# UK retail market share dynamics using Lotka–Volterra models

The Rise of the discounters

Competition in the UK retail marketplace is dynamic and supermarkets change their competitive behaviour over time. Using integrable nonautonomous Lotka–Volterra (LV) models I can model this behaviour using Python.

This system has a number of useful properties.

i)The analytical solutions of this system are known.

ii)The analytical solutions only depend on the utility functions of the competing supermarkets.

iii)The model has a strong connection with the logit model.

The logit model is used in economics to describe market demand, so this approach has a sound theoretical basis.

The outer good can be calculated as,

outer good = (1-sum of market shares) at time t.

The utility function of the supermarkets can be calculated using,

utility = ln(marketshare) - ln(outer goods)

The utility function can then be fitted using a variety of models, in this work I have used a quadratic polynomial.

The market shares are then calculated from the utility function by using,

marketshare of firm = exp(polynomial )/sum for all firms(exp(polynomial))

It is a general feature of logit models of market shares that the time evolution is related to the firms’ utility functions. In the nonautonomous Lotka-Volterra model, the market shares are dependant on the supermarket utility functions. This allows me to both predict and explain the market dynamics and account for the dynamic nature of the competitive environment using the signs of the time-dependent coefficients in the nonautonomous Lotka-Volterra equations.

As indicated the competitive roles are deduced from the signs of g(t) which is obtained from the derivative of the fitted quadratic polynomial. For example, the type of interaction indicated by ++ is pure competition, whilst that indicated by -+ would be a predator-prey interaction.

Let’s look at the community matrix for the competitive interactions

For Tesco and Asda, Tesco and Aldi, and Tesco and Iceland the signs are -+ so there is a Predator-Prey interaction. Tesco’s other competitive interactions are mutualism — in which both supermarkets involved benefit to some extent with neither supermarket being damaged.

Aldi’s competitive interactions are generally of the predator-prey type and it has a strong positive g(i). Aldi is capturing market share most other supermarkets

In order to evaluate the fitting and forecasting performance of the nonautonomous Lokta-Volterra model, I used the mean square error (MSE), mean square error (RMSE) and the mean absolute percentage error (MAPE), to compare the historical values of the market shares with the predicted values.

The standard MAPE prediction capability levels indicate that a score < 10 is highly accurate, whilst a score in the region 10−20 is considered good.

What I did next was to look at a 5-year forecast assuming that the competitive strategies of the supermarkets remained on a business as usual basis. The initial analysis indicates that Aldi will consistently increase its market share. It would be expected that other supermarkets would make adaptations to their competitive strategies in light of the evolutionary pressure from the rise of discounters and also online shopping in the grocery retail sector.

This post has been substantially revised since it’s the first draft. In addition to this nontechnical post, I have completed a more technically rigorous and in-depth analysis which is due to be published as a preprint on Arxiv and as an article in the journal, Technological Forecasting & Social Change.

The Jupyter notebook and datasets associated with this work can be found here: https://github.com/davidsdatascience/marketshare

Written by

## David Horgan

#### I am a theoretical physicist with a data science background. At present, I am developing a UK retail market using ABM, ML and computational econometrics.

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