Alexey Savvateev and game theory: “What is the probability that an atomic bomb will be dropped in the next five years?”

Interpretation of the lecture video.

Game theory is a discipline that is firmly established between mathematics and the social sciences. One rope to mathematics, the other rope to the social sciences, is firmly attached.
')
There are theorems in it, quite serious ones (the equilibrium existence theorem), the film “Mind Games” was made about it, the game theory is manifested in many works of art. If you look around, every now and then you meet the game situation. I collected several plots.

All my presentations are done by a wife. All presentations can be freely distributed, I will be very happy if you lecture on it. This is completely free stuff .

Some subjects are controversial. Models can be different, you can not agree with my model.

• Game Theory in the Talmud.
• Theory of games in the Russian classics.
• Telegame or the task of parking spaces.
• Luxembourg in the European Union.
• Shinzo Abe and North Korea
• Brayes Paradox in Metrogorodka (Moscow)
• Two Paradoxes of Donald Trump
• Rational madness (North Korea again)

(At the end of the post - a survey about the bomb.)

Talmud: the problem of inheritance

Once polygamy was allowed (3-4 thousand years ago). The Jew, when married, signed a marriage contract, how much will pay his wife when he dies. Situation: a Jew dies who has three wives. The first was bequeathed 100 coins, the second - 200, the third - 300. But when the inheritance was opened, there were less than 600 coins. What to do?

Oftop about the approach of the Jews to the solution of questions:

Shabbat begins with the first star. And beyond the Arctic Circle?

1. "Go down" along the meridian and navigate the area where everything is fine. (does not work with the North Pole)
2. Start at 00-00 and do not bathe. (also does not work with the North Pole), therefore:
3. A Jew has nothing to do outside the Arctic Circle and do not need to go there.

1. In the Talmud it is written that if the inheritance is less than 100 coins, then divide equally.
2. If up to 300 coins, then divide 50-100-150
3. If 200 coins, divide 50-75-75

How can these three conditions be glued together in a single formula?

The principle of how to solve cooperative games.

We write out the claims of each wife, the claims of couples of wives, provided that the third one “extinguished”. We receive the list of claims, not only single, but also "companies". Such a decision is taken, such a division of inheritance, so that the heaviest claim is the minimum possible (maximin). In game theory, it was studied, called the " nucleolus ". Robert Alman proved that all three scenarios from the Talmud are strictly nucleolus!

How can it be? 3,000 years ago? Neither I, no one else understands how it can be. (Did the Lord dictate? Or was their mathematics much more difficult than we think?)

Nikolai Vasilyevich Gogol

Ikharev Let me ask you one question: what did you do before to use the deck? It’s not always possible to bribe servants.

Comforting. God save Yes, and dangerous. That means sometimes selling yourself. We do it differently. Once we did this: our agent arrives at the fair, stops under the name of a merchant in a city tavern. The shops had not yet been hired; chests and packs while in the room. He lives in a tavern, he keeps himself, eats, drinks - and suddenly disappears in the middle of nowhere, without paying. Boss fumbles in the room. He sees only one pack left; unpacks a hundred dozen cards. Cards, naturally, this very hour are sold at public auction. Let the ruble cheaper, the merchants instantly snapped up in their shops. And in four days the whole city was lost!

This is a purely numerical two-move theorem. I, too, recently had a two-way walk in my life, in Tyumen. Going by train. I study the situation and ask me to take me the top seat in the compartment. They say to me: "Do not save, take the bottom, money is not a question." I say: "Top."

Why did I request the top spot? (Hint: I completed the task at 3/4)

It's time to play

Here is my phone. In the phone itself there is not a single unread text message, the sound is turned off. Within a minute, you either send SMS, or do not send. Chocolates will be received by those who sent SMS, but only if there are no more than two senders. Time has gone.

A minute has passed. 11 cm:

• Chocolate!
• Chocolate
• Izi
• Shhh
• 123
• Hello Alexey Vladimirovich
• Hello Alexey
• Chocolate :)
• +
• Combo-breacker
• BUT

In Maykop, I had a head of the republic of Adygea at my lecture and asked a meaningful question.

In Krasnoyarsk in the hall were 300 motivated students. 138 cm. I began to read them, the fifth was obscene.

Let's take this game. Of course it's a scam. Never in the history of raffles (closer to 100 rounds) was it possible for someone to get the chocolate.

