5 major games that mathematicians play with our mind and conscience.

Mathematicians are always trying to describe our life as a formula. Sometimes it’s very convincing. But this is only until purely human variables come into play – conscience, trust, lust for justice, egoism, altruism. Here the mathematics stops working and at least psychology begins. We have selected the ten brightest intellectual games, based on life in all its diversity of manifestations.

“Prisoner dilemma”: knock or silence? Math 1st grade

Rules .

Imagine you tried to rob a bank. But, alas, you and your accomplice were caught and placed in different cells. The investigator offers a deal: You testify against your partner and then get a chance to be released for helping the investigation. You have four options.

1. You agree and testify. Your partner is silent. Then he gets ten years, and you go free.
2. You hesitate, and your partner hesitates. Then you both get two years.
3. You’re proud to be quiet, but your partner is testifying. Then he goes free, and you get ten years.
4. You both remain silent, and six months later you’re released for lack of evidence.

And what do you choose? The investigator already opens your cell door…

History and application

The basic model of “Prisoner Dilemma” was proposed in 1950 by American mathematicians Merrill Flood and Melvin Drescher, who worked for the research corporation RAND. This game was needed to predict a nuclear arms race – the USSR and the United States acted as prisoners.

Since then the game has become very popular among math 1st grade games, philosophers and psychologists. Leaving the military criminal cover, you can find many other examples. The same arms race is now very similar to advertising campaigns. Constant competition in the amount of advertising content steadily increases the costs of firms. Accordingly, all parties would benefit from stopping the race. But if someone treacherously breaks the truce, he will win the war for the consumer, and the rest lose.

In the U.S., there were even competitions between teams of universities for the best programmed strategy for the game in the “Prisoner’s Dilemma”, where the victory was awarded for the minimum term of imprisonment after several rounds of questioning. The program based on the “eye for an eye” principle won: it treated each of its partners exactly as it had been treated in the past. But it is still mathematics, human life is much more complicated.

Human qualities.

In reality, you’re unlikely to rob banks. And the Russian investigators won’t offer such deals. It’s a model, a parable, a metaphor for human relations.

It’s more profitable for both of you to remain silent so that the total penalty is the lowest. Then the total sentence will be only a year – much less than in any other scenario. But how much do you trust the man you’ve been working with? Does he trust you? And what does his interests mean to you? To what extent are you willing to take risks?

The main problem with Prisoner Dilemma is trust. It is because of an unwillingness to trust another that a conflict of interest is created, which was raised to the absolute, for example, the writers of the series of chorror “Saw”. So, in the fifth part, the heroes can literally get off with a little blood to get out of the trap. But instead, they begin to compete, which leads to the death of most of them.

2. “Ultimatum”: How much are you willing to pay for justice?

The rules. Two players are invited to share a certain amount of money with each other, say 1000 rubles. The first of them, the applicant, offers his option of sharing, for example, each of 500 rubles, or 800 rubles, and his partner – 200, etc. The second player, accepting, can either agree to the proposed conditions and get your share, or reject the scheme section. In the second case, no one gets the money – it goes back to the bank.

History and application

The rules of this game were first formulated in 1982 in the Journal of Economic Behaviour and Organization to describe the negotiation process. Simple to model and paradoxical in its results, it quickly became a favorite subject for scientists around the world. The game “Ultimatum” is suitable for many life situations. For example, when it is decided what part of the profit to let the employees’ salary, and what part to give to the owners of the company.

Human qualities

What would you do if you were the host? On the basis of rationality, you must agree to any option of sharing money. Even if the pitcher wants to take 990 rubles, it’s still not worth arguing: 10 rubles is still more than zero. But besides rationality, there is also justice.

In hundreds of experiments conducted, the pitcher most often offer their partners from 50 to 30%. Somewhere in the range from 30 to 20% of the receiver begins to refuse the deal, choosing the principle “So do not get you anyone!

The understanding of justice depends on culture. The Peruvian Indians, for example, were more inclined to accept almost any offer, while Asians were much more sensitive and intransigent than Americans. In one experiment in Indonesia, the test subjects refused even to pay a few monthly salaries.

In general, psychologists have much to say about the game “Ultimatum. It turned out that the results of the experiment affects many factors: sexual arousal, age, degree of aggressiveness, testosterone levels, and so on.

