Hello, Habr!
My name is Asya. I found a very cool lecture, I can not help but share.
I bring to your attention a synopsis of a video lecture on social conflicts in the language of theoretical mathematicians. The full lecture is available at the link: Model of social split: a ternary choice game on interaction networks (A.V. Leonidov, A.V. Savvateev, A.G. Semenov). 2016.
Aleksey Vladimirovich Savvateev - Ph.D. in Economics, Doctor of Physics and Mathematics, 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 voting for Britain's exit from the European Union (
Brexit) , the phenomenon of a deep social split in Russia after the
Maidan , and the
election in the USA with a sensational outcome .
How can such situations be modeled so that they contain echoes of reality? To understand the phenomenon, it is necessary to study it comprehensively, but in this lecture there will be a model.
Social schism means
In these three scenarios, the general thing is that a person somehow adjoins some camp, or refuses to participate and discuss his choice. Those. The choice of each person is ternary - of three values:
- 0 - refuse to participate in the conflict;
- 1 - to participate in the conflict on one side;
- -1 - to participate in the conflict on the opposite side.
There are direct consequences that are actually related to your own attitude to the conflict. There is an assumption that everyone has some 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 a number line somewhere around zero, for example, 0.1. When a person is 100% sure that someone is right, then his internal parameter will be -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 he 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 gain in the game, you have abandoned the conflict.
If you choose something that is inconsistent with your position, then a minus appears before vi, for example, v
i = - 3. If your internal position coincides with the side of the conflict at which you are speaking, and your position is σ
i = -1, then v
i = +3.
Then the question arises, for what reasons sometimes you have to choose the wrong side that is in your soul? This can happen under pressure from your social environment. And this is a postulate.
The postulate is that you are affected by consequences that are beyond your control. The expression a
ji is a material parameter of the degree and sign of influence on you by j. You are number i, and the person who affects you is person number j. Then there will be a whole matrix of such a
ji .
This person j can even influence you negatively. For example, this is how you can describe the speech of an unpleasant politician on the opposite side of the conflict. When you look at a speech and think: "This idiot, and look what he says, I said that he is 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 mismatch of the accepted positions.
Those. if σ
i , σ
j , is a positive sign, and a
ji is also a 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 is not.
Thus, we tried to take into account all the effects of social influence.
Next is the next point. There are many such models of social interaction described from different angles (models for making a threshold decision, many foreign models). They consider a standard game theory concept called Nash equilibrium. There is a deep dissatisfaction with this concept for games with a large number of participants, as in the examples with the UK and the USA mentioned above, that is, many millions of people.
In this situation, the correct solution to the problem goes through approximation using the continuum. The number of players is a kind of continuum, a “cloud” playing, with a certain space of important parameters. There is a theory of continuum games,
Lloyd Shapley“Importance for non-atomic games.” This is an approach to cooperative game theory.
There is no non-cooperative game theory with a continuous number of participants, as a theory. There are separate classes that are studied, but this knowledge has not yet been formed in the 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 past few years, there is some agreement that the concept developed by
Palfrey and McKelvey, which sounds in English as “
Quantal response equilibrium ”, or “
Discrete response equilibrium, ” as we translated with Zakharov. The translation belongs to us, and since nobody translated it into Russian before us, we imposed this translation on the Russian-speaking world.
We meant by this name that each specific person 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 selected, and in response, I see how a person plays, but I don’t know where he is in this cloud, that is, there is hidden information there, I perceive person in the "cloud" as the probability with which he will go in one way or another. This is a static concept. The mutually enriching symbiosis of physicists and player theorists, it seems to me, will determine the theory of games of the 21st century.
We generalize the available experience of modeling such situations with completely arbitrary initial data and write out a system of equations that is remote from the equilibrium of the discrete response. That's all, then, in order to solve the equations, it is necessary to make a reasonable approximation of the situations. But this is still ahead, this is a huge direction in science.
Discrete response equilibrium is an equilibrium in which, in fact, we play
incomprehensibly with anyone . In this case, ε is added to the gain from the pure strategy. There are three winnings, some three numbers that mean “drown” on one side, “drown” on the other side and abstain, and there is ε, which is added to these three. Moreover, the combination of these ε is unknown. The combination can be estimated only a priori, knowing the probability of distribution 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 coordination is the equilibrium of the discrete response. We will return to this moment.
