Hey Habr!
My name is Asya. I found a very cool lecture, I can not help but share.
I bring to your attention a summary of a video lecture on social conflicts in the language of theoretical mathematicians. The full lecture is available here: (A. V. Leonidov, A. V. Savvateev, A. G. Semyonov). 2016.
Aleksey Vladimirovich Savvateev — PhD in Economics, Doctor of Physical and Mathematical Sciences, Professor at Moscow Institute of Physics and Technology, Leading Researcher at NES.
In this lecture, I will talk about how mathematicians and game theorists look at a recurring social phenomenon, examples of which are the vote to leave the UK (, the phenomenon of a deep social split in Russia after , with sensational results.
How can such situations be simulated so that they have echoes of reality? To understand the phenomenon, it is necessary to study it comprehensively, but in this lecture there will be a model.
social division means
In these three scenarios, the common thing is that a person one way or another adjoins some kind of camp, or else refuses to participate and discuss his choice. Those. The choice of each person is ternary - from three values:
- 0 - refuse to participate in the conflict;
- 1 - participate in the conflict on one side;
- -1 - participate in the conflict on the opposite side.
There are direct consequences that are related to your own attitude towards the conflict in fact. There is an assumption that each person has some kind of a priori feeling of who is right here. And this is a real variable.
For example, when a person really does not understand who is right, the point is located on the number line somewhere around zero, for example, at 0,1. When a person is 100% sure that someone is right, then his internal parameter will be already -3 or +15, depending on the strength of beliefs. That is, there is a certain material parameter that a person has in his head, and it expresses his attitude to the conflict.
It is important that if you choose 0, then this does not entail any consequences for you, there is no winning in the game, you have abandoned the conflict.
If you choose something that is not consonant with your position, then a minus will appear before vi, for example vi = - 3. If your internal position coincides with the side of the conflict on which you speak, and your position is σi = -1, then vi = +3.
Then the question arises, for what reasons sometimes you have to choose the wrong side that is in your soul? This may be under pressure from your social environment. And this is a postulate.
The postulate is that you are affected by consequences beyond your control. The expression aji is a real parameter of the degree and sign of the influence on you from j. You are number i, and the person who influences you is person number j. Then there will be a whole matrix of such aji.
This person j may even influence you negatively. For example, this is how you can describe the speech of a political figure unpleasant to you on the opposite side of the conflict to you. When you look at a performance and think, "That idiot, look what he says, I told you he was an idiot."
However, if we consider the influence of a person close or respected by you, then it turns out to be immediately one player j on all players i. And this influence is multiplied by the coincidence or non-coincidence of the adopted positions.
Those. if σi, σj is of positive sign, and at the same time aji is also of positive sign, then this is a plus to your payoff function. If you or a person who is very important to you have taken a zero position, then this term does not exist.
Thus, we tried to take into account all the effects of social influence.
Next next moment. There are many such models of social interaction described from different angles (models for making a threshold decision, many foreign models). They consider a concept that is standard in game theory, called the Nash equilibrium. There is a deep dissatisfaction with this concept for games with a large number of participants, as in the UK and US examples mentioned above, i.e. many millions of people.
In this situation, the correct solution of the problem passes through the continuum approximation. The number of players is some kind of continuum, a “cloud” playing, with a certain space of important parameters. There is a theory of continual games,
"Meaning for non-atomic games". This is an approach to cooperative game theory.
There is no non-cooperative game theory with a continual number of participants as a theory yet. There are separate classes that are being studied, but this knowledge has not yet been formed into a general theory. And one of the main reasons for its absence is that in a particular case, the Nash equilibrium is wrong. Essentially the wrong concept.
What then is the correct concept? In the last few years, there is some agreement that the concept developed in the works Palfrey and McKelvey which sounds like "", or "“as Zakharov and I translated it. The translation belongs to us, and since no one has translated it into Russian before us, we have imposed this translation on the Russian-speaking world.