In 2003, the journal Science has an article about the study, in which the brain players in “Ultimatum” continuously monitored by MRI. It turned out that the receiver after receiving the proposal activated island proportion of the brain, the upper frontal cortex and cingulate gyrus. The first of these areas is considered responsible for processing and forming negative emotional information, while the other two are responsible for cognitive processes of self-control and choice. The outcome of this confrontation between the ancient mechanism of emotions and the acquired rational thought and determines the final decision.

Experiments gave unexpected results. Test subjects were artificially blocked by the rational frontal cortex. It would seem that the emotions released should reject all unjust proposals in a frenzy. But it turned out the other way round: players became much more accommodating and compliant, emotions of anger and resentment gave way to the innate sense of profit. It turns out that the same rational activity of the frontal cortex and leads to a deviation from a reasonable grade 2 math games strategy, and the ideas of honor and justice force people to make a balanced and unfavorable decision. It is not without reason that in experiments conducted on autistic groups, the failure rate was much lower. Deprived of social prejudice, they were much more likely to follow an ideal mathematical model.

• The interaction of cognitive and emotional mechanisms of decision-making determines the rational behavior of the individual, and failure in any of them leads to the choice of non-optimal strategies. These two systems can also conflict, resulting in many examples where utilitarian thinking has had horrific consequences and directly contradicted moral standards,” explains Anna Shestakova, Senior Researcher at the Center for Neurocognitive Research at the Moscow State Pedagogical University.

3. “Community field tragedy”: if everyone does this.

Rules

The villagers own a common pasture. If everyone grazes one cow on it, that’s all right, there’s enough grass. If someone wants to have a second one, it seems to be okay too: the field is big. But if everyone starts to graze two cows, there will be not enough grass on the field, the pasture will be exhausted and hunger will begin.

History and application

This model was proposed by William Forster Lloyd in 1833 in his book on overpopulation.

• This tragedy of communities often occurs in life – a classic scenario from game theory is played. There are plenty of examples of this: environmental problems, traffic jams – any place where a person seems to be able to make a profit from society without being noticed,” NES Professor Alexey Savvateyev explains.

There is no need to go far for examples. In Moscow, where traffic jams have become a huge problem and the environmental situation is deteriorating year after year, residents persistently ignore the public action “Day without a Car”, held since 2008. Moreover, according to some reports, this is the day when traffic jams are particularly high.

Articles with various modifications of this game appear in leading scientific journals like Science and in our time. For example, there is a version of the experiment called “Public Good”. Here is how it is described by scientists from the Higher School of Economics Dilyara Valeeva and Maria Yudkevich: “Each of the participants is initially given a certain amount of money. Everyone should decide privately what share of this personal money they can invest in the public good. The money invested in the public good is increased by several times and divided equally. The group will gain maximum benefit if each participant invests the entire initial amount of money. However, players may evade investing their money in public enterprises. In the balance predicted by the theory, each participant makes a zero contribution. In real experiments, the outcome is usually different: players invest a certain amount of money in the public good.

Human qualities

We do not consider it a sin to do little harm to nature or society. “The world will not collapse from a single piece of abandoned paper,” says the passerby, and cities are overgrown with mountains of garbage.

Sociologists and psychologists have long been trying to understand how to make people more altruistic. One of the methods is to involve people in the process, giving them a sense of pride in the good or harm reduction they have done. For example, at the Russian Reporter Summer School, students are offered to make a contribution of 150 to 600 rubles a day, depending on financial possibilities. If some part of the participants make a minimum contribution, nothing terrible will happen. But if everyone does so, the Summer School will be doomed to lack of food and other problems. It seems that we are saved by a sense of belonging: “This is my project, I am also responsible for it. At least for the last few years, the average fee has been double the minimum.

In the same series, the distribution of music over the Internet. Some bands offer to download their works for free, and then, after listening to them, pay any amount. If no one pays, the band has nothing to record a new album on.

Some economists believe that these are the schemes that have the future, at least in the distribution of music, books and movies. For example, Alexander Dolgin, professor at the Higher School of Economics, introduces the concept of “postfactum gratitude payments”. In his scheme, the economics of the future will be able to defeat the freeloaders through public appraisal. If I read a book or watched a movie, I have to make a personal assessment – to what extent I liked it. And it would be illogical if I put the highest score without donating a significant amount to the author.

