Payoff interdependence and the multi-store paradox

Abstract

We solve the multi-store paradox by introducing interdependent payoffs between the firms. We show that firms set up multiple stores unless the degree of payoff interdependence is low. We also show that multiple equilibria, namely intertwined and neighboring location equilibria, exist if the degree of payoff interdependence is intermediate.

Keywords:

• Interdependent payoff
• relative profit
• circular-city model
• spatial competition

Acknowledgements

We thank participants of the seminars at The University of Tokyo, Chonnam National University, and The University of Tokyo and National Taiwan University Joint Conference on Industrial Organization at National Taiwan University for their helpful comments. We are responsible for any remaining errors.

Notes

The authors have no material interests related to this research to disclose.

1 See Gibbons and Murphy (1990) for empirical evidence.

2 See Armstrong and Huck (2010). In the real world, people are concerned about relative performance, but not because of the monetary incentives. See Grieco, Powell, and Snidal (1993) and Mastanduno (1991) in the context of political science and Ariely (2009) in the context of behavioral economics. The payoff function based on relative wage or relative wealth status has also been intensively discussed in the macroeconomics context. Keynes (1936) discussed the rigidity of nominal wage based on relative wage. See also Akerlof and Yellen (1988) and Corneo and Jeanne (1997, 1999), and Futagami and Shibata (1998). We believe that this is because concern about relative performance is realistic and thus has attracted the interest of many macroeconomists.

3 See Vega-Redondo (1997) for the model formulation of a related evolutionary game. He considered a quantity-setting model in a homogeneous product market and showed that if firms myopically imitate the most profitable firm’s strategy, the industry converges to a highly competitive outcome.

4 For a general discussion of this approach and useful applications, see Matsumura, Matsushima, and Cato (2013) and Matsumura and Okamura (2015). The former (latter) discussed the relationship between competition and R&D activities (optimal privatization policy). We can show that given the locations, the ratio between the profit margin (i.e. price minus marginal cost) and the price, known as the Lerner index, is decreasing in . This index is intensively adopted in the empirical literature as a measure of the intensity of market competition in product markets. Furthermore, Matsumura and Matsushima (2012) showed that collusion is less stable when is larger under moderate conditions. In this sense, a larger again indicates a more competitive market.

5 As pointed out by Hendel and Neiva de Figueiredo (1997), in the context of a spatial model, the strategic effect of investment in a duopoly is much stronger than that in an oligopoly () and duopoly and triopoly models often yield contrasting results. Thus, it is important to solve this paradox under the strongest strategic effect case (duopoly case) because the multi-product case appears under competition (e.g. between Coca-Cola and Pepsi or between Aderans and Artnature.

6 Because of the symmetry of the circular market, a similar principle can also apply in the case where .

7 Regarding its store location strategy, Seven-Eleven Japan states that ‘high-density, concentrated store openings . . . are a vital trajectory toward realizing close and convenient stores.’ (Seven-Eleven Japan Corporate Profile 2013–2014, p. 15).

Additional information

Funding

This work was supported by Murata Science Foundation and JSPS KAKENHI [grant number 15k03347].