Nash equilibrium

We have to confess that, if until now we had not used the scene from the much-mentioned movie "A Beautiful Mind" to write a note for this section, it was simply because it almost seemed to us, a little bit, like cheating.

It happens that in the next scene, from the time when Russell Crowe kept winning Oscars... the relationship is so explicit that the note became too easy. But the lesson is so important that in the end it was not a viable option to avoid the topic... and we decided. We estimate that any reader over 30 will have seen this scene

We also assume that many know who John Nash is, precisely thanks to the movie. Well, strictly speaking, the film industry provides sugary versions of generally more complex people (to put it diplomatically). The video is almost self-explanatory.

Individual action, with all individuals seeking their greatest benefit, does not always lead to the best result in its collective form. Adam Smith was wrong!

 This theory, which has been widely perfected to date, is commonly known as “Game Theory.”

However, the most important line of the film is not Nash's explanation, but his partner's response. “Hey, isn't it a stratagem yours to stay with the blonde?” And the main point of Nash's theory: it does not point to the possibility of reaching a better equilibrium by coordinating actions, but to the difficulties that arise for this to occur.

Let's assume that Nash likes women more than his theory, unlike what is shown in the movie. And suppose he manages to convince all of his friends of his strategy. Each one of them asks one of the brunettes to dance. At the end of the round they would be left: Nash, the blonde who doesn't understand why no one invited her to dance, and the last of her friends.

What is the most beneficial option for John? From what is seen in the scene, ask the blonde to dance and win the biggest prize. Why would he accept the game when everyone else has already decided on the choice? Can John convince his colleagues to do the same one night at the bar?

Let's imagine that everyone agrees to go with the “least pretty” girls. The moment the blonde girl gives a clear signal that she is bored with her, she would end up paying attention to anyone who talks to her.

What should happen if everyone seeks their individual benefit? As they barely perceive that, they have incentives to stop talking to their corresponding girl, and be able to start a conversation with the prettiest one. Obviously, if that happened at the same time, it would be back to the initial situation, where everyone loses their chance.

El sweet spot where everyone ends up with a lady is very difficult to sustain, given that temptation exists for everyone at all times.

If there is not strict coordination, everyone will probably end up courting the blonde. The result that sought the gain of all participants, through a point of stability in which none of the parties obtains a benefit if they move their positions (given the actions of the others).

If they remain in the same situation, both obtain moderate profits, which is commonly known as “Nash equilibrium".

Now, given the ease with which the economy-film relationship is explained in this case, we feel that we owe them something more. And the way to compensate for it is to make the relationship inverse to the usual one.

Just as we normally show how a scene seemingly unrelated to our social science is closely linked to some theory, today we are going to show how game theory can be applied to practically any area of ​​life.

I hope that the following examples we have selected explain the enormous importance of Nash's contribution.

Simple Nash equilibrium: global warming and 24° air

I once remember a comment like this about global warming: “Look at how irrational human beings are that we are heading towards disaster, and yet we continue producing, polluting and consuming energy.”

Let's imagine the case of an individual who has to choose between saving energy by setting the air at 24 degrees or enjoying, as he really likes, sleeping under a blanket and with the device set to 16°.

If what he really likes is 16° air, why should they worry about the environment? The answer is obvious: irresponsible behavior with respect to ecology is leading us to global warming, climate change... and subsequent chaos, and the end of the world for the most alarmist.

Going to an example a little closer to Argentina: excess consumption could generate an overload in the network and therefore a power outage throughout the neighborhood, greatly harming some of its neighbors, who may lose merchandise or be lost. the final of the football championship, which is ending.

Well, then everything seems to indicate that the most logical action is to worry about the environment (or your neighbors) and not doing so is, as my computer says, irrational. But this is where game theory comes in.

Let us keep in mind that this individual does not live alone in the world and that, therefore, global warming does not depend on what she does but also on the choices of other people. What are the options then and the result of each one? There are four possible scenarios:

• That the person saves energy and the rest of the people do not. In this case, the person makes the effort to refrain from setting the air at 16° but... in the end, if everyone wastes energy, the light on the block goes off or the temperature of the earth rises.
• That the person and the rest of the people save energy. A sacrifice today, but the world safe.
• That the person and the rest of the world waste energy. The world is going to destruction, but at least today we can have the temperature we want.
• That the person does not save energy, and the rest of the world does. In this case the world is saved because this person's energy expenditure is minimal. And we get all the benefits.

What is interesting here? If the rest of the world saves energy, the other person's best option is to waste it, because nothing bad will happen anyway. If the rest of the world wastes energy, man must do the same, since all is lost anyway. So... if everyone thinks like this... is the final result the best for everyone?

