suppose there are two supermarkets, A and B. If they sell goods at high prices at the same time, they can both lie down and make a lot of money. But as long as one party starts to cut prices, the other party must immediately follow suit in order to compete for consumers. As any party in the game, whether to lower the price will lead to four endings (only the simplified situation is explained here): First, if neither party lowers the price, they will make a lot of money; second, if you do not lower the price and the other party lowers the price, you will go bankrupt; third, lower the price yourself If the other party does not reduce the price, the other party will go bankrupt; fourth, if both you and the other party reduce the price, both parties will make a small profit. It can be seen that you can guarantee your 100% survival by lowering the price, and there is a certain risk of bankruptcy if you do not lower the price. Therefore, any rational operator will choose to lower the price. In this way, price wars between peers are inevitable.
Selection and income table for both sides of the price war
Because each party in the game cannot control the behavior of all other participants, to ensure that their own profits are maximized, each party must adopt a compromise but a prudent plan, and it is impossible to achieve the most favorable outcome for everyone. It is the core idea of Nash equilibrium.
The wise pig game: how do the strong and the weak coexist?
According to different game scenarios, there are several specific forms of Nash equilibrium, among which the “smart pig game” is particularly special. What is the wise pig game? In a long and narrow pigpen, there lived a big pig and a little pig. One end of the pigsty is the feeding port. If you press the switch on the other end of the pigsty, a certain amount of food will be released from the feeding port. Both pigs have two choices: either they run to the feeding port after pressing the button, or they walk slowly to the feeding port and wait for the other to press the button. Big pigs eat pig food much faster than little pigs, so regardless of whether the pigs actively press the button or not, the big pigs will eventually eat most of the pig food. And if the little pig chooses to wait, then the big pig can only choose to actively press the button in order not to be starved to death. In this case, the Nash equilibrium means that the big pig presses the button, and the little pig waits for the pig’s food at the feeding port.
The wise pig game can be said to be the most widely used game model in real life. In business competition, the tasks of market development and innovation are often undertaken by large companies with strong funds, while small companies are more willing to choose to follow and imitate large companies. This kind of Nash equilibrium can ensure that large companies that pay more can get more benefits, while small companies that pay less can get good benefits back with minimal investment. In a company, employees who can think and work hard are like big pigs in the wise pig game, and employees who are lazy and unwilling to think are like piglets. Generally speaking, although business owners will find ways to make employees work harder, as long as there are big pigs, there will be room for pigs to live.
Gunners Game: Victory is not only based on strength
On the platform of the water park, A, B, and C are playing cockfighting games. They stand on one foot, hit each other with the other leg, and fall into the water even if they fail. Among them, A plays the best, with an 80% probability that the opponent will fall into the water with each impact; B has a second time, 60%; C is the worst, only 40%. The rules of the game are: three people start to act at the same time in each round, and each person can only collide once per round. So, who wins the game in the end? Judging from the hit rate, it should be A, but the result is not necessarily the case.
For example, A may decide to hit B first on the grounds that B is more threatening to himself. If B and C think the same, then they will hit A first. B and C must collide with each other at the same time, so that A will not be hit into the water. The probability of this happening is 24% (60%×40%). The probability that B will not be hit into the water is 20% (1-80%). And because no one will hit C in the first round, C won’t be knocked into the water.
If the rules are changed and the three people take turns to collide, in order to take care of the worst C, and start with C first, then what will the outcome be? In this case, if C wants to prevent himself from being knocked into the water, the best strategy is to neither hit A nor B, but pretend to miss. In the next round, B may give priority to hitting A who is more threatening to him. This will lead to two possible situations: B hits A into the water, and C hits B in turn; if B fails to hit A into the water, A will hit B in the next round… When the third round is over, A continues to stand on the platform The probability of is 55.6%, B is 66.7%, and C is 83.3% (the derivation process is omitted).
In this way, C, the worst player, is the most likely to win the game. This is the famous “Gunsman Game”. It tells us that the ability to win in the game does not entirely depend on the strength of the participating parties, but more on the mutual constraints caused by the strength of the parties.
Three-person cockfighting diagram
Competition, weed out species and protect species
In order to survive, creatures in nature are also engaged in fierce games. During the game, a kind of “rock-paper-scissors” Nash equilibrium will be formed between species.
Scientists have always wondered why many species can be accommodated in the same habitat? Why is there no super-species that can completely squeeze out all other species? Take coral as an example. Corals in the same sea area compete fiercely with each other. Some corals grow very large to get as much light as possible and block the light of other corals; some corals will use poisonous stinging cells in their tentacles or spraying mucus containing stinging cells to kill competitors. However, even if some corals compete very viciously, no coral species can beat all other corals. More interestingly, the competition between corals not only does not limit their growth, but instead allows corals to occupy the niche of many other species-as the main competing species of corals, macroalgae generally only appear in areas with low coral diversity. .
The competition between corals is like a “rock-paper-scissors” game. There is no absolute strong, and different coral species will take turns to occupy the dominant position. As coral species become more abundant, their survival strategies are becoming more and more complex, and the stability of the entire large coral reef ecosystem is getting higher and higher. The more stable the coral reef, the less likely a single coral species will die completely. therefore. Competition and mutual restraint are one of the secrets of coral prosperity.
Faced with limited space and resources, competition and confrontation are inevitable. This is why the study of game theory is so important. By understanding the various Nash equilibriums, we can know that sometimes the best results are not pursued: sometimes we need to allow others to free ride, sometimes we need to see the relationship between ourselves and others. At the same time, game thinking is complex and changing. For example, the main premise of the gunman game is that everyone has information about the level of himself and the opponent, but what happens if a master deliberately hides his strength?