4. “The problem of the wagon”: Is it possible to kill a man out of humanism

Rules

There’s going to be an accident on the railroad. A carriage filled with passengers is rolling into an abyss. You have the opportunity to save it. To do that, you have to push a fat road worker, who happened to be nearby, onto the rails with your own hands. The man will die. But dozens of lives will be saved. Are you ready?

History and application

The original formulation of this painful dilemma was proposed in 1967 by the British philosopher Philippe Foote as a mental experiment in ethics. Over the years, there have been many modifications. You kill one and save three. You kill a child and keep ten alive. There’s even a piercing short film in which the shooter has to choose whether to crush his own son with the structures of the bridge or allow the train with hundreds of passengers to crash.

The most common application of this dilemma is, of course, military action. Leaving the platoon to cover the retreat of the regiment, the commander sends to the right death of thirty people, but gives a chance of a thousand. But this situation can happen on a real railway. Or during a fire. Or somewhere else.

It doesn’t have to be about life and death. Imagine you’re the head of a department that needs to fire one employee in order to save the whole team. Or you’re teaching a lesson at school and you have to yell at one kid so the rest of the class can study in peace.

Human qualities.

There is very little math in this game: ten is more than one, even a first grader knows it. But a psychology with ethics in this bulk dilemma. The commandment “Don’t Kill!” is in conflict with the value of saving life. By the way, in a short film about the archer, the protagonist still sacrifices his son and the train with unsuspecting passengers calmly goes on.

In the experiment, conducted by psychologists from the University of Michigan, the subjects were offered a realistic three-dimensional model with a wagon, ways and the need to kill one to save the five. About 90% of the participants transferred the shooter and killed a person for the passengers of the carriage. But this is computer reality, not real life.

5. “Hawks and Pigeons”: attack or run.

Rules

In the same animal population, two groups coexist with different strategies to fight for resources. The first, the “hawks”, are always in conflict and go all the way to the end when they meet a competitor. As a result, they either win and assign all resources in the area (+50 points), or lose and get seriously injured in a fight (-100 points). “The Pigeons, on the other hand, are peaceful. Seeing the “hawk”, they immediately retreat (0 points “pigeon” and 50 points “hawk”), and when you meet with their congeners only depict the willingness to fight. After a lengthy exchange of threats (-10 points for both pigeons), the luckier pigeon gets the resources (+50 points).

There are many other variations in the rules, but the main features of the game remain the same: winning brings any bird the average number of points, getting injured by hawks is equated to a huge fine, and ritual battles of the “pigeons” also require some minimum cost.

The aim of the game is extremely simple: to earn the maximum number of points, no matter what is hidden behind them – food, money, females or “representation of individual genes in the gene pool of the population”, as Richard Dawkins expresses in his book “Selfish Gene”.

History and application

The rules were first published in Nature magazine in 1973. The authors proposed to formalize animal conflicts over resources, territory or sexual partners in this way. The model allows the ratio of strategies in the population to calculate the amount of resources spent and received by individuals in a particular variant of interaction. The bird metaphor was borrowed from the geopolitical slang of the time (“hawks” for tough confrontation with the enemy, “pigeons” for détente and compromises).

“Hawks and Pigeons” appeared as a development of the game in which two drivers are rushing towards each other. The loser was considered to be the one who would be the first to be afraid of frontal collision and turn aside.

Human qualities.

• We have tried to move away from classical game theory, in which the set of possible strategies is small and rigidly defined, – says Mikhail Burtsev, head of the Laboratory of Neurointellect and Neuromorphic Systems at the Kurchatov Institute. In 2007, he and his supervisor Peter Turchin published an article in Nature that described how strategies of “hawks” and “pigeons” appear naturally in the process of computer model evolution.
• We have created a virtual world inhabited by agents performing primitive actions that could be combined into more complex strategies. The behavior of an individual agent was controlled by its own neural network. This allowed us to discover such strategies, which in standard game theory did not come to mind to investigate, – explains Mikhail.

So in the process of evolution of this computer world in it appeared its own peaceful “pigeons”, “hawks” attacking all the strangers, and even “starlings” who gather in packs in the face of danger. But the most interesting thing is that these mathematical agents began to show sublime human feelings: care for relatives, sacrifice and altruism.