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

Transcript of the video lecture.

Game theory is a discipline that is firmly stuck between mathematics and the social sciences. One rope to math, another rope to the social sciences, firmly attached.

It contains theorems that are quite serious (the equilibrium existence theorem), the film “Mind Games” was shot about it, and game theory is manifested in many literary works. If you look around, then you come across a game situation. I have collected some plots.

All my presentations are made by my wife. All presentations can be freely distributed, I will be extremely glad if you will lecture on it. This is completely free material .

Some stories are controversial. Models may be different, you may not agree with my model.

• Game Theory in the Talmud.
• Game Theory in Russian Classics.
• A television game or a task about parking spaces.
• Luxembourg in the European Union.
• Shinzo Abe and North Korea
• The Braies Paradox in Metrogorodok (Moscow)
• Two paradoxes of Donald trump
• Rational Madness (North Korea again)

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

Talmud: inheritance problem

Polygamy was once permitted (3-4 thousand years ago). A Jew, when married, signed a prenuptial agreement on how much he would pay his wife when he died. Situation: a Jew dies with three wives. The first 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 Jewish approach to resolving issues:

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

1. “Go down” along the meridian and navigate through the area where everything is fine. (does not work with the North Pole)
2. Start at 00-00 and do not steam. (also does not work with the North Pole), therefore:
3. A Jew has nothing to do beyond the Arctic Circle and there is no need to go there.

1. It is written in the Talmud, 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 into one formula?

The principle of how to solve cooperative games.

We write down the claims of each wife, the claims of the pairs of wives, provided that the third has “paid off” everything. We get a list of claims, not only single, but also "companies". Such a decision is made, such a division of inheritance, so that the heaviest claim is the least possible (maximin). In game theory, this was studied, called the " nucleolus ". Robert Alman proved that all three scenarios from the Talmud are strictly in nucleolus!

How can it be? 3000 years ago? Neither I nor anyone else understands how this can be. (Did the Lord dictate? Or was their mathematics much more complicated than we think?)

Nikolai Vasilyevich Gogol

Ikharev. Let me ask you one question: what have you done so far to use decks? After all, bribing servants is not always possible.

Comforting. God save! Yes, and dangerous. This sometimes means 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 benches have not yet had time to hire; chests and packs while in the room. He lives in a tavern, is wasted, eats, drinks - and suddenly disappears in the middle of nowhere, without paying. The owner fumbles in the room. He sees there is only one pack left; unpacks - one hundred dozen cards. Cards, naturally, this hour are sold at public auction. They let them go in cheaper, merchants snapped them into their shops for a moment. And in four days the whole city was lost!

This is a purely theoretical number two-step. I also recently had a two-run in my life, in Tyumen. I'm going by train. I study the situation and ask me to take 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 ask for the top spot? (Hint: I completed the task by 3/4)

It's time to play

Here is my telephone number. There are no unread sms in the phone itself, the sound is turned off. Within a minute, you either send SMS or do not send. Chocolate will be received by those who sent SMS, but only if there are no more than two who have sent. The time has gone.

A minute has passed. 11 sms:

• Chocolate!
• Chocolate
• Easy
• Shhh
• 123
• Hello Alexey Vladimirovich
• Hello Alex
• Chocolate :)
• +
• Combo breacker
• A

In Maykop, I had a head of the Republic of Adygea at a lecture and asked a substantive question.

In Krasnoyarsk, 300 motivated students sat in the hall. 138 sms. I began to read them, the fifth turned out to be obscene.

Let's take this game apart. Of course this is a scam. Not once in the history of the draws (closer to 100 rounds) was it not for someone to get a chocolate bar.

There is a balance when the hall agrees on some two people. The contract should be one in which everyone benefits from participating.

Equilibrium is such a joke when it is possible to announce strategies out loud, and from this they will not change.

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

Mixed equilibrium. Each of you doubts and does not know how to play. And he gives his turn to chance. For example, roulette 1/6. A person decides that in 1/6 of cases (with multiple games) he will send SMS.

Question: which roulette will be in equilibrium?

We want to find a symmetrical balance. We give everyone a 1 / p. We need to make sure that people want to play such a roulette.

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

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

If p is too large, for example 1/2. Then the right decision would be to not send SMS and save the ruble. Surely you will not be second, but most likely forty-second.

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

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

We calculate this probability.

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


In the video, the analysis of formulas in the 33rd minute.

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

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

With a price ratio of chocolate / sms = 100, the number of sms will be 7, with 1000 - 10.

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

One of the results of game theory - the idea of ​​a free market that he himself will fix everything - is completely wrong. If allowed to drift - it will be worse than if agreed.