Formalization through Discrete Response Equilibrium
Here is the gain in this model:
It brackets all the influence that is exerted on you if you have chosen a side, or will be multiplied by zero if you have not chosen any side. Further it will be with a “+” sign if σ
1 = 1, and with a “-” sign if σ
1 = -1. And to this is added ε. That is, σ
i is multiplied by your internal state, and all the people who influence you.
At the same time, a specific person can influence millions of people, just as media personalities, actors or even the president affect 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 each, this gain is the sum of a small number of terms, but a
ij (the influence of a person on someone) can be nonzero for a huge number of j, and the influence of a
ji (the influence of someone on a person) is not so great, often limited to a hundred. Here a very large asymmetry arises.
Network Participant Examples
We tried to interpret the initial data of the model in sociological terms. For example, who is a "conformist-careerist"? This is a person who is not internally involved 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 at any equilibrium.
Further, a “passionary” is a person with a strong inner conviction on the side of the conflict.
Its a
ij (influence on someone) is great, unlike the previous version, where a
ji (influence of someone on man) is great.
Further, autistic is a person who is not involved in the game. His beliefs are near zero, and no one is influencing him.
And finally, a “fanatic” is a person whom
no one influences
at all .
Perhaps, from the point of view of linguistics, the present terminology is incorrect, but work still remains in this direction.
This suggests that he, like that of the “passionary,” vi is much greater than zero, but aji = 0. I draw your attention to the fact that “passionary” can be “fanatic” at the same time.
We assume that within such nodes it will be important what decision is made by the “passionate / fanatic”, because this decision will be distributed around the cloud. But this is not knowledge, but only an assumption. So far we cannot solve this problem in any approximation.
And there’s a TV. What is a television? This is a shift in your internal state, a kind of “magnetic field”.
In this case, 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 model of interaction itself, it needs to be discussed. We will call this an external source, if not information, then some kind of noise.
We describe three possible strategies for σ
i = 0, σ
i = 1, σ
i = -1:
How is the interaction going? In the beginning, all participants are “clouds”, and each person only knows about everyone else that it is a “cloud”, and assumes an a priori distribution of the probabilities of these “clouds”. As soon as a specific person begins to interact, he recognizes to himself the whole three ε, i.e. a specific point, and at the moment the person makes a decision that gives him a larger number (from those where ε is added to the gain, he chooses the one that is more than the other two), the rest do not know what point he is in, therefore they cannot predict .
Next, a person chooses (σ
i = 0 / σ
i = 1 / σ
i = -1), and to choose, he needs to know σ
j for everyone else. Pay attention to the bracket, the bracket [∑
j ≠ i a
ji σ
j ], that is, what a person does not know. He must foresee this in equilibrium, but in equilibrium he does not perceive σ
j as numbers, he perceives them as probabilities.
This is the essence of the difference between the equilibrium of the discrete response and the Nash equilibrium. A person must predict the probability, thus a system of equations based on probability. Suppose we imagine a system of equations for 100 million people, we multiply by another 2. because there is a probability of choosing “+”, the probability of choosing “-” (the probability of staying away does not take into account, because this is a dependent parameter). As a result, 200 million variables. And 200 million equations. Solving this is unrealistic. And to collect such information exactly is also impossible.
But sociologists tell us: "Wait, friends, we will tell you how to typify a society." They ask how many types we can solve. I say, we will still solve 50 equations, the computer can solve a system where 50 equations, even 100 are nothing. They say that it’s no problem. And then they disappeared, bastards.
We really had a meeting with psychologists and sociologists from HSE, they said that we can write a breakthrough revolutionary project, our model, their data. And did not come.
If you want to ask me why this happens through the ass, I tell you because psychologists and sociologists do not come to our meetings. They would come together, the mountains would turn.
As a result, a person must choose from three possible strategies, but cannot, because he does not know σj. Then we change σ
j for probability.
