Transcription of the lecture video.
Game theory is a discipline that is firmly stuck between mathematics and the social sciences. One rope to mathematics, another rope to the social sciences, firmly attached.
It has theorems that are quite serious (the equilibrium existence theorem), a film “A Beautiful Mind” was made about it, game theory appears in many works of art. If you look around, every now and then you meet a game situation. I have collected several stories.
All my presentations are made by my wife. All presentations can be freely distributed, I will be extremely glad if you lecture on it. .
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.
- TV game or problem about parking spaces.
- Luxembourg in the European Union.
- Shinzo Abe and North Korea
- Bries paradox in Metrogorodok (Moscow)
- Two paradoxes of Donald Trump
- Rational Madness (North Korea again)
(At the end of the post is a poll about the bomb.)
Talmud: the task of inheritance
Once polygamy was allowed (3-4 thousand years ago). A Jew, when he got married, signed a marriage contract, how much would be paid to his wife when he died. Situation: A Jew who has three wives dies. 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 solving issues:
Shabbat begins with the first star. And beyond the Arctic Circle?
- "Go down" along the meridian and navigate the area where everything is fine. (does not work with North Pole)
- Start at 00-00 and do not bathe. (does not work with the North Pole either), so:
- A Jew has nothing to do beyond the Arctic Circle and there is no need to go there.
- The Talmud says that if the inheritance is less than 100 coins, then it should be divided equally.
- If up to 300 coins, then divide 50-100-150
- 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 out the claims of each wife, the claims of pairs of wives, provided that the third one “repaid” everything. We get a list of claims, not only individual, but also "companies". Such a decision is taken, such a division of the inheritance, so that the most severe claim is the minimum possible (maximin). In game theory, this was studied, called "". Robert Alman proved that all three scenarios from the Talmud are strictly according to the nucleolus!
How can it be? 3000 years ago? Neither I nor anyone else understands how it can be. (God dictated? Or was their math much more complicated than we think?)
Nikolai Vasilyevich Gogol
Ikharev. Let me ask you one question: what have you done up to now to put the decks into play? It is not always possible to bribe servants.
Comforting. Save God! yes, it is dangerous. This means sometimes selling yourself. We do it differently. Once we acted as follows: our agent comes to 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, spends his money, eats, drinks - and suddenly disappears, no one knows where, without paying. The owner is rummaging around in the room. He sees that there is only one pack left; unpacks - one hundred dozen cards. Cards, of course, this hour sold at a public auction. They let the ruble cheaper, the merchants snapped up the moment in their shops. And in four days the whole city lost!
This is purely a number-theoretic two-move move. I also recently had a two-move in my life, in Tyumen. I'm on the train. I study the situation and ask you to take me the top seat in the compartment. They tell me: “Don’t save, take the lower one, money is not an issue.” I say: "Upper".
Why did I ask for the top spot? (Hint: I completed the task by 3/4)
answerAs a result, I had two places - upper and lower.
The lower one is one and a half times more expensive. Expensive places are not taken. I saw that almost all the top ones were bought, and the bottom ones were almost all empty. So I randomly took the top one. Only on the Yekaterinburg-Tyumen section was there a neighbor.
It's time to play
Here is my phone number. There is not a single unread SMS in the phone itself, the sound is turned off. Within a minute you either send SMS or don't send it. Those who sent SMS will receive chocolate, but only if there are no more than two senders. Time has gone.
A minute has passed. 11 sms:
- Chocolate!
- Chocolate
- Izi
- Shshshsh
- 123
- Hello Alexey Vladimirovich
- Hello Alex
- Chocolate :)
- +
- combo breaker
- А
In Maikop, I had a lecture at the head of the Republic of Adygea and asked a meaningful question.
In Krasnoyarsk, 300 motivated schoolchildren sat in the hall. 138 sms. I began to read them, the fifth turned out to be obscene.
Let's take a look at this game. Of course it's a scammer. Never in the history of the draws (closer to 100 rounds) has anyone ever got a chocolate bar.
There is a balance when the hall agrees on some two people. The agreement should be one in which it is beneficial for everyone to participate.
Equilibrium is such a draw when you can announce strategies out loud, and this will not change them.
Let a chocolate bar be 100 times more expensive than an SMS (if it is 1000, then the result will be a little 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 move to chance. For example, roulette 1/6. A person decides that in 1/6 of the cases (with a multiple game) he will send SMS.
Question: what "roulette" will be the equilibrium?
We want to find a symmetrical equilibrium. We distribute roulette 1 / r to everyone. You have to make sure that people want to play this kind of roulette.
Essential detail. If you understand it, consider that you have already met with game theory. I argue that only one "p" is compatible with equilibrium.
Suppose "r" 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 "r" is too big, like 1/2. Then the right decision would be not to send SMS and save a ruble. You will definitely not be second, but most likely forty-second.
There is a balance calculation with simultaneous deep thinking. But now we are not talking about them.
The values of "p" should be such that your winnings from what you send SMS, on average, will be equal to the winnings from what you do not send.
Let's calculate this probability.
N+2 is the number of people in the audience.
On 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 wheel is such that its independent launch by all other participants leads to the probability of receiving a chocolate bar if you send an SMS (equal to 0,01).
With a price ratio of chocolate bar / sms = 100, the number of sms will be 7, at 1000 - 10.
You see that collective rationality suffers. We are looking for a balance where everyone behaves rationally, but as an outcome, there will almost certainly be more texts. Only collusion will give more results.
One of the results of game theory - the idea of the free market that it will fix everything itself - is completely wrong. If they let it go by itself, it will be worse than if they agreed.
Luxembourg in the European Union
Get ready 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).