We meant by this name that each individual does not play a mixed strategy, he plays a pure one. But in this “cloud” there are zones in which this or that clean one is chosen, and in response, I see how a person plays, but I don’t know where he is in this cloud, i.e. there is hidden information there, I perceive person in the "cloud" as the probability with which he will go one way or another. This is a statistical concept. The mutually enriching symbiosis of physicists and player theorists, I think, will define game theory in the 21st century.
We generalize the existing experience in modeling such situations with completely arbitrary initial data and write out a system of equations that will settle to the discrete response equilibrium. That's all, further, in order to solve the equations, it is necessary to make a reasonable approximation of situations. But this is still ahead, this is a huge direction in science.
The discrete response equilibrium is the equilibrium in which we actually play not sure with whom. In this case, ε is added to the gain from the pure strategy. There are three payoffs, some three numbers that mean sink one side, sink the other side, and abstain, and there's ε added to those three. In this case, the combination of these ε is unknown. The combination can only be estimated a priori by knowing the distribution probability for ε. In this case, the probabilities of the combination ε should be dictated by the person's own choices, i.e., his estimates of other people and estimates of their probabilities. This mutual consistency is the equilibrium of the discrete response. We will return to this point.
Formalization via discrete response equilibrium
Here is what the payoff looks like in this model:
It brackets all the influence that is on you if you have chosen any side, or will be multiplied by zero if you have not chosen any side. Further, it will be with a “+” sign if σ1 = 1, and with a “-” sign if σ1 = -1. And to this is added ε. That is, σi is multiplied by your inner state and all the people who influence you.
At the same time, a particular person can influence millions of people, just as media personalities, actors, or even the president influence millions of people. It turns out that the influence matrix is terribly asymmetric, vertically it can contain a huge number of non-zero entries, and horizontally, out of 200 million people in the country, for example, 100 non-zero numbers. For everyone, this gain is the sum of a small number of terms, but aij (the influence of a person on someone) can be non-zero for a huge amount of j, and the influence of aji (the influence of someone on a person) is not so large, more often limited to a hundred. This is where a lot of asymmetry comes in.
Examples of network members
We have tried to interpret in sociological terms the initial data of the model. For example, who is a "conformist-careerist"? This is a person who does not internally participate in the conflict, but there are people who strongly influence him, for example, the boss.
One can predict how his choice is related to the choice of the boss in any equilibrium.
Further, a “passionary” is a person with a strong inner conviction in the direction of the conflict.
Its aij (influence on someone) is large, in contrast to the previous version, where aji (influence of someone on a person) is large.
Further, "autistic" is a person who does not participate in games. His beliefs are near zero, and no one exerts influence on him.
And finally, a "fanatic" is a person who no one at all does not affect.
Perhaps from a linguistic point of view, the present terminology is incorrect, but there is still work to be done in this direction.
This suggests that he, like the “passionary”, has vi much more than zero, but aji = 0. I draw your attention to the fact that the “passionary” can be a “fanatic” at the same time.
We assume that inside such nodes it will be important what decision the “passionary / fanatic” makes, since this decision will spread around like a cloud. But this is not knowledge, but only an assumption. So far, we cannot solve this problem in any approximation.
And there is also a TV. What is a TV? This is a shift in your inner state, a kind of "magnetic field".
At the same time, the influence of the TV, in contrast to the physical "magnetic field" on all "social molecules" can be different both in magnitude and in sign.
Can I replace the TV with the Internet?
Rather, the Internet is the very model of interaction that needs to be discussed. Let's call it an external source, if not information, then some kind of noise.
Let's write down three possible strategies for σi=0, σi=1, σi=-1:
How do you interact. In the beginning, all participants are “clouds”, and each person knows only that this is a “cloud” about everyone else, and assumes an a priori probability distribution of these “clouds”. As soon as a particular person begins to interact, he recognizes the whole triple ε, i.e. a specific point, and at the moment a person makes a decision that gives him a larger number (of those where ε is added to the gain, he chooses the one that is greater than the other two), the rest do not know at what point he is, therefore they cannot predict .
