Wisdom of the crowd

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.

The model

In the previous post I have mentioned that opinion is encoded by a real number between zero and unity. Actually, opinion could be any real number and the model still works as fine. In this post we won't restrict the opinions.

In this post we also increase the number of agents acting in the system to \( 300 \). So, naturally, you won't be able to adjust trust matrix manually. Instead the trust matrix will be filled-in randomly:

1. Diagonal elements of the trust matrix will be set to \( t_{\mathrm{self}} \). Note that this value will change during the normalization procedure, but in general it allows partial control of how self-confident the agents are.
2. Off-diagonal elements will be set to zero with probability \( 1 - p_{\mathrm{trust}} \). The smaller \( p_{\mathrm{trust}} \), the smaller social networks of the agents will be.
3. Non-zero elements will be set to a uniformly distributed random value in \( [0, 1] \) interval.

Wisdom of the crowd

As is commonly known, the wisdom of the crowd is an observation that the aggregated opinion of a diverse and independent group of individuals would be closer to the true answer than the opinion of a single expert.

In DeGroot model this observation works a bit differently. Let initial opinion of the agents be distribution according to some arbitrary distribution (in the app we use normal distribution) centered around some true opinion \( \mu \). No matter the initial diversity of the opinions, let \( \sigma \) be the standard deviation, as long as the trust network is strongly connected (there are no separated components), the system should reach a consensus state close to \( \mu \). An important requirement to observe this effect is that influence of the most influential agent should diminish to zero as the number of agents grows to infinity.

For the network to be strongly connected \( p_{\mathrm{trust}} \gg \frac{\log(N)}{N} \) condition should be satisfied. This condition can be derived from the connectivity of Erdos-Renyi network. As in our case \( N = 300 \), so \( p_{\mathrm{trust}} \gg 0.02 \).

Interactive app

The app below allows you to explore the wisdom of the crowd effect in the context of the DeGroot model. Initial distribution of opinions is shown as gray dots, while the current (evolving) opinion distribution is shown as red dots. Observe that the current opinion distribution quickly shrinks around value close to \( \mu \). Unless \( p_{\mathrm{trust}} \) becomes small - then most agent will converge towards consensus, but outliers (not connected to large components) will retain their own opinions.

Note that the app will stop automatically when the current opinions become too similar.

References

• M. H. DeGroot. Reaching a Consensus. Journal of the American Statistical Association 69: 118-121 (1974). doi: 10.2307/2285509.