Explaining Opinion Polarization with Opinion Copulas

Opinions are important in economics just as they are in social sciences in general. Bubbles, manias and information cascades form in opinion spaces about expectations and they evolve in a self-organizing manner. They are the outgrowth of adaptive dynamics whose understanding is of pressing significance. Clustering and polarisation, even without artificial communication obstacles, appear to persist and abound even as we are able to communicate information to each other without geographic, temporal or medial obstacles. In fact it is, perhaps paradoxically perhaps not, in this time of ease of communication that we are witnessing a pronounced misalignment of perception and fact or such utterly illogical phenomena as “post-truth politics”. We are thus reminded of the importance of social psychology in opinion formation.

The appearance of the first speculative bubbles coincides with the advent of the newspapers (“Irrational Exuberance” Shiller (p. 101)), the volatility of the stock market in the 1920s with the proliferation of the telephone (“Irrational Exuberance” Shiller pp. 181- 182), the stock market boom of the 1990 and the subsequent collapse of the new economy is related to the widespread adoption of the internet, and the emergence of social media after 2000 lead us to the collapse of 2008.

The World Economic Forum in its 2013 report identified “digital wildfires in a hyperconnected world” as one of the major global economic risks.

The more enhanced the capacity for interpersonal communication becomes the more accelerated the “contagion of ideas” and the more prominent their role in the formation of speculative bubbles.

Which, if any, aspects of our internet-centred society might help explain the observed opinion cleavages given that there is no longer an exogenous structure (e.g. distance or access) obstructing or limiting the exchange of ideas?

Besides decreasing the diameter of the graph of connections among economic agents, technological innovation increases the ability to communicate on more than one topics at the same time and this is what my new paper is about (forthcoming in PLOS ONE).

It discusses opinions dynamics in the presence of more than one “topics” or “issues” and gives shape to the thought that whether we will agree with each other on a given topic is certainly not completely immune to our differences on other issues, a fact which is well documented in Social Psychology. Our opinions are influenced by the context in which we formulate them and are some times influenced by what we erroneously think others think, thus often producing unexpected aggregate effects (e.g. pluralistic ignorance). So it is natural to conjecture that the more opinions we reveal to each other the more likely it becomes to get cross-topic contamination or to reveal an opinion gap too large to bridge.

In Social Media, where Shiller proposes that the contagion of ideas now takes place, we are more than just our opinion on whether stock or housing prices will go up. We reveal just about everything to the other making issue spill-over possible and its study relevant. I propose therefore that while in the hyper-connected world the diameter of the information graph decreases, the distances in opinion space increases in tandem with the fact that we turn agents from opinion scalars to opinion vectors. Ie highlight the role of heterogeneous priors in opinion dynamics and show how “dimensionality” matters and in fact appears to be an elusive confounding factor which might help explain the persistence of opinion cleavages.

By using the idea of opinion copulas (which formalizes topic spill-over and topic contagion i.e. covariance of opinions on different topics) I show theoretically, by example, analytically and by simulation that in a world of uniform, univariate opinion distributions, polarization arises if agents reveal, care about and are able to track more than one topics at a time something which becomes more likely in the context of Social Media and an increasingly algorithm-driven society.

The toy example opinion copula which demonstrates the idea is given in the figure below. it depicts two uniform, univariate opinion distributions (the marginals shown in the attached histograms) joined with a copula known as the ordinal sum copula. Notice how the marginals are uniform and hence under stochastic homophily will produce consensus but the opinion copula is almost polarized and in fact will produce polarization i.e. at least bimodal distribution under stochastic homophily with the same parameters. What this example shows is that in a world of uniform distributions polarization emerges spontaneously via opinion copulas. This example might appear artificial at first but it is less so than one thinks. It simply says that people keep on the same side of the center on two topics but are otherwise uniformly free to choose where they stand. This requirement appears mild and in fact it can be empirically verified in many contexts.

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