Infrastructure topology optimization under competition through cross-entropy

Abstract

In this article, we study a two-level non-cooperative game between providers acting on the same geographic area. Each provider has the opportunity to set up a network of stations so as to capture as many consumers as possible. Its deployment being costly, the provider has to optimize both the number of settled stations as well as their locations. In the first level each provider optimizes independently his infrastructure topology while in the second level they price dynamically the access to their network of stations. The consumers’ choices depend on the perception (in terms of price, congestion and distances to the nearest stations) that they have of the service proposed by each provider. Each providers' market share is then obtained as the solution of a fixed point equation since the congestion level is supposed to depend on the market share of the provider, which increases with the number of consumers choosing the same provider. We prove that the two-stage game admits a unique equilibrium in price at any time instant. An algorithm based on the cross-entropy method is proposed to optimize the providers' infrastructure topology and it is tested on numerical examples providing economic interpretations.

Keywords:

• non-cooperative game
• implicit function
• cross-entropy method

Acknowledgements

The author thanks anonymous reviewers for their comments.

Notes

1 It is assumed that online informations regarding the total number of EV drivers wishing to reload, prices, congestion levels and distance to stations are publicly accessible both for the EVs, through electronic technologies like smart phones for instance, and for the service providers.

2 The analytical results obtained in this article can be extended to the case where we choose another density such as normal, truncated exponential, of gamma type, khi-deux, etc. However, under such density assumptions, it might not be possible to derive the analytical expressions of the service providers’ market shares and a numerical approach should be envisaged.

3 Case 2 can be solved similarly by reversing the role played by both service providers.

4 In economics, the total market share that is, the sum of the rival providers’ market share is also called penetration or market coverage rate (Laffont and Tirole, 2001).

5 The discount factor is supposed identical for both providers. It means that both of them has the same risk aversion level for the future (Ross, 1983; Yildizoglu, 2011). The introduction of a discount factor is classical in (stochastic) dynamic programming and repeated game theory where the players consider their long-term utilities and have uncertainties on the future. However, extensions where it differs between the rival providers could be considered in a companion paper more devoted to simulation results.