Soon we will kick off with a new topic entirely. The topic will not be outside the scope of Physics of Risk, but to a completely different subject still within our research interests. Namely, we will move on to cover some of the most well known or otherwise interesting models of the opinion dynamics.

Note that we have already covered a few opinion dynamics models. Such as Galam's referendum model, Voter model, AB model and Axelrod's culture dissemination model. Though we haven't discussed it in detail Bass diffusion model as well as Kirman's model could be also seen as opinion dynamics models. Though we have applied in a very different context, in their pure form both of them are about opinions than other rational arguments. Recently I even have published a generalized Kirman's model to describe N-state dynamics and applied the generalized model to reproduce Lithuanian parliamentary election results [1, 2].

References

• A. Kononovicius. Empirical analysis and agent-based modeling of Lithuanian parliamentary elections. Complexity 2017: 7354642 (2017). doi: 10.1155/2017/7354642. arXiv: 1704.02101 [physics.soc-ph].
• A. Kononovicius. Modeling of the parties' vote share distributions. Acta Physica Polonica A 133 (6): 1450 (2018). doi: 10.12693/APhysPolA.133.1450. arXiv: 1709.07655 [physics.soc-ph].

This is the final post in our continuing series on the order book models. Though we do not entirely neglect this topic. It is quite likely that in the near future we will come back to discuss more of the order book models.

So this time we will finally talk about an order book model we (Aleksejus Kononovicius and Julius Ruseckas) have decided to propose. This will be only a brief introduction into the model as it will recycle couple of ideas and concepts we have discussed earlier. More details on the model are available in [1].

So, our order book model involves elements from the agent-based herding model, about which we often write about here on Physics of Risk, [2]. We have extended our earlier approach [3] by defining how different types of agents implement their strategies in order book setting. This implementation heavily relies on the core ideas of the empirical high-frequency trader's behavior model [4, 5] we have discussed recently.

We continue our series of posts on order book models by considering an order book model proposed by a group of scientists from Japan [1, 2, 3], which is based on high resolution data from foreign exchange market. Their work is extremely interesting as it starts from the empirical observations at the lowest level observable and is built up to reproduce some empirical observations at the higher levels. Also the model is analytically tractable using kinetic theory.

How could you estimate population size? Catch random sample of individuals to be "marked" (capture). Catch another random sample and count proportion of the "marked" (recapture). The proportion of the "marked" in the second sample is statistically equivalent to the proportion of the "marked" in whole population.

Given that we have marked \( S_1 \) individuals during capture phase and have later recaptured \( M_2 \) of them together with \( S_2 - M_2 \) unmarked individuals, the total size of population \( N \) is given by \begin{equation} N = \frac{S_1 S_2}{M_2} . \end{equation} This formula should give a rather good approximation of \( N \) if samples sizes are large and enough and the sampling is truly random.

You can find two exemplary experiments in a video by Matt Parker (one of the most well-known standup mathematicians).

We continue our series of posts on order book models by considering extremely simple order book model. In this model most agents submit only market orders, while there is also a single market maker, which provides liquidity for them.