Agents playing Hotelling’s game: an agent-based approach to a game theoretic model

Ann Reg Sci (2016) 57:393–411 DOI 10.1007/s00168-015-0711-z SPECIAL ISSUE PAPER Agents playing Hotelling’s game: an agent-based approach to a game theoretic model Eveline van Leeuwen1 · Mark Lijesen1 Received: 11 February 2015 / Accepted: 24 September 2015 / Published online: 19 October 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract This paper combines game theory and agent-based modelling, two powerful tools that economists use to understand the behavior of economic agents. We construct an agent-based version of Hotelling’s two-stage game of spatial competition and explore the possibilities of creating synergies between the two approaches. Game theoretic insights into strategic behavior and equilibrium states can provide useful theoretic underpinnings for agent-based approaches in regional science. By combining the two, we can model micro-based social order as it emerges out of local interactions. The use of agent-based modelling in the context of a multistage game is new and hence provides a valuable contribution to both streams of the literature. We show that combining the two approaches is feasible, also in the context of a more complex two-stage game. The model correctly reproduces the analytical results and also allows for more complex situations. As an example, we show the effect of different levels of consumer tastes for variety in Main Street. The reconstruction of Hotelling’s model of spatial competition opens up a wide variety of possibilities for further extensions that can lead to a better understanding of the variations we observe in reality. For some extensions, the use of a single-stage model would probably be more feasible though. JEL Classification C63 · C72 · D21 · L13 1 Introduction Hotelling’s metaphor of spatial competition on Main Street (Hotelling 1929) is widely accepted as one of the most important models in understanding strategic product B Eveline van Leeuwen 1 Department of Spatial Economics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands 123 394 E. van Leeuwen, M. Lijesen differentiation. The known standard outcomes allow us to evaluate the results of the basic model. A vast literature has evolved, discussing extensions of the model with respect to the nature of transport costs (Economides 1986), distributions of consumers (or preferences) (Tabuchi and Thisse 1995; Meagher and Zauner 2005), dimensions (Irmen and Thisse 1998; Chen and Riordan 2007), and so on. Although being widely used, game theory has to deal with several challenges, both from a theoretic perspective and from the perspective of translating its outcomes to the real world. Many economic games assume all actors to be fully rational and perfectly informed, which may be practical in finding a solution, but is unlikely to hold in real life. Although game theory as such allows for relaxing these assumptions, doing so often leads to intractable results or the nonexistence of equilibria (Halpern and Pass 2015). Likewise, allowing for alternative underlying distributions (e.g., of consumers over space in the Hotelling model) might yield a model without equilibria in pure strategies (Caplin and Nalebuff 1991). Studying states and processes evolving from the behavior of non-rational agents is at the core of agent-based modelling. The number of researchers involved in modelling agents’ behavior has increased significantly over the last years (Crooks and Heppenstall, 2012). The so-called agentbased models are developed to better understand the theories of political identity and stability (Lustick 2002); economic processes as dynamic systems of interacting agents (Tesfatsion 2006); company size and growth rate distributions (citealtA99); burglary activities (Malleson et al. 2010); size-frequency distributions for traffic jams (Nagel and Rasmussen 1994); spatial patterns of unemployment (Topa 2001); size distributions of cities (Mansury and Gulyás 2007); land system dynamics and wellbeing (Robinson et al. 2012); and so on. However, until recently, agent-based models are not well established in spatial economic research mainly due to the lack of microfoundations underlying individual motivation and behavior. In general, it is not easy to translate micro-foundations into agent behavior in complex (spatial) situations. Our contribution to the literature is to combine game theory and agent-based modelling in such a way that both tools strengthen each other. It is striking to see that the strengths of agent-based modelling correspond with the weaknesses of game theory and vice versa. We develop an agent-based version of Hotelling’s model of spatial competition to explore the potential for synergy of using these tools simultaneously. Finding the optimal location of a shop essentially depends on the trade-off between market power (shops far away from each other) and market share (shops close to the center of the market). The players of this game are thus the shops and their payoff of the profit based on the number of customers they attract at a certain price. In an earlier approach, Ottino et al. (2009) developed an agent-based model that uses the concepts of Hotelling in both a one- and a two-dimensional world. In this model, the shops change their location to gain market share and then change their price to see whether they can also improve their profit at that specific location. By construction, this puts the emphasis on the market-stealing effect, which, as can be seen in the model, always results in shops ending up being located close to each other with very low revenues. Furthermore, the shops do not interact with each other; after they have moved simultaneously, they respond to the new situation. Since the shops do not search for best responses (to the other’s shop strategy), this agent-based version lacks the strategic interaction that characterizes game theoretic models. 123 Agents playing Hotelling’s game: an agent-based approach… 395 Our agent-based model goes a step further than the model of Ottino et al. (2009), as it aims to capture the full game theoretic nature of Hotelling’s model. Since this model is a multistage game, an important challenge is to find a computational alternative for the analytical tools used. Alternatives for backward induction have to our knowledge not been discussed in the literature on economic games. The use of computational models in multistage games is well developed in a broader context however, the most obvious example being chess computers (or software). Chess computers calculate the payoff of each possible sequence of moves to select the move with the highest expected payoff. We use the same basic mechanism. First of all, we incorporate strategic behavior by asking the shops to respond to each other’s moves in order to maximize their profit. Furthermore, the model maintains the two-stage nature of Hotelling’s model in which the interaction between (...truncated)