Next, a person chooses (σi=0/ σi=1/ σi=-1), and in order to choose, he needs to know σj for everyone else. Let's pay attention to the bracket, the bracket contains the expression [∑ j ≠ i aji σj], i.e. what the person does not know. He must predict this in equilibrium, but in equilibrium he does not perceive σj as numbers, he perceives them as probabilities.
This is the essence of the difference between the discrete response equilibrium and the Nash equilibrium. A person must predict probabilities, thus a system of equations for probabilities arises. Let's imagine a system of equations for 100 million people, multiply by 2 more, since there is a probability of choosing "+", a probability of choosing "-" (the probability of staying away is not taken into account, since this is a dependent parameter). As a result, 200 million variables. And 200 million equations. Solving this is unrealistic. And it is also impossible to collect such information exactly.
But sociologists tell us: "Wait, friends, we will tell you how to typify society." They ask how many types of problem we can solve. I say, we still solve 50 equations, the computer can solve a system where 50 equations, even 100 are still nothing. They say it's no problem. And then they disappeared, you bastards.
We really had a meeting with HSE psychologists and sociologists, they said that we could write a breakthrough revolutionary project, our model, their data. And they didn't come.
If you want to ask me why everything is happening like this through the ass, I tell you, because psychologists and sociologists do not come to our meetings. If we got together, we would move mountains.
As a result, a person must choose from three possible strategies, but cannot, because he does not know σj. Then we change σj to probabilities.
Gains in Discrete Response Equilibrium
Instead of the unknown σj, we substitute the difference in the probabilities that some person takes one or another of the parties in the conflict. When we know at what vector ε we get to what point of three-dimensional space. At these points (winnings) "clouds" appear, and we can integrate them and find the weight of each of the 3 "clouds".
As a result, we find the probabilities from the side of an external observer that a particular person will choose one or the other, before he knows his true position. That is, it will be a formula that, in response to knowing all the other p, will give its own. And such a formula can be written for each i and leave a system of equations from it, which will be familiar to those who have studied the Ising and Potz models. Statistical physics is rigidly based on the fact that aij = aji, the interaction cannot be asymmetric.
But there are some "miracles" here. Mathematical “miracles” are that the formulas almost coincide with the formulas from the corresponding statistical models, despite the fact that there is no game interaction, but there is a functional that is optimized on a variety of various fields.
With arbitrary input data, the model behaves as if someone is optimizing something in it. Such models are called "potential games" when it comes to Nash equilibrium. When the game is arranged in such a way that Nash equilibria are given by optimizing some functional on the space of all choices. What is potentiality in discrete response equilibrium has not yet been finally formulated. (Although Fedor Sandomirsky may be able to answer this question. This would definitely be a breakthrough).
This is what the complete system of equations looks like:
The probabilities with which you choose this or that are consistent with the forecast for you. The idea is the same as in the Nash equilibrium, but it is implemented through probabilities.
A special distribution of ε, namely the Gumbel distribution, which is a fixed point of taking the maximum of a large number of independent random variables.
A normal distribution is obtained by averaging a large number of independent random variables with variance in acceptable values. And if we take the maximum from a large number of independent random variables, then we get such a special distribution.
By the way, the parameter of randomness of decisions λ is missing in the equation, I forgot to write it down.
Understanding how to solve this equation will help you understand how to cluster society. In the theoretical aspect, the potentiality of games from the point of view of the discrete response equation.
We need to try a real social graph, which is distinguished by a set of properties:
- small diameter;
- power law of distribution of vertex degrees;
- high clustering.
That is, you can try to rewrite the properties of a real social network inside this model. While no one has tried, maybe then something will work out.
Now I can try to answer your questions. At least I can definitely listen to them.
How does this explain the Brexit mechanism and the US elections?
