Life is like a game

Is it possible to solve business or personal difficulties using math? We have gathered ten bright examples from the Game theory that simulate people's behavior and help solving problems of modern society through mathematical algorithms.

1. The prisoner's dilemma

In 1950, American researchers Merrill Flood and Melvin Dresher framed a basic model of the prisoner's dilemma. There is a version that it was developed for forecasting a nuclear arms race between the US and the USSR.

It shows why two completely rational individuals might not cooperate, even if it is beneficial for them. The task is as follows: when robbing a bank, two criminals are caught. The police interrogate everyone separately, and until then the suspects are staying in different places and cannot communicate. There are four possible outcomes:

1. You testify against your partner. Your partner is silent. Then you are free, and he gets ten years in prison.

2. You are silent, but your partner gives testimony against you. Then he is free, and you get ten years in prison.

3. Both you and your partner give testimony against each other. Then you both get two years.

4. No one confesses, then both are free after six months for the lack of evidence.

The matrix of this problem looks like this:

It is more advantageous to remain silent, so both suspects get just a short term in jail. If the partner is silent, it is better to give a testimony and avoid the punishment. If the partner testifies against you, it is more preferable to confess. The problem is that suspects cannot be sure about decision their partner makes in the same situation.

Of course, in real life, such a situation is impossible; it is just a metaphor for human relations. Conflict of interest in this example arises from the lack of trust and the rejection of a beneficial arrangement for the favor of rivalry.

This dilemma is very similar to the modern market scheme, in which companies cannot agree to keep the same prices and prefer to compete. Nowadays advertising campaigns definitely seem like the arms race. All parties would benefit from general agreements, but the winner in this fight is always the one who is violating agreements and follows only its own selfish purposes.

2. Ultimatum

This well-known game for describing the negotiation process was formed by American economists in 1982. The essence is as follows:

Two participants have to share a certain amount of money, for example, 100 dollars. The first player offers how much he to keep, and how much to give to another person. If the second player agrees with this condition, then the money is divided between them as follows. If he refuses, then the money goes back to the bank.

The first player should propose such an option, so that the second participant agrees on it, and at the same time, the former maximizes his benefits. The first player can offer the second player just one dollar, and the consent in this case would look logical. If the second player refuses, he does not get anything. However, the experiment shows that often the second player rejects the proposal that is less than 20% of the whole amount. Thus, the second player punishes the greed. However, first players usually make more generous offers – at least 30-50% of the whole amount.

Cultural differences are quite evident in this game. The Peruvian Indians accept almost any proposal, and the Asians seem less flexible. Psychologists say that age, testosterone level and sexual arousal also influence the experiment.

In 2003, scientists conducted a study that showed a brain of a player when receiving a proposal. The brain of the second player had three areas activated: one that is responsible for processing negative information, one for self-control and one for making decisions. The final decision is always comes out of the confrontation between emotions and rational thinking.

During the experiment, people with temporarily blocked rational thinking became more flexible. It turned out that the rational part of the brain rejects a reasonable strategy in order to protect its perceptions of honor and justice, even if it requires a loss. Autistics that lack social prejudices were less likely to refuse the proposal and more often followed the ideal mathematical scheme of the game.

3. The tragedy of the commons

There is a rural community, which has only one pasture. Members of the community can graze any number of cattle on it. Increasing the number cattle raises the income of its owner, but if everyone does so, the pasture will be destroyed and no one will benefit. Less cattle will make the pasture more fertile, but the income of community members in this case will be much lower.

The tragedy of the commons is based on the model of overpopulation proposed by William Forster Lloyd, and is suitable for many modern situations: resources, environmental problems and even traffic jams. In fact, it describes the contradiction between the interests of individuals in relation to the common goods. Excessive use and free access cause depleting, but members of the community do not want to be responsible for the costs of its maintenance.

In 2009, Elinor Ostrom was awarded the Nobel Prize in Economics for a possible alternative solution to the Tragedy of the commons. The researcher offered algorithms for collective use of the resources. Control and strict quotas, in her opinion, will not only rationalize the use of the resources, but also will renew it. Ostrom's work received a lot of criticism. Her critics say it is better to reward those who refuse to desecrate the common resources. Making people more interested and involved in the process is more reliable solution than just pressuring them.

