Opinions and How They Change

An individual forms opinions based on how he can himself assimilate the facts around him and also based on what opinions his friends and other people he interacts with hold. This makes things complicated and intriguing enough that this has been an active area of research since decades.

One question is, can we formulate simple enough models that match with the data we get from real life experiments? If we could, then we would get some insight into human behavior and a tool for making useful predictions.

The simplest model that has been studied is this:

An opinion is just a real number. Each person starts with an initial opinion. Next, in each time step, he looks at the opinions held by his friends and updates his own opinion to the average of his old opinion and the opinions of his friends. It doesn’t have to be a simple average. A person may have different trusts for different friends and thus he might want to take a weighted average instead. However the model doesn’t allow individuals to change the weights at any step. The weights chosen in the beginning have to be the weights always.

Using simple linear algebra tricks and borrowing known results from the Markov Chain literature, it can be shown that this kind of system converges to an equilibrium in most natural cases. An equilibrium here means a set of opinions for which the averaging step doesn’t lead to any change, i.e., for all individuals, the new opinion remains the same as the old one. In fact, it can be proved that the equilibrium that’s reached is a consensus, i.e., every individual has the same opinion.

An objection to this model that one might have is the simple representation of an opinion. Can it really be represented by just a single real number?

Anyway, DeGroot, the person who introduced this model also showed that the same thing happens if the opinions are drawn from any vector space and in each time step a person updates his opinion to some convex combination of the opinions of his friends (including himself).

That’s something.

The only issue is that in real life, people don’t reach consensus. So what’s going on?

Of course, the model seems too simple to resemble real life accurately. For one, the weights (or trust) we assign to people changes over time depending on various factors. For example, if a person seems to be changing his mind every minute, we will probably assign a lesser weight to his opinion.

Also, even though this process of repeated averaging has been shown to always converge to a consensus, we don’t really know the amount of time it takes to reach there. From what I know by quickly glancing through Bernard Chazelle’s new work on bird flocking, the time taken by a community whose size is close to the population of a country to reach a consensus is probably way more than the age of the universe.

Anyway. Friedkin and Johnsen modified this model a bit to make it more realistic. In their model, an individual has a fixed internal opinion that doesn’t change with time and during an averaging step, he takes a weighted average of the opinions of his friends (including himself) and the fixed internal opinion. Because the internal opinion can be different for different people, this system will obviously not reach a consensus always.

The system does have an equilibrium though and Friedkin and Johnsen proved that the equilibrium is almost always reached.

However, their model is different from DeGroot’s simpler model in a fundamental way. Let me explain.

Given a set of numbers {a_1, \ldots, a_n}, the mean is the number that minimizes the function {(z-a_1)^2+\ldots +(z-a_n)^2}. Thus the averaging step above can be seen as a step where a person is trying to minimize the cost incurred with respect to the cost function {\sum_{j\in N(i)} (z_i-z_j)^2}. Here {N(i)} represents the neighborhood of {i}, i.e., the set of friends of {i}.

With the above definition of cost, we can measure the quality of a certain opinion vector. For example, we can say that the sum of costs incurred by each person is the social cost of the whole group. And then given an opinion vector, we can decide how good it is by measuring how far it is from the opinion vector that minimizes the social cost. In particular, we can measure the quality of the opinion vector that the group converges to in equilibrium.

The fundamental difference between DeGroot’s model and Friedkin and Johnsen’s model is that in DeGroot’s model, the equilibrium reached also minimizes the total social cost but in Friedkin and Johnsen’s model, it does not necessarily.

David Bindel, Jon Kleinberg and Sigal Oren prove in their FOCS ’11 paper that the situation is not that bad. Even though the total cost at equilibrium may not minimize the total social cost, it can be worse at most by a factor of 9/8. That’s pretty cool.

Advertisements

Building the perfect world in four extremely difficult steps

(I had written this for Goodblogs a long time ago.)

A perfect world would be one where everyone just did whatever they wanted to and lived happily doing that.