So so. It doesn't explain anything. But it does give a hint as to why opinion polls are consistently wrong in their forecasts. Because people publicly answer what their social environment requires them to answer, and in private they vote for their inner conviction. And if we can solve this equation, that the solution will be what the sociological survey gave us, and vi is what will be on the vote.
And this model can be considered as a separate factor not a person, but a social stratum?
That is exactly what I would like to do. But we do not know the structure of social strata. That is why we are trying to keep up with sociologists and psychologists.
Can your model somehow be applied to explain the mechanism of various kinds of social crises that are observed in Russia? Let us assume a discrepancy between the operation of formal institutions?
No, it's not about that. This is about the conflict of people. I do not think that the crisis of institutions here can be somehow explained. On this topic, I have my own idea that the institutions created by mankind are too complex, they will not be able to hold on to such a degree of complexity and will be forced to degrade. This is my understanding of reality.
Is it possible to somehow investigate the phenomenon of polarization of society? You already have v in it, how good is it for anyone ...
Not really, we have a TV there, v+h. This is comparative static.
Yes, but the polarization happens gradually. I mean that the participation of a society with a pronounced position is 10% v-positive, 6% v-negative, and the gap is getting wider and wider between these values.
I do not know what will be in the dynamics at all. In correct dynamics, apparently, v will take the values of the previous σ. But whether this will work, I don't know. There is no panacea, there is no universal model of society. This model is some insight that may be useful. I believe that if we solve this problem, we will see how opinion polls steadily diverge from the reality of voting. The society is in great chaos. Even measuring a certain parameter gives different results.
Is it somehow related to the classical theory of matrix games?
These are the matrix games. It's just that the matrices here are 200 million by 200 million in size. This is a game of everyone with everyone, the matrix is written as a function. This is connected with matrix games in the following way: matrix games are games of two people, and here 200 million are playing. Therefore, this is a tensor, which has a dimension of 200 million. Not even a matrix, but a cube, with a dimension of 200 million. But they consider an unusual solution concept.
Is there a concept of the price of the game?
The value of the game is possible only in the antagonistic game of two players, i.e. with zero sum. This notantagonistic game of a huge number of players. Instead of the price of the game, there are equilibrium payoffs, and not in the Nash equilibrium, but in the discrete response equilibrium.
What about the term "strategy"?
The strategies are, 0, -1, 1. This came out of the classical concept of the Nash-Bayes equilibrium, the equilibrium And in this particular case, the Bayes-Nash equilibrium is put on the regular game data. Due to this, a combination is obtained, called the discrete response equilibrium. And this is infinitely far from the matrix games of the middle of the XNUMXth century.
Something is doubtful that with a million players you can do something ...
This is the question of how to cluster society, it is impossible to solve a game with so many players, you are right.
Literature in related areas in statistical physics and sociology
- Dorogovtsev SN, Goltsev AV, and Mendes JFF Critical phenomena in complex networks // Reviews of Modern Physics. 2008 Vol. 80.Pp. 1275-1335.
- Lawrence E. Blume, Steven Durlauf Equilibrium Concepts for Social Interaction Models // International Game Theory Review. 2003 Vol. 5, (3). pp. 193-209.
- Gordon MB et. al., Discrete Choices under Social Influence: generic Perspectives // Mathematical Models and methods in Applied Science. 2009 Vol. 19.Pp. 1441-1381.
- Bouchaud J.-P. Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges // Journal of Static Physics. 2013. Vol. 51(3). pp. 567-606.
- Sornette D. Physics and financial economics (1776—2014): puzzles, lsing, and agent-based models // Reports on Progress in Physics. 2014. Vol. 77, (6). pp. 1-287
Only registered users can participate in the survey. , you are welcome.
(purely for an example) Your position in relation to Igor Sysoev:
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62,1%+1 (participate in the conflict on the side of Igor Sysoev)175
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1,4%-1 (participate in conflict on opposite side)4
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28,7%0 (refuse to participate in the conflict)81
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7,8%try to use the conflict for personal interests22
282 users voted. 63 users abstained.
Source: habr.com