4. The trolley problem

There is a runaway trolley barreling down the railway tracks. Ahead, on the tracks, there are five people tied up and unable to move. The trolley is headed straight for them. You are standing some distance off in the train yard. If you pull a lever, the trolley will switch to a different set of tracks. However, you notice that there is one person tied up on the side track. You have two options:

1) Do nothing, and the trolley kills the five people on the main track.

2) Pull the lever, diverting the trolley onto the side track where it will kill one person.

Which will you do?

Both options take away at least one life, but switching the lever or pushing the person onto the rails makes you responsible for the lives of others. If you refuse to act, it shifts the blame to someone else. But inaction is also an action. You consciously choose to stay away, and it makes you responsible for the death of people.

While choosing what will hurt less, many prefer to save the lives of five persons and indirectly kill only one. The stopping factor from this scheme is a thought that in this case you commit a deliberate murder, which is never fully justified in our society.

About 90% of the people agree to pull the lever of the trolley and kill one person to save five, but this experiment significantly differs from real life. In such an extreme situation, it is hard to be logical. Only instincts and emotions are involved, so it is quite difficult to foresee your own actions. Such examples are possible during military operations or emergency situations.

5. Hawks and Pigeons

Two groups of animals with different strategies compete for resources. Hawks behave aggressively and when meeting a competitor they fight to the end. Pigeons tend to suppress the opponent but avoid real fights. The names of groups are not relevant to these particular species, they just show stereotyped strategies of behavior.

When meeting a hawk, a pigeon will immediately step back. In the fight against another pigeon, there will be no fight really, just threatening, and the luckiest one will win. In both cases, pigeons lose nothing and stay alive. Two hawks always fight – one of them gets all the resources, the second one gets serious injuries.

The description of this model first appeared in 1973. The game allows you to learn about conflicts and interactions for territory, resources, sexual partners etc.

In 2007, the Kurchatov Institute created a virtual reality with primitive agents. The behavior of the agents was controlled by neural networks, which could also turn simple actions into more complex behavior strategies. During the evolution of this virtual reality, the agents divided into Pigeons, Hawks and Starlings groups. The last group always solved all problems together and gathered in a flock before the conflict. The experiment had proved that these strategies of competitive relations are the most stable and are the result of the evolution.

In the study conducted by the Kurchatov Institute, it was also proved that altruism and other sublime human qualities contributed to the development of a virtual society along with selfishness.

6. The Pascal's Wager

The Pascal's Wager is an argument in philosophy presented by French philosopher Blaise Pascal. It posits that humans bet with their lives that God either exists or does not. Pascal was deeply religious, and many believers see him as an example of combining the scientific mind and faith. In reality, he could not combine these two areas and when in 1654 he had a religious insight he decided to quit science.

If God does exist, all money you spent in the church allegedly will pay off when you die, and an atheist who does not spend a cent on it, will lose everything.

1. The high probability that there is no God multiplied by a small value of the prize is a large finite value.

2. The small probability of God's mercy multiplied by the infinite greater value of the prize is an infinitely greater value.

Blaise Pascal concludes that the second option is much more attractive, since infinite values are preferable to finite ones. However, choosing a humble life, a person rejects many pleasures, such as fame, wealth etc. The game theory looks like this:

The Pascal's Wager emphasizes a desire to explain irrational things from the position of reasons and benefits: "It is more reasonable to believe than not to believe in what the Christian religion teaches us." His critics point out that the faith is not a real faith when it is "just in case". Nevertheless, in reality, many believers do rituals quite formally. Pascal had said: "If your religion is false, you risk nothing, considering it to be true; if it is true, you risk everything, considering it false."

This scheme does not evaluate the truth or the falsity of both statements. This game strategy with uncertain conditions allows you to get much more than the required costs, or makes you lose everything in the future, while having all the desirable right now.