The countries were different and therefore:
- France Germany Italy - 4 votes each,
- 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 exceeded. Quota = 12…
There is no potential situation where Luxembourg can change the course of a decision with its vote. A man sits at the table for 15 years and never decides anything.
When I found out about this, I asked my German acquaintances (there were no acquaintances from Luxembourg) to comment. They have replyed:
- Don't compare Luxembourg with your Soviet camp, where mathematics is well known. They have no concept of even/odd.
- How, the whole country?!??!?
“Yeah, except maybe a couple of teachers.
I asked another German who is married to a Luxembourger. 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 is happening in their backyard.
Abe
I was driving to a lecture on game theory and saw the news:
I got an alarm bell. That this can't be. No way. North Korea is capable of making an atomic bomb, but it is unlikely to deliver it.
Why introduce deliberate misinformation?
The truth is that missiles can reach Japan. It's scary for the Japanese. But if this is reported to NATO, it will not lead to anything, but by scaring “Europe” it will.
I do not insist on being right, maybe there are other analyzes of this news.
metrogorodok
Once upon a time, jokers called the street "Open Highway", because it is a dead end and rests on the forest. The same jokers called the area "Metrogorodok" because there will never be a metro there.
In the early 90s, there were no traffic jams yet, and the following story played out.
The metro-town is marked with the letter "M".
Shchelkovskoye Highway connects a giant cluster of cities. 700 people, according to the latest census.
A small winding path leads from Metrogorodok to VDNKh, without a single traffic light. On the highway to go an hour, on the track - 20 minutes. Part of the people from the highway begins to “cut off” - the result is a 30-minute traffic jam.
It's definitely from game theory. If the traffic jam is much less than 30 minutes, this is known, and then even more cars turn off to “cut off”. If it is much more, the people stop "cutting off".
The equilibrium value of the traffic jam time is purely the result of the number-theoretic interaction of motorists who decide where to go. Wardrop principle.
For drivers, it was the same hour as it was, and for residents of Metrogorodok, 20 minutes turned into 50. Without a "connector" 1 hour and 20 minutes, with a "connector" - 1 hour and 50 minutes. Pure paradox of Braess.
And here is an example that cost . 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 perhaps some kind of ban can lead to an improvement. And He portrayed the specifics when it happens.
There is point "A" and point "B" and in the middle is a point that cannot be avoided.
As a result, everyone travels 1 hour and 20 minutes. Nesterov proposed putting up a "change of road" sign.
As a result, the cars were divided into two categories: those who drove straight and then took a detour (4000) and those who went to the detour and then straight (4000) and there were no traffic jams on the narrow straight road. And as a result, all road users travel 1 hour.
Tramp
Fewer people voted for Trump than against him.
Electors.
There are 8 million people in the first state, all are “against” Trump. 2 electors.
The second state has 12 million people, 8 are for, 4 are against. 3 electors and all must vote for Trump.
As a result, 2:3 on electors in favor of Trump, although 8 million voted for him and 12 million against him.
Scandalous Candidate
It happens that some candidate does not pass the polls. Or about Brexit, according to the polls, it should not have happened. There are poor-quality surveys (when objectionable opinions are cut out of the sample), but professional sociologists rarely do this.
A person lives as if in a caftan, says one thing, but in front of the ballot box he throws off his caftan and votes in a different way. It is comfortable to live in a caftan, he has a certain social environment: employer, family, parents.
Here is the model of my friend, because I don't have facebook. All these people influence him in one way or another.
The opinions of 500 people matter. And if we discuss politics with him and disagree strongly, this represents some small uncomfortable component.
Model of social division.
examples:
- Broomweight
- Russian-Ukrainian split
- US elections
There are people who, on principle, do not participate in disputes, this is their position, not because they do not have their own 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 aij (many millions by many millions). In each cell it is written how each person influences each and with what sign. Strongly nonsymmetric matrix. One can influence so many, but one is influenced by 200 people.
We multiply the person's internal state vi by what he said aloud σi.
Equilibrium is when everyone has decided which σ to broadcast out loud.
They can even think about one thing at the same time, and speak out loud at the same time another. Both lie, but solidarize.
Added more noise. And it is calculated with what probability you will remain silent, say “for” or “against”. Equations arise for this set of probabilities.
With passionaries and fanatics, one must begin to calculate the balance.
TV is a magnetic field that shifts the inner mind.
The probability that you will drown "for" any 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. The result is a system of equations.
With the white noise modeling formula:
It turns out two equations for each person, 100 million people - 200 million equations. So many.
Perhaps there will come a time when you can take poll data, examine the quantitative indicators of a 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 don't know how many obstacles there will be on the way there.
Conclusions
People are ashamed of their support for a "scandalous" candidate (Zhirinovsky, Navalny, etc.), but at the ballot box "they give an outlet to the protest." By solving this system of equations, we would be able to quantify the deviations of poll results from real voting results. But we are hindered by the complexity of the device of social networks.
Model of rational insanity
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 considering the fate of Gaddafi, Saddam Hussein and others. Is Kim Jong-un out of his mind? However, there may well be a rational grain in his "crazy" behavior.
This is a model of Caesar burning bridges.
In the event of war, a country with nuclear weapons will be completely destroyed. If it does not have nuclear weapons, it can be won without complete destruction. If the leader of the country knows that “either pan or go”, then huge resources will be used for the war. And if so, then the opposite side will be afraid of these large resources, because they themselves will have a big loss from the war.
Game tree and forecast.
PS
Raise your hand, who thinks that the atomic bomb will be dropped in the next five years?
I think 50%. I would raise my hand.
Only registered users can participate in the survey. , you are welcome.
What is the probability that an atomic bomb will be dropped in the next five years?
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less than 5%
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5-20%
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20-40%
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50%
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60-80%
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more 95%
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other
256 users voted. 76 users abstained.
Source: habr.com