Winnings in Discrete Response Equilibrium
Together with the unknown σ
j, we substitute the difference in the probabilities that a person occupies one or another of the parties to the conflict. When we know at which vector ε at which point in three-dimensional space we fall. “Clouds” appear at these points (wins), and we can integrate them and find the weight of each of the 3 “clouds”.
As a result, we find the probabilities on the part of an external observer that a particular person will choose one or another before he finds out his true position. That is, it will be a formula that, in response to the knowledge of all the other p, will give its own. And such a formula can be written for each i and leave out of it a system of equations that will be familiar to those who worked on Ising and Pots models. Statphysics firmly assumes that a
ij = a
ji , the interaction cannot be asymmetric.
But there are some "miracles." The mathematical “miracles” are that the formulas almost coincide with the formulas from the corresponding static physical models, despite the fact that there is no game interaction, but there is a functional that is optimized on a variety of all kinds of fields.
With arbitrary input data, the model behaves as if someone is optimizing something in it. Such models are called “potential games,” in the case of Nash equilibrium. When a game is designed so that Nash equilibria are determined by optimizing some kind of functional in the space of all choices. What is potentiality in the equilibrium of a discrete response has not yet been finalized. (Although Fedor Sandomierz may be able to answer this question. This would definitely be a breakthrough).
Here is the complete system of equations:
The probabilities with which you choose one or another are consistent with the forecast for you. The idea is the same as in Nash equilibrium, but it is realized through probabilities.
A special distribution of ε, namely the Humbel 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 the permissible values. And if you take the maximum from a large number of independent random variables, you get such a special distribution.
By the way, the equation omits the randomness parameter λ of the decisions made, I forgot to write it.
Understanding how to solve this equation will help you understand how to cluster a society. In the theoretical aspect, the potentiality of games in terms of the discrete response equation.
We must try a real social graph, which differs in a set of properties:
- small diameter;
- power law of the distribution of degrees of vertices;
- high clustering.
That is, the properties of a real social network can be rewritten inside this model. So far 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 election?So it is. This does not explain anything. But this gives a hint why sociological polls are consistently mistaken in their forecasts. Because people are publicly responding to what their social environment requires, and in private they give their vote for inner conviction. And if we can solve this equation, the solution will be what the sociological survey gave us, and v
i is what will be on the ballot.
And this model can be considered as a separate factor not a person, but a social stratum?This 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 be used somehow to explain the mechanism of various kinds of social crises that are observed in Russia? Suppose the discrepancy between the actions of formal institutions?No, it's not about that. This is about the conflict of people. I don’t think that the crisis of institutions here can somehow be explained. On this subject, I have my own idea that the institutions created by mankind are too complex, they will not be able to hold onto 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 this in v, how good is it for anyone ...Not really, we have a TV there, v + h. This is comparative statics.
Yes, but polarization is gradual. I mean that the participation of a society with a pronounced position is 10% v-positive, 6% v-negative, and the gap is widening more and more between these values.I do not know what will be the dynamics in general. In the correct dynamics, apparently, v will take the values of previous σ. But whether such an effect is obtained, I do not know. There is no panacea; there is no universal model of society. This model is some look that may be useful. I believe that if we solve this problem, we will see how opinion polls stably diverge from the reality of voting. There is huge chaos in society. Even measuring a specific parameter gives different results.
Is this somehow related to the classical theory of matrix games?These are 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 this way: matrix games are two-person games, and here they play 200 million. Therefore, it is a tensor that 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 a game?The price of the game is possible only in the antagonistic game of two players, i.e. with zero amount. This is not an antagonistic game of a huge number of players. Instead of the price of the game, there are equilibrium wins, not in the Nash equilibrium, but in the equilibrium of the discrete response.And the concept of "strategy"?There are strategies, 0, -1, 1. It came out of the classical concept of Nash-Bayes equilibrium, the equilibrium of games with incomplete information.And in a particular case, the Bayes-Nash equilibrium is put on the data of a regular game. Due to this, a combination is obtained, called the equilibrium of the discrete response. And this is infinitely far from the matrix games of the mid-20th century.Something is doubtful that you can do something with a million players ...This is also a question of how to cluster a society, it is impossible to solve a game with so many players, you are right.Literature in related areas in statistics 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