Spatial competition with endogenous location choices: An application to discount retailing

Abstract

This paper examines the importance of geographical differentiation in store location decisions of firms in the retail discount industry. Using a novel data set that includes the store locations and accompanying market conditions for all stores belonging to the Wal-Mart, Kmart, and Target chains, we study the factors that influence the entry and location decisions of these firms. The model involves an incomplete information game between the three players where each firm has private information about its own profitability. A key feature of our modeling approach is that it permits asymmetries across firms in the impact of exogenous market characteristics and competitive interaction effects. Variations in the exogenous firm specific characteristics, such as the distances from the market to firms’ headquarters and the nearest distribution centers, serve as exclusion restrictions and provide the source for model identification. Parameter estimates of the payoff functions are used to predict the equilibrium market structure under a variety of market conditions that provide insights into the competitive landscape of the industry. Results show that all firms exert a strong negative impact on competitors when they are in close proximity, but the effect decreases with distance to rivals suggesting strong returns to spatial differentiation in this industry. Target stores fare well under competition except when these competitors are in close proximity. Wal-Mart’s supercenter format is found to be the most formidable player as it substantially impacts competitors even at a large distance. We also find significant asymmetries across players in their response to market conditions and competition interactions.

Appendix: Equilibrium properties of incomplete information games

A well-known problem with multi-agent discrete games is that they have non-existence of pure strategy equilibria and multiple equilibria in which certain values of the underlying latent payoffs could simultaneously be consistent with more than one equilibrium outcome. For existence, researchers generally assume that the data is generated by a pure strategy equilibrium and that the model satisfies restrictions sufficient to guarantee the existence of at least one such equilibrium. In the current application, since Eq. (6) is a continuous mapping from a probability space to itself, the existence of a solution of (6) can be proved from Brouwer’s Fixed Point Theorem. However, the uniqueness of such a solution is not guaranteed. The game will have multiple equilibria if, for given parameter θ, more than one value of P satisfies (6) in which case the likelihood function is not well defined. Aradillas-Lopez (2005) discusses the sufficient conditions for the uniqueness of equilibrium of incomplete information game and points out that the conditions for existence of a well-defined likelihood function are generally weaker in incomplete information game than in the complete information version of the game.

Heckman (1978) studies the general class of simultaneous equation qualitative response models and shows that if the distribution of the unobservables is continuous with unbounded support, a necessary and sufficient condition for the model to have a unique equilibrium for any values of exogenous variables is that the model is recursive. Though useful in some applications, this condition implies setting some strategic interactions parameters (which are of main interest) to zero. Instead, researchers have generally adopted one of the two approaches to deal with multiple equilibria problem. First, models sometimes yield a prediction for some outcome that is unique across all equilibria. For instance, Bresnahan and Reiss (1990, 1991a, b) and Berry (1992) ignore the identities of the players that enter a market and only consider the total number of entrants, an outcome for which their models yield a unique prediction. Alternatively, some papers impose additional structure on the model that guarantees equilibrium uniqueness. For example, Zhu et al. (2008) consider sequential move games that yields additional conditions for an outcome to be a subgame perfect Nash equilibrium.

In this Appendix, we study the equilibrium properties for the model proposed in Section 2. We begin with a proof on the sufficient conditions of uniqueness for the two player–two action game and then rely on simulation techniques for a more complex game, i.e., when the action space or the number of players becomes larger.

Consider a simple 2×2 game with two players and two possible actions: in or out. Firm i’s payoff of entry is

$$ \pi_{i}=x_{i}\beta_{i}-\delta_{i}p_{-i}+\varepsilon_{i}\text{, }i=1,2, $$

(14)

where p _− i is the entry probability of i’s competitor, and ε _i is the private information held by firm i and is type 1 extreme value distributed. Similar to the discussion on Eq. (7), the BNE is obtained by solving the following two simultaneous equations:

$$ p_{1}=\frac{\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }{1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }, $$

and

$$ p_{2}=\frac{\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }{1+\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }. $$