Nash equilibrium in the long and short run

Are there cases in which actions are coordinated to achieve the greatest benefit of the group? Of course. But the results are not always very beneficial. In some cases, avoiding Nash equilibrium is even prohibited. This is the case of posters.

 The cartel is a collusive agreement between a number of companies, which is designed to restrict production and increase the price of the good in question.

The best known worldwide is the cartel of oil producing countries, OPEC. In 1973, this organization, which involves most of the world's largest oil producers, decided to abruptly cut crude oil production, greatly affecting world supply. Prices rose from just over $3 per barrel to over $40 in a matter of months, unleashing a global economic crisis.

This measure to keep prices high requires great coordination. Any country that decides to increase production by doing things overboard, with an enormous immediate economic benefit, given that it could sell a large quantity at an artificially high price. What maintains these agreements? The expectation that everyone will be better off in the long run if they stick to the rules. That is to say... after all, what is not a Nash equilibrium in the short term ends up being one in the long term.

So let's go back to the movie example. How can the boys make sure Nash doesn't go with the blonde? Very easy. If there is some type of social punishment for whoever breaks the rule (for example, we no longer go out to drink beer together) it is possible that the incentive will be reversed and no one will end up benefiting in the long run from breaking the agreement.

As you will see, society's own morality is an agreement to avoid individual benefit. As Thomas Hobbes pointed out in “The Leviathan”: why aren't human beings killing each other to obtain individual benefit? Because in the long run it doesn't suit us. If everyone does that... anyone could end up murdered one night. But avoiding that requires a significant coordination effort, which leads to granting the state a monopoly on force.

This is one of the most important lessons also when doing business: if a supplier believes that it can generate a stable, long-term relationship with a company, it is much more likely that it will make an effort to fulfill the order. In one-time relationships... you have to be more careful.

What if it is not fulfilled? The centipede game.

The best-known example of non-compliance with Nash equilibrium is what is known as the “centipede game.” Here the rules of the game, from the definition of Wikipedia:

• Two players
• Two piles of coins, first pile has 2 coins, second pile has 0
• In turn each player must choose between:

1.- Keep the largest amount and give the smallest, on the contrary.

2.- Pass both piles, on the contrary.

• Each time a player chooses option 2, the piles grow by 1 coin
• If the game reaches 100 turns and neither player chooses option 1, the game ends and no one wins anything.

What should happen according to Nash's theory? The game is solved by what is known as reverse induction.

In the last turn, all players, logically, choose option 1. Now, knowing that this is going to be a fact, in the previous turn the other player will have incentives to also choose option 1. And knowing that, he himself The player whose turn it was to finish the game will finish it on the previous turn. Then, the Nash equilibrium is such that the first player chooses to end the game on the first turn and takes the two coins.

What happens in practice? Players usually advance until they get close to the end. That is, the Nash equilibrium is not met in practice.

Researchers have tried to give several explanations, but our favorite is that the balance is not met because the players quickly demonstrate that they are not completely rational in the terms proposed by the theory, or at least, that they are not going to play rationally.

This may occur because they do not fully understand the game, or because they want to make the other believe that they do not understand it, or simply because there are other cultural variables, such as the predominance of certain values, that have not been rationalized by the theory. In this regard, Joshua Greene's research captured in the book Moral Tribes (not yet translated into Spanish) shows that social mechanisms partially break Nash equilibria.

But one way or another, the first player to pass is giving an important message to the other: I'm not going to play according to what would be rational. Whatever the reason (I don't understand it or I don't care) the other player no longer has an incentive to play Nash equilibrium, because there is no “rational” player on the other side.

Nash equilibrium is even less likely in other situations

A group of people write a number between 1 and 100 on a piece of paper and put it in an urn. The winner will be the one who comes as close as possible to half of the average of the votes. Logical reasoning (you can think of it yourself) leads to everyone tying at 1.

But if only one of the players does not understand the game completely well, or decides not to comply with its rules (like the joke about the woodpecker that is drilling into Noah's ark, before the storm) and chooses, for For example, 50 or 25, the average rises significantly. And, what's more, if some players believe that someone can “get it wrong” in this way, they will give a larger answer and raise the average even more. There are people who simply cannot handle his genius.

Strangely, this can even lead to some benefit for those who do not play rationally. This is the case, for example, of the bee, whose strength lies in not knowing that it dies when it stings. It is precisely this ignorance that makes it feared by other animals... otherwise, the Nash equilibrium would be for Winnie the Pooh to attack the honeycomb, and for the bees... to escape without stinging.

By way of conclusion You see, game theory is important for studying countless situations that go beyond economics, both when the result is the equilibrium proposed by Nash, and when for some reason we move away from it.