If tomorrow, everyone just decides to do whatever they want to, the world will turn into a chaos. For example, no one will want to clean the garbage and as a result, we will eventually rot in our own filth. To keep the world working, it is necessary that at least some people do things that they don’t particularly like doing. Can we repair this? In general, can we design a perfect world, a world where you never have to feel guilty, a world where you can just do whatever you feel like at any given moment and that will be the best thing to do for you and for the society? If yes, then what are the steps we need to take?

To understand this, we first need to understand what is the best thing for the society. There are things that are good for some people and bad for the others and there are things, that are good for the society right now, but in the long run, will lead to the decay of mankind. Let’s, for the time being, define ‘best’ as the thing that has the best average over people and over time. That is, we take the average happiness level of the world at this time and then take the average of this average over time. The best thing then, would be the thing that maximizes this average.

Once we have this out of our way, we can understand that the essential problem is to align what an individual feels like doing at a given moment and what’s the best thing to do for the society at that moment. Since our definition of ‘best’ depends on happiness levels of people, there are two extreme approaches to solving this problem. One extreme is to reprogramme the human brain so that it feels happy or sad in a more controlled way. This extreme is slightly trivial. All we need to do is to build the perfect mood enhancing drug and make it compulsory for everyone to take it. This will suddenly boost up the total happiness level of the world. The other extreme is to leave the human brain untouched and reengineer the world in such a way that whatever we want at this moment is made possible immediately. This is perhaps impossible. It’s easy to imagine an individual getting so angry at another person that he genuinely wants to kill him, or harm him severely in some other way. If then, this were made possible immediately for him, it would create more grief to the person being harmed than happiness to the person inflicting the harm. It’s similarly easy to imagine completely outrageous, or even physically impossible wishes that a person can make. One might have to break some laws of physcis to make that possible immediately. Since this approach seems impossible and the other extreme is kind of sad, the ideal should lie somewhere between the two extremes.

One possible midway approach is the following.

Step 1 – Build robots to do all the dirty work that no one in the world wants to do but is necessary to be done. This will leave out the kinds of work that at least some people in the world like doing. Let them do that work. But there might still be problems. The person who likes doing the work X might be living in Japan and the place where X needs to be done might be in Canada. Moreover, if we consider one person who likes doing X, then he may not want to do X all the time. The time when X needs to be done in Canada, he might be in a mood to go swimming with his kids.

Step 2 – Build a global work organizer. This will be some huge global device that will take the help of the internet. It will monitor what each person in the world is in the mood of doing right now and the things that need to be done at this moment in different parts of the world. Then, it will match the tasks to the suitable people. Since the world is such a large place, we can assume that what a random person wants to do at a given moment is useful for someone somewhere in the world and what’s useful for a given person is being wanted to be done by someone somewhere in the world. If there is some task, where this doesn’t happen, we already have step 1 to take care of it. Such things are already being done. There are several outsourcing services online for tasks that do not require physical presence. For example, Mechanical Turk and oDesk are websites that are designed exactly for this purpose. Even GoodBlogs is similar. The previous post I wrote was originally intended to be an email to my mother. But somehow somewhere in the world, there was a group of people who agreed to give me $20 for it. However, building this global work organizer will still not build the perfect world. For that, we will need the next step.

Step 3 – Upgrade the human mind so that its emotions are in control. For example, no one should ever feel like seriously inflicting any harm on anyone else. Not just this, but one will also need to make sure to not inflict any harm on the future self. Even if everyone is full of kindness and love towards others, one still might want to smoke a cigarette and thus suffer with lung cancer later in life. If not, then one may simultaneously want to learn how to play the piano and to not practise, or, to get a girl and to not develop the skills to woo women etc etc.

Step 4 – Make learning easy. Even if we are the masters of our own emotions, have robot servants and are never forced to do something that we don’t want to do, there might be a situation where we want to learn something quickly but we can’t. For example, one one may be craving good food, but not know how to cook. Then it will be good to have a plugin that they can install on their mind to give them that feature in a few seconds.