There is a balance when the hall agrees on some two people. The contract must be one in which it is profitable for all to participate.

Equilibrium is such a joke, when you can declare strategies out loud, and this will not change them.

Let the chocolate is 100 times more expensive than sms (if at 1000, the result will be slightly different). The number of people in the hall plays almost no role.

Mixed equilibria. Each of you doubts and does not know how to play. And he gives his turn to the occasion. For example, a 1/6 roulette. The person decides that in 1/6 cases (with repeated play) he will send SMS.

Question: what "roulette" will be equilibrium?

We want to find symmetrical equilibrium. We distribute roulette 1 / p to everyone. We must make sure that people want to play such a roulette.

Essential detail. If you understand it, consider that you have already become acquainted with the theory of games. I contend that only one “p” is compatible with equilibrium.

Suppose that the "p" is very small. For example, 1/1000. Then, having received such a roulette, you will quickly guess that you can’t see chocolates and throw out such a roulette and send SMS.

If "p" is too large, for example, 1/2. Then the correct solution would be not to send SMS and save the ruble. Similarly, you will not be the second, but most likely forty second.

There is a calculation of equilibrium with simultaneous deep thinking. But now it is not about them.

The values ​​of "p" should be such that your gain from what you send SMS, on average, will be equal to the gain from what you do not send.

Calculate this probability.

N + 2 - the number of people in the audience.


In the video analysis of the formulas at the 33rd minute.

(1 + pn) (1 + p) ^ n = 1/100 (probability of chocolates = price of SMS)

If the roulette is such that its independent launch by all other participants leads to the probability of getting a chocolate bar in case of your sending SMS (equal to 0.01).

If the price is the ratio of prices chocolate / sms = 100, the number of sms will be 7, at 1000 - 10.

You see that collective rationality suffers. We are looking for a balance when everyone behaves rationally, but as an outcome, there will almost certainly be more sms. Only collusion will give more results.

One of the results of game theory is the idea of ​​a free market, that he himself will fix everything - is completely wrong. If you let it flow, it will be worse than if we agreed.

Luxembourg in the European Union

Prepared to laugh.

Luxembourg was part of the European Union.

The Council of Ministers of the European Union consisted of 6 representatives, one from each EU country (from 1958 to 1973).

Countries were different and therefore:

• France Germany Italy - by 4 votes,
• Belgium, the Netherlands - 2 votes,
• Luxembourg - 1 vote.

Six people made decisions on all issues for 15 years in a row. The decision is made if the quota is reached or exceeded (more or equal). Quota = 12 ...

There is no potential situation where Luxembourg can change the course of a decision with its voice. A man sits 15 years at the table and never decides anything.

When I found out, I asked my German acquaintances (there were no acquaintances from Luxembourg) to comment. They have replyed:
- You do not compare Luxembourg with your Soviet camp, where mathematics is well known. They have no idea about evenness / oddness.
- How, the whole country?! ??!?
- Well, yes, except maybe a couple of teachers.

He asked another German who was married to a luxemburg. He said:
- Luxembourg is a country that is completely apolitical, and does not follow foreign policy at all. In Luxembourg, people are only interested in what happens in their yard.

Shinzo Abe

I was driving to a lecture on game theory and saw the news:


I played the alarm bell. That this can not be. No North Korea can make an atomic bomb, but it is unlikely to deliver it.

Why enter into deliberate disinformation?

The truth is that rockets can fly to Japan. This is scary for the Japanese. But if this is communicated to NATO, it will lead nowhere, and frightening with “Europe” will lead.

I do not insist on the correctness, there may be other analyzes of this news.

Metro town

Once upon a time, pranksters called the street “Open Highway” because it is a dead end and rests against a forest. The same jokers called the district "Metrogorodok" because there will never be a subway there. "

In the early 90s there were no traffic jams and the next story was played out.


Metropolitan marked with the letter "M".

Schelkovskoe highway connects a giant cluster of cities. 700,000 people, according to the latest census.

From Metrogorodka to VDNH a small winding path leads, without a single traffic light. On the highway go an hour, on the track - 20 minutes. Part of the people from the highway begins to "cut off" - the result - a 30-minute traffic jam.

This is exactly from game theory. If the plug is much less than 30 minutes - this is known, and then even more cars are turned off to “cut off”. If much more, the people cease to "cut off."

The equilibrium value of traffic jams is purely the result of a number-theoretic interaction between motorists who decide where to go. Wardrop principle.