7. The paradox of a blonde

A group of friends notice some girls, among whom the blond one attracts the most attention. If all guys flirt with the blonde, other girls will be offended, and as a result no one will get a blonde. If guys do not compete with each other, then everyone will get a girl.

You always get the best result when you act in your own interests. "In practice, the results are optimal if each member of the group acts considering both, his own and group interests" – says the main character from the movie "Mind Games" which prototype was the Nobel laureate John Nash.

Nash Equilibrium is a situation involving two or more players in which each player is assumed to know the equilibrium strategies of the others, and no player has anything to gain by changing only their own strategy. If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium.

Initially, this formula was developed for business, but it can be applied to other areas, including personal life, as can be seen from the described example.

8. A Duel for three

Three men with guns stand at the tops of an equilateral triangle. Each of them is entitled to one shot. Who will shoot at whom, and who will survive?

Let's say, two players (A and B) can handle the gun very well, and the third one (C) is bad in this. The third player is almost insignificant, and the game interaction takes place between the two first participants.

This scheme is contrary to evolution: here, unlike Darwin's theory, the weakest survives. Players A and B are busy destroying each other and forget about player C.

Each player has an ideal pattern of behavior that brings the best outcome. However, if players A and B simultaneously use their best strategies, then they will find themselves in a terrible situation. It is better for both if they make an agreement to act as a team.

This game theory is also used in military, business and marketing. Huge corporations are fighting with each other, and as a result, some medium-sized attract more customers.

You can see it in sports as well, especially in short-distance running. Two athletes go ahead, and the third athlete outruns them at the finish line. They did not pay attention to him, trying to compete with a strong opponent.

Having two rivals in front of you, you should not focus only on one of them, even if he is stronger. Pay attention to both of them, because even a weak player can cause a lot of trouble when not considered seriously.

9.  Twenty dollars

Professor Max Bazerman teaches his students how to sell 20 dollars for 200. It is an interesting game theory about the irrationality of thinking, which is widely used in financial sector.

He shows the bill to the class and says that he would give 20 dollars a person, who will give for it the most money. A condition is that a person offering the second biggest sum is the one who has to pay the sum offered for the bill. If two highest bids were 15 dollars and 16 dollars, then a winner gets 20 dollars and a loser have to pay professor 15 dollars.

Initially, everyone makes small offers. During the auction, the highest betting players understand that they are trapped, since if they lose, they lose a substantial amount of money. They increase stakes not for the desire to win, but because of the fear of lose. A record Bazerman got during his teaching career was 204 dollars.

In this game theory, Bazerman reveals a mechanism of gambling, stock exchanges and auctions. Losses make people behave irrationally when they effort to reduce their losses.

The scientist explains that it is necessary to decide what will be the maximum loss that you agree to before starting the game. After reaching this limit, it is necessary to stop and leave the game, despite the excitement and desire to recoup.

10. Stable marriage problem

In 1962, American mathematicians David Gale and Lloyd Shapley published an article about the stable marriage problem. Given n men and n women, where each person has ranked all members of the opposite sex in order of preference. Depending on personal preferences, each participant ranks the opposite sex. The task is to make couples, so that none of the participants has a mutual attraction to someone else's partner, then the set of marriages is deemed stable. Is it possible?

In the original algorithm, each man makes an offer to the first woman from his list. Women respond "yes" to the most attractive candidates, "maybe" to candidates they are not so sure about, "no" to the rest. Then the suitors who are still single go to their vice-favorites, who also make their choices. Women can leave the first suitor if the second candidate proposed has a higher rank in their rating. The script is repeated until all men get a bride.

The scheme does not work if we are considering homosexual relations. This game exists only with the elements of two sets, it would be unstable with only one. However, it still works if women and men change their roles, and women can choose first.

In 2012, Lloyd Shapley received the Nobel Prize in Economics for this work, to be exactly for the theory of sustainable distribution and the practice of market modeling. Now this algorithm is used for recruitment of employees and athletes, distribution of donor organs and college assignments, but nobody actually tried to find a soul mate this way.