In an earlier post, we invited you to coordinate presidential campaign within a simple web game. We have also mentioned that the framework of the web game relates to game theory and the Colonel Blotto game. Let us turn our attention for the next few posts to this classic game. As we will see it contains rich strategic landscape, and has numerous practical applications in various fields, including political campaigning (as previously discussed), warfare (as suggested by its original context), marketing, and even sports.

Here, in this post, will start by discussing the original formulation of the game.

In a few recent posts we have looked at the DeGroot model and its generalizations. In those posts I have mentioned that while the models are somewhat old and could be considered to be classical, they have resurfaced recently in the opinion dynamics literature.

This recent Master thesis is an example of a study conducted using the DeGroot model. Its novelty lies in an exploration of in-group and out-group biases within the framework of the DeGroot model. We invite you to watch the recording of the defense presentation shared by IFISC.

Another model I have seen recently featured a lot in the literature is the Friedkin-Johnsen model. This model is a generalization of the DeGroot model we have discussed earlier, but it adds a novel twist which allowed it to become a ground-breaking model on its own [1]. Namely, it has added stubbornness into the DeGroot model [2].

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

In the previous post we have talked about the core ideas in the DeGroot model. In this post let me show how wisdom of the crowd effect emerges in the model.

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.