Location games with references

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Gaëtan Fournier, Amaury Francou, 2023, Games and Economic Behavior, Volume 142, Pages 17-32.



Agents are often influenced by a reference situation determined by their prior decisions or the choices made by their predecessors. These references significantly affect the competition outcome.


Spatial competition explores the strategic interactions among economic agents who select characteristics to attract the greatest demand possible. Primary examples include competition between firms informed of consumer preferences, choosing product characteristics to maximize their clientele, and electoral competition among political parties selecting their platforms to maximize their electorate.

This literature tends to overlook the fact that agents are limited in their choices of characteristics, as they are often influenced by a reference situation determined by their prior decisions or the choices made by their predecessors. Although these references are not always controlled by the agents, they significantly affect the competition’s outcome.

We study these anchoring effects comprehensively, covering a wide range of application domains. The two above competitive situations can be re-examined by introducing an anchoring effect:

Example 1: In a competitive market, we assume that firms can change the characteristics of the goods they historically produce to gain new clients through research and development investments. When these firms know consumers’ preferences and the characteristics chosen by their competitors, they can infer the optimal changes to implement in their products. With changes involving increasing and convex costs, the firms face a trade-off between potential gains in market shares and differentiation costs.

Example 2: In the classical Downs model, political parties only have office motivations, treating policy as an opportunistic instrument for electoral success. It is sometimes more realistic to suppose that political parties also have policy motivations, such as not wanting to betray their genuine policy objectives. The political party must find a compromise between a purely competitive political platform and a platform closer to their sincere convictions.




Introducing a reference point ensures that the competition admits at most one pure Nash equilibrium. This result contrasts with standard location games that generally admit a continuum of equilibria, making qualitative analysis and comparative statics challenging. This simplification makes it possible to analyze the competition between any number of economic agents and any vector of reference points. We exhibit the unique equilibrium candidate and describe the necessary and sufficient conditions for its existence. A key property of this equilibrium is that only peripheral agents can significantly differ from their references. Using the vocabulary of the first example above, our results indicate that firms producing goods with extreme characteristics have more freedom to adjust their positions than firms with moderate characteristics.

In the specific case of two firms, we mitigate the principle of minimal differentiation, which states that two firms have no interest in choosing different characteristics for their products. This results in a single good available to consumers and decreased welfare, as consumers buy a product they like less in general. We characterize how this inefficiency disappears with the anchoring effect. The sequence of graphs below shows the nature of the equilibrium in a duopoly competition.

Description of the figure: from left to right, we increase the magnitude of the anchoring effect. On each sub-figure, we represent the square of all possible reference locations, one for each player of the duopoly. We then draw in orange the situations where the equilibrium displays no differentiation, in blue situations where the equilibrium displays differentiation, and in white situations where no equilibrium exists.


This paper assumes references are non-strategic. A future research direction could consist in exploring the case of farsighted agents who, in a repeated interaction scenario, choose an adaptation path by gradually changing their characteristics while controlling their competitors’ reactions.


→ This article was issued in AMSE Newletter, Winter 2023.


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