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Tax evasion is a serious problem for many governments. In Greece, the more or less ubiquitous avoidance of tax has brought the country to its knees and the European Union to the verge of collapse. So there’s more than a little interest in understanding the way this behaviour spreads and how to combat it.

In recent years, economists have turned to computational models to help. The basic idea is that tax payers can declare their full income, part of it or none at all. The rate of tax and the way it changes for different levels of income is an important factor. In general, however, risk-averse tax payers are more likely to pay tax.

But the perceived risk depends on many factors such as the probability of being audited and the penalty for being caught. There are also social factors at work that change the ‘temperature’ of the environment, making it more or less likely that an individual will avoid tax.

Much of the work on tax avoidance has focused on modelling this perception of risk and how it changes. The most successful models consist of a number of different agents each with particular behaviours, such as a certain aversion to risk. However, this behaviour can change when the agents interact with each other.

This has given economists much insight into the problem. The models suggest, for instance, that even low rates of audit can lead to a high fraction of tax payers, although it may take along time to reach this state.

One problem, however, is the number of the agents that can be included in these models, which is limited by the richness of the agents’ behaviour and the complexity of the results. Often these models simulate the behaviour of a few thousand agents. Clearly, economists would like to model entire societies, indeed entire continents (we looked at one proposal to model the entire planet here).

Today, Michael Pickhardt and Goetz Seibold at the Brandenburg University of Technology Cottbus in Germany suggest a different way to model populations that can be scaled relatively easily to the size of countries.

To do this, these guys turn to a well known type of physics simulation called an Ising model. This consists of a lattice in which each node can be in one of two different states. Each node influences the state of its nearest neighbours. These lattices can be vast, consisting of millions of nodes and can accurately simulate the kind of phase changes seen in many natural systems, such as ferromagnets.

In recent years, social scientists have also begun using Ising models to show how social phenomena can spread through society as individuals try to adjust their behaviour to conform with their neighbours. That would suggest that Ising models ought to be able to capture the behaviour of tax avoiders.

But the results from agent -based and Ising models have always differed significantly.

The problem is that Ising models tend to encode only very simple behaviour since the nodes can only be in one of two states. So individuals are either tax avoiders or tax payers, with no middle ground allowed.

Pickhardt and Seibold have found a way round this problem. To understand their approach, it’s helpful to think of the example of ferromagnets, which consist of a lattice of particles with a spin that can be up or down. The spins tend to align, particularly if there is an external magnetic field but the system also has a temperature that tends to randomise the alignments.

So this gives them two more parameters to play with–the temperature and the magnetic field in addition to the strength of the interaction between neighbours. In Pickhardt and Seibold’s model, the temperature and magnetic fields are kinds of social pressures that tend to prevent or promote copying behaviour.

So what these guys have done is allow the temperature to vary throughout their model. So each node effectively has its own little microenvironment with its own magnetic field, temperature and interaction with its neighbours. When the model runs, these environments interact, allowing certain kinds of behaviours to spread. This is like an agent-based model, in that every node has its own behaviour, but with the computational power of an Ising model.

Pickhardt and Seibold’s econophysics model makes a variety of predictions. For example, in addition to penalising tax avoiders with fines, it suggests that other non-monetary penalties can help to reduce avoidance. These might be processes such as monitoring people after they’ve been audited, which has the effect of spreading the perception of risk among the population.

That’s significant because this kind of tinkering with social pressures is exactly the prediction made by agent-based models.

“This is the first time that such a reproduction has been achieved with an econophysics model,” say Pickhardt and Seibold. Their implication is clear: they clearly believe that Ising models are becoming as capable as agent-based models in capturing tax avoiding behaviour.

If that’s true, it could lead to a much easier way of carrying out larger simulations that more accurately capture the behaviour of entire societies. Pickhardt and Seibold’s model consists of a population of 1 million, that’s a decent-sized city and a small country (Greece has a population of about 11 million). And bigger simulations would be straightforward to do.

Of course, the predictions will have to be tested on real populations to see whether they work as described. It’s quite possible that behaviours won’t spread in quite the way suggested by this Ising model or indeed the agent-based ones.

Clearly, there’s work ahead for econophysicists.

Ref: arxiv.org/abs/1112.0233: Income Tax Evasion Dynamics: Evidence from an Agent-based Econophysics Model

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