While briefly going back to the topic of opinion dynamics let us become familiar with one possibly unsolvable real-life problem. In other words, did you know that democracy is mathematically impossible?

For a democratic decision making mechanism (e.g., elections) to work properly, we must ensure that:

1. Choice rankings should be transitive. If choice A is preferred over choice B, and B is preferred over C, then A should be preferred over C.
2. Some decision should always be made. The mechanism simply can't "give up" under any however unusual circumstance.
3. There should be no dictators. The decision must not depend only on one voter's decision.
4. The mechanism should be unbiased, it should not impose any preference on the voters. If voters prefer choice A over choice B, this should be taken into account.
5. Voter preference between two alternatives should depend only on the two alternatives. For example, if a group of voter prefers choice A over choice B, then introduction choice C should not change the relative ranking of A and B.

Seems reasonable, but it is impossible as per the Arrow's impossibility theorem [1]. Well, at least for ordinal (ranked) voting systems. Watch the video below to get a popular summary on "why?".

By the end of the video a possible solution is discussed. Instead of using ranked voting systems we should use rated voting systems, as first shown by D. Black [2]. I feel a bit skeptical about this, as lots of people would likely get confused over what should they write in their ballots. Furthermore, I am not sure if such voting is impervious to strategic voting and would encourage truthful statement of the preferences. Still, maybe?

References

• K. J. Arrow. A difficulty in the concept of social welfare. Journal of Political Economy 58: 328-346 (1950). doi: 10.1086/256963.
• D. Black. On the Rationale of Group Decision-making. Journal of Political Economy 56(1): 23-34 (1948). doi: 10.1086/256633.

While looking through the recent opinion dynamics literature I have started noticing papers exploring various extensions of the DeGroot model [1, 2]. Prior to those papers I haven't even heard or paid much attention to it. So I felt a bit curious.

At a first glance DeGroot model appears to be similar to the trust and suspicion models we have discussed few years ago. Also, in the broadest strokes it likely inspired bounded confidence models.

But let us see what it is actually about!

One of the things I wondered about the previous summer was the difference between stationarity (balance of all inflows and outflows from the state) and detailed balance (balance of flows between two particular states). For example, when looking for the stationary distribution of the noisy voter model it is sufficient to solve equations for the detailed balance condition. But why? Do all stationary stochastic models satisfy detailed balance condition?

An excellent introduction to complex systems by professor emeritus at IFISC Maxi San Miguel. We invite you to watch it!

Have you ever heard about the hypergeometric distribution? I haven't up to at least a few weeks ago. It is related to the binomial distribution in a sense that both of these distributions describe the probability to have certain number of successes after a given number of experiments. The difference between them being that binomial distribution assumes experiments to be independent (drawing the balls from the box with replacement), while hypergeometric distribution assumes dependence (the balls are drawn without replacement).

Let us construct a simple model for the hypergeometric distribution, and run simulations!