Define $f=\left(\begin{array} [c]{c}f_{1}\\ f_{2}\end{array} \right)$, where

$$ \begin{array}{rcl} f_{1} & = & p_{1}-\frac{\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1} p_{2}\right) }{1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) },\\ f_{2} & = & p_{2}-\frac{\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2} p_{1}\right) }{1+\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }, \end{array} $$

then taking the derivates of f ₁ with respect to p ₂ yields

$$ \begin{array}{rcl} \frac{\partial f_{1}}{\partial p_{2}} &=& \frac{\delta_{1}\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) \left[ 1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) \right] -\delta_{1} \exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) \exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }{\left[ 1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) \right] ^{2}} \\ \\ & = & \frac{\delta_{1}\exp\left( x_{1}^{\prime}\beta-\delta_{1}p_{2}\right) }{\left[ 1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) \right] ^{2}}, \end{array} $$

and similarly

$$ \frac{\partial f_{2}}{\partial p_{1}}=\frac{\delta_{2}\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }{\left[ 1+\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) \right] ^{2}}. $$

Hence the Jacobian matrix f ^′ is

$$ \left( \begin{array} [c]{cc} 1 & \frac{\delta_{1}\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1} p_{2}\right) }{\left[ 1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1} p_{2}\right) \right] ^{2}}\\ \frac{\delta_{2}\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }{\left[ 1+\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) \right] ^{2}} & 1 \end{array} \right). $$

From the Gale-Nikaido Theorem: Suppose f: ℝⁿ→ℝⁿ is continuously differentiable function, and define Ω as the rectangle $\Omega=\left\{ x\in\mathbb{R}^{n},a\leq x\leq b\right\},$ where a and b are given vectors in ℝⁿ. Then f is one-to-one in Ω if for all x ∈ Ω, Jacobian matrix $f^{\prime }\left( x\right)$ has only strictly positive principal minors. The sufficient condition that (1) has unique solution for $\left( p_{1},p_{2}\right)$ is

$$ 1-\left[ \frac{\delta_{1}\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1} p_{2}\right) }{\left[ 1+\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1} p_{2}\right) \right] ^{2}}\right] \left[ \frac{\delta_{2}\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }{\left[ 1+\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) \right] ^{2}}\right] >0\text{,} $$

or equivalently,

$$ \begin{array}{rcl} \delta_{1}\delta_{2} & < &\frac{\left[ 1+\exp\left( x_{1}^{\prime}\beta _{1}-\delta_{1}p_{2}\right) \right] ^{2}\left[ 1+\exp\left( x_{2}^{\prime }\beta_{2}-\delta_{2}p_{1}\right) \right] ^{2}}{\exp\left( x_{1}^{\prime }\beta_{1}-\delta_{1}p_{2}\right) \exp\left( x_{2}^{\prime}\beta_{2} -\delta_{2}p_{1}\right) }\\ \\ & = & 4+\frac{2}{\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }+2\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) +\frac{2} {\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }\\ \\ && +2\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) +\frac{\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) } {\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }+\frac {\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}\right) }{\exp\left( x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }\\ \\ && +\frac{1}{\exp\left( x_{1}^{\prime}\beta_{1}-\delta_{1}p_{2}+x_{2} ^{\prime}\beta_{2}-\delta_{2}p_{1}\right) }+\exp\left( x_{1}^{\prime} \beta_{1}-\delta_{1}p_{2}+x_{2}^{\prime}\beta_{2}-\delta_{2}p_{1}\right) \\ \\ & \geq & 16 \end{array} $$

Hence as long as the interaction term of the competition intensity δ ₁ δ ₂ is below 16, the uniqueness of the Bayesian Nash equilibrium is ensured in this 2×2 game.

The message from the simple example above is that as long as the strength of strategic interaction is within a certain range, equilibrium is unique. As the game becomes more complex, i.e., the action space or the number of players gets larger, the matrix algebra becomes cumbersome. We rely on simulation technique to investigate the uniqueness property of incomplete information games. In the simulation, We are interested in how the equilibrium uniqueness property is related to $\left(1\right)$ the variation of market characteristics across locations (choice alternatives), $\left(2\right)$ the heterogeneity across players, and $\left(3\right)$ the magnitude of competition effect. This simulation exercise is motivated by Bayer and Timmins (2004). We assume

• There are two locations in the market, location 1 and location 2.

• There are two players A and B who decide whether to enter the market or not and which location to choose.

• The competition between players at the same location is stronger than the competition between players from different locations.