Luxembourg in the European Union

Prepared to laugh.

Luxembourg was a member 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).

The countries were different and therefore:

• France Germany Italy - 4 votes each
• Belgium, 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 (greater than or equal to). Quota = 12 ...

There is not a single potential situation where Luxembourg can change the course of the decision with its own voice. A man of 15 years sits at a table and never decides anything.

When I found out, I asked my acquaintances Germans (there were no acquaintances from Luxembourg) to comment. They replied:
“Do not compare Luxembourg with your Soviet camp, where mathematics is well known.” They have no idea of ​​parity / oddness.
- How, the whole country?! ??!?
“Well, yes, except maybe a couple of teachers.”

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

Shinzo Abe

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


An alarming bell began to play. That this cannot be. No way. North Korea is capable of making an atomic bomb, but hardly delivering it.

Why introduce into deliberate misinformation?

The truth is that missiles can reach Japan. This is scary for the Japanese. But if you tell NATO, it will not lead to anything, but scaring it with Europe will lead to it.

I do not insist on rightness, maybe there are other analyzes of this news.

Metro town

Once upon a time, pranksters called the Open Highway, because it is a dead end and rests on a forest. The same pranksters called the metro town area, because there will never be a metro there. "

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


The metro town is marked with the letter "M".

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

A small winding path leads from Metro to VDNH without a single traffic light. Go on the highway for an hour, on the track - 20 minutes. Part of the people from the highway begins to "cut" - the result is a 30-minute traffic jam.

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

The equilibrium value of the traffic jam time is purely the result of the number-theoretical interaction of motorists who decide where to go. Wardrop principle.

For drivers, as it was an hour, it remained, but for residents of Metrogorodok 20 minutes turned into 50. Without an “interconnector,” 1 hour and 20 minutes, with an “interconnector,” 1 hour and 50 minutes. The pure Braes paradox.

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

The idea is this. If the emergence of a new road can lead to a deterioration of the road situation, then, probably, some kind of ban can lead to improvement. And He depicted the specifics when this happens.

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


As a result, everyone goes 1 hour and 20 minutes. Nesterov proposed to put a sign "change of road."
As a result, the cars were divided into two categories, those who drove straight, and then bypass (4000) and those who rode bypass, and then directly (4000), while there were no traffic jams on a narrow straight road. And in the end, all road users travel 1 hour.

Trump

Trump voted less than against him.

Electors.


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

Scandalous candidate

It happens that a polling candidate fails. Or about "Brexit" according to polls should not have happened. There are poor-quality polls (when they cut objectionable opinions from a sample), but professional sociologists rarely do this.

A person lives as if in a caftan, says one thing, and before the urn he drops the caftan and votes differently. It is convenient to live in a caftan; it has a certain social environment: the 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 diverge greatly, this represents some small uncomfortable component.

Model of social split.

Examples:

• Brexit
• Russian-Ukrainian schism
• US election

There are people who in principle do not participate in disputes, this is their position, not because they do not have their own opinions, but because the costs of expressing their point of view are very high.

You can write a payoff function:


There is a matrix of interactions a _ij (many millions per many millions). In each cell it is written how each person influences everyone and with what sign. Strongly asymmetric matrix. One can affect very many, but one is affected by 200 people.

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

Equilibrium is when everyone has decided which σ to broadcast aloud.

They may even think about one thing at the same time, and speak out loud at the same time another thing. Both lie, but are in solidarity.

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


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


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


The probability that you will drown “for” a 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 formula for modeling white noise:


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

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

Theoretically, it could be. I don’t know how many obstacles there will be.

conclusions

People are embarrassed by their support for the “scandalous” candidate (Zhirinovsky, Navalny, etc.), but at the ballot box “they give way to protest.” By solving this system of equations, we could quantify the deviations of the survey results from the results of real voting. But we are disturbed by the complexity of the device of social networks.

Rational Madness Model

Many people are amazed at the "fearlessness" of the North Korean leadership, which is testing its nuclear weapons "under the nose" of the United States. Especially, given the fate of Gaddafi, Saddam Hussein and others. Is Kim Jong-un gone crazy? However, in his "crazy" behavior there may well be a rational kernel.

This is a model of Caesar burning bridges.


In case of war, a country with nuclear weapons will be completely destroyed. If she does not have nuclear weapons, she can be defeated without complete destruction. If the head of the country knows that “either pan or disappeared”, then enormous resources will be put into the war. And if so, then the opposite side will be scared of these large resources, because she herself will have a big loss from the war.


Game tree and forecast.

PS

Raise your hand, who believes that an atomic bomb will be dropped in the next five years?
I think 50%. I’d raise my hands.