For the drivers, both the hour and the time remained, and for the residents of the Metro town, 20 minutes turned into 50. Without a connector, 1 hour and 20 minutes, with a connector, 1 hour and 50 minutes. Pure paradox Braes.

And here is an example that cost the Danzig Prize . Yuri Evgenievich Nesterov received the highest award in the field of mathematical programming.

The idea is this. If the appearance of a new road can lead to a worsening of the traffic situation, then, probably, a ban can lead to an improvement. And He portrayed specifics when this happens.

There is point “A” and point “B” and in the middle is a point that cannot be avoided.


In the end, all go 1 hour and 20 minutes. Nesterov proposed to put a sign "change the road."
As a result, the cars were divided into two categories, who drove straight, and then detour (4000) and those who traveled around, and then straight (4000) and at the same time traffic jams did not arise on a narrow straight road. And in the end, all road users go 1 hour.

Trump

Trump voted less than against him.

Electors.


In the first state of 8 million people, all "against" Trump. 2 electors.
In the second state 12 million people, 8 - “for”, 4 - “against”. 3 electors and all are required to vote for Trump.
As a result, 2: 3 for the electors in favor of Trump, although 8 voted for him and 12 million against him.

Scandalous candidate

It happens that some candidate does not pass through polls. Or about the "Brexit" on polls should not have happened. There are poor-quality polls (when they cut out disagreeable opinions from the sample), but professional sociologists rarely do so.

A person lives as if in a caftan, says one thing, and in front of an urn, throws a caftan and votes differently. It is convenient to live in a caftan, he has a certain social environment: employer, family, parents.

Here is a model of my friend, because I do not have facebook. All these people, one way or another, affect him.


The opinions of 500 people are important. And if we discuss politics with him and disagree strongly, this represents some small uncomfortable component.

The social split model.

Examples:

• Brexit
• Russian-Ukrainian split
• US Election

There are people who in principle do not participate in disputes, this is their position, not because they have no opinion, but because the costs of expressing their point of view are very high.

You can write a win function:


There is a matrix of interactions a _ij (many millions to many millions). Each cell says how each person influences each person and with what sign. Strongly asymmetric matrix. One can affect so many, but one affects 200 people.

We multiply the internal state v _{i of a} person by what he said out loud σ _i .

Equilibrium is when everyone has made a decision on which σ to broadcast out loud.

They may even think at the same time about one thing, and speak out loud at the same time another. Both lie, but they identify with each other.

More noise is added. And it is calculated with what probability you keep silent, say “for” or “against”. On this set of probabilities, equations arise.


With the drive and fanatics, one must begin to calculate the balance.


A television is a magnetic field that shifts the inner mind.


The probability that you will drown “for” some particular side is equal to the probability that the difference in white noise will be greater than the gain. Everything is determined by the value inside the brackets, and this is obtained depending on the rest. As a result, the system of equations.

With the white noise simulation formula:


It turns out two equations for each person, 100 million people - 200 million equations. Lots of.

Perhaps the moment will come when it will be possible to take data from the polls, explore the quantitative indicators of the social network of acquaintances and say: “In this system, a poll will reduce the number of votes for this candidate by 7%.”

Theoretically it could be. I do not know how many obstacles there will be on the way there.

findings

People are embarrassed about their support for the “scandalous” candidate (Zhirinovsky, Navalny, etc.), but at the ballot box “they give a protest.” Having solved this system of equations, we would be able to quantify the deviations of the results of the polls from the results of the actual voting. But we are hampered by the complexity of the device social networks.

Model of Rational Madness

Many people are amazed at the “fearlessness” of the leadership of North Korea, which is conducting tests of its nuclear weapons “under the nose” of the United States. Especially, given the fate of Gaddafi, Saddam Hussein, and others. Is Kim Jong-un really out of his mind? However, in his "crazy" behavior may well be rational.

This is a model of Caesar burning bridges.


In the event of war, the country with nuclear weapons will be completely destroyed. If she does not have a nuclear weapon, you can win without complete destruction. If the leader of the country knows that “either pan or disappeared”, then huge resources will be put to war. And if so, then the opposite side will be frightened by these great resources, because she herself will have a big loss from the war.


Tree game and prediction.

PS

Raise your hand, who thinks that in the next five years the atomic bomb will be thrown off?
I count 50%. Paul hands I would raise.