Players’ payoff functions are assumed to be:

$$ \pi_{il}=x_{l}\beta_{i}-\sum_{k=1}^{2}\delta_{lk}^{i}P_{i^{\prime} l}+\varepsilon_{il},i,i^{\prime}\in\left\{ A,B\right\} \text{ and }i\neq i^{\prime},l,k=1,2 $$

where $\delta_{lk}^{i}$ denotes the how firm i’s payoff at location l is affected by its competitor’s entry of location k and $\delta_{ll}^{i} \geq\delta_{ll^{\prime}}^{i}$, where l ^′ ≠ l. ε _il are iid type one extreme value distributed.

We normalize x ₁ = 1, β _A  = 1, and change x ₂ from 1 to 10 and β _B from 1 to 2. The ratio x ₂/x ₁ captures the variation of market conditions across locations. The larger the ratio, the more different the two locations are. Similarly, the ratio β _B /β _A represents the level of firm heterogeneity. Given the same market conditions, firms’ profitability will be different due to their preferences of those market factors. Given x ₁, x ₂, β ₁, β ₂ and $\delta_{lk}^{i}$, the fixed points of the probability vector $\left( P_{1A},P_{2A,}P_{1B,}P_{2B}\right)$ are solved from different starting values: $\left( 0\text{ }0\text{ }0\text{ }0\right) ^{\prime}$, $\left( 0\text{ }0\text{ }1\text{ }0\right) ^{\prime}$, $\left( 0\text{ }0\text{ }0\text{ }1\right) ^{\prime}$, $\left( 1\text{ }0\text{ }0\text{ }0\right) ^{\prime}$, $\left( 1\text{ }0\text{ }1\text{ }0\right) ^{\prime}$, $\left( 1\text{ }0\text{ }0\text{ }1\right) ^{\prime}$, $\left( 0\text{ }1\text{ }0\text{ }0\right) ^{\prime}$, $\left( 0\text{ }1\text{ }1\text{ }0\right) ^{\prime}$, $\left( 0\text{ }1\text{ }0\text{ }1\right) ^{\prime}$. The discussion on why one should choose those points as starting points to test equilibrium multiplicity can be found in Bayer and Timmins (2004). As we change $\delta_{11}^{A}$ from 0 to 100, and find $\bar{\delta}_{ll}^{i}$, such that as $\delta_{ll}^{i}>\bar{\delta}_{ll}^{i}$, we start to observe multiple fixed points for a particular combination of x ₁, x ₂, β ₁, β ₂. In Figs. 4 and 5 we plot how $\bar{\delta }_{ll}^{i}$ changes as x ₂/x ₁ and β ₂/β ₁ changes under two scenarios:

• Case 1: Symmetric competition , $\delta_{lk}^{A}=\delta_{lk}^{B}$, $\delta_{ll^{\prime}}^{i}=\frac{1}{2}\delta_{ll}^{i}$

• Case 2: Asymmetric competition: $\delta_{lk}^{A}=2\delta_{ik}^{B}$, $\delta_{ll^{\prime}}^{i}=\frac{1}{2}\delta_{ll}^{i}$

Fig. 4

Unique equilibrium under symmetric competition

Fig. 5

Unique equilibrium under asymmetric competition

In the first case, firms are symmetric in terms of competition, while in the second scenario Player B is stronger than player A in the sense that it exerts stronger negative impact on its competitor. The competition intensity is weaker if two players choose different locations. Comparing the plot for case 2 and that for case 1, we can find that the threshold for case 2 is larger. The reason is that the uniqueness conditions are determined by the interaction terms of the competition intensity. Hence the range for the competition intensity of $\bar{\delta}_{11}^{1}$ is larger if $\delta_{lk} ^{A}>\delta_{lk}^{B}$.

The results from simulation can be summarized as follows. First, as long as the strength of competition effect is moderate, there is a unique equilibrium in the incomplete information game. Second, there exists a threshold $\bar{\delta}_{lk}^{A}$ such that when $\delta_{lk}^{A}>\bar{\delta}_{lk}^{A}$, one starts to observe multiple equilibrium given the location characteristics and firms’ preference of those market characteristics. We find that as the variation of market characteristics increases, the threshold $\bar{\delta}_{lk}^{A}$ increases. Put another way, the multiple equilibria is less likely to be observed when there is larger variation across locations. This finding is consistent with Seim (2001). Finally, as the asymmetry between players increases both in terms of their preference to market conditions $\left( \beta_{B}/\beta_{A}\right)$ and the asymmetric of competition effect, the threshold $\bar{\delta}_{lk}^{A